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Contents

Preface xi

Notation and conventions xvii

1 Fundamentals 11.1 Vectors, dual vectors, and tensors . . . . . . . . . . . . . 21.2 Covariant differentiation . . . . . . . . . . . . . . . . . . 41.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Lie differentiation . . . . . . . . . . . . . . . . . . . . . . 71.5 Killing vectors . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Local flatness . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Metric determinant . . . . . . . . . . . . . . . . . . . . . 121.8 Levi-Civita tensor . . . . . . . . . . . . . . . . . . . . . . 131.9 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 141.10 Geodesic deviation . . . . . . . . . . . . . . . . . . . . . 161.11 Fermi normal coordinates . . . . . . . . . . . . . . . . . 18

1.11.1 Geometric construction . . . . . . . . . . . . . . . 181.11.2 Coordinate transformation . . . . . . . . . . . . . 191.11.3 Deviation vectors . . . . . . . . . . . . . . . . . . 201.11.4 Metric on γ . . . . . . . . . . . . . . . . . . . . . 211.11.5 First derivatives of the metric on γ . . . . . . . . 211.11.6 Second derivatives of the metric on γ . . . . . . . 221.11.7 Riemann tensor in Fermi normal coordinates . . . 23

1.12 Bibliographical notes . . . . . . . . . . . . . . . . . . . . 231.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Geodesic congruences 272.1 Energy conditions . . . . . . . . . . . . . . . . . . . . . . 28

2.1.1 Introduction and summary . . . . . . . . . . . . . 282.1.2 Weak energy condition . . . . . . . . . . . . . . . 292.1.3 Null energy condition . . . . . . . . . . . . . . . . 292.1.4 Strong energy condition . . . . . . . . . . . . . . 302.1.5 Dominant energy condition . . . . . . . . . . . . 302.1.6 Violations of the energy conditions . . . . . . . . 31

2.2 Kinematics of a deformable medium . . . . . . . . . . . . 31

i

ii Contents

2.2.1 Two-dimensional medium . . . . . . . . . . . . . 312.2.2 Expansion . . . . . . . . . . . . . . . . . . . . . . 322.2.3 Shear . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.4 Rotation . . . . . . . . . . . . . . . . . . . . . . . 332.2.5 General case . . . . . . . . . . . . . . . . . . . . . 332.2.6 Three-dimensional medium . . . . . . . . . . . . . 34

2.3 Congruence of timelike geodesics . . . . . . . . . . . . . 342.3.1 Transverse metric . . . . . . . . . . . . . . . . . . 352.3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . 352.3.3 Frobenius’ theorem . . . . . . . . . . . . . . . . . 362.3.4 Raychaudhuri’s equation . . . . . . . . . . . . . . 382.3.5 Focusing theorem . . . . . . . . . . . . . . . . . . 392.3.6 Example . . . . . . . . . . . . . . . . . . . . . . . 392.3.7 Another example . . . . . . . . . . . . . . . . . . 402.3.8 Interpretation of θ . . . . . . . . . . . . . . . . . 41

2.4 Congruence of null geodesics . . . . . . . . . . . . . . . . 432.4.1 Transverse metric . . . . . . . . . . . . . . . . . . 442.4.2 Kinematics . . . . . . . . . . . . . . . . . . . . . 452.4.3 Frobenius’ theorem . . . . . . . . . . . . . . . . . 462.4.4 Raychaudhuri’s equation . . . . . . . . . . . . . . 482.4.5 Focusing theorem . . . . . . . . . . . . . . . . . . 482.4.6 Example . . . . . . . . . . . . . . . . . . . . . . . 482.4.7 Another example . . . . . . . . . . . . . . . . . . 492.4.8 Interpretation of θ . . . . . . . . . . . . . . . . . 50

2.5 Bibliographical notes . . . . . . . . . . . . . . . . . . . . 512.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Hypersurfaces 573.1 Description of hypersurfaces . . . . . . . . . . . . . . . . 58

3.1.1 Defining equations . . . . . . . . . . . . . . . . . 583.1.2 Normal vector . . . . . . . . . . . . . . . . . . . . 583.1.3 Induced metric . . . . . . . . . . . . . . . . . . . 593.1.4 Light cone in flat spacetime . . . . . . . . . . . . 61

3.2 Integration on hypersurfaces . . . . . . . . . . . . . . . . 623.2.1 Surface element (non-null case) . . . . . . . . . . 623.2.2 Surface element (null case) . . . . . . . . . . . . . 633.2.3 Element of two-surface . . . . . . . . . . . . . . . 65

3.3 Gauss-Stokes theorem . . . . . . . . . . . . . . . . . . . 673.3.1 First version . . . . . . . . . . . . . . . . . . . . . 673.3.2 Conservation . . . . . . . . . . . . . . . . . . . . 683.3.3 Second version . . . . . . . . . . . . . . . . . . . 69

3.4 Differentiation of tangent vector fields . . . . . . . . . . . 703.4.1 Tangent tensor fields . . . . . . . . . . . . . . . . 703.4.2 Intrinsic covariant derivative . . . . . . . . . . . . 713.4.3 Extrinsic curvature . . . . . . . . . . . . . . . . . 72

Contents iii

3.5 Gauss-Codazzi equations . . . . . . . . . . . . . . . . . . 733.5.1 General form . . . . . . . . . . . . . . . . . . . . 733.5.2 Contracted form . . . . . . . . . . . . . . . . . . 743.5.3 Ricci scalar . . . . . . . . . . . . . . . . . . . . . 75

3.6 Initial-value problem . . . . . . . . . . . . . . . . . . . . 763.6.1 Constraints . . . . . . . . . . . . . . . . . . . . . 763.6.2 Cosmological initial values . . . . . . . . . . . . . 773.6.3 Moment of time symmetry . . . . . . . . . . . . . 783.6.4 Stationary and static spacetimes . . . . . . . . . . 783.6.5 Spherical space, moment of time symmetry . . . . 793.6.6 Spherical space, empty and flat . . . . . . . . . . 793.6.7 Conformally-flat space . . . . . . . . . . . . . . . 80

3.7 Junction conditions and thin shells . . . . . . . . . . . . 813.7.1 Notation and assumptions . . . . . . . . . . . . . 813.7.2 First junction condition . . . . . . . . . . . . . . 823.7.3 Riemann tensor . . . . . . . . . . . . . . . . . . . 833.7.4 Surface stress-energy tensor . . . . . . . . . . . . 833.7.5 Second junction condition . . . . . . . . . . . . . 853.7.6 Summary . . . . . . . . . . . . . . . . . . . . . . 86

3.8 Oppenheimer-Snyder collapse . . . . . . . . . . . . . . . 863.9 Thin-shell collapse . . . . . . . . . . . . . . . . . . . . . 893.10 Slowly rotating shell . . . . . . . . . . . . . . . . . . . . 903.11 Null shells . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.11.1 Geometry . . . . . . . . . . . . . . . . . . . . . . 943.11.2 Surface stress-energy tensor . . . . . . . . . . . . 973.11.3 Intrinsic formulation . . . . . . . . . . . . . . . . 993.11.4 Summary . . . . . . . . . . . . . . . . . . . . . . 1003.11.5 Parameterization of the null generators . . . . . . 1013.11.6 Imploding spherical shell . . . . . . . . . . . . . . 1043.11.7 Accreting black hole . . . . . . . . . . . . . . . . 1053.11.8 Cosmological phase transition . . . . . . . . . . . 108

3.12 Bibliographical notes . . . . . . . . . . . . . . . . . . . . 1103.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4 Lagrangian and Hamiltonian formulations of general rel-ativity 1154.1 Lagrangian formulation . . . . . . . . . . . . . . . . . . . 116

4.1.1 Mechanics . . . . . . . . . . . . . . . . . . . . . . 1164.1.2 Field theory . . . . . . . . . . . . . . . . . . . . . 1174.1.3 General relativity . . . . . . . . . . . . . . . . . . 1184.1.4 Variation of the Hilbert term . . . . . . . . . . . 1194.1.5 Variation of the boundary term . . . . . . . . . . 1214.1.6 Variation of the matter action . . . . . . . . . . . 1214.1.7 Nondynamical term . . . . . . . . . . . . . . . . . 1224.1.8 Bianchi identities . . . . . . . . . . . . . . . . . . 123

iv Contents

4.2 Hamiltonian formulation . . . . . . . . . . . . . . . . . . 1244.2.1 Mechanics . . . . . . . . . . . . . . . . . . . . . . 1244.2.2 3 + 1 decomposition . . . . . . . . . . . . . . . . 1254.2.3 Field theory . . . . . . . . . . . . . . . . . . . . . 1274.2.4 Foliation of the boundary . . . . . . . . . . . . . 1304.2.5 Gravitational action . . . . . . . . . . . . . . . . 1324.2.6 Gravitational Hamiltonian . . . . . . . . . . . . . 1344.2.7 Variation of the Hamiltonian . . . . . . . . . . . . 1364.2.8 Hamilton’s equations . . . . . . . . . . . . . . . . 1404.2.9 Value of the Hamiltonian for solutions . . . . . . 141

4.3 Mass and angular momentum . . . . . . . . . . . . . . . 1424.3.1 Hamiltonian definitions . . . . . . . . . . . . . . . 1424.3.2 Mass and angular momentum for stationary, ax-

ially symmetric spacetimes . . . . . . . . . . . . . 1434.3.3 Komar formulae . . . . . . . . . . . . . . . . . . . 1454.3.4 Bondi-Sachs mass . . . . . . . . . . . . . . . . . . 1474.3.5 Distinction between ADM and Bondi-Sachs masses:

Vaidya spacetime . . . . . . . . . . . . . . . . . . 1474.3.6 Transfer of mass and angular momentum . . . . . 150

4.4 Bibliographical notes . . . . . . . . . . . . . . . . . . . . 1514.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5 Black holes 1595.1 Schwarzschild black hole . . . . . . . . . . . . . . . . . . 159

5.1.1 Birkhoff’s theorem . . . . . . . . . . . . . . . . . 1595.1.2 Kruskal coordinates . . . . . . . . . . . . . . . . . 1605.1.3 Eddington-Finkelstein coordinates . . . . . . . . . 1625.1.4 Painleve-Gullstrand coordinates . . . . . . . . . . 1635.1.5 Penrose-Carter diagram . . . . . . . . . . . . . . 1645.1.6 Event horizon . . . . . . . . . . . . . . . . . . . . 1665.1.7 Apparent horizon . . . . . . . . . . . . . . . . . . 1665.1.8 Distinction between event and apparent horizons:

Vaidya spacetime . . . . . . . . . . . . . . . . . . 1685.1.9 Killing horizon . . . . . . . . . . . . . . . . . . . 1715.1.10 Bifurcation two-sphere . . . . . . . . . . . . . . . 171

5.2 Reissner-Nordstrom black hole . . . . . . . . . . . . . . . 1725.2.1 Derivation of the Reissner-Nordstrom solution . . 1725.2.2 Kruskal coordinates . . . . . . . . . . . . . . . . . 1735.2.3 Radial observers in Reissner-Nordstrom spacetime.1765.2.4 Surface gravity . . . . . . . . . . . . . . . . . . . 179

5.3 Kerr black hole . . . . . . . . . . . . . . . . . . . . . . . 1825.3.1 The Kerr metric . . . . . . . . . . . . . . . . . . . 1825.3.2 Dragging of inertial frames: ZAMOs . . . . . . . 1835.3.3 Static limit: static observers . . . . . . . . . . . . 1835.3.4 Event horizon: stationary observers . . . . . . . . 184

Contents v

5.3.5 The Penrose process . . . . . . . . . . . . . . . . 1865.3.6 Principal null congruences . . . . . . . . . . . . . 1875.3.7 Kerr-Schild coordinates . . . . . . . . . . . . . . . 1895.3.8 The nature of the singularity . . . . . . . . . . . . 1905.3.9 Maximal extension of the Kerr spacetime . . . . . 1915.3.10 Surface gravity . . . . . . . . . . . . . . . . . . . 1935.3.11 Bifurcation two-sphere . . . . . . . . . . . . . . . 1945.3.12 Smarr’s formula . . . . . . . . . . . . . . . . . . . 1955.3.13 Variation law . . . . . . . . . . . . . . . . . . . . 195

5.4 General properties of black holes . . . . . . . . . . . . . 1965.4.1 General black holes . . . . . . . . . . . . . . . . . 1965.4.2 Stationary black holes . . . . . . . . . . . . . . . 1985.4.3 Stationary black holes in vacuum . . . . . . . . . 200

5.5 The laws of black-hole mechanics . . . . . . . . . . . . . 2015.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . 2015.5.2 Zeroth law . . . . . . . . . . . . . . . . . . . . . . 2025.5.3 Generalized Smarr formula . . . . . . . . . . . . . 2035.5.4 First law . . . . . . . . . . . . . . . . . . . . . . . 2055.5.5 Second law . . . . . . . . . . . . . . . . . . . . . 2065.5.6 Third law . . . . . . . . . . . . . . . . . . . . . . 2075.5.7 Black-hole thermodynamics . . . . . . . . . . . . 208

5.6 Bibliographical notes . . . . . . . . . . . . . . . . . . . . 2095.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

Bibliography 219

List of figures

1.1 A tensor at P lives in the manifold’s tangent plane at P . 31.2 Differentiation of a tensor. . . . . . . . . . . . . . . . . . 41.3 Deviation vector between two neighbouring geodesics. . . 161.4 Geometric construction of the Fermi normal coordinates. 19

2.1 Two-dimensional deformable medium. . . . . . . . . . . . 322.2 Effect of the shear tensor. . . . . . . . . . . . . . . . . . 332.3 Deviation vector between two neighbouring members of

a congruence. . . . . . . . . . . . . . . . . . . . . . . . . 352.4 Family of hypersurfaces orthogonal to a congruence of

timelike geodesics. . . . . . . . . . . . . . . . . . . . . . . 372.5 Geodesics converge into a caustic of the congruence. . . . 402.6 Congruence’s cross section about a reference geodesic. . . 422.7 Family of hypersurfaces orthogonal to a congruence of

null geodesics. . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1 A three-dimensional hypersurface in spacetime. . . . . . 583.2 A null hypersurface and its generators. . . . . . . . . . . 603.3 Proof of the Gauss-Stokes theorem. . . . . . . . . . . . . 673.4 Two spacelike surfaces and their normal vectors. . . . . . 683.5 Two regions of spacetime joined at a common boundary. 813.6 The Oppenheimer-Snyder spacetime. . . . . . . . . . . . 87

4.1 The boundary of a region of flat spacetime. . . . . . . . 1234.2 Foliation of spacetime into spacelike hypersurfaces. . . . 1264.3 Decomposition of tα into lapse and shift. . . . . . . . . . 1274.4 The region , its boundary ∂, and their foliations. . . . . . 1284.5 A radiating spacetime. . . . . . . . . . . . . . . . . . . . 150

5.1 Spacetime diagram based on the (u, v) coordinates. . . . 1615.2 Kruskal diagram. . . . . . . . . . . . . . . . . . . . . . . 1625.3 Spacetime diagram based on the (v, r) coordinates. . . . 1635.4 Compactified coordinates for the Schwarzschild spacetime.1645.5 Penrose-Carter diagram of the Schwarzschild spacetime. . 1655.6 Trapped surfaces and apparent horizon of a spacelike

hypersurface. . . . . . . . . . . . . . . . . . . . . . . . . 167

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viii List of figures

5.7 Black hole irradiated with ingoing null dust. . . . . . . . 1705.8 Kruskal patches for the Reisser-Nordstrom spacetime. . . 1745.9 Penrose-Carter diagram of the Reisser-Nordstrom space-

time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1765.10 Effective potential for radial motion in Reisser-Nordstrom

spacetime. . . . . . . . . . . . . . . . . . . . . . . . . . . 1775.11 Eddington-Finkelstein patches for the Reisser-Nordstrom

spacetime. . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.12 Static limit and event horizon of the Kerr spacetime. . . 1855.13 Kruskal patches for the Kerr spacetime. . . . . . . . . . . 1925.14 Penrose-Carter diagram of the Kerr spacetime. . . . . . . 1935.15 Causal future and past of an event p. . . . . . . . . . . . 1975.16 Event and apparent horizons of a black-hole spacetime. . 1985.17 A spacelike hypersurface in a black-hole spacetime. . . . 204

List of tables

2.1 Energy conditions. . . . . . . . . . . . . . . . . . . . . . 28

4.1 Geometric quantities of Σt, St, and . . . . . . . . . . . . 132

5.1 Boundaries of the compactified Schwarzschild spacetime. 166

ix

Preface

Does the world really need a new textbook on general relativity? Ifeel that my first duty in presenting this book should be to provide aconvincing affirmative answer to this question.

There is already a vast array of available books. I will not attempthere to make an exhaustive list, but I will mention three of my favorites.For its unsurpassed pedagogical presentation of the elementary aspectsof general relativity, I like Schutz’ A first course in general relativity.For its unsurpassed completeness, I like Gravitation, by Misner, Thorne,and Wheeler. And for its unsurpassed elegance and rigour, I like Wald’sGeneral relativity. In my view, a serious student could do no better thanstart with Schutz for an outstanding introductory course, then moveon to Misner, Thorne, and Wheeler to get a broad coverage of manydifferent topics and techniques, and then finish off with Wald to gainaccess to the more modern topics and the mathematical standard thatWald has since imposed on this field. This is a long route, but withthis book I hope to help the student along: I see my place as beingsomewhere between Schutz and Wald — more advanced than Schutzbut less sophisticated than Wald — and I cover some of the few topicsthat are not handled by Misner, Thorne, and Wheeler.

In the winter of 1998 I was given the responsibility of creating anadvanced course in general relativity. The course was intended forgraduate students working in the Gravitation Group of the Guelph-Waterloo Physics Institute, a joint graduate program in Physics sharedby the Universities of Guelph and Waterloo. I thought long and hardbefore giving the first offering of this course, in an effort to round upthe most useful and interesting topics, and to create the best possiblecourse. I came up with a few guiding principles. First, I wanted to letthe students in on a number of results and techniques that are part ofevery relativist’s arsenal, but are not adequately covered in the populartexts. Second, I wanted the course to be practical, in the sense thatthe students would learn how to compute things, not just a bunch ofabstract concepts. And third, I wanted to put these techniques to workin a really cool application of the theory, so that this whole enterprisewould seem to have purpose.

As I developed the course it became clear that it would not match

xi

xii Preface

the material covered in any of the existing textbooks; to meet my re-quirements I would have to form a synthesis of many texts, I wouldhave to consult review articles, and I would have to go to the technicalliterature. This was a long but enjoyable undertaking, and I learneda lot. It gave me the opportunity to homogenize the various separatetreatments, consolidate the various different notations, and present thissynthesis as a unified whole. During this process I started to type uplecture notes that would be distributed to the students. These haveevolved into this book.

In the end, the course was designed around my choice of “reallycool application”. There was no contest: the immediate winner wasthe mathematical theory of black holes, surely one of the most ele-gant, successful, and relevant applications of general relativity. Thisis covered in Chapter 5 of this book, which offers a thorough reviewof the solutions to the Einstein field equations that describe isolatedblack holes, a description of the fundamental properties of black holesthat are independent of the details of any particular solution, and anintroduction to the four laws of black-hole mechanics. In the next para-graphs I outline the material covered in the other chapters, and describethe connections with the theory of black holes.

The most important aspect of black-hole spacetimes is that theycontain an event horizon, a null hypersurface that marks the boundaryof the black hole and shields external observers from events going oninside. On this hypersurface there runs a network (or congruence) ofnonintersecting null geodesics; these are called the null generators ofthe event horizon. To understand the behaviour of the horizon as awhole it proves necessary to understand how the generators themselvesbehave, and in Chapter 2 of this book we develop the relevant tech-niques. The description of congruences is concerned with the motionof nearby geodesics relative to a given reference geodesic; this motionis described by a deviation vector that lives in a space orthogonal tothe reference geodesic’s tangent vector. This transverse space is easy toconstruct when the geodesics are timelike, but the case of null geodesicsis subtle. This has to do with the fact that the transverse space is thentwo-dimensional — the null vector tangent to the generators is orthog-onal to itself and this direction must be explicitly removed from thetransverse space. I show how this is done in Chapter 2. While nullcongruences are treated in other textbooks (most notably in Wald),the student is likely to find my presentation [which I have adaptedfrom Carter (1979)] better suited for practical computations. WhileChapter 2 is concerned mostly with congruences of null geodesics, Ipresent also a complete treatment of the timelike case. There are tworeasons for this. First, this forms a necessary basis to understand thesubtleties associated with the null case. Second, and more importantly,the mathematical techniques involved in the study of congruences of

Preface xiii

timelike geodesics are used widely in the general relativity literature,most notably in the field of mathematical cosmology. Another topiccovered in Chapter 2 are the standard energy conditions of general rel-ativity; these constraints on the stress-energy tensor ensure that undernormal circumstances, gravity acts as an attractive force — it tends tofocus geodesics. Energy conditions appear in most theorems governingthe behaviour of black holes.

Many quantities of interest in black-hole physics are defined by inte-gration over the event horizon. An obvious example is the hole’s surfacearea. Another example is the gain in mass of an accreting black hole;this is obtained by integrating a certain component of the accretingmaterial’s stress-energy tensor over the event horizon. These integra-tions require techniques that are introduced in Chapter 3 of this book.In particular, we shall need a notion of surface element on the eventhorizon. If the horizon were a timelike or a spacelike hypersurface, theconstruction of a surface element would pose no particular challenge,but once again there are interesting subtleties associated with the nullcase. I provide a complete treatment of these issues in Chapter 3; Ibelieve that my presentation is more systematic, and more practical,than what can be found in the popular textbooks. Other topics cov-ered in Chapter 3 include the initial-value problem of general relativity(which involves the induced metric and extrinsic curvature of a space-like hypersurface), and the Darmois-Lanczos-Israel-Barrabes formalismfor junction conditions and thin shells (which constrains the possiblediscontinuities in the induced metric and extrinsic curvature). Theinitial-value problem is discussed at a much deeper level in Wald, but Ifelt it was important to include this material here: it provides a usefulillustration of the physical meaning of the extrinsic curvature, an objectthat plays an important role in Chapter 4 of this book. Junction con-ditions and thin shells, on the other hand, are not covered adequatelyin any textbook, in spite of the fact that the Darmois-Lanczos-Israel-Barrabes formalism is used very widely in the literature. (Junctionconditions and thin shells are touched upon in Misner, Thorne, andWheeler, but I find that their treatment is too brief to do justice to theformalism.)

Among the most important quantities characterizing black holesare their mass and angular momentum, and the question arises as tohow the mass and angular momentum of an isolated object is to bedefined in general relativity. I find that the most compelling defini-tions come from the gravitational Hamiltonian, whose value for a givensolution to the Einstein field equation depends on a specifiable vectorfield. If this vector corresponds to a time translation at spatial infinity,then the Hamiltonian gives the total mass of the spacetime; if, on theother hand, the vector corresponds to an asymptotic rotation aboutan axis, then the Hamiltonian gives the spacetime’s total angular mo-

xiv Preface

mentum in the direction of this axis. This connection is both deepand beautiful, and in this book it forms the starting point for definingblack-hole mass and angular momentum. Chapter 4 of this book isdevoted to a systematic treatment of the Lagrangian and Hamiltonianformulations of general relativity, with this goal in mind of arriving atwell-motivated notions of mass and angular momentum. What sets mypresentation apart from what can be found in other texts, includingMisner, Thorne, and Wheeler and Wald, is that I pay careful attentionto the “boundary terms” that must be included in the gravitationalaction to produce a well-posed variational principle. These boundaryterms have been around for a very long time, but it is only fairly re-cently that their importance has been fully recognized. In particular,they are directly involved in defining the mass and angular momentumof an asymptotically-flat spacetime.

To set the stage, I review the fundamentals of differential geome-try in Chapter 1 of this book. The collection of topics is standard:vectors and tensors, covariant differentiation, geodesics, Lie differenti-ation, Killing vectors, curvature tensors, geodesic deviation, and a fewothers. The goal here is not to provide an introduction to these top-ics; although some may be new, I assume that for the most part, thestudent will have encountered them before (in an introductory courseat the level of Schutz, for example). Instead, my objective with thisChapter is to refresh the student’s memory and establish the style andnotation that I use throughout the book.

As I have indicated, I have tried to present this material as a unifiedwhole, using a consistent notation and maintaining a fairly uniformlevel of precision and rigour. While I have tried to be somewhat preciseand rigourous, I have deliberately avoided putting too much emphasison this. My attitude is that it is more important to illustrate how atheorem works and can be used in a practical situation, than it is toprovide all the fine print that goes into a rigourous proof. The proofsthat I do provide are informal; they may sometimes be incomplete, butthey should be sufficient to convince the student that the theorems aretrue. They may, however, leave the student wanting for more; in thiscase I shall have to refer her to a more authoritative text such as Wald.

I have also indicated that I wanted this book to be practical — Ihope that after studying this book, the student will be able to use whatshe has learned to compute things of direct relevance to her. To helpwith this purpose I have inserted a large number of examples withinthe text. I also provide problem sets at the end of each chapter; herethe student’s understanding will be put to the test. The problems varyin difficulty, from the plug-and-grid type designed to make the studentfamiliar with a new technique, to the more challenging type that issupposed to make the student think. Some of the problems require alarge amount of tensor algebra, and I strongly encourage the student to

Preface xv

let the computer perform the most routine operations. (My favouritepackage for tensor manipulations is GRTensorII, available free of chargeat http://grtensor.phy.queensu.ca/.)

Early versions of this book have been used by graduate students whotook my course over the years. A number of them have expressed greatpraise by involving some of the techniques covered here in their own re-search. This is extraordinarily gratifying, and it has convinced me thata wider release of this book might do more than just service my vanity.A number of students have carefully checked through the manuscriptfor errors (typographical or otherwise), and some have made usefulsuggestions for improvements. For this I thank Daniel Bruni, SeanCrowe, Luis de Menezes, Paul Kobak, Karl Martel, Sanjeev Seahra,and Katrin Rohlf. Of course, I accept full responsibility for whatevererrors remain. The reader is invited to report any error she may find(poisson@physics.uoguelph.ca), and can look up those already re-ported at http://www.physics.uoguelph.ca/poisson/book/.

This book is dedicated to Werner Israel, my teacher, mentor, andfriend, whose influence on me, both as a relativist and as a humanbeing, runs deep. His influence, I trust, will be felt throughout thebook. Every time I started the elaboration of a new topic I would askmyself: “How would Werner approach this?”. I do not believe for onesecond that the answers I came up with would even come close to hislevel of pedagogical excellence, but there is no doubt that to ask thequestion has made me try harder to reach that level.

Notation and conventions

We use the sign conventions of Misner, Thorne, and Wheeler (1973),with a metric of signature (−1, 1, 1, 1), a Riemann tensor defined byRα

βγδ = Γαβδ,γ + · · ·, and a Ricci tensor defined by Rαβ = Rµ

αµβ. Greekindices (α, β, . . . ) run from 0 to 3, lower-case latin indices (a, b, . . . )run from 1 to 3, and upper-case latin indices (A, B, . . . ) run from 2 to3. Geometrized units, in which G = c = 1, are employed.

Here’s a list of frequently occurring symbols:

xvii

xviii Notation and conventions

Symbol Description

xα Arbitrary coordinates on manifoldya Arbitrary coordinates on hypersurface ΣθA Arbitrary coordinates on two-surface S∗= Equals in specified coordinateseα

a = ∂xα/∂ya, eαA = ∂xα/∂θA Holonomic basis vectors

eαµ, eα

A Orthonormal basis vectorsgαβ Metric on

hab = gαβeαaeβ

b Induced metric on Σ

σAB = gαβeαAeβ

B Induced metric on Sg, h, σ Metric determinantsA(αβ) = 1

2(Aαβ + Aβα) Symmetrization

A[αβ] = 12(Aαβ − Aβα) Antisymmetrization

Γαβγ Christoffel symbols constructed from gαβ

Γabc Christoffel symbols constructed from hab

Rαβγδ, Rαβ, R As constructed from gαβ

Rabcd, Rab,3R As constructed from hab

ψ,α = ∂αψ Partial differentiation with respect to xα

ψ,a = ∂aψ Partial differentiation with respect to ya

Aα;β = ∇βAα Covariant differentiation (gαβ-compatible)

Aa|b = DbA

a Covariant differentiation (hab-compatible)

£uAα Lie derivative of Aα along uα

ξα Killing vector: £ξgαβ = 0[α β γ δ] Permutation symbolεαβγδ =

√−g [α β γ δ] Levi-Civita tensor

dΣµ = εµαβγ eα1 eβ

2eγ3 d3y Directed surface element on Σ

dSµν = εµναβ eα2 eβ

3 d2θ Directed surface element on Snα Unit normal on Σ (if timelike or spacelike)ε = nαnα +1 if Σ is timelike, −1 if Σ is spacelike

Kab = nα;β eαaeβ

b Extrinsic curvature of Σθ, σαβ, ωαβ Expansion, shear, and rotationdΩ2 = dθ2 + sin2 θ dφ2 Line element on unit two-sphere

Chapter 1

Fundamentals

This first chapter is devoted to a brisk review of the fundamentals ofdifferential geometry. The collection of topics presented here is fairlystandard, and most of these topics should have been encountered in aprevious introductory course on general relativity. Some, however, maybe new, or may be treated here from a different point of view, or withan increased degree of completeness.

We begin in Sec. 1.1 by providing definitions for tensors on a differ-entiable manifold. The point of view adopted here, and throughout thetext, is entirely unsophisticated: We do without the abstract formu-lation of differential geometry and define tensors in the old-fashionedway, in terms of how their components transform under coordinatetransformations. While the abstract formulation (in which tensors aredefined as multilinear mappings of vectors and dual vectors into realnumbers) is decidedly more elegant and beautiful, and should be anintegral part of an education in general relativity, the old approachhas the advantage of economy, and this motivated its adoption here.Also, the old-fashioned way of defining tensors produces an immedi-ate distinction between tensor fields in spacetime (four-tensors) andtensor fields on a hypersurface (three-tensors); this distinction will beimportant in later chapters of this book.

Covariant differentiation is reviewed in Sec. 1.2, Lie differentiationin Sec. 1.4, and Killing vectors are introduced in Sec. 1.5. In Sec. 1.3we develop the mathematical theory of geodesics. The theory is basedon a variational principle and employs an arbitrary parameterizationof the world line. The advantage of this approach (over one in whichgeodesics are defined by parallel transport of the tangent vector) isthat the limiting case of null geodesics can be treated more naturally.Also, it is often convenient, especially with null geodesics, to use aparameterization that is not affine; we will do so in later portions ofthis book.

In Sec. 1.6 we review a fundamental theorem of differential geome-try, the local flatness theorem. Here we prove the theorem in the stan-dard way, by counting the number of functions required to go from an

1

2 Fundamentals

arbitrary coordinate system to a locally Lorentzian frame. In Sec. 1.11we extend the theorem to an entire geodesic, and we prove it by erectingFermi normal coordinates in a neighbourhood of this geodesic.

Useful results involving the determinant of the metric tensor arederived in Sec. 1.7. The metric determinant is used in Sec. 1.8 to definethe Levi-Civita tensor, which will be put to use in later parts of thisbook (most notably in Chapter 3). The Riemann curvature tensor andits contractions are introduced in Sec. 1.9, along with the Einstein fieldequations. The geometrical meaning of the Riemann tensor is exploredin Sec. 1.10, in which we derive the equation of geodesic deviation.

1.1 Vectors, dual vectors, and tensors

Consider a curve γ on a manifold. The curve is parameterized by λ andis described in an arbitrary coordinate system by the relations xα(λ).We wish to calculate the rate of change of a scalar function f(xα) alongthis curve:

df

dλ=

∂f

∂xα

dxα

dλ= f,αuα.

This procedure allows us to introduce two types of objects on the man-ifold: uα = dxα/dλ is a vector which is everywhere tangent to γ, andf,α = ∂f/∂xα is a dual vector, the gradient of the function f . These ob-jects transform as follows under an arbitrary coordinate transformationfrom xα to xα′ :

f,α′ =∂f

∂xα′ =∂f

∂xα

∂xα

∂xα′ =∂xα

∂xα′ f,α

and

uα′ =dxα′

dλ=

∂xα′

∂xα

dxα

dλ=

∂xα′

∂xαuα.

From these equations we recover the fact that df/dλ is an invariant:f,α′u

α′ = f,αuα.Any object Aα which transforms as

Aα′ =∂xα′

∂xαAα (1.1.1)

under a coordinate transformation will be called a vector. On the otherhand, any object pα which transforms as

pα′ =∂xα

∂xα′ pα (1.1.2)

under the same coordinate transformation will be called a dual vector.The contraction Aαpα between a vector and a dual vector is invariantunder the coordinate transformation, and is therefore a scalar.

1.1 Vectors, dual vectors, and tensors 3

Figure 1.1: A tensor at P lives in the manifold’s tangent plane at P .

uα

P

γ

Generalizing these definitions, a tensor of type (n,m) is an objectTα···β

γ···δ which transforms as

Tα′···β′γ′···δ′ =

∂xα′

∂xα· · · ∂xβ′

∂xβ

∂xγ

∂xγ′ · · ·∂xδ

∂xδ′ T α···βγ···δ (1.1.3)

under a coordinate transformation. The integer n is equal to the num-ber of superscripts, while m is equal to the number of subscripts. Itshould be noted that the order of the indices is important; in general,T β···α

γ···δ 6= T α···βγ···δ. By definition, vectors are tensors of type (1, 0),

and dual vectors are tensors of type (0, 1).

A very special tensor is the metric tensor gαβ, which is used to definethe inner product between two vectors. It is also the quantity thatrepresents the gravitational field in general relativity. The metric or itsinverse gαβ can be used to lower or raise indices. For example, Aα ≡gαβAβ and pα ≡ gαβpβ. The inverse metric is defined by the relationsgαµgµβ = δα

β. The metric and its inverse are symmetric tensors.

Tensors are not actually defined on the manifold itself. To illustratethis, consider the vector uα tangent to the curve γ, as represented inFig. 1.1. The diagram makes it clear that the tangent vector actually“sticks out” of the manifold. In fact, a vector at a point P on themanifold is defined in a plane tangent to the manifold at that point;this plane is called the tangent plane at P . Similarly, tensors at apoint P can be thought of as living in this tangent plane. Tensorsat P can be added and contracted, and the result is also a tensor.However, a tensor at P and another tensor at Q cannot be combinedin a tensorial way, because these tensors belong to different tangentplanes. For example, the operations Aα(P )Bβ(Q) and Aα(Q)−Aα(P )are not defined as tensorial operations. This implies that differentiationis not a straightforward operation on tensors. To define the derivativeof a tensor, a rule must be provided to carry the tensor from one pointto another.

4 Fundamentals

Figure 1.2: Differentiation of a tensor.

γ

P : xα

Q : xα + dxαAα(P )

Aα(Q)

1.2 Covariant differentiation

One such rule is parallel transport. Consider a curve γ, its tangent vec-tor uα, and a vector field Aα defined in a neighbourhood of γ (Fig. 1.2).Let point P on the curve have coordinates xα, and point Q have coor-dinates xα + dxα. As was stated previously, the operation

dAα ≡ Aα(Q)− Aα(P )

= Aα(xβ + dxβ)− Aα(xβ)

= Aα,β dxβ

is not tensorial. This is easily checked: under a coordinate transforma-tion,

Aα′,β′ =

∂

∂xβ′∂xα′

∂xαAα =

∂xα′

∂xα

∂xβ

∂xβ′ Aα,β +

∂2xα′

∂xα∂xβ

∂xβ

∂xβ′ Aα,

which is not a tensorial transformation. To be properly tensorial, thederivative operator should have the form DAα = Aα

T(P )−Aα(P ), whereAα

T(P ) is the vector that is obtained by “transporting” Aα from Q to P .We may write this as DAα = dAα + δAα, where δAα ≡ Aα

T(P )−Aα(Q)is also not a tensorial operation. The precise rule for parallel transportmust now be specified. We demand that δAα be linear in both Aµ anddxβ, so that δAα = Γα

µβ Aµ dxβ for some (nontensorial) field Γαµβ called

the connection. A priori, this field is freely specifiable.We now have DAα = Aα

,βdxβ + Γαµβ Aµdxβ, and dividing through

by dλ, the increment in the curve’s parameter, we obtain

DAα

dλ= Aα

;βuβ, (1.2.1)

where uβ = dxβ/dλ is the tangent vector, and

Aα;β ≡ Aα

,β + Γαµβ Aµ. (1.2.2)

1.2 Covariant differentiation 5

This is the covariant derivative of the vector Aα. Other standard no-tations are Aα

;β ≡ ∇βAα and DAα/dλ ≡ ∇uAα.

The fact that Aα;β is a tensor allows us to deduce the transformation

property of the connection. Starting from ΓαµβAµ = Aα

;β − Aα,β, it is

easy to show that

Γα′µ′β′A

µ′ =∂xα′

∂xα

∂xβ

∂xβ′ ΓαµβAµ − ∂2xα′

∂xµ∂xβ

∂xβ

∂xβ′ Aµ.

Expressing Aµ′ in terms of Aµ on the left-hand side and using the factthat Aµ is an arbitrary vector field, we obtain

Γα′µ′β′

∂xµ′

∂xµ=

∂xα′

∂xα

∂xβ

∂xβ′ Γαµβ −

∂2xα′

∂xµ∂xβ

∂xβ

∂xβ′ .

Multiplying through by ∂xµ/∂xγ′ and rearranging the indices, we arriveat

Γα′µ′β′ =

∂xα′

∂xα

∂xβ

∂xβ′∂xµ

∂xµ′ Γαµβ −

∂2xα′

∂xµ∂xβ

∂xβ

∂xβ′∂xµ

∂xµ′ . (1.2.3)

Covariant differentiation can be extended to other types of tensorsby demanding that the operator D obey the product rule of differentialcalculus. (For scalars, it is understood that D ≡ d.) For example, wemay derive an expression for the covariant derivative of a dual vectorfrom the requirement

d(Aαpα) ≡ D(Aαpα) = (DAα)pα + AαD(pα).

Writing the left-hand side as Aα,βpαdxβ+Aαpα,βdxβ and using Eqs. (1.2.1)

and (1.2.2), we obtainDpα

dλ= pα;βuβ, (1.2.4)

where

pα;β ≡ pα,β − Γµαβpµ. (1.2.5)

This procedure generalizes easily to tensors of arbitrary type. For ex-ample, the covariant derivative of a type-(1, 1) tensor is given by

Tαβ;γ = T α

β,γ + ΓαµγT

µβ − Γµ

βγTαµ. (1.2.6)

The rule is that there is a connection term for each tensorial index; itcomes with a plus sign if the index is a superscript, or with a minussign if the index is a subscript.

Up to now the connection has been left completely arbitrary. Aspecific choice is made by demanding that it be symmetric and metriccompatible,

Γαγβ = Γα

βγ, gαβ;γ = 0. (1.2.7)

6 Fundamentals

In general relativity, these properties come as a consequence of Ein-stein’s principle of equivalence. It is easy to show that Eqs. (1.2.7)imply

Γαβγ =

1

2gαµ(gµβ,γ + gµγ,β − gβγ,µ). (1.2.8)

Thus, the connection is fully determined by the metric. In this context,Γα

βγ are called the Christoffel symbols.We conclude this section with some terminology: A tensor field

Tα···β··· is said to be parallel transported along a curve γ if its covariant

derivative along the curve vanishes: DT α···β···/dλ = T α···

β···;µuµ = 0.

1.3 Geodesics

A curve is a geodesic if it extremizes the distance between two fixedpoints.

Let a curve γ be described by the relations xα(λ), where λ is anarbitrary parameter, and let P and Q be two points on this curve. Thedistance between P and Q along γ is given by

` =

∫ Q

P

√±gαβxαxβ dλ, (1.3.1)

where xα ≡ dxα/dλ. In the square root, the positive (negative) signis chosen if the curve is spacelike (timelike); it is assumed that γ isnowhere null. It is clear that ` is invariant under a reparameterizationof the curve, λ → λ′(λ).

The curve for which ` is an extremum is determined by substitutingthe “Lagrangian” L(xµ, xµ) = (±gµν x

µxν)1/2 into the Euler-Lagrangeequations,

d

dλ

∂L

∂xα− ∂L

∂xα= 0.

A straightforward calculation shows that xα(λ) must satisfy the differ-ential equation

xα + Γαβγx

βxγ = κ(λ)xα (arbitrary parameter), (1.3.2)

where κ ≡ d ln L/dλ. The geodesic equation can also be written asuα

;βuβ = κuα, in which uα = xα is tangent to the geodesic.A particularly useful choice of parameter is proper time τ when the

geodesic is timelike, or proper distance s when the geodesic is spacelike.(It is important that this choice be made after extremization, and notbefore.) Because dτ 2 = −gαβdxαdxβ for timelike geodesics and ds2 =gαβdxαdxβ for spacelike geodesics, we have that L = 1 in either case,and this implies κ = 0. The geodesic equation becomes

xα + Γαβγx

βxγ = 0 (affine parameter), (1.3.3)

1.4 Lie differentiation 7

or uα;βuβ = 0, which states that the tangent vector is parallel trans-

ported along the geodesic. These equations are invariant under repa-rameterizations of the form λ → λ′ = aλ + b, where a and b are con-stants. Parameters related to s and τ by such transformations are calledaffine parameters. It is useful to note that Eq. (1.3.3) can be recoveredby substituting L′ = 1

2gαβxαxβ into the Euler-Lagrange equations; this

gives rise to practical method of computing the Christoffel symbols.By continuity, the general form uα

;βuβ = κuα for the geodesic equa-tion must be valid also for null geodesics. For this to be true, the pa-rameter λ cannot be affine, because ds = dτ = 0 along a null geodesic,and the limit is then singular. However, affine parameters can nev-ertheless be found for null geodesics. Starting from Eq. (1.3.2) it isalways possible to introduce a new parameter λ∗ such that the geodesicequation will take the form of Eq. (1.3.3). It is easy to check that theappropriate transformation is

dλ∗

dλ= exp

[∫ λ

κ(λ′)dλ′]. (1.3.4)

(You will be asked to provide a proof of this statement in Sec. 1.13,Problem 2.) It should be noted that while the null version of Eq. (1.3.2)was obtained by a limiting procedure, the null version of Eq. (1.3.3)cannot be considered to be a limit of the same equation for timelike orspacelike geodesics: the parameterization is highly discontinuous.

We conclude this section with the following remark: Along an affinelyparameterized geodesic (timelike, spacelike, or null), the scalar quantityε = uαuα is a constant. The proof requires a single line:

dε

dλ= (uαuα);βuβ = (uα

;βuβ)uα + uα(uα;βuβ) = 0.

If proper time or proper distance is chosen for λ, then ε = ∓1, respec-tively. For a null geodesic, ε = 0.

1.4 Lie differentiation

In Sec. 1.2, covariant differentiation was defined by introducing a rule totransport a tensor from a point Q to a neighbouring point P , at whichthe derivative was to be evaluated. This rule involved the introductionof a new structure on the manifold, the connection. In this sectionwe define another type of derivative — the Lie derivative — withoutintroducing any additional structure.

Consider a curve γ, its tangent vector uα = dxα/dλ, and a vectorfield Aα defined in a neighbourhood of γ (Fig. 1.2). As before, the pointP shall have the coordinates xα, while the point Q shall be at xα +dxα.The equation

x′α ≡ xα + dxα = xα + uα dλ

8 Fundamentals

can be interpreted as an infinitesimal coordinate transformation fromthe system x to the system x′. Under this transformation, the vectorAα becomes

A′α(x′) =∂x′α

∂xβAβ(x)

= (δαβ + uα

,β dλ)Aβ(x)

= Aα(x) + uα,βAβ(x) dλ.

In other words,

A′α(Q) = Aα(P ) + uα,βAβ(P ) dλ.

On the other hand, Aα(Q), the value of the original vector field at thepoint Q, can be expressed as

Aα(Q) = Aα(x + dx)

= Aα(x) + Aα,β(x) dxβ

= Aα(P ) + uβAα,β(P ) dλ.

In general, A′α(Q) and Aα(Q) will not be equal. Their difference definesthe Lie derivative of the vector Aα along the curve γ:

£uAα(P ) ≡ Aα(Q)− A′α(Q)

dλ.

Combining the previous three equations yields

£uAα = Aα

,βuβ − uα,βAβ. (1.4.1)

Despite an appearance to the contrary, £uAα is a tensor: It is easy to

check that Eq. (1.4.1) is equivalent to

£uAα = Aα

;βuβ − uα;βAβ, (1.4.2)

whose tensorial nature is evident.The definition of the Lie derivative extends to all types of tensors.

For scalars, £uf ≡ df = f,αuα. For dual vectors, the same steps revealthat

£upα = pα,βuβ + uβ,αpβ

(1.4.3)

= pα;βuβ + uβ;αpβ.

As another example, the Lie derivative of a type-(1, 1) tensor is givenby

£uTαβ = T α

β,µuµ − uα

,µTµβ + uµ

,βT αµ

(1.4.4)

= T αβ;µu

µ − uα;µT

µβ + uµ

;βT αµ.

1.5 Killing vectors 9

Further generalizations are obvious. It may be verified that the Liederivative obeys the product rule of differential calculus. For example,the relation

£u(Aαpβ) = (£uA

α)pβ + Aα(£upβ) (1.4.5)

is easily established.

A tensor field T α···β··· is said to be Lie transported along a curve γ

if its Lie derivative along the curve vanishes: £uTα···

β··· = 0, where uα

is the curve’s tangent vector. Suppose that the coordinates are chosenso that x1, x2, and x3 are all constant on γ, while x0 ≡ λ varies on γ.In such a coordinate system,

uα =dxα

dλ∗= δα

0,

where the symbol “∗=” means “equals in the specified coordinate sys-

tem”. It follows that uα,β

∗= 0, so that

£uTα···

β···∗= Tα···

β···,µuµ ∗

=∂

∂x0T α···

β···.

If the tensor is Lie transported along γ, then the tensor’s componentsare all independent of x0 in the specified coordinate system.

We have established the following theorem:

If £uTα···

β··· = 0, that is, if a tensor is Lie transported along a curveγ with tangent vector uα, then a coordinate system can be constructedsuch that uα ∗

= δα0 and Tα···

β···,0∗= 0. Conversely, if in a given coordi-

nate system the components of a tensor do not depend on a particularcoordinate x0, then the Lie derivative of the tensor in the direction ofuα vanishes.

Thus, the Lie derivative is the natural construct to express, covari-antly, the invariance of a tensor under a change of position.

1.5 Killing vectors

If, in a given coordinate system, the components of the metric do notdepend on x0, then by the preceding theorem, £ξgαβ = 0, where ξα ∗

=δα

0. The vector ξα is then called a Killing vector. The condition for ξα

to be a Killing vector is that

0 = £ξgαβ = ξα;β + ξβ;α. (1.5.1)

Thus, the tensor ξα;β is antisymmetric if ξα is a Killing vector.

Killing vectors can be used to find constants associated with themotion along a geodesic. Suppose that uα is tangent to a geodesic

10 Fundamentals

affinely parameterized by λ. Then

d

dλ(uαξα) = (uαξα);βuβ

= uα;βuβξα + ξα;βuαuβ

= 0.

In the second line, the first term vanishes by virtue of the geodesicequation, and the second term vanishes because ξα;β is an antisymmet-ric tensor and uαuβ is symmetric. Thus, uαξα is constant along thegeodesic.

As an example, consider a static, spherically symmetric spacetimewith metric

ds2 = −A(r) dt2 + B(r) dr2 + r2 dΩ2,

where dΩ2 = dθ2 + sin2 θ dφ2. Because the metric does not depend ont nor φ, the vectors

ξα(t) =

∂xα

∂t, ξα

(φ) =∂xα

∂φ

are Killing vectors. This implies that along timelike geodesics, thequantities

E = −uαξα(t), L = uαξα

(φ)

are constants. These can be interpreted as energy and angular momen-tum per unit mass, respectively. It should also be noted that sphericalsymmetry implies the existence of two additional Killing vectors,

ξα(1)∂α = sin φ ∂θ + cot θ cos φ ∂φ, ξα

(2)∂α = − cos φ ∂θ + cot θ sin φ ∂φ.

It is straightforward to show that these do indeed satisfy Killing’s equa-tion (1.5.1). (To prove this is the purpose of Sec. 1.13, Problem 5.)

1.6 Local flatness

For a given point P in spacetime, it is always possible to find a coordi-nate system xα′ such that

gα′β′(P ) = ηα′β′ , Γα′β′γ′(P ) = 0, (1.6.1)

where ηα′β′ = diag(−1, 1, 1, 1) is the Minkowski metric. Such a coordi-nate system will be called a local Lorentz frame at P . We note that itis not possible to also set the derivatives of the connection to zero if thespacetime is curved. The physical interpretation of the local-flatnesstheorem is that free-falling observers see no effect of gravity in theirimmediate vicinity, as required by Einstein’s principle of equivalence.

1.6 Local flatness 11

We now prove the theorem. Let xα be an arbitrary coordinatesystem, and let us assume, with no real loss of generality, that P is atthe origin of both coordinate systems. Then the coordinates of a pointnear P are related by

xα′ = Aα′βxβ + O(x2), xα = Aα

β′xβ′ + O(x′2),

where Aα′β and Aα

β′ are constant matrices. It is easy to check that oneis in fact the inverse of the other:

Aα′µA

µβ′ = δα′

β′ , Aαµ′A

µ′β = δα

β.

Under this transformation, the metric becomes

gα′β′(P ) = Aαα′A

ββ′gαβ(P ).

We demand that the left-hand side be equal to ηα′β′ . This gives us10 equations for the 16 unknown components of the matrix Aα

α′ . Asolution can always be found, with 6 undetermined components. Thiscorresponds to the freedom of performing a Lorentz transformation (3rotation parameters and 3 boost parameters) which does not alter theform of the Minkowski metric.

Suppose that a particular choice has been made for Aαα′ . Then Aα′

α

is found by inverting the matrix, and the coordinate transformation isknown to first order. Let us proceed to second order:

xα′ = Aα′βxβ +

1

2Bα′

βγxβxγ + O(x3),

where the constant coefficients Bα′βγ are symmetric in the lower indices.

Recalling Eq. (1.2.3), we have that the connection transforms as

Γα′β′γ′(P ) = Aα′

αAββ′A

γγ′Γ

αβγ(P )−Bα′

βγAββ′A

γγ′ .

To put the left-hand side to zero, it is sufficient to impose

Bα′βγ = −Aα′

αΓαβγ(P ).

These equations determine Bα′βγ uniquely, and the coordinate transfor-

mation is now known to second order. Irrespective of the higher-orderterms, it enforces Eqs. (1.6.1).

We shall return in Sec. 1.11 with a more geometric proof of thelocal-flatness theorem, and its extension from a single point P to anentire geodesic γ.

12 Fundamentals

1.7 Metric determinant

The quantity√−g, where g ≡ det[gαβ], occurs frequently in differ-

ential geometry. We first note that√

g′/g, where g′ = det[gα′β′ ], isthe Jacobian of the transformation xα → xα′(xα). To see this, recallfrom ordinary differential calculus that under such a transformation,d4x = J d4x′, where J = det[∂xα/∂xα′ ] is the Jacobian. Now considerthe transformation of the metric,

gα′β′ =∂xα

∂xα′∂xβ

∂xβ′ gαβ.

Because the determinant of a product of matrices is equal to the productof their determinants, this equation implies g′ = gJ2, which proves theassertion.

As an important application, consider the transformation from xα′ ,a local Lorentz frame at P , to xα, an arbitrary coordinate system.The four-dimensional volume element around P is d4x′ = J−1 d4x =√

g/g′ d4x. But since g′ = −1, we have that

√−g d4x (1.7.1)

is an invariant volume element around the arbitrary point P . Thisresult generalizes to a manifold of any dimension with a metric of anysignature; in this case, |g|1/2dnx is the invariant volume element, wheren is the dimension of the manifold.

We shall now derive another useful result,

Γµµα =

1

2gµνgµν,α =

1√−g(√−g),α. (1.7.2)

Consider, for any matrix M , the variation of ln |detM | induced by avariation of M ’s components. Using the product rule for determinants,we have

δ ln |detM | ≡ ln |det(M + δM)| − ln |detM |= ln

det(M + δM )

detM= ln detM−1(M + δM)

= ln det(1 + M−1δM).

We now use the identity det(1 + ε) = 1 + Tr ε + O(ε2), valid for any“small” matrix ε. (Try proving this for 3× 3 matrices.) This gives

δ ln |detM | = ln(1 + Tr M−1δM )

= Tr M−1δM .

1.8 Levi-Civita tensor 13

Substituting the metric tensor in place of M gives δ ln |g| = gαβδgαβ,or

∂

∂xµln |g| = gαβgαβ,µ.

This establishes Eq. (1.7.2).Equation (1.7.2) gives rise to the divergence formula: For any vector

field Aα,

Aα;α =

1√−g(√−gAα),α. (1.7.3)

A similar result holds for any antisymmetric tensor field Bαβ:

Bαβ;β =

1√−g(√−gBαβ),β. (1.7.4)

These formulae are useful for the efficient computation of covariantdivergences.

1.8 Levi-Civita tensor

The permutation symbol [α β γ δ], defined by

[α β γ δ] =

+1 if αβγδ is an even permutation of 0123−1 if αβγδ is an odd permutation of 0123

0 if any two indices are equal,

(1.8.1)is a very useful, non-tensorial quantity. For example, it can be used togive a definition for the determinant: For any 4× 4 matrix Mαβ,

det[Mαβ] = [α β γ δ]M0αM1βM2γM3δ

(1.8.2)

= [α β γ δ]Mα0Mβ1Mγ2Mδ3.

Either equality can be established by brute-force computation. Thewell-known property that det[Mβα] = det[Mαβ] follows directly fromEq. (1.8.2).

We shall now show that the combination

εαβγδ =√−g [α β γ δ] (1.8.3)

is a tensor, called the Levi-Civita tensor. Consider the quantity

[α β γ δ]∂xα

∂xα′∂xβ

∂xβ′∂xγ

∂xγ′∂xδ

∂xδ′ ,

which is completely antisymmetric in the primed indices. This musttherefore be proportional to [α′ β′ γ′ δ′]:

[α β γ δ]∂xα

∂xα′∂xβ

∂xβ′∂xγ

∂xγ′∂xδ

∂xδ′ = λ[α′ β′ γ′ δ′],

14 Fundamentals

for some proportionality factor λ. Putting α′β′γ′δ′ = 0123 yields

λ = [α β γ δ]∂xα

∂x0′∂xβ

∂x1′∂xγ

∂x2′∂xδ

∂x3′ ,

which determines λ. But the right-hand side is just the determinantof the matrix ∂xα/∂xα′ , that is, the Jacobian of the transformationxα′(xα). So λ =

√g′/g, and we have

√−g [α β γ δ]∂xα

∂xα′∂xβ

∂xβ′∂xγ

∂xγ′∂xδ

∂xδ′ =√−g′ [α′ β′ γ′ δ′].

This establishes the fact that εαβγδ does indeed transform as a type-(0, 4) tensor.

The preceding proof could have started instead with the relation

[α β γ δ]∂xα′

∂xα

∂xβ′

∂xβ

∂xγ′

∂xγ

∂xδ′

∂xδ= λ′[α′ β′ γ′ δ′],

implying λ′ =√

g/g′ and showing that

εαβγδ = − 1√−g[α β γ δ] (1.8.4)

transforms as a type-(4, 0) tensor. (The minus sign is important.)It is easy to check that this is also the Levi-Civita tensor, obtainedfrom εαβγδ by raising all four indices. Alternatively, we may show thatεαβγδ = gαµgβνgγλgδρε

µνλρ. This relation implies

ε0123 = − 1√−g[µ ν λ ρ]g0µg1νg2λg3ρ = − 1√−g

g =√−g,

which is evidently compatible with Eq. (1.8.3).The Levi-Civita tensor is used in a variety of contexts in differential

geometry. We will meet it again in Chapter 3.

1.9 Curvature

The Riemann tensor Rαβγδ may be defined by the relation

Aµ;αβ − Aµ

;βα = −RµναβAν , (1.9.1)

which holds for any vector field Aα. Evaluating the left-hand side ex-plicitly yields

Rαβγδ = Γα

βδ,γ − Γαβγ,δ + Γα

µγΓµβδ − Γα

µδΓµβγ. (1.9.2)

The Riemann tensor is obviously antisymmetric in the last two indices.Its other symmetry properties can be established by evaluating Rα

βγδ in

1.9 Curvature 15

a local Lorentz frame at some point P . A straightforward computationgives

Rαβγδ∗=

1

2(gαδ,βγ − gαγ,βδ − gβδ,αγ + gβγ,αδ),

and this implies the tensorial relations

Rαβγδ = −Rβαγδ = −Rαβδγ = Rγδαβ (1.9.3)

andRµαβγ + Rµγαβ + Rµβγα = 0, (1.9.4)

which are valid in any coordinate system. A little more work alongthe same lines reveals that the Riemann tensor satisfies the Bianchiidentities,

Rµναβ;γ + Rµνγα;β + Rµνβγ;α = 0. (1.9.5)

In addition to Eq. (1.9.1), the Riemann tensor satisfies the relations

pµ;αβ − pµ;βα = Rνµαβpν (1.9.6)

andT µ

ν;αβ − T µν;βα = −Rµ

λαβT λν + Rλ

ναβT µλ, (1.9.7)

which hold for arbitrary tensors pα and T αβ. Generalization to tensors

of higher ranks is obvious: the number of Riemann-tensor terms on theright-hand side is equal to the number of tensorial indices.

Contractions of the Riemann tensor produce the Ricci tensor Rαβ

and the Ricci scalar R. These are defined by

Rαβ = Rµαµβ, R = Rα

α. (1.9.8)

It is easy to show that Rαβ is a symmetric tensor. The Einstein tensoris defined by

Gαβ = Rαβ − 1

2Rgαβ; (1.9.9)

this is also a symmetric tensor. By virtue of Eq. (1.9.5), the Einsteintensor satisfies

Gαβ;β = 0, (1.9.10)

the contracted Bianchi identities.The Einstein field equations,

Gαβ = 8π Tαβ, (1.9.11)

relate the spacetime curvature (as represented by the Einstein tensor)to the distribution of matter (as represented by Tαβ, the stress-energytensor). Equation (1.9.10) implies that the stress-energy tensor musthave a zero divergence: Tαβ

;β = 0. This is the tensorial expression forenergy-momentum conservation. Equation (1.9.10) implies also that of

16 Fundamentals

the ten equations (1.9.11), only six are independent. The metric cantherefore be determined up to four arbitrary functions, and this reflectsour complete freedom in choosing the coordinate system. We note thatthe field equations can also be written in the form

Rαβ = 8π(Tαβ − 1

2Tgαβ

), (1.9.12)

where T ≡ Tαα is the trace of the stress-energy tensor.

1.10 Geodesic deviation

The geometrical meaning of the Riemann tensor is best illustrated byexamining the behaviour of neighbouring geodesics. Consider two suchgeodesics, γ0 and γ1, described by relations xα(t) in which t is an affineparameter; the geodesics can be either spacelike, timelike, or null. Wewant to develop the notion of a deviation vector between these twogeodesics, and derive an evolution equation for this vector.

For this purpose we introduce, in the space between γ0 and γ1, anentire family of interpolating geodesics (Fig. 1.3). To each geodesic weassign a label s ∈ [0, 1], such that γ0 comes with the label s = 0 andγ1 with s = 1. We collectively describe these geodesics with relationsxα(s, t), in which s serves to specify which geodesic and t is an affineparameter along the specified geodesic. The vector field uα = ∂xα/∂tis tangent to the geodesics, and it satisfies the equation uα

;βuβ = 0.

If we keep t fixed in the relations xα(s, t) and vary s instead, weobtain another family of curves, labelled by t and parameterized by s; ingeneral these curves will not be geodesics. The family has ξα = ∂xα/∂sas its tangent vector field, and the restriction of this vector to γ0, ξα|s=0,gives a meaningful notion of a deviation vector between γ0 and γ1. We

Figure 1.3: Deviation vector between two neighbouring geodesics.

t t

s

s

s

γ0 γ1

uα

ξα

1.10 Geodesic deviation 17

wish to derive an expression for its acceleration,

D2ξα

dt2≡ (ξα

;βuβ);γuγ, (1.10.1)

in which it is understood that all quantities are to be evaluated on γ0.In flat spacetime, the geodesics γ0 and γ1 are straight, and althoughtheir separation may change with t, this change is necessarily linear:D2ξα/dt2 = 0 in flat spacetime. A nonzero result for D2ξα/dt2 willtherefore reveal the presence of curvature, and indeed, this vector willbe found to be proportional to the Riemann tensor.

It follows at once from the relations uα = ∂xα/∂t and ξα = ∂xα/∂sthat

£uξα = £ξu

α = 0 ⇒ ξα;βuβ = uα

;βξβ. (1.10.2)

We also have at our disposal the geodesic equation, uα;βuβ = 0. These

equations can be combined to prove that ξαuα is constant along γ0:

d

dtξαuα = (ξαuα);βuβ

= ξα;βuβuα + ξαuα;βuβ

= uα;βξβuα

=1

2(uαuα);βξβ

= 0,

because uαuα ≡ ε is a constant. The parameterization of the interpo-lating geodesics can therefore be tuned so that on γ0, ξα is everywhereorthogonal to uα:

ξαuα = 0. (1.10.3)

This means that the curves t = constant cross γ0 orthogonally. Thisadds weight to the interpretation of ξα as a deviation vector.

We may now calculate the relative acceleration of γ1 with respectto γ0. Starting from Eq. (1.10.1) and using Eqs. (1.9.1) and (1.10.2),we obtain

D2ξα

dt2= (ξα

;βuβ);γuγ

= (uα;βξβ);γu

γ

= uα;βγξ

βuγ + uα;βξβ

;γuγ

= (uα;γβ −Rα

µβγuµ)ξβuγ + uα

;βuβ;γξ

γ

= (uα;γu

γ);βξβ − uα;γu

γ;βξβ −Rα

µβγuµξβuγ + uα

;βuβ;γξ

γ.

The first term vanishes by virtue of the geodesic equation, while thesecond and fourth terms cancel out, leaving

D2ξα

dt2= −Rα

βγδuβξγuδ. (1.10.4)

18 Fundamentals

This is the geodesic deviation equation. It shows that curvature pro-duces a relative acceleration between two neighbouring geodesics; evenif they start parallel, curvature prevents the geodesics from remainingparallel.

1.11 Fermi normal coordinates

The proof of the local-flatness theorem presented in Sec. 1.6 gives verylittle indication as to how one might construct a coordinate system thatwould enforce Eqs. (1.6.1). Our purpose in this section is to return tothis issue, and provide a more geometric proof of the theorem. In fact,we will extend the theorem from a single point P to an entire geodesicγ. For concreteness we will take the geodesic to be timelike.

We will show that we can introduce coordinates xα = (t, xa) suchthat near γ, the metric can be expressed as

gtt = −1−Rtatb(t)xaxb + O(x3),

gta = −2

3Rtbac(t)x

bxc + O(x3), (1.11.1)

gab = δab − 1

3Racbd(t)x

cxd + O(x3).

These coordinates are known as Fermi normal coordinates, and t isproper time along the geodesic γ, on which the spatial coordinates xa

are all zero. In Eq. (1.11.1), the components of the Riemann tensorare evaluated on γ, and they depend on t only. It is obvious thatEq. (1.11.1) enforces gαβ|γ = ηαβ and Γµ

αβ|γ = 0. The local-flatnesstheorem therefore holds everywhere on the geodesic.

1.11.1 Geometric construction

We will use xα = (t, xa) to denote the Fermi normal coordinates, andxα′ will refer to an arbitrary coordinate system. We imagine that weare given a spacetime with a metric gα′β′ expressed in these coordinates.

We consider a timelike geodesic γ in this spacetime. Its tangentvector is uα′ , and we let t be proper time along γ. On this geodesicwe select a point O at which we set t = 0. At this point we erect anorthonormal basis eα′

µ (the subscript µ serves to label the four basis

vectors), and we identify eα′t with the tangent vector uα′ at O. From

this we construct a basis everywhere on γ by parallel transporting eα′µ

away from O. Our basis vectors therefore satisfy

eα′µ;β′u

β′ = 0, eα′t = uα′ , (1.11.2)

as well asgα′β′ e

α′µ eβ′

ν = ηµν , (1.11.3)

1.11 Fermi normal coordinates 19

everywhere on γ. Here, ηµν = diag(−1, 1, 1, 1) is the Minkowski metric.Consider now a spacelike geodesic β originating at a point P on γ,

at which t = tP . This geodesic has a tangent vector vα′ , and we let sdenote proper distance along β; we set s = 0 at P . We assume that atP , vα′ is orthogonal to uα′ , so that it admits the decomposition

vα′|γ = Ωa eα′a . (1.11.4)

To ensure that vα′ is properly normalized, the expansion coefficientsmust satisfy δabΩ

aΩb = 1. By choosing different coefficients Ωa we canconstruct new geodesics β that are also orthogonal to γ at P . We shalldenote this entire family of spacelike geodesics by β(tP , Ωa).

The Fermi normal coordinates of a point Q located off the geodesicγ are constructed as follows (Fig. 1.4). First we find the unique geodesicthat passes through Q and intersects γ orthogonally. We label the inter-section point P , and we call this geodesic β(tP , Ωa

Q), with tP denotingproper time at the intersection point, and Ωa

Q the expansion coefficients

of vα′ at that point. We then assign to Q the new coordinates

x0 = tP , xa = ΩaQ sQ, (1.11.5)

where sQ is proper distance from P to Q. These are the Fermi normalcoordinates of the point Q. Generically, therefore, xα = (t, Ωas), andwe must now figure out how these coordinates are related to xα′ , theoriginal system.

1.11.2 Coordinate transformation

For this purpose, we note first that we can describe the family ofgeodesics β(t, Ωa) by relations of the form xα′(t, Ωa, s). In these, theparameters t and Ωa serve to specify which geodesic, and s is properdistance along this geodesic. If we substitute s = 0 in these relations,

Figure 1.4: Geometric construction of the Fermi normal coordinates.

γ

P

uα

t

s

Q vα

β

20 Fundamentals

we recover the description of the timelike geodesic γ in terms of itsproper time t; the parameters Ωa are then irrelevant. The tangent tothe geodesics β(t, Ωa) is

vα′ =(∂xα′

∂s

)t,Ωa

; (1.11.6)

the notation explicitly indicates that the derivative with respect to s istaken while keeping t and Ωa fixed. This vector satisfies the geodesicequation and is subjected to the initial condition vα′ |s=0 = Ωaeα′

a . Butthe geodesic equation is invariant under a rescaling of the affine param-eter, s → s/c, in which c is a constant. Under this rescaling, vα′ → c vα′

and as a consequence, we have that Ωa → c Ωa. We have therefore es-tablished the identity xα′(t, Ωa, s) = xα′(t, c Ωa, s/c), and as a specialcase, we find

xα′(t, Ωa, s) = xα′(t, Ωas, 1) ≡ xα′(xα). (1.11.7)

By virtue of Eqs. (1.11.5), this relation is the desired transformationbetween xα′ and the Fermi normal coordinates.

Now, as a consequence of Eqs. (1.11.4), (1.11.6), and (1.11.7), wehave

Ωaeα′a = vα′|γ =

∂xα′

∂s

∣∣∣s=0

=∂xα′

∂xa

∣∣∣s=0

Ωa,

which shows that∂xα′

∂xa

∣∣∣γ

= eα′a . (1.11.8)

From our previous observation that the relations xα′(t, Ωa, 0) describethe geodesic γ, we also have

∂xα′

∂t

∣∣∣γ

= uα′ ≡ eα′t . (1.11.9)

Equations (1.11.8) and (1.11.9) tell us that on γ, ∂xα′/∂xµ = eα′µ .

1.11.3 Deviation vectors

Suppose now that in the relations xα′(t, Ωa, s), the parameters Ωa arevaried while keeping t and s fixed. This defines new curves that connectdifferent geodesics β at the same proper distance s from their commonintersection point P on γ. This is very similar to the constructiondescribed in Sec. 1.10, and the vectors

ξα′a =

(∂xα′

∂Ωa

)t,s

(1.11.10)

1.11 Fermi normal coordinates 21

are deviation vectors relating geodesics β(t, Ωa) with different coeffi-cients Ωa. Similarly,

ξα′t =

(∂xα′

∂t

)s,Ωa

(1.11.11)

is a deviation vector relating geodesics β(t, Ωa) that start at differentpoints on γ, but share the same coefficients Ωa. The four vectors definedby Eqs. (1.11.10) and (1.11.11) satisfy the geodesic deviation equation,Eq. (1.10.4). (It must be kept in mind that in this equation, the tangentvector is vα′ , not uα′ , and the affine parameter is s, not t.)

1.11.4 Metric on γ

The components of the metric in the Fermi normal coordinates arerelated to the old components by the general relation

gαβ =∂xα′

∂xα

∂xβ′

∂xβgα′β′ .

Evaluating this on γ yields gαβ|γ = eα′α eβ′

β gα′β′ , after using Eqs. (1.11.8)and (1.11.9). Substituting Eq. (1.11.3), we arrive at

gαβ|γ = ηαβ. (1.11.12)

This states that in the Fermi normal coordinates, the metric is Minkowskieverywhere on the geodesic γ.

1.11.5 First derivatives of the metric on γ

To evaluate the Christoffel symbols in the Fermi normal coordinates, werecall from Eq. (1.11.5) that the curves x0 = t, xa = Ωas are geodesics,so that these relations must be solutions to the geodesic equation,

d2xα

ds2+ Γα

βγ

dxβ

ds

dxγ

ds= 0.

This gives Γαbc(x

α)ΩbΩc = 0. On γ, the Christoffel symbols are func-tions of t only, and are therefore independent of Ωa. Since these co-efficients are arbitrary, we conclude that Γα

bc|γ = 0. To obtain theremaining components, we recall that the basis vectors eα

µ are paralleltransported along γ, so that

deαµ

dt+ Γα

βγ|γ eβµeγ

t = 0,

since eγt = uα. By virtue of Eqs. (1.11.8) and (1.11.9), we have that

eαµ = δα

µ in the Fermi normal coordinates, and the parallel-transport

22 Fundamentals

equation implies Γαβt|γ = 0. The Christoffel symbols are therefore all

zero on γ. We shall write this as

gαβ,γ|γ = 0. (1.11.13)

This proves that the Fermi normal coordinates enforce the local-flatnesstheorem everywhere on the timelike geodesic γ.

1.11.6 Second derivatives of the metric on γ

We next turn to the second derivatives of the metric, or the first deriva-tives of the connection. From the fact that Γα

βγ is zero everywhere onγ, we obtain immediately

Γαβγ,t|γ = 0. (1.11.14)

From the definition of the Riemann tensor, Eq. (1.9.2), we also get

Γαβt,γ|γ = Rα

βγt|γ. (1.11.15)

The other components are harder to come by. For these we must involvethe deviation vectors ξα

µ introduced in Eqs. (1.11.10) and (1.11.11).These vectors satisfy the geodesic deviation equation, Eq. (1.10.4),which we write in full as

d2ξα

ds2+2Γα

βγvβ dξγ

ds+

(Rα

βγδ +Γαβγ,δ−Γα

γµΓµβδ +Γα

δµΓµβγ

)vβξγvδ = 0.

According to Eqs. (1.11.5), (1.11.6), (1.11.10), and (1.11.11), we havethat vα = Ωaδα

a, ξαt = δα

t, and ξαa = sδα

a in the Fermi normal coordi-nates. If we substitute ξα = ξα

t in the geodesic deviation equation andevaluate it at s = 0, we find Γα

bt,c|γ = Rαbct|γ, which is just a special

case of Eq. (1.11.15).To learn something new, let us substitute ξα = ξα

a instead. In thiscase we find

2ΓαabΩ

b + s(Rα

bad + Γαab,d − Γα

aµΓµbd + Γα

dµΓµab

)ΩbΩd = 0.

Before evaluating this on γ (which would give 0 = 0), we expand thefirst term in a power series in s:

Γαab = Γα

ab|γ + sΓαab,µ|γvµ + O(s2) = sΓα

ab,d|γΩd + O(s2).

Dividing through by s and then evaluating on γ, we arrive at

(Rαbad + 3Γα

ab,d)|γΩbΩd = 0.

1.12 Bibliographical notes 23

Because the coefficients Ωa are arbitrary, we conclude that the quantitywithin the brackets, properly symmetrized in the indices b and d, mustvanish. A little algebra finally reveals

Γαab,c|γ = −1

3(Rα

abc + Rαbac)|γ. (1.11.16)

Equations (1.11.14), (1.11.15), and (1.11.16) give the complete set ofderivatives of the Christoffel symbols on γ.

It is now a simple matter to turn these equations into statementsregarding the second derivatives of the metric at γ. Because the metricis Minkowski everywhere on the geodesic, only the spatial derivativesare nonzero. These are given by

gtt,ab = −2Rtatb|γ,gta,bc = −2

3(Rtbac + Rtcab)|γ, (1.11.17)

gab,cd = −1

3(Racbd + Radbc)|γ.

From Eqs. (1.11.12), (1.11.13), and (1.11.17) we recover Eqs. (1.11.1),the expansion of the metric about γ, to second order in the spatialdisplacements xa.

1.11.7 Riemann tensor in Fermi normal coordinates

To express a given metric as an expansion in Fermi normal coordinates,it is necessary to evaluate the Riemann tensor on the reference geodesic,and write it as a function of t in this coordinate system. This is not ashard as it may seem. Because the Riemann tensor is evaluated on γ,we need to know the coordinate transformation only at γ; as was notedabove, this is given by ∂xα′/∂xµ = eα′

µ . We therefore have, for example,

Rtabc(t) = Rµ′α′β′γ′ eµ′t eα′

a eβ′b eγ′

c .

The difficult part of the calculation is therefore the determination ofthe orthonormal basis (which is parallel transported on the referencegeodesic). Once this is known, the Fermi components of the Riemanntensor are obtained by projection, and these will naturally be expressedin terms of t.

1.12 Bibliographical notes

Nothing in this text can be claimed to be entirely original, and thebibliographical notes at the end of each chapter intend to give credit

24 Fundamentals

where credit is due. During the preparation of this chapter I have re-lied on the following references: d’Inverno (1992); Manasse and Misner(1963); Misner, Thorne, and Wheeler (1973); Wald (1984); and Wein-berg (1972).

More specifically:Sections 1.2, 1.4, and 1.6 are based on Secs. 6.3, 6.2, and 6.11 of

d’Inverno, respectively. Sections 1.7 and 1.8 are based on Secs. 4.7and 4.4 of Weinberg, respectively. Section 1.10 is based on Sec. 3.3 ofWald. Finally, Sec. 1.11 and Problem 10 below are based on the paperby Manasse and Misner.

1.13 Problems

Warning: The results derived in Problem 9 are used in later portionsof this book.

1. The surface of a two-dimensional cone is embedded in three-dimensional flat space. The cone has an opening angle of 2α.Points on the cone which all have the same distance r from theapex define a circle, and φ is the angle that runs along the circle.

a) Write down the metric of the cone, in terms of the coordinatesr and φ.

b) Find the coordinate transformation x(r, φ), y(r, φ) that bringsthe metric into the form ds2 = dx2 + dy2. Do these coordi-nates cover the entire two-dimensional plane?

c) Prove that any vector parallel transported along a circle ofconstant r on the surface of the cone ends up rotated by anangle β after a complete trip. Express β in terms of α.

2. Show that if tα = dxα/dλ obeys the geodesic equation in the formtα;βtβ = κtα, then uα = dxα/dλ∗ satisfies uα

;βuβ = 0 if λ∗ and λare related by dλ∗/dλ = exp

∫κ(λ) dλ.

3. a) Let xα(λ) describe a timelike geodesic parameterized by a non-affine parameter λ, and let tα = dxα/dλ be the geodesic’stangent vector. Calculate how ε ≡ −tαtα changes as a func-tion of λ.

b) Let ξα be a Killing vector. Calculate how p ≡ ξαtα changesas a function of λ on that same geodesic.

c) Let bα be such that in a spacetime with metric gαβ, £bgαβ =2c gαβ, where c is a constant. (Such a vector is called homo-thetic.) Let xα(τ) describe a timelike geodesic parameterizedby proper time τ , and let uα = dxα/dτ be the four-velocity.Calculate how q ≡ bαuα changes with τ .

1.13 Problems 25

4. Prove that the Lie derivative of a type-(0, 2) tensor is given by£uTαβ = Tαβ;µu

µ + uµ;αTµβ + uµ

;βTαµ.

5. Prove that ξα(1) and ξα

(2), as given in Sec. 1.5, are indeed Killingvectors of spherically symmetric spacetimes.

6. A particle with electric charge e moves in a spacetime with metricgαβ in the presence of a vector potential Aα. The equations ofmotion are uα;βuβ = eFαβuβ, where uα is the four-velocity andFαβ = Aβ;α − Aα;β. It is assumed that the spacetime possesses aKilling vector ξα, so that £ξgαβ = £ξAα = 0.

Prove that

(uα + eAα)ξα

is constant on the world line of the charged particle.

7. In flat spacetime, all Cartesian components of the Levi-Civitatensor can be obtained from εtxyz = 1 by permutation of theindices. Using its tensorial property under coordinate transfor-mations, calculate εαβγδ in the following coordinate systems:

a) Spherical coordinates (t, r, θ, φ).

b) Spherical-null coordinates (u, v, θ, φ), where u = t − r andv = t + r.

Show that your results are compatible with the general relationεαβγδ =

√−g [α β γ δ] if [t r θ φ] = 1 in spherical coordinates, while[u v θ φ] = 1 in spherical-null coordinates.

8. In a manifold of dimension n, the Weyl curvature tensor is definedby

Cαβγδ = Rαβγδ− 2

n− 2

(gα[γRδ]β−gβ[γRδ]α

)+

2

(n− 1)(n− 2)R gα[γgδ]β.

Show that it possesses the same symmetries as the Riemann ten-sor. Also, prove that any contracted form of the Weyl tensorvanishes identically. This shows that the Riemann tensor can bedecomposed into a tracefree part given by the Weyl tensor, and atrace part given by the Ricci tensor. The Einstein field equationsimply that the trace part of the Riemann tensor is algebraicallyrelated to the distribution of matter in spacetime; the tracefreepart, on the other hand, is algebraically independent of the mat-ter. Thus, it can be said that the Weyl tensor represents the truegravitational degrees of freedom of the Riemann tensor.

26 Fundamentals

9. Prove that the relations

ξα;µν = Rα

µνβξβ, ¤ξα = −Rαβξβ

are satisfied by any Killing vector ξα. Here, ¤ ≡ ∇α∇α is thecurved-spacetime d’Alembertian operator. [Hint: Use the cyclicidentity for the Riemann tensor, Rµαβγ + Rµγαβ + Rµβγα = 0.]

10. Express the Schwarzschild metric as an expansion in Fermi normalcoordinates about a radially infalling, timelike geodesic.

11. Construct a coordinate system in a neighbourhood of a point Pin spacetime, such that gαβ|P = ηαβ, gαβ,µ|P = 0, and

gαβ,µν |P = −1

3(Rαµβν + Rανβµ)|P .

Such coordinates are called Riemann normal coordinates.

12. A particle moving on a circular orbit in a stationary, axially sym-metric spacetime is subjected to a dissipative force which drivesit to another, slightly smaller, circular orbit. During the transi-tion, the particle loses an amount δE of orbital energy (per unitrest-mass), and an amount δL of orbital angular momentum (perunit rest-mass). You are asked to prove that these quantities arerelated by δE = Ω δL, where Ω is the particle’s original angularvelocity.

By “circular orbit” we mean that the particle has a four-velocitygiven by

uα = γ(ξα(t) + Ω ξα

(φ)),

where ξα(t) and ξα

(φ) are the spacetime’s timelike and rotationalKilling vectors, respectively; Ω and γ are constants.

You may proceed along the following lines: First, express γ interms of E and L. Second, find an expression for δuα, the changein four-velocity as the particle goes from its original orbit to itsfinal orbit. Third, prove the relation

uαδuα = γ(δE − Ω δL),

from which the theorem follows.

Chapter 2

Geodesic congruences

Our purpose in this chapter is to develop the mathematical techniquesrequired in the description of congruences, the term designating an en-tire system of nonintersecting geodesics. We will consider separatelythe cases of timelike geodesics and null geodesics. (The case of space-like geodesics does not require a separate treatment, as it is virtuallyidentical to the timelike case; it is also less interesting from a physicalpoint of view.) We will introduce the expansion scalar, as well as theshear and rotation tensors, as a means of describing the congruence’sbehaviour. We will derive a useful evolution equation for the expan-sion, known as Raychaudhuri’s equation. On the basis of this equation,we will show that gravity tends to focus geodesics, in the sense thatan initially diverging congruence (geodesics flying apart) will be foundto diverge less rapidly in the future, and that an initially convergingcongruence (geodesics coming together) will converge more rapidly inthe future. And we will present Frobenius’ theorem, which states that acongruence is hypersurface orthogonal — the geodesics are everywhereorthogonal to a family of hypersurfaces — if and only if its rotationtensor vanishes.

The chapter begins (Sec. 2.1) with a review of the standard energyconditions of general relativity, since some of these are required in theproof of the focusing theorem. It continues (Sec. 2.2) with a simpleintroduction to the expansion scalar, shear tensor, and rotation ten-sor, based on the kinematics of a deformable medium. Congruences oftimelike geodesics are then presented in Sec. 2.3, and the case of nullgeodesics is treated in Sec. 2.4.

The techniques presented in this chapter are used in many differ-ent areas of gravitational physics. Most notably, they are used in themathematical description of event horizons, a topic covered in Chapter5. They also play a key role in the formulation of the singularity the-orems of general relativity, a topic that (unfortunately) is not coveredin this book.

27

28 Geodesic congruences

Table 2.1: Energy conditions.

Name Statement Conditions

Weak Tαβvαvβ ≥ 0 ρ ≥ 0, ρ + pi > 0

Null Tαβkαkβ ≥ 0 ρ + pi ≥ 0

Strong (Tαβ − 12Tgαβ)vαvβ ≥ 0 ρ +

∑i pi ≥ 0, ρ + pi ≥ 0

Dominant −T αβvβ future directed ρ ≥ 0, ρ ≥ |pi|

2.1 Energy conditions

2.1.1 Introduction and summary

In the context of classical general relativity, it is reasonable to expectthat the stress-energy tensor will satisfy certain conditions, such as pos-itivity of the energy density and dominance of the energy density overthe pressure. Such requirements are embodied in the energy conditions,which are summarized in Table 2.1.

To put the energy conditions in concrete form it is useful to assumethat the stress-energy tensor admits the decomposition

T αβ = ρ eα0 eβ

0 + p1 eα1 eβ

1 + p2 eα2 eβ

2 + p3 eα3 eβ

3 , (2.1.1)

in which the vectors eαµ form an orthonormal basis; they satisfy the

relationsgαβ eα

µ eβν = ηµν , (2.1.2)

where ηµν = diag(−1, 1, 1, 1) is the Minkowski metric. (It goes withoutsaying that the basis vectors are functions of the coordinates.) Equa-tions (2.1.1) and (2.1.2) imply that the quantities ρ (energy density)and pi (principal pressures) are eigenvalues of the stress-energy tensor,and eα

µ are the normalized eigenvectors.The inverse metric can neatly be expressed in terms of the basis

vectors. It is easy to check that the relation

gαβ = ηµν eαµ eβ

ν , (2.1.3)

where ηµν = diag(−1, 1, 1, 1) is the inverse of ηµν , is compatible withEq. (2.1.2). Equations such as (2.1.3) are called completeness relations.

If the stress-energy tensor is that of a perfect fluid, then p1 = p2 =p3 ≡ p. Substituting this into Eq. (2.1.1) and using Eq. (2.1.3) yields

T αβ = ρ eα0 eβ

0 + p(eα1 eβ

1 + eα2 eβ

2 + eα3 eβ

3 )

= ρ eα0 eβ

0 + p(gαβ + eα0 eβ

0 )

= (ρ + p) eα0 eβ

0 + p gαβ.

2.1 Energy conditions 29

The vector eα0 is identified with the four-velocity of the perfect fluid.

Some of the energy conditions are formulated in terms of a normal-ized, future-directed, but otherwise arbitrary timelike vector vα; thisrepresents the four-velocity of an arbitrary observer in spacetime. Interms of the orthonormal basis, such a vector can be expressed as

vα = γ(eα0 + a eα

1 + b eα2 + c eα

3 ), γ = (1− a2 − b2 − c2)−1/2, (2.1.4)

where a, b, and c are arbitrary functions of the coordinates, such thata2 + b2 + c2 < 1. We will also need an arbitrary, future-directed nullvector kα. This we shall express as

kα = eα0 + a′ eα

1 + b′ eα2 + c′ eα

3 , (2.1.5)

where a′, b′, and c′ are arbitrary functions of the coordinates, such thata′2 + b′2 + c′2 = 1. Recall that the normalization of a null vector isalways arbitrary.

2.1.2 Weak energy condition

The weak energy condition states that the energy density of any mat-ter distribution, as measured by any observer in spacetime, must benonnegative. Because an observer with four-velocity vα measures theenergy density to be Tαβvαvβ, we must have

Tαβvαvβ ≥ 0 (2.1.6)

for any future-directed timelike vector vα. To put this in concrete formwe substitute Eqs. (2.1.1) and (2.1.4), which gives

ρ + a2p1 + b2p2 + c2p3 ≥ 0.

Because a, b, c, are arbitrary, we may choose a = b = c = 0, and thisgives ρ ≥ 0. Alternatively, we may choose b = c = 0, which givesρ + a2p1 ≥ 0. Recalling that a2 must be smaller than unity, we obtain0 ≤ ρ + a2p1 < ρ + p1. So ρ + p1 > 0, and similar expressions hold forp2 and p3. The weak energy condition therefore implies

ρ ≥ 0, ρ + pi > 0. (2.1.7)

2.1.3 Null energy condition

The null energy condition makes the same statement as the weak form,except that vα is replaced by an arbitrary, future-directed null vectorkα. Thus,

Tαβkαkβ ≥ 0 (2.1.8)

30 Geodesic congruences

is the statement of the null energy condition. Substituting Eqs. (2.1.1)and (2.1.5) gives

ρ + a′2p1 + b′2p2 + c′2p3 ≥ 0.

Choosing b′ = c′ = 0 enforces a′ = 1, and we obtain ρ + p1 ≥ 0, withsimilar expressions holding for p2 and p3. The null energy conditiontherefore implies

ρ + pi ≥ 0. (2.1.9)

Notice that the weak energy condition implies the null form.

2.1.4 Strong energy condition

The statement of the strong energy condition is

(Tαβ − 1

2Tgαβ

)vαvβ ≥ 0, (2.1.10)

or Tαβvαvβ ≥ −12T , where vα is any future-directed, normalized, time-

like vector. Because Tαβ − 12Tgαβ = Rαβ/8π by virtue of the Einstein

field equations, the strong energy condition is really a statement aboutthe Ricci tensor. Substituting Eqs. (2.1.1) and (2.1.4) gives

γ2(ρ + a2p1 + b2p2 + c2p3) ≥ 1

2(ρ− p1 − p2 − p3).

Choosing a = b = c = 0 enforces γ = 1, and we obtain ρ+p1+p2+p3 ≥0. Alternatively, choosing b = c = 0 implies γ2 = 1/(1− a2), and aftersome simple algebra we obtain ρ + p1 + p2 + p3 ≥ a2(p2 + p3 − ρ− p1).Because this must hold for any a2 < 1, we have ρ+p1 ≥ 0, with similarrelations holding for p2 and p3. The strong energy condition thereforeimplies

ρ + p1 + p2 + p3 ≥ 0, ρ + pi ≥ 0. (2.1.11)

It should be noted that the strong energy condition does not imply theweak form.

2.1.5 Dominant energy condition

The dominant energy condition embodies the notion that matter shouldflow along timelike or null world lines. Its precise statement is that ifvα is an arbitrary, future-directed, timelike vector field, then

−Tαβvβ is a future-directed, timelike or null, vector field. (2.1.12)

The quantity −Tαβvβ is the matter’s momentum density as measured by

an observer with a four-velocity vα, and this is required to be timelike or

2.2 Kinematics of a deformable medium 31

null. Substituting Eqs. (2.1.1) and (2.1.4) and demanding that −Tαβvβ

not be spacelike gives

ρ2 − a2p12 − b2p2

2 − c2p32 ≥ 0.

Choosing a = b = c = 0 gives ρ2 ≥ 0, and demanding that −Tαβvβ

be future directed selects the positive branch: ρ ≥ 0. Alternatively,choosing b = c = 0 gives ρ2 ≥ a2p1

2. Because this must hold for anya2 < 1, we have ρ ≥ |p1|, having taken the future direction for −Tα

βvβ.Similar relations hold for p2 and p3. The dominant energy conditiontherefore implies

ρ ≥ 0, ρ ≥ |pi|. (2.1.13)

2.1.6 Violations of the energy conditions

While the energy conditions typically hold for classical matter, theycan be violated by quantized matter fields. A well-known example isthe Casimir vacuum energy between two conducting plates separatedby a distance d:

ρ = − π2

720

~d4

.

Although quantum effects allow for a localized violation of the energyconditions, recent work suggests that there is a limit to the amount bywhich the energy conditions can be violated globally. In this context itis useful to formulate averaged versions of the energy conditions. Forexample, the averaged null energy condition states that the integral ofTαβkαkβ along a null geodesic γ must be nonnegative:

∫

γ

Tαβkαkβ dλ ≥ 0.

Such averaged energy conditions play a central role in the theory oftraversable wormholes. The averaged null energy condition is known toalways hold in flat spacetime, for noninteracting scalar and electromag-netic fields in arbitrary quantum states; this is true in spite of the factthat Tαβkαkβ can be negative somewhere along the geodesic. Its statusin curved spacetimes is not yet fully settled. A complete discussion, asof 1994, can be found in Matt Visser’s book.

2.2 Kinematics of a deformable medium

2.2.1 Two-dimensional medium

As a warmup for what is to follow, consider, in a purely Newtoniancontext, the internal motion of a two-dimensional deformable medium.

32 Geodesic congruences

Figure 2.1: Two-dimensional deformable medium.

Oξα P

t = t0

O Pt = t1

(Picture this as a thin film of Jell-O; see Fig. 2.1.) How the mediumactually moves depends on its internal dynamics, which will remain un-specified for the purpose of this discussion. From a purely kinematicalpoint of view, however, we may always write that for a sufficiently smalldisplacement ξa about a reference point O,

dξa

dt= Ba

b(t)ξb + O(ξ2),

for some tensor Bab. The time dependence of this tensor is determined

by the medium’s dynamics. For short time intervals,

ξa(t1) = ξa(t0) + ∆ξa(t0),

where∆ξa = Ba

b(t0)ξb(t0) ∆t + O(∆t2),

and ∆t = t1 − t0. To describe the action of Bab we will consider the

simple figure — a circle — described by ξa(t0) = r0(cos φ, sin φ).

2.2.2 Expansion

Suppose first that Bab is proportional to the identity matrix, so that

Bab =

(12θ 00 1

2θ

),

where θ ≡ Baa. Then ∆ξa = 1

2θr0∆t(cos φ, sin φ), which corresponds to

a change in the circle’s radius: r1 = r0 + 12θr0∆t. The corresponding

change in area is then ∆A ≡ A1 − A0 = πr02θ∆t, so that

θ =1

A0

∆A

∆t.

The quantity θ is therefore the fractional change of area per unit time;we shall call it the expansion parameter. This is actually a function, asθ may depend on time and on the choice of reference point O.

2.2 Kinematics of a deformable medium 33

Figure 2.2: Effect of the shear tensor.

σ× = 0 σ+ = 0

2.2.3 Shear

Suppose next that Bab is symmetric and tracefree, so that

Bab =

(σ+ σ×σ× −σ+

).

Then ∆ξa = r0∆t(σ+ cos φ + σ× sin φ,−σ+ sin φ + σ× cos φ). The para-metric equation describing the new figure is r1(φ) = r0(1+σ+∆t cos 2φ+σ×∆t sin 2φ). If σ× = 0, this represents an ellipse with major axis ori-ented along the φ = 0 direction (Fig. 2.2). If, on the other hand, σ+ = 0,then the ellipse’s major axis is oriented along φ = π/4. The generalsituation is an ellipse oriented at an arbitrary angle. It is easy to checkthat the area of the figure is not affected by the transformation. Whatwe have, therefore, is a shearing of the figure, and σ+ and σ× are calledthe shear parameters. These may also vary over the medium.

2.2.4 Rotation

Finally, we suppose that Bab is antisymmetric, so that

Bab =

(0 ω−ω 0

).

Then ∆ξa = r0ω∆t(sin φ,− cos φ), and the new displacement vector isξa(t1) = r0(cos φ′, sin φ′), where φ′ = φ − ω∆t. This clearly representsan overall rotation of the original figure, which also leaves its areaunchanged; ω is called the rotation parameter.

2.2.5 General case

The most general matrix Bab has 2× 2 = 4 components, and it may be

expressed as

Bab =

(12θ 00 1

2θ

)+

(σ+ σ×σ× −σ+

)+

(0 ω−ω 0

).

34 Geodesic congruences

The action of this most general tensor is a linear combination of ex-pansion, shear, and rotation. The tensor can also be expressed as

Bab =1

2θ δab + σab + ωab,

where θ = Baa (the expansion scalar) is the trace part of Bab, σab =

B(ab) − 12θδab (the shear tensor) is the symmetric-tracefree part of Bab,

and ωab = B[ab] (the rotation tensor) is the antisymmetric part of Bab.

2.2.6 Three-dimensional medium

In three dimensions, the tensor Bab would be expressed as

Bab =1

3θ δab + σab + ωab,

where θ = Baa is the expansion scalar, σab = B(ab) − 1

3θδab the shear

tensor, and ωab = B[ab] the rotation tensor. In the three-dimensionalcase, the expansion is the fractional change of volume per unit time:

θ =1

V

∆V

∆t.

To see this, treat the three-dimensional relation

ξa(t1) = (δab + Ba

b∆t)ξb(t0)

as a coordinate transformation from ξa(t0) to ξa(t1). The Jacobian ofthis transformation is

J = det[δab + Ba

b∆t]

= 1 + Tr[Bab∆t]

= 1 + θ∆t.

This implies that volumes at t0 and t1 are related by V1 = (1+ θ∆t)V0,so that V0θ = (V1−V0)/∆t. This argument shows also that the volumeis not affected by the shear and rotation tensors.

2.3 Congruence of timelike geodesics

Let be an open region in spacetime. A congruence in is a family ofcurves such that through each point in there passes one and only onecurve from this family. (The curves do not intersect; picture this asa tight bundle of copper wires.) In this section we will be interestedin congruences of timelike geodesics, which means that each curve inthe family is a timelike geodesic; congruences of null geodesics will be

2.3 Congruence of timelike geodesics 35

Figure 2.3: Deviation vector between two neighbouring members of acongruence.

τ0τ0

ξα

τ1

τ1ξα

considered in the following section. We wish to determine how sucha congruence evolves with time. More precisely stated, we want todetermine the behaviour of the deviation vector ξα between two neigh-bouring geodesics in the congruence (Fig. 2.3), as a function of propertime τ along the reference geodesic. The geometric setup is the sameas in Sec. 1.10, and the relations

uαuα = −1, uα;βuβ = 0, uα

;βξβ = ξα;βuβ, uαξα = 0,

where uα = dxα/dτ is tangent to the geodesics, will be assumed tohold. Notice in particular that ξα is orthogonal to uα: the deviationvector points in the directions transverse to the flow of the congruence.

2.3.1 Transverse metric

Given the congruence and the associated timelike vector field uα, thespacetime metric gαβ can be decomposed into a longitudinal part−uαuβ

and a transverse part hαβ given by

hαβ = gαβ + uαuβ. (2.3.1)

The transverse metric is purely “spatial”, in the sense that it is orthog-onal to uα: uαhαβ = 0 = hαβuβ. It is effectively three-dimensional:in a comoving Lorentz frame at some point P within the congruence,uα

∗= (−1, 0, 0, 0), gαβ

∗= diag(−1, 1, 1, 1), and hαβ

∗= diag(0, 1, 1, 1). We

may also note the relations hαα = 3 and hα

µhµβ = hα

β.

2.3.2 Kinematics

We now introduce the tensor field

Bαβ = uα;β. (2.3.2)

36 Geodesic congruences

Like hαβ, this tensor is purely transverse, as uαBαβ = uαuα;β = 12(uαuα);β =

0 and Bαβuβ = uα;βuβ = 0. It determines the evolution of the deviationvector: from ξα

;βuβ = uα;βξβ we immediately obtain

ξα;βuβ = Bα

βξβ, (2.3.3)

and we see that Bαβ measures the failure of ξα to be parallel transported

along the congruence.Equation (2.3.3) is directly analogous to the first equation of Sec. 2.2.

We may decompose Bαβ into trace, symmetric-tracefree, and antisym-metric parts. This gives

Bαβ =1

3θ hαβ + σαβ + ωαβ, (2.3.4)

where θ = Bαα = uα

;α is the expansion scalar, σαβ = B(αβ) − 13θ hαβ

the shear tensor, and ωαβ = B[αβ] the rotation tensor. These quantitiescome with the same interpretation as in Sec. 2.2. In particular, thecongruence will be diverging (geodesics flying apart) if θ > 0, and itwill be converging (geodesics coming together) if θ < 0.

2.3.3 Frobenius’ theorem

Some congruences have a vanishing rotation tensor, ωαβ = 0. Theseare said to be hypersurface orthogonal, meaning that the congruence iseverywhere orthogonal to a family of spacelike hypersurfaces foliating(Fig. 2.4). We now provide a proof of this statement.

The congruence will be hypersurface orthogonal if uα is everywhereproportional to nα, the normal to the hypersurfaces. Supposing thatthese are described by equations of the form Φ(xα) = c, where c is aconstant specific to each hypersurface, then nα ∝ Φ,α and

uα = −µΦ,α,

for some proportionality factor µ. (We suppose that Φ increases towardthe future, and the positive quantity µ can be determined from the nor-malization condition uαuα = −1.) Differentiating this equation givesuα;β = −µΦ;αβ − Φ,αµ,β. Consider now the completely antisymmetrictensor

u[α;βuγ] ≡ 1

3!(uα;βuγ + uγ;αuβ + uβ;γuα − uβ;αuγ − uα;γuβ − uγ;βuα).

Direct evaluation of the right-hand side, using Φ;βα = Φ;αβ, returnszero. We therefore have

hypersurface orthogonal ⇒ u[α;βuγ] = 0. (2.3.5)

2.3 Congruence of timelike geodesics 37

Figure 2.4: Family of hypersurfaces orthogonal to a congruence oftimelike geodesics.

γ1 γ2γ3

Σ1

Σ2

Σ3

The converse of this statement, that u[α;βuγ] = 0 implies the existenceof a scalar field Φ such that uα ∝ Φ,α, is also true (but harder to prove).

Equation (2.3.5) is a useful result, because whether or not uα ishypersurface orthogonal can be decided on the basis of the vector fieldalone, without having to find Φ explicitly. We note that the geodesicequation uα

;βuβ = 0 was never used in the derivation of Eq. (2.3.5). Wealso never used the fact that uα was normalized. Equation (2.3.5) istherefore quite general: A congruence of curves (timelike, spacelike, ornull) is hypersurface orthogonal if and only if u[α;βuγ] = 0, where uα istangent to the curves. This statement is known as Frobenius’ theorem.

We now return to our geodesic congruence, and use Eqs. (2.3.2) and(2.3.4) to calculate

3! u[α;βuγ] = 2(u[α;β]uγ + u[γ;α]uβ + u[β;γ]uα)

= 2(B[αβ]uγ + B[γα]uβ + B[βγ]uα)

= 2(ωαβuγ + ωγαuβ + ωβγuα).

If we put the left-hand side to zero and multiply the right-hand side byuγ, we obtain ωαβ = 0, because ωγαuγ = 0 = ωβγu

γ. (Recall the purelytransverse property of Bαβ.) Therefore,

hypersurface orthogonal ⇒ ωαβ = 0. (2.3.6)

This concludes the proof of our initial statement.

Notice that Eq. (2.3.6) holds for timelike geodesics only, whereasEq. (2.3.5) is general. In fact, Eq. (2.3.6) could have been derivedmuch more directly, but in doing so we would have bypassed the moregeneral formulation of Frobenius’ theorem. The direct proof goes asfollows.

If uα is hypersurface orthogonal, then uα = −µΦ,α for some scalars

38 Geodesic congruences

µ and Φ. It follows from ωαβ = u[α;β] and the symmetry of Φ;αβ that

ωαβ = −Φ[,αµ,β] =1

µu[αµ,β].

But we know that ωαβ must be orthogonal to uα, and the relationωαβuβ = 0 implies µ,α = −(µ,βuβ)uα. This, in turn, establishes thatthe rotation tensor vanishes identically: ωαβ = 0.

We have learned that µ must be constant on each hypersurface,because it varies only in the direction orthogonal to the hypersurfaces.Thus, µ can be expressed as a function of Φ, and defining a new scalarΨ =

∫µ(Φ) dΦ, we find that uα is not only proportional to a gradient,

it is equal to one: uα = −Ψ,α. Notice that if uα can be expressed in thisform, then it automatically satisfies the geodesic equation: uα;βuβ =Ψ;αβΨ,β = Ψ;βαΨ,β = 1

2(Ψ,βΨ,β);α = 1

2(uβuβ);α = 0.

In summary:A vector field uα (timelike, spacelike, or null, and not necessarily

geodesic) is hypersurface orthogonal if there exists a scalar field Φ suchthat uα ∝ Φ,α, which implies u[α;βuγ] = 0. If the vector field is timelikeand geodesic, then it is hypersurface orthogonal if there exists a scalarfield Ψ such that uα = −Ψ,α, which implies ωαβ = u[α;β] = 0.

2.3.4 Raychaudhuri’s equation

We now want to derive an evolution equation for θ, the expansion scalar.We begin by developing an equation for Bαβ itself:

Bαβ;µuµ = uα;βµu

µ

= (uα;µβ −Rανβµuν)uµ

= (uα;µuµ);β − uα;µu

µ;β −Rανβµu

νuµ

= −BαµBµβ −Rαµβνu

µuν .

The equation for θ is obtained by taking the trace:

dθ

dτ= −BαβBβα −Rαβuαuβ.

It is then easy to check that BαβBβα = 13θ2 +σαβσαβ−ωαβωαβ. Making

the substitution, we arrive at

dθ

dτ= −1

3θ2 − σαβσαβ + ωαβωαβ −Rαβuαuβ. (2.3.7)

This is Raychaudhuri’s equation for a congruence of timelike geodesics.We note that since the shear and rotation tensors are purely spatial,σαβσαβ ≥ 0 and ωαβωαβ ≥ 0, with the equality sign holding if and onlyif the tensor is identically zero.

2.3 Congruence of timelike geodesics 39

2.3.5 Focusing theorem

The importance of Eq. (2.3.7) for general relativity is revealed by thefollowing theorem: Let a congruence of timelike geodesics be hypersur-face orthogonal, so that ωαβ = 0, and let the strong energy conditionhold, so that (by virtue of the Einstein field equations) Rαβuαuβ ≥ 0.Then the Raychaudhuri equation implies

dθ

dτ= −1

3θ2 − σαβσαβ −Rαβuαuβ ≤ 0.

The expansion must therefore decrease during the congruence’s evo-lution. Thus, an initially diverging (θ > 0) congruence will divergeless rapidly in the future, while an initially converging (θ < 0) congru-ence will converge more rapidly in the future. This is the statementof the focusing theorem. Its physical interpretation is that gravitationis an attractive force when the strong energy condition holds, and thegeodesics get focused as a result of this attraction.

It also follows from Raychaudhuri’s equation that under the condi-tions of the focusing theorem, dθ/dτ ≤ −1

3θ2. This can be integrated

at once, giving

θ−1(τ) ≥ θ0−1 +

τ

3,

where θ0 ≡ θ(0). This shows that if the congruence is initially con-verging (θ0 < 0), then θ(τ) → −∞ within a proper time τ ≤ 3/|θ0|.The interpretation of this result is that the congruence will develop acaustic, a point at which some of the geodesics come together (Fig. 2.5).Obviously, a caustic is a singularity of the congruence, and equationssuch as (2.3.7) lose their meaning at such points.

2.3.6 Example

As an illustrative example, let us consider the congruence of comovingworld lines in an expanding universe with metric

ds2 = −dt2 + a2(t)(dx2 + dy2 + dz2),

where a(t) is the scale factor. The tangent vector field is uα = −∂αt,and a quick calculation reveals that

Bαβ = uα;β =a

ahαβ,

where an overdot indicates differentiation with respect to t. This showsthat the shear and rotation tensors are both zero for this congruence.The expansion, on the other hand, is given by

θ = 3a

a=

1

a3

d

dta3.

40 Geodesic congruences

Figure 2.5: Geodesics converge into a caustic of the congruence.

caustic

This illustrates rather well the general statement (made in Sec. 2.3.8below) that the expansion is the fractional rate of change of the con-gruence’s cross-sectional volume (which is here proportional to a3).

2.3.7 Another example

As a second example, we consider a congruence of radial, marginallybound, timelike geodesics of the Schwarzschild spacetime. The metricis

ds2 = −f dt2 + f−1 dr2 + r2 dΩ2,

where f = 1− 2M/r and dΩ2 = dθ2 + sin2 θ dφ2. For radial geodesics,uθ = uφ = 0, and the geodesics are marginally bound if 1 = E ≡−uαξα

(t) = −ut. This means that the conserved energy is precisely equal

to the rest-mass energy, and this gives us the equation ut = 1/f . Fromthe normalization condition gαβuαuβ = −1 we also get ur = ±

√2M/r;

the upper sign applies to outgoing geodesics, and the lower sign appliesto ingoing geodesics.

The four-velocity is therefore given by

uα ∂α = f−1 ∂t ±√

2M/r ∂r, uα dxα = −dt± f−1√

2M/r dr.

It follows that uα is equal to a gradient: uα = −Φ,α, where

Φ = t∓ 4M

[√r/2M +

1

2ln

(√r/2M − 1√r/2M + 1

)].

This means that the congruence is everywhere orthogonal to the space-like hypersurfaces Φ = constant.

The expansion is calculated as

θ = uα;α =

1√−g(√−g uα),α =

1

r2(r2ur)′,

2.3 Congruence of timelike geodesics 41

where a prime indicates differentiation with respect to r. Completingthe calculation gives

θ = ±3

2

√2M

r3.

Not surprisingly, the congruence is diverging (θ > 0) if the geodesics areoutgoing, and converging (θ < 0) if the geodesics are ingoing. The rateof change of the expansion is calculated as dθ/dτ = (dθ/dr)(dr/dτ) =θ′ur, and the result is

dθ

dτ= −9M

2r3.

As dictated by the focusing theorem, dθ/dτ is negative in both cases.

2.3.8 Interpretation of θ

We now prove that θ is equal to the fractional rate of change of δV ,the congruence’s cross-sectional volume:

θ =1

δV

d

dτδV. (2.3.8)

Although this may already be obvious from Eqs. (2.3.3) and (2.3.4),it is still instructive to go through a formal proof. The first step is tointroduce the notions of cross section, and cross-sectional volume.

Select a particular geodesic γ from the congruence, and on thisgeodesic, pick a point P at which τ = τP . Construct, in a smallneighbourhood around P , a small set δΣ(τP ) of points P ′ such that(i) through each of these points there passes another geodesic fromthe congruence, and (ii) at each point P ′, τ is also equal to τP . Thisset forms a three-dimensional region, a small segment of the hyper-surface τ = τP (Fig. 2.6). We assume that the parameterization hasbeen adjusted so that γ intersects δΣ(τP ) orthogonally. (There is norequirement that other geodesics do, as the congruence may not behypersurface orthogonal.) We shall call δΣ(τP ) the congruence’s crosssection around the geodesic γ, at proper time τ = τP . We want tocalculate the volume of this hypersurface segment, and compare it withthe volume of δΣ(τQ), where Q is a neighbouring point on γ.

We introduce coordinates on δΣ(τP ) by assigning a label ya (a =1, 2, 3) to each point P ′ in the set. Recalling that through each ofthese points there passes a geodesic from the congruence, we see thatwe may use ya to label the geodesics themselves. By demanding thateach geodesic keep its label as it moves away from δΣ(τP ), we simul-taneously obtain a coordinate system ya in δΣ(τQ) or any other crosssection. This construction therefore defines a coordinate system (τ, ya)in a neighbourhood of the geodesic γ, and there exists a transforma-tion between this system and the one originally in use: xα = xα(τ, ya).

42 Geodesic congruences

Figure 2.6: Congruence’s cross section about a reference geodesic.

γ

P P ′δΣ(τP )

Q Q′δΣ(τQ)

Because ya is constant along the geodesics, we have

uα =(∂xα

∂τ

)ya

. (2.3.9)

On the other hand, the vectors

eαa =

(∂xα

∂ya

)τ

(2.3.10)

are tangent to the cross sections. These relations imply £ueαa = 0, and

we also have uα eαa = 0 holding on γ (and only γ).

We now introduce a three-tensor hab, defined by

hab = gαβ eαaeβ

b . (2.3.11)

(A three-tensor is a tensor with respect to coordinate transformationsya → ya′ , but a scalar with respect to transformations xα → xα′ .) Thisacts as a metric tensor on δΣ(τ): For displacements contained withinthe set (so that dτ = 0), xα = xα(ya), and

ds2 = gαβ dxαdxβ

= gαβ

(∂xα

∂yadya

)(∂xβ

∂ybdyb

)

= (gαβ eαaeβ

b ) dyadyb

= hab dyadyb.

Thus, hab is the three-dimensional metric on the congruence’s crosssections. Because γ is orthogonal to its cross sections (uα eα

a = 0), wehave that hab = hαβ eα

aeβb on γ, where hαβ = gαβ +uαuβ is the transverse

metric. If we define hab to be the inverse of hab, then it is easy to checkthat

hαβ = hab eαaeβ

b (2.3.12)

2.4 Congruence of null geodesics 43

on γ.The three-dimensional volume element on the cross sections, or

cross-sectional volume, is δV =√

h d3y, where h ≡ det[hab]. Becausethe coordinates ya are comoving (since each geodesic moves with a con-stant value of its coordinates), d3y does not change as the cross sectionδΣ(τ) evolves from τ = τP to τ = τQ. A change in δV therefore comes

entirely from a change in√

h:

1

δV

d

dτδV =

1√h

d

dτ

√h =

1

2hab dhab

dτ.

We must now calculate the rate of change of the three-metric:

dhab

dτ≡ (gαβ eα

aeβb );µu

µ

= gαβ(eαa;µu

µ)eβb + gαβ eα

a (eβb;µu

µ)

= gαβ(uα;µe

µa)eβ

b + gαβ eαa (uβ

;µeµb )

= uβ;α eαaeβ

b + uα;β eαaeβ

b

= (Bαβ + Bβα)eαaeβ

b . (2.3.13)

Multiplying by hab and evaluating on γ, so that Eq. (2.3.12) may beused, we obtain

hab dhab

dτ= (Bαβ + Bβα)(hab eα

aeβb )

= 2Bαβhαβ

= 2Bαβgαβ

= 2θ.

This proves that

θ =1√h

d

dτ

√h, (2.3.14)

which is the same statement as in Eq. (2.3.8).

2.4 Congruence of null geodesics

We now turn to the case of null geodesics. The geometric setup isthe same as in the preceding section, except that the tangent vectorfield, denoted kα, is null. We assume that the geodesics are affinelyparameterized by λ, so that kα = dxα/dλ. The deviation vector willagain be denoted ξα, and we again take it to be orthogonal to, andLie transported along, the geodesics. The following equations thereforehold:

kαkα = 0, kα;βkβ = 0, kα

;βξβ = ξα;βkβ, kαξα = 0.

44 Geodesic congruences

As in the preceding section, we will be interested in the transverseproperties of the congruence, which are described by the deviation vec-tor ξα. We can, however, anticipate some difficulties, because here thecondition kαξα = 0 fails to remove an eventual component of ξα in thedirection of kα. One of our first tasks, therefore, will be to isolate thepurely transverse part of the deviation vector. This we will do with thehelp of hαβ, the transverse metric.

2.4.1 Transverse metric

To isolate the part of the metric that is transverse to kα is not entirelystraightforward when kα is null. The expression h′αβ = gαβ + kαkβ does

not work, because h′αβkβ = kα 6= 0. To see what must be done, let usgo to a local Lorentz frame at some point P , and let us introduce thenull coordinates u = t− x and v = t + x. The line element can then beexpressed as ds2 ∗

= −du dv + dy2 + dz2. Supposing that kα is tangentto the curves u = constant, we see that the transverse line element isds2 ∗

= dy2 + dz2: the transverse metric is two-dimensional. This clearlyhas to do with the fact that ds2 = 0 for displacements along the vdirection.

To isolate the transverse part of the metric we need to introduceanother null vector field Nα, such that Nαkα 6= 0. Because the nor-malization of a null vector is arbitrary, we may always impose kαNα =−1. If kα

∗= −∂αu in the local Lorentz frame, then we might choose

Nα = −12∂αv. Now consider the object hαβ = gαβ +kαNβ +Nαkβ. This

is clearly orthogonal to both kα and Nα: hαβkβ = hαβNβ = 0. Fur-

thermore, hαβ∗= diag(0, 0, 1, 1) in the local Lorentz frame, and hαβ is

properly transverse and two-dimensional. This, therefore, is the objectwe seek.

Let us now summarize the preceding discussion. For a congruenceof null geodesics, the transverse metric is obtained as follows: Given thenull vector field kα, select an auxiliary null vector field Nα and chooseits normalization to be such that kαNα = −1. Then the transversemetric is given by

hαβ = gαβ + kαNβ + Nαkβ. (2.4.1)

It satisfies the relations

hαβkβ = hαβNβ = 0, hαα = 2, hα

µhµβ = hα

β, (2.4.2)

which confirm that hαβ is purely transverse (orthogonal to both kα andNα) and effectively two-dimensional.

Evidently, the conditions NαNα = 0 and kαNα = −1 do not de-termine Nα uniquely. This implies that the transverse metric is notunique. As we shall see, however, quantities such as the expansion of

2.4 Congruence of null geodesics 45

the congruence will turn out to be the same for all choices of auxil-iary null vector. Further aspects of this non-uniqueness are explored inSec. 2.6, Problem 6.

2.4.2 Kinematics

As before, we introduce the tensor field

Bαβ = kα;β (2.4.3)

as a measure of the failure of ξα to be parallel transported along thecongruence:

ξα;βkβ = Bα

βξβ. (2.4.4)

As before, Bαβ is orthogonal to the tangent vector field: kαBαβ = 0 =Bαβkβ. However, Bαβ is not orthogonal to Nα, and Eq. (2.4.4) has anon-transverse component that should be removed.

We begin by isolating the purely transverse part of the deviationvector, which we denote ξα. Because hαβ is itself purely transverse, itis easy to see that

ξα ≡ hαµξ

µ = ξα + (Nµξµ)kα (2.4.5)

is the desired object. Its covariant derivative in the direction of kα

represents the relative velocity of two neighbouring geodesics. It isgiven by

ξµ;βkβ = hµ

νBνβξβ + hµ

ν;βξνkβ,

where we have inserted Eq. (2.4.4) in the first term of the right-handside. Calculating the second term gives

ξµ;βkβ = hµ

νBνβξβ + (Nν;βξνkβ)kµ,

and we see that the vector ξµ;βkβ has a component along kµ. Once again

we remove this by projecting with hαµ. Using the last of Eqs. (2.4.2),

we obtain

(ξα;βkβ ) ≡ hα

µ(ξµ;βkβ) = hα

µBµνξ

ν

= hαµB

µν ξ

ν

= hαµh

νβBµ

ν ξβ

for the transverse components of the relative velocity. In the first linewe have replaced ξν with ξν because Bµ

νkν = 0. In the third line we

have inserted the relation ξν = hνβ ξβ; this holds because ξν is already

purely transverse.We have obtained

(ξα;βkβ ) = Bα

β ξβ, (2.4.6)

46 Geodesic congruences

whereBαβ = hµ

αhνβBµν (2.4.7)

is the purely transverse part of Bµν = kµ;ν . This can be expressed in amore explicit form by using Eq. (2.4.1):

Bαβ = (g µα + kαNµ + Nαkµ)(g ν

β + kβN ν + Nβkν) Bµν

= (g µα + kαNµ + Nαkµ)(Bµβ + kβBµνN

ν)

= Bαβ + kαNµBµβ + kβBαµNµ + kαkβBµνN

µN ν . (2.4.8)

Equation (2.4.6) governs the purely transverse behaviour of the nullcongruence, and the vector Bα

β ξβ can be interpreted as the transverserelative velocity between two neighbouring geodesics.

As before, the evolution tensor Bαβ will be decomposed into itsirreducible parts:

Bαβ =1

2θ hαβ + σαβ + ωαβ, (2.4.9)

where θ = Bαα is the expansion scalar, σαβ = B(αβ) − 1

2θ hαβ the shear

tensor, and ωαβ = B[αβ] the rotation tensor. The expansion is givenmore explicitly by

θ = gαβBαβ

= gαβBαβ,

which follows from Eq. (2.4.8) and the fact that Bαβ is orthogonal tokα. From this we obtain

θ = kα;α. (2.4.10)

We see explicitly that θ does not depend on the choice of auxiliarynull vector Nα: the expansion is unique. The geometric meaning ofthe expansion will be considered in detail below; we will show that θis the fractional rate of change (per unit affine-parameter distance) ofthe congruence’s cross-sectional area. (Recall that here, the transversespace is two-dimensional.)

2.4.3 Frobenius’ theorem

We now show that if the vector field kα is such that ωαβ = 0, thenthe congruence is hypersurface orthogonal, in the sense that kα mustbe proportional to the normal Φ,α of a family of hypersurfaces de-scribed by Φ(xα) = c. These hypersurfaces must clearly be null:gαβΦ,αΦ,β ∝ gαβkαkβ = 0. Furthermore, because kα is at once par-allel and orthogonal to Φ,α (kαΦ,α = 0), the vector kα is also tangent tothe hypersurfaces. The null geodesics therefore lie within the hypersur-faces (Fig. 2.7); they are called the null generators of the hypersurfacesΦ(xα) = c.

2.4 Congruence of null geodesics 47

Figure 2.7: Family of hypersurfaces orthogonal to a congruence of nullgeodesics.

Φ = constant

kα

We begin with the general statement of Frobenius’ theorem derivedin Sec. 2.3.3: the congruence is hypersurface orthogonal if and only ifk[α;βkγ] = 0. This condition implies B[αβ]kγ + B[γα]kβ + B[βγ]kα = 0,and transvecting with Nγ gives

B[αβ] = B[γα]kβNγ + B[βγ]kαNγ

= 12(Bγαkβ −Bαγkβ + Bβγkα −Bγβkα)Nγ

= Bγ[αkβ]Nγ + k[αBβ]γN

γ.

But from Eq. (2.4.8) we also have

B[αβ] = B[αβ] −Bµ[αkβ]Nµ − k[αBβ]µN

µ,

and it follows immediately that B[αβ] = 0. We therefore have

hypersurface orthogonal ⇒ ωαβ = 0, (2.4.11)

and this concludes the proof. (In Sec. 2.6, Problem 6 we show that ifωαβ = 0 for a specific choice of auxiliary null vector Nα, then ωαβ = 0for all possible choices.)

The congruence is hypersurface orthogonal if there exists a scalarfield Φ(xα) which is constant on the hypersurfaces and kα = −µΦ,α

for some scalar µ. In this form, kα automatically satisfies the geodesicequation:

kα;βkβ = −(µΦ;αβ + Φ,αµ,β)kβ

= −(µ,βΦ,β) kα,

where we have used Φ;αβΦ,β = Φ;βαΦ,β = 12(Φ,βΦ,β),α = 0. This is

the general form of the geodesic equation, corresponding to a param-eterization that is not affine. Affine parameterization is recovered ifµ,αkα = 0, that is, if µ does not vary along the geodesics.

48 Geodesic congruences

2.4.4 Raychaudhuri’s equation

The derivation of the null version of Raychaudhuri’s equation proceedsmuch as in Sec. 2.3.4. In particular, the equation

dθ

dλ= −BαβBβα −Rαβkαkβ

follows from the same series of steps. It is then easy to check thatBαβBβα = BαβBβα = 1

2θ2 + σαβσαβ − ωαβωαβ, which gives

dθ

dλ= −1

2θ2 − σαβσαβ + ωαβωαβ −Rαβkαkβ. (2.4.12)

This is Raychaudhuri’s equation for a congruence of null geodesics.It should be noted that this equation is invariant under a change ofauxiliary null vector Nα; this is established in Sec. 2.6, Problem 6.We also note that because the shear and rotation tensors are purelytransverse, σαβσαβ ≥ 0 and ωαβωαβ ≥ 0, with the equality sign holdingif and only if the tensor vanishes.

2.4.5 Focusing theorem

The null version of the focusing theorem goes as follows: Let a con-gruence of null geodesics be hypersurface orthogonal, so that ωαβ = 0,and let the null energy condition hold, so that (by virtue of the Ein-stein field equations) Rαβkαkβ ≥ 0. Then the Raychaudhuri equationimplies

dθ

dλ= −1

2θ2 − σαβσαβ −Rαβkαkβ ≤ 0,

which means that the geodesics are focused during the evolution of thecongruence.

Integrating dθ/dλ ≤ −12θ2 yields

θ−1(λ) ≥ θ0−1 +

λ

2,

where θ0 ≡ θ(0). This shows that if the congruence is initially converg-ing (θ0 < 0), then θ(λ) → −∞ within an affine parameter λ ≤ 2/|θ0|.As in the case of a timelike congruence, this generally signals the oc-currence of a caustic.

2.4.6 Example

As an illustrative example, let us consider the congruence formed bythe generators of a null cone in flat spacetime. The geodesics emanatefrom a single point P (which we place at the origin of the coordinatesystem) and they radiate in all directions. (Note that P is a caustic of

2.4 Congruence of null geodesics 49

the congruence.) In spherical coordinates, the geodesics are describedby the relations t = λ, r = λ, θ = constant, and φ = constant, in whichλ is the affine parameter. The tangent vector field is

kα = −∂α(t− r).

We must find an auxiliary null vector field Nα that satisfies kαNα = −1.If we assume that Nα lies entirely within the (t, r) plane, the unique so-lution is Nα = −1

2∂α(t+r). With this choice we find that the transverse

metric is given by hαβ = diag(0, 0, r2, r2 sin2 θ). A straightforward cal-culation gives Bαβ = kα;β = diag(0, 0, r, r sin2 θ), and we see that Bαβ

is already transverse for this choice of Nα. We have found

Bαβ =1

rhαβ,

and this shows that the shear and rotation tensors are both zero forthis congruence. The expansion, on the other hand, is given by

θ =2

r=

1

4πr2

d

dλ(4πr2).

This verifies the general statement (made in Sec. 2.4.8 below) that theexpansion is the fractional rate of change of the congruence’s cross-sectional area.

We might ask how making a different choice for Nα would affect ourresults. It is easy to check that the vector Nα dxα = −dt + r sin θ dφsatisfies both NαNα = 0 and Nαkα = −1. It is therefore an acceptablechoice of auxiliary null vector field. This choice leads to a complicatedexpression for the transverse metric, which now has components alongt and r. And while the expression for Bαβ does not change, we find thatBαβ is no longer equal to Bαβ, and is much more complicated that theexpression given previously. You may check, however, that the relationBαβ = hαβ/r is not affected by the change of auxiliary null vector. Ourresults for θ, σαβ, and ωαβ are therefore preserved.

2.4.7 Another example

As a second example, we consider the radial null geodesics of Schwarzschildspacetime. For dθ = dφ = 0, the Schwarzschild line element reduces to

ds2 = −f dt2 + f−1 dr2 = −f(dt− f−1 dr)(dt + f−1 dr),

where f = 1− 2M/r. The displacements will be null if ds2 = 0. If wedefine

u = t− r∗, v = t + r∗,

50 Geodesic congruences

where r∗ =∫

f−1 dr = r +2M ln(r/2M −1), we find that u = constanton outgoing null geodesics, while v = constant on ingoing null geodesics.The vector fields

koutα = −∂αu, kin

α = −∂αv

are null, and they both satisfy the geodesic equation, with +r as anaffine parameter for kα

out, and −r as an affine parameter for kαin. As

their labels indicate, koutα is tangent to the outgoing geodesics, while

kinα is tangent to the ingoing geodesics. The congruences are clearly

hypersurface orthogonal. Their expansions are easily calculated:

θ = ±2

r,

where the positive (negative) sign refers to the outgoing (ingoing) con-gruence. We also have

dθ

dλ= − 2

r2,

which is properly negative.

2.4.8 Interpretation of θ

We shall now give a formal proof of the statement that θ is the fractionalrate of change of the congruence’s cross-sectional area:

θ =1

δA

d

dλδA, (2.4.13)

where δA is measured in the purely transverse directions. The proofis very similar to what was presented in Sec. 2.3.8; the only crucialdifference concerns the dimensionality of the transverse space.

We pick a particular geodesic γ from the congruence, and on thisgeodesic we select a point P at which λ = λP . We then consider the nullcurves to which Nα is tangent, and we let µ be the parameter on theseauxiliary curves; we adjust the parameterization so that µ is constant onthe null geodesics. The auxiliary curve that passes through P is calledβ, and we have that at P , µ = µγ. The cross section δS(λP ) is defined tobe a small set of points P ′ in a neighbourhood of P such that (i) througheach of these points there passes another geodesic from the congruenceand another auxiliary curve, and (ii) at each point P ′, λ is also equal toλP and µ is equal to µγ. This set forms a two-dimensional region, theintersection of small segments of the hypersurfaces λ = λP and µ = µγ.We assume that the parameterization has been adjusted so that bothγ and β intersect δS(λP ) orthogonally. (There is no requirement thatother curves do.)

We introduce coordinates in δS(λP ) by assigning a label θA (A =1, 2) to each point in the set. Recalling that through each of these

2.5 Bibliographical notes 51

points there passes a geodesic from the congruence, we see that wemay use θA to label the geodesics themselves. By demanding that eachgeodesic keep its label as it moves away from δS(λP ), we simultane-ously obtain a coordinate system θA in any other cross section δS(λ).This construction therefore produces a coordinate system (λ, µ, θA) ina neighbourhood of the geodesic γ, and there exists a transformationbetween this system and the one originally in use: xα = xα(λ, µ, θA).Because µ and θA are constant along the geodesics, we have

kα =(∂xα

∂λ

)µ,θA

.

On the other hand, the vectors

eαA =

(∂xα

∂θA

)λ,µ

are tangent to the cross sections. These relations imply £keαA = 0, and

we have also that on γ (and only γ), kα eαA = Nα eα

A = 0.The remaining steps are very similar to those carried out in Sec. 2.3.8,

and it will suffice to present a brief outline. The two-tensor

σAB = gαβ eαAeβ

B

acts as a metric on δS(λ). The cross-sectional area is therefore definedby δA =

√σ d2θ, where σ = det[σAB]. The inverse σAB of the two-

metric is such that on γ, hαβ = σAB eαAeβ

B, where hαβ = gαβ + kαNβ +Nαkβ is the transverse metric. The relation

dσAB

dλ= (Bαβ + Bβα) eα

AeβB

follows, and taking its trace yields

θ =1√σ

d

dλ

√σ.

This statement is equivalent to Eq. (2.4.13).

2.5 Bibliographical notes

During the preparation of this chapter I have relied on the followingreferences: Carter (1979); Visser (1995); and Wald (1984).

More specifically:Section 2.1 is based on Sec. 9.2 of Wald and Chapter 12 of Visser.

Sections 2.3 and 2.4 are based partially on Sec. 9.2 and Appendix B ofWald, as well as Sec. 6.2.1 of Carter.

52 Geodesic congruences

2.6 Problems

Warning: The results derived in Problem 8 are used in later portionsof this book.

1. Consider a curved spacetime with metric

ds2 = −dt2 + d`2 + r2(`) dΩ2,

where the function r(`) is such that (i) it is minimum at ` = 0,with a value r0, and (ii) it asymptotically becomes equal to |`| as` → ±∞.

a) Argue that this spacetime contains a traversable wormholebetween two asymptotically-flat regions, with a throat ofradius r0.

b) Find which energy conditions are violated at ` = 0.

2. We examine the congruence of comoving world lines of a Fried-mann-Robertson-Walker spacetime, whose metric is

ds2 = −dt2 + a2(t)( dr2

1− kr2+ r2 dΩ2

),

where a(t) is the scale factor, and k a constant normalized toeither ±1 or zero. The vector tangent to the congruence is uα =∂xα/∂t.

a) Show that the congruence is geodesic.

b) Calculate the expansion, shear, and rotation of this congru-ence.

c) Use the Raychaudhuri equation to deduce

a

a= −4π

3(ρ + 3p),

where ρ is the energy density of a perfect fluid with four-velocity uα, and p is the pressure.

3. In this problem we consider the vector field

uα∂α =1√

1− 3M/r

(∂t +

√M/r3 ∂θ

)

in Schwarzschild spacetime; the vector is expressed in terms ofthe usual Schwarzschild coordinates, and M is the mass of theblack hole.

2.6 Problems 53

a) Show that the vector field is timelike and geodesic. Describethe geodesics to which uα is tangent.

b) Calculate the expansion of the congruence. Explain why theexpansion is positive in the northern hemisphere and nega-tive in the southern hemisphere. Explain also why the ex-pansion is singular at the north and south poles.

c) Compute the rotation tensor for this congruence. Check thatits square is given by

ωαβωαβ =M

8r3

(1− 6M/r

1− 3M/r

)2

.

d) Calculate dθ/dτ and check that Raychaudhuri’s equation issatisfied.

4. Derive the following evolution equations for the shear and rotationtensors of a congruence of timelike geodesics:

σαβ;µuµ = −2

3θ σαβ − σαµσ

µβ − ωαµω

µβ +

1

3(σµνσµν − ωµνωµν)hαβ

− Cαµβν uµuν +1

2RTT

αβ ,

ωαβ;µuµ = −2

3θ ωαβ − σαµω

µβ − ωαµσ

µβ.

Here, Cαµβν is the Weyl tensor (Sec. 1.13, Problem 8), and RTTαβ ≡

RTαβ− 1

3(hµνRT

µν)hαβ is the “transverse-tracefree” part of the Riccitensor; its transverse part is RT

αβ ≡ h µα h ν

β Rµν .

5. In this problem we consider a spacetime with metric

ds2 = −dt2+r2 + a2 cos2 θ

r2 + a2dr2+(r2+a2 cos2 θ) dθ2+(r2+a2) sin2 θ dφ2,

where a is a constant, together with a congruence of null geodesicswith tangent vector field

kα ∂α = ∂t + ∂r +a

r2 + a2∂φ.

a) Check that kα is null, that it satisfies the geodesic equation,and that r is an affine parameter.

b) Find a suitable auxiliary null vector Nα and calculate thecongruence’s expansion, shear, and rotation. In particular,verify the following results:

θ =2r

r2 + a2 cos2 θ, σαβ = 0, ωαβωαβ =

2a2 cos2 θ

(r2 + a2 cos2 θ)2.

These show that the congruence is diverging, shear-free, andthat it is not hypersurface orthogonal.

54 Geodesic congruences

c) Show that the coordinate transformation

x =√

r2 + a2 sin θ cos φ, y =√

r2 + a2 sin θ sin φ, z = r cos θ

brings the metric to the standard Minkowski form for flatspacetime. Express kα in this coordinate system.

6. The auxiliary null vector field Nα introduced in Sec. 2.4 is notunique, and in this problem we examine various consequences ofthis fact. For the purpose of this discussion we introduce vectorseα

A (A = 1, 2) pointing in the two directions orthogonal to bothkα and Nα, and we choose them to be orthonormal, so that theysatisfy gαβ eα

AeβB = δAB. We also introduce the 2× 2 matrix

BAB = Bαβ eαAeβ

B,

the projection of the tensor Bαβ = kα;β in the transverse spacespanned by the vectors eα

A. In the following we shall use δAB

and δAB to lower and raise uppercase latin indices; for example,BAB = δAMδBNBMN .

a) Derive the following relations:

hαβ = δAB eαAeβ

B, Bαβ = BAB eαAeβ

B,

θ = δABBAB, σαβ = σAB eαAeβ

B, ωαβ = ωAB eαAeβ

B,

where σAB = 12(BAB + BBA − θ δAB) and ωAB = 1

2(BAB −

BBA). These confirm that the tensors hαβ, Bαβ, σαβ, andωαβ are all orthogonal to both kα and Nα. We now mustdetermine how a change of auxiliary null vector field affectsthese results.

b) The vector Nα must satisfy the relations NαNα = 0 andkαNα = −1. Prove that the transformation

Nα → N ′α = Nα + c kα + cA eαA,

where c = 12cAcA, is the only one that preserves the defining

relations for the auxiliary null vector. (The coefficients cA

are arbitrary.)

c) Calculate how hαβ changes under this transformation.

d) Calculate how Bαβ changes.

e) Show that θ is invariant under the transformation.

f) Prove that σαβ changes according to

σ′αβ = (cAcBσAB) kαkβ+(cAσ BA ) kαeβ

B+(cBσ AB ) eα

Akβ+σAB eαAeβ

B.

This shows that if σαβ = 0 for one choice of Nα, then σαβ = 0for any other choice. Prove that σαβσαβ is invariant underthe transformation.

2.6 Problems 55

g) Prove that ωαβ changes according to

ω′αβ = (cAω BA ) kαeβ

B − (cBω AB ) eα

Akβ + ωAB eαA eβ

B.

This shows that if ωαβ = 0 for one choice of Nα, then ωαβ = 0for any other choice. Prove that ωαβωαβ is invariant underthe transformation.

These results imply that the Raychaudhuri equation is invariantunder a change of auxiliary null vector field. They also show thatωαβ = 0 implies hypersurface orthogonality for any choice of Nα.

7. We now want to derive evolution equations for the shear and ro-tation tensors of a congruence of null geodesics. For this purposeit is useful to refer back to the basis kα, Nα, eα

A, and the 2 × 2matrix BAB = Bαβ eα

AeβB, introduced in Problem 6. We shall also

need

RAB = Rαµβν eαAkµeβ

Bkν , ΓAB = eBµeµA;νk

ν .

Notice that RAB is a symmetric matrix, while ΓAB is antisym-metric. Notice also that it is possible to set ΓAB = 0 by choosingeα

A to be parallel transported along the congruence.

a) First, derive the main evolution equation,

dBAB

dλ= −BACBC

B −RAB + Γ CA BCB + Γ C

B BAC .

b) Second, decompose the various matrices into their irreducibleparts, as

BAB =1

2θ δAB + σAB + ωAB, RAB =

1

2δAB + CAB,

where σAB and CAB are both symmetric and tracefree, whileωAB is antisymmetric. Prove that = Rαβkαkβ and CAB =

Cαµβν eαAkµeβ

Bkν , where Cαµβν is the Weyl tensor (Sec. 1.13,Problem 8). Then introduce the parameterization

σAB =

(σ+ σ×σ× −σ+

), CAB =

(C+ C×C× −C+

)

for the symmetric-tracefree matrices, and

ωAB =

(0 ω−ω 0

), ΓAB =

(0 Γ−Γ 0

)

for the antisymmetric matrices.

56 Geodesic congruences

c) Third, and finally, derive the following explicit forms for theevolution equations,

dθ

dλ= −1

2θ2 − 2(σ+

2 + σ×2) + 2ω2 − ,

dσ+

dλ= −θ σ+ − C+ + 2Γ σ×,

dσ×dλ

= −θ σ× − C× − 2Γ σ+,

dω

dλ= −θ ω.

Check that the equation for θ agrees with the form of Ray-chaudhuri’s equation given in the text. Recall that we canalways set Γ = 0 by taking eα

A to be parallel transportedalong the congruence; this eliminates the coupling betweenthe shear parameters.

8. Retrace the steps of Sec. 2.4, but without the assumption thatthe null geodesics are affinely parameterized. Show that:

a) Equation (2.4.8) stays unchanged.

b) The expansion is now given by θ = kα;α−κ, where κ is defined

by the relation kα;βkβ = κ kα.

c) Raychaudhuri’s equation now takes the form

dθ

dλ= κ θ − 1

2θ2 − σαβσαβ + ωαβωαβ −Rαβkαkβ.

Chapter 3

Hypersurfaces

This chapter covers three main topics that can all be grouped underthe rubric of hypersurfaces, the term designating a three-dimensionalsubmanifold in a four-dimensional spacetime.

The first part of the chapter (Secs. 3.1 to 3.3) is concerned withthe intrinsic geometry of a hypersurface, and it examines the followingquestions: Given that the spacetime is endowed with a metric tensorgαβ, how does one define an induced, three-dimensional metric hab ona particular hypersurface? And once this three-metric has been intro-duced, how does one define a vectorial surface element that allows vec-tor fields to be integrated over the hypersurface? While these questionsadmit straightforward answers when the hypersurface is either timelikeor spacelike, we will see that the null case requires special care.

The second part of the chapter (Secs. 3.4 to 3.6) is concerned withthe extrinsic geometry of a hypersurface, or how the hypersurface isembedded in the enveloping spacetime manifold. We will see how thespacetime curvature tensor can be decomposed into a purely intrinsicpart — the curvature tensor of the hypersurface — and an extrinsicpart that measures the bending of the hypersurface in spacetime; thisbending is described by a three-dimensional tensor Kab known as theextrinsic curvature. We will see what constraints the Einstein fieldequations place on the induced metric and extrinsic curvature of a hy-persurface.

The third part of the chapter (Secs. 3.7 to 3.11) is concerned withpossible discontinuities of the metric and its derivatives across a hy-persurface. We will consider the following question: Suppose that ahypersurface partitions spacetime into two regions, and that we aregiven a distinct metric tensor in each region; does the union of the twometrics form a valid solution to the Einstein field equations? We willsee that the conditions for an affirmative answer are that the inducedmetric and the extrinsic curvature must be the same on both sides ofthe hypersurface. Failing this, we will see that a discontinuity in theextrinsic curvature can be explained by the presence of a thin distribu-tion of matter — a surface layer — at the hypersurface. (The induced

57

58 Hypersurfaces

Figure 3.1: A three-dimensional hypersurface in spacetime.

Φ = 0

metric can never be discontinuous: the hypersurface would not have awell-defined intrinsic geometry.) We will first develop the mathematicalformalism of junction conditions and surface layers, and then considersome applications.

3.1 Description of hypersurfaces

3.1.1 Defining equations

In a four-dimensional spacetime manifold, a hypersurface is a three-dimensional submanifold that can be either timelike, spacelike, or null.A particular hypersurface Σ is selected either by putting a restrictionon the coordinates,

Φ(xα) = 0, (3.1.1)

or by giving parametric equations of the form

xα = xα(ya), (3.1.2)

where ya (a = 1, 2, 3) are coordinates intrinsic to the hypersurface. Forexample, a two-sphere in a three-dimensional flat space is describedeither by Φ(x, y, z) = x2 + y2 + z2 − R2 = 0, where R is the sphere’sradius, or by x = R sin θ cos φ, y = R sin θ sin φ, and z = R cos θ, whereθ and φ are the intrinsic coordinates. Notice that the relations xα(ya)describe curves contained entirely in Σ (Fig. 3.1).

3.1.2 Normal vector

The vector Φ,α is normal to the hypersurface, because the value of Φchanges only in the direction orthogonal to Σ. A unit normal nα canbe introduced if the hypersurface is not null. This is defined by

nαnα = ε ≡ −1 if Σ is spacelike

+1 if Σ is timelike, (3.1.3)

3.1 Description of hypersurfaces 59

and we demand that nα point in the direction of increasing Φ: nαΦ,α >0. It is easy to check that nα is given by

nα =εΦ,α

|gµνΦ,µΦ,ν |1/2(3.1.4)

if the hypersurface is either spacelike or timelike.The unit normal if not defined when Σ is null, because gµνΦ,µΦ,ν is

then equal to zero. In this case we let

kα = −Φ,α (3.1.5)

be the normal vector; the sign is chosen so that kα is future-directedwhen Φ increases toward the future. Because kα is orthogonal to it-self (kαkα = 0), this vector is also tangent to the null hypersurface Σ(Fig. 3.2). In fact, by computing kα

;βkβ and showing that it is propor-tional to kα, we can prove that kα is tangent to null geodesics containedin Σ. We have kα;βkβ = Φ;αβΦ,β = Φ;βαΦ,β = 1

2(Φ,βΦ,β);α; because

Φ,βΦ,β is zero everywhere on Σ, its gradient must be directed along kα,and we have that (Φ,βΦ,β);α = 2κkα for some scalar κ. We have foundthat the normal vector satisfies

kα;βkβ = κkα,

the general form of the geodesic equation. The hypersurface is there-fore generated by null geodesics, and kα = dxα/dλ is tangent to thegenerators. In general, the parameter λ is not affine, but in specialsituations in which the relations Φ(xα) = constant describe a wholefamily of null hypersurfaces (so that Φ,βΦ,β is zero not only on Σ butalso in a neighbourhood around Σ), κ = 0 and λ is an affine parameter.

When the hypersurface is null, it is advantageous to install on Σa coordinate system that is well adapted to the behaviour of the gen-erators. We therefore let the parameter λ be one of the coordinates,and we introduce two additional coordinates θA (A = 2, 3) to labelthe generators; these are constant on each generator, and they spanthe two-dimensional space transverse to the generators. Thus, we shalladopt

ya = (λ, θA) (3.1.6)

when Σ is null; varying λ while keeping θA constant produces a displace-ment along a single generator, and changing θA produces a displacementacross generators.

3.1.3 Induced metric

The metric intrinsic to the hypersurface Σ is obtained by restricting theline element to displacements confined to the hypersurface. Recalling

60 Hypersurfaces

Figure 3.2: A null hypersurface and its generators.

kα

Φ = 0

the parametric equations xα = xα(ya), we have that the vectors

eαa =

∂xα

∂ya(3.1.7)

are tangent to curves contained in Σ. (This implies that eαanα = 0 in the

non-null case, and eαakα = 0 in the null case.) Now, for displacements

within Σ,

ds2Σ = gαβ dxαdxβ

= gαβ

(∂xα

∂yadya

)(∂xβ

∂ybdyb

)

= hab dyadyb, (3.1.8)

where

hab = gαβ eαaeβ

b (3.1.9)

is called the induced metric, or first fundamental form, of the hyper-surface. It is a scalar with respect to transformations xα → xα′ of thespacetime coordinates, but it transforms as a tensor under transforma-tions ya → ya′ of the hypersurface coordinates. We will refer to suchobjects as three-tensors.

These relations simplify when the hypersurface is null and we usethe coordinates of Eq. (3.1.6). Then eα

1 = (∂xα/∂λ)θA ≡ kα, and itfollows that h11 = gαβkαkβ = 0 and h1A = gαβkαeβ

A = 0, because byconstruction, eα

A ≡ (∂xα/∂θA)λ is orthogonal to kα. In the null case,therefore,

ds2Σ = σAB dθAdθB, (3.1.10)

where

σAB = gαβ eαAeβ

B, eαA =

(∂xα

∂θA

)λ. (3.1.11)

Here the induced metric is a two-tensor.

3.1 Description of hypersurfaces 61

We conclude by writing down completeness relations for the inversemetric. In the non-null case,

gαβ = εnαnβ + habeαaeβ

b , (3.1.12)

where hab is the inverse of the induced metric. Equation (3.1.12) isverified by computing all inner products between nα and eα

a . In thenull case we must introduce, everywhere on Σ, an auxiliary null vectorfield Nα satisfying Nαkα = −1 and Nαeα

A = 0 (see Sec. 2.4). Then theinverse metric can be expressed as

gαβ = −kαNβ −Nαkβ + σABeαAeβ

B, (3.1.13)

where σAB is the inverse of σAB. Equation (3.1.13) is verified by com-puting all inner products between kα, Nα, and eα

A.

3.1.4 Light cone in flat spacetime

An example of a null hypersurface in flat spacetime is the future lightcone of an event P , which we place at the origin of a Cartesian co-ordinate system xα. The defining relation for this hypersurface isΦ ≡ t − r = 0, where r2 = x2 + y2 + z2. The normal vector iskα = −∂α(t − r) = (−1, x/r, y/r, z/r). A suitable set of parametricequations is t = λ, x = λ sin θ cos φ, y = λ sin θ sin φ, and z = λ cos θ,in which ya = (λ, θ, φ) are the intrinsic coordinates; λ is an affine pa-rameter on the light cone’s null generators, which move with constantvalues of θA = (θ, φ).

From the parametric equations we compute the hypersurface’s tan-gent vectors,

eαλ =

∂xα

∂λ= (1, sin θ cos φ, sin θ sin φ, cos θ) = kα,

eαθ =

∂xα

∂θ= (1, λ cos θ cos φ, λ cos θ sin φ,−λ sin θ),

eαφ =

∂xα

∂φ= (1,−λ sin θ sin φ, λ sin θ cos φ, 0).

You may check that these vectors are all orthogonal to kα. Inner prod-ucts between eα

θ and eαφ define the two-metric σAB, and we find

σAB dθAdθB = λ2(dθ2 + sin2 θ dφ2).

Not surprisingly, the hypersurface has a spherical geometry, and λ isthe areal radius of the two-spheres.

It is easy to check that the unique null vector Nα that satisfies the re-lations Nαkα = −1 and Nαeα

A = 0 is Nα = 12(1,− sin θ cos φ,− sin θ sin φ,− cos θ).

You may also verify that the vectors kα, Nα, and eαA combine as in

Eq. (3.1.13) to form the inverse Minkowski metric.

62 Hypersurfaces

3.2 Integration on hypersurfaces

3.2.1 Surface element (non-null case)

If Σ is not null, thendΣ ≡ |h|1/2 d3y, (3.2.1)

where h ≡ det[hab], is an invariant three-dimensional volume elementon the hypersurface. To avoid confusing this with the four-dimensionalvolume element

√−g d4x, we shall refer to dΣ as a surface element.The combination nαdΣ is a directed surface element that points in thedirection of increasing Φ. In the null case these quantities are notdefined, because h = 0 and nα does not exist.

To see how Eq. (3.2.1) must be generalized to incorporate also thenull case, we consider the infinitesimal vector field

dΣµ = εµαβγ eα1 eβ

2eγ3 d3y, (3.2.2)

where εµαβγ =√−g[µα β γ] is the Levi-Civita tensor. We will show

below thatdΣα = εnαdΣ (3.2.3)

when the hypersurface is not null. Thus, apart from a factor ε = ±1,dΣα is a directed surface element on Σ. Notice that when Σ is spacelike,the factor ε = −1 makes dΣα a past-directed vector; this is a potentialsource of confusion. Notice also that Eq. (3.2.2) remains meaningfuleven when the hypersurface is null. By continuity, therefore, dΣα isalso a directed surface element on a null hypersurface.

Because dΣα is proportional to the completely antisymmetric Levi-Civita tensor, its sign depends on the ordering of the coordinates y1,y2, and y3. But this ordering is a priori arbitrary, and we need aconvention to remove the sign ambiguity. We shall choose an orderingthat makes the scalar f ≡ εµαβγn

µeα1 eβ

2eγ3 a positive quantity. Notice

that this convention was already adopted when we went from Eq. (3.2.2)to Eq. (3.2.3): nα dΣα = dΣ > 0.

As a first example of how this works, consider a hypersurface ofconstant t in Minkowski spacetime. If Φ = t, then nα = −∂αt is thefuture-directed normal vector. If we choose the ordering ya = (x, y, z),we find that f = εtxyz = 1 has the correct sign. Equation (3.2.2) impliesdΣµ = δt

µ dxdydz = −nµ dxdydz, which is compatible with Eq. (3.2.3).As a second example, we take a surface of constant x in Minkowski

spacetime. We take Φ = x, and nα = ∂αx points in the direction ofincreasing Φ. We choose the ordering ya = (y, t, z) because f = εxytz =−εxtyz = εtxyz = 1 has then the correct sign. [Notice that the moretempting ordering ya = (t, y, z) would produce the wrong sign.] Withthis choice, Eq. (3.2.2) implies dΣµ = δx

µ dtdydz = nµ dtdydz, which iscompatible with Eq. (3.2.3).

3.2 Integration on hypersurfaces 63

We now turn to the derivation of Eq. (3.2.3). It is clear that dΣµ

must be proportional to nµ, because eµaεµαβγ eα

1 eβ2e

γ3 = 0 by virtue of

the antisymmetric property of the Levi-Civita tensor. So we may write

εµαβγ eα1 eβ

2eγ3 = εfnµ,

where f = εµαβγnµeα

1 eβ2e

γ3 . Because f is a scalar, we may evaluate it

in any convenient coordinate system xα. We choose our coordinates sothat x0 ≡ Φ, and on Σ we identify xa with the intrinsic coordinatesya. Then f

∗=√−g nΦ. In these coordinates, gΦΦ ∗

= gαβΦ,αΦ,β, and

nΦ∗= ε |gΦΦ|−1/2 is the only nonvanishing component of the normal.

It follows that nΦ ∗= gΦαnα

∗= gΦΦnΦ

∗= |gΦΦ|1/2, and we have that

f∗= |ggΦΦ|1/2. We now use the definition of the matrix inverse to write

gΦΦ ∗= cofactor(gΦΦ)/g, where the cofactor of a matrix element is the

determinant obtained after eliminating the row and column to whichthe element belongs. This determinant is clearly h, and we concludethat

f = |h|1/2.

While this result was obtained in the special coordinates xα, it is validin all coordinate systems because h, like hab, is a scalar with respectto four-dimensional coordinate transformations. This result shows thatwhen Σ is not null, Eq. (3.2.3) is indeed equivalent to Eq. (3.2.2).

3.2.2 Surface element (null case)

As we have seen in Sec. 3.1.2, when Σ is null we identify y1 with λ,the parameter on the hypersurface’s null generators, and the remainingcoordinates, denoted θA, are constant on the generators. Then eα

1 = kα,d3y = dλ d2θ, and we may write the directed surface element as

dΣµ = kνdSµνdλ, (3.2.4)

where

dSµν = εµνβγ eβ2e

γ3 d2θ (3.2.5)

is interpreted as an element of two-dimensional surface area. We willshow below that this can also be expressed as

dSαβ = 2k[αNβ]

√σ d2θ, (3.2.6)

where Nα is the auxiliary null vector field introduced in Eq. (3.1.13),and σ = det[σAB], with σAB the two-metric defined by Eq. (3.1.11).Combining Eq. (3.2.6) with Eq. (3.2.4) yields

dΣα = −kα

√σ d2θdλ. (3.2.7)

64 Hypersurfaces

The interpretation of this result is clear: Apart from a minus sign, thesurface element is directed along kα, the normal to the null hypersur-face; the factor dλ represents an element of parameter-distance alongthe null generators, and

√σ d2θ is an element of cross-sectional area —

an element of two-dimensional surface area in the directions transverseto the generators.

There is also an ordering issue with the coordinates θA, and ourconvention shall be that the scalar f ≡ εµνβγN

µkνeβ2e

γ3 must be a pos-

itive quantity. Notice that this convention was already adopted whenwe went from Eq. (3.2.5) to Eq. (3.2.6): Nαkβ dSαβ =

√σ d2θ > 0.

As an example, consider a surface u = constant in Minkowski space-time, where u = t− x. The normal vector is kα = −∂α(t− x), and wemay choose the ordering θA = (y, z). Then Nα = −1

2∂α(t + x) satisfies

all the requirements for an auxiliary null vector field. It is easy to checkthat with these choices, f = 1 (which is properly positive). We obtaindStx = dydz = −dSxt, and since t can be identified with the affineparameter λ, Eq. (3.2.4) implies dΣt = dtdydz = −dΣx. These resultsare compatible with Eq. (3.2.7).

Let us consider a more complicated example: the light cone ofSec. 3.1.4. The vectors kα, Nα, and eα

A are displayed in that section, andthe cone’s intrinsic coordinates are ya = (λ, θ, φ). We want to computedΣµ for this hypersurface, starting with the definition of Eq. (3.2.2).We know that dΣµ must point in the direction of the normal, so that

dΣµ = −fkµ dθdφdλ, where f = εµνβγNµkνeβ

2eγ3 . If we let Nµ ≡ eµ

0

and kν ≡ eν1, we can write this as f = [µ ν β γ]eµ

0eν1e

β2e

γ3 ≡ det E, where

E is the matrix constructed by lining up the four basis vectors. Itsdeterminant is easy to compute, and we obtain f = λ2 sin θ =

√σ. We

therefore have dΣµ = −kµ

√σ d2θdλ, which is just the same statement

as in Eq. (3.2.7).

We must now give a proper derivation of Eq. (3.2.6). The steps aresomewhat similar to those leading to Eq. (3.2.3). We begin by notingthat the tensor εµνβγ eβ

2eγ3 is orthogonal to eα

A and antisymmetric in theindices µ and ν. It may be expressed as

εµνβγ eβ2e

γ3 = 2fk[µNν] = f(kµNν −Nµkν),

where f = εµνβγNµkνeβ

2eγ3 > 0. To evaluate f we choose our coordi-

nates xα such that x0 ≡ Φ, and xa ≡ ya = (λ, θA) on Σ. In these

coordinates, kΦ∗= 1 and kλ ∗

= 1 are the only nonvanishing componentsof the normal vector, NΦ ∗

= 1 comes as a consequence of the normaliza-tion condition Nαkα = −1, and gΦΦ ∗

= 0 follows from the fact that kα

is null. Using this information, we deduce that f∗=√−g, and we must

now compute the metric determinant in the specified coordinates. Forthis purpose, we note that the completeness relations of Eq. (3.1.13)

3.2 Integration on hypersurfaces 65

imply the following structure for the inverse metric:

g−1 =

0 1 01 −2Nλ −NA

0 −NA σAB

;

this immediately implies det g−1 = −det[σAB], or√−g

∗=√

σ. Wetherefore have

f =√

σ,

which holds in any coordinate system xα. This shows that Eq. (3.2.6)is indeed equivalent to Eq. (3.2.5), and this implies that Eq. (3.2.7) isequivalent to Eq. (3.2.2) when Σ is null and coordinates ya = (λ, θA)are placed on the hypersurface.

3.2.3 Element of two-surface

The interpretation of

dSµν = εµνβγ eβ2e

γ3 d2θ

as a directed element of two-dimensional surface area is not limitedto the consideration of null hypersurfaces. Here we consider a typi-cal situation, in which a two-dimensional surface S is imagined to beembedded in a three-dimensional, spacelike hypersurface Σ.

The hypersurface Σ is described by an equation of the form Φ(xα) =0, and by parametric relations xα(ya); nα ∝ ∂αΦ is the future-directedunit normal, and the vectors eα

a = ∂xα/∂ya are tangent to the hyper-surface. The metric on Σ, induced from gαβ, is hab = gαβ eα

aeβb , and we

have the completeness relations gαβ = −nαnβ + hab eαaeβ

b .The two-surface S is introduced as a submanifold of Σ. It is de-

scribed by an equation of the form ψ(ya) = 0, and by parametric re-lations ya(θA), in which θA are coordinates intrinsic to S; ra ∝ ∂aψ isthe outward unit normal, and the three-vectors ea

A = ∂ya/∂θA are tan-gent to the two-surface. The metric on S, induced from hab, is σAB =hab ea

AebB, and we have the completeness relations hab = rarb+σAB ea

AebB.

The parametric relations ya(θA) and xα(ya) can be combined togive the relations xα(θA), which describe how S is embedded in thefour-dimensional spacetime. The vectors

eαA =

∂xα

∂θA=

∂xα

∂ya

∂ya

∂θA= eα

a eaA

are tangent to S, and

rα ≡ ra eαa , rαnα = 0

66 Hypersurfaces

is normal to S. The vector nα is also normal to S, and we have thatthe two-surface admits two normal vectors: a timelike normal nα and aspacelike normal rα. We note that the spacelike normal can be relatedto a gradient, rα ∝ ∂αΨ, if we introduce, in a neighbourhood of Σ, afunction Ψ(xα) such that Ψ|Σ ≡ ψ. In this description, the inducedmetric on S is still

σAB = hab eaAeb

B

= (gαβ eαaeβ

b ) eaAeb

B

= gαβ(eαaea

A)(eβb eb

B)

= gαβ eαAeβ

B,

and the completeness relations

gαβ = −nαnβ + rαrβ + σAB eαAeβ

B

are easily established from our preceding results.We want to show that dSαβ can be expressed neatly in terms of the

timelike normal nα, the spacelike normal rα, and√

σ d2θ, the inducedsurface element on S. The expression is

dSαβ = −2n[αrβ]

√σ d2θ, (3.2.8)

where σ = det[σAB]. The derivation of this result involves familiarsteps. We first note that because εµνβγ eβ

2eγ3 is orthogonal to eα

A andantisymmetric in µ and ν, it may be expressed as

εµνβγ eβ2e

γ3 = −2fn[µrν] = −f(nµrν − rµnν),

where f = εµνβγnµrνeβ

2eγ3 > 0. To evaluate f we adopt coordinates

x0 ≡ Φ, x1 ≡ Ψ, and on S we identify xA with θA. In these coordi-nates, nΦ

∗= −(−gΦΦ)−1/2 is the only nonvanishing component of the

timelike normal, rΨ∗= (gΨΨ)−1/2 is the only nonvanishing component

of the spacelike normal, and from the fact that these vectors are or-thogonal we infer gΦΨ ∗

= 0. From all this we find that f 2 ∗= ggΦΦgΨΨ,

which we rewrite as f 2 ∗= cofactor(gΦΦ) cofactor(gΨΨ)/g. We also have

cofactor(gΦΨ)∗= 0, and these two equations give us enough information

to deducef =

√σ.

This result is true in any coordinate system xα.As a final remark, we note that the vectors nα and rα can be com-

bined to form null vectors kα and Nα. The appropriate relations are

kα =1√2

(nα + rα), Nα =1√2

(nα − rα),

and these vectors are the null normals of the two-surface S. It is easyto check that with these substitutions, Eq. (3.2.8) takes the form ofEq. (3.2.6).

3.3 Gauss-Stokes theorem 67

Figure 3.3: Proof of the Gauss-Stokes theorem.

∂

x0 = const

x0 = 1

3.3 Gauss-Stokes theorem

3.3.1 First version

We consider a finite region of the spacetime manifold, bounded by aclosed hypersurface ∂ (Fig. 3.3). The signature of the hypersurface isnot restricted; it may have segments that are timelike, spacelike, ornull. We will show that for any vector field Aα defined within ,

∫Aα

;α

√−g d4x =

∮

∂

Aα dΣα, (3.3.1)

where dΣα is the surface element defined by Eq. (3.2.2).

To prove this result, known as Gauss’ theorem, we construct thefollowing coordinate system in . We imagine a nest of closed hyper-surfaces foliating , with the boundary ∂ forming the outer layer of thenest. (Picture this as the many layers of an onion.) We let x0 be aconstant on each one of these hypersurfaces, with x0 = 1 designating∂, and x0 = 0 the zero-volume hypersurface at the “centre” of . Whilex0 grows “radially outward” from this “centre”, we take the remainingcoordinates xa to be angular coordinates on the closed hypersurfacesx0 = constant. The coordinates ya on ∂ are then identified with theseangular coordinates.

Using such coordinates, the left-hand side of Eq. (3.3.1) becomes

∫Aα

;α

√−g d4x =

∫(√−g Aα),α d4x

∗=

∫dx0

∮(√−g A0),0 d3x +

∫dx0

∮(√−gAa),a d3x

∗=

∫dx0 d

dx0

∮ √−g A0 d3x

∗=

∮ √−g A0 d3x∣∣∣x0=1

x0=0

68 Hypersurfaces

Figure 3.4: Two spacelike surfaces and their normal vectors.

Σ1

nα1

Σ2

nα2

∗=

∮

∂

√−g A0 d3y.

In the first line we have used the divergence formula for the vector fieldAα. The second integral of the second line vanishes because xa are angu-lar coordinates and the integration is over a closed three-dimensionalsurface. (Understanding this statement requires some thought. Tryworking through a three-dimensional version of the proof, using spher-ical coordinates in flat space.) In the fourth line, the contribution atx0 = 0 vanishes because the “hypersurface” x0 = 0 has zero volume.

It is easy to check that in the specified coordinates, dΣα∗= δ0

α

√−g d3y,giving ∮

∂

Aα dΣα∗=

∮

∂

A0√−g d3y

for the right-hand side of Eq. (3.3.1). The two sides are therefore equalin the specified coordinate system; because Eq. (3.3.1) is a tensorialequation, this suffices to establish the validity of the theorem.

3.3.2 Conservation

Gauss’ theorem has many useful applications. An example is the fol-lowing conservation statement.

Suppose that a vector field jα has a vanishing divergence,

jα;α = 0.

Then∮

Σjα dΣα = 0 for any closed hypersurface Σ. Supposing now that

jα vanishes at spatial infinity, we can choose Σ to be composed of twospacelike hypersurfaces, Σ1 and Σ2, extending all the way to infinity(Fig. 3.4), and of a three-cylinder at infinity, on which jα = 0. Then

∫

Σ1

jα dΣα +

∫

Σ2

jα dΣα = 0.

3.3 Gauss-Stokes theorem 69

On each of the spacelike hypersurfaces, dΣα = −nα

√h d3y, where nα is

the outward normal to the closed surface Σ, and h is the determinant ofthe induced metric on the spacelike hypersurfaces. Letting nα ≡ n2α onΣ2 and nα ≡ −n1α on Σ1, where n1α and n2α are both future directed,we have

jα;α = 0 ⇒

∫

Σ1

jαn1α

√h d3y =

∫

Σ2

jαn2α

√h d3y. (3.3.2)

The interpretation of this result is clear: If jα is a divergence-free vector,then the “total charge”

∫jαnα dΣ is independent of the hypersurface

on which it is evaluated. This is obviously a statement of “charge”conservation.

3.3.3 Second version

Another version of Gauss’ theorem (usually called Stokes’ theorem)involves a three-dimensional region Σ bounded by a closed two-surface∂Σ. It states that for any antisymmetric tensor field Bαβ in Σ,

∫

Σ

Bαβ;β dΣα =

1

2

∮

∂Σ

Bαβ dSαβ, (3.3.3)

where dSαβ is the two-surface element defined by Eq. (3.2.5).The derivation of this identity proceeds along familiar lines. We

construct a coordinate system such that (i) x0 is constant on the hy-persurface Σ, (ii) x1 = constant describes a nest of closed two-surfacesin Σ (with x1 = 1 representing ∂Σ and x1 = 0 the zero-area surface atthe “centre” of Σ), and (iii) xA are angular coordinates on the closedsurfaces (with θA = xA on ∂Σ).

It is easy to check that with such coordinates, dΣα = δ0α

√−g dx1dx2dx3.The left-hand side of Eq. (3.3.3) becomes

∫

Σ

Bαβ;β dΣα =

∫

Σ

1√−g(√−g Bαβ),β dΣα

∗=

∫

Σ

(√−g B0β),β dx1dx2dx3

∗=

∫dx1

∮(√−g B01),1 dx2dx3 +

∫dx1

∮(√−g B0A),A dx2dx3

∗=

∮ √−g B01 dx2dx3∣∣∣x1=1

x1=0

∗=

∮

∂Σ

√−g B01 d2θ.

In the first line we have used the divergence formula for an antisym-metric tensor field. The explicit expression for dΣα was substituted in

70 Hypersurfaces

the second line. The second integral of the third line vanishes becausexA are angular coordinates and the domain of integration is a closedtwo-surface. In the fourth line, the lower limit of integration does notcontribute because the “surface” x1 = 0 has zero area.

It is easy to check that in the specified coordinate system, Bαβ dSαβ =(B01−B10)

√−g d2θ = 2B01√−g d2θ. The right-hand side of Eq. (3.3.3)

therefore reads

1

2

∮

∂Σ

Bαβ dSαβ∗=

∮

∂Σ

B01√−g d2θ.

Equation (3.3.3) follows from the equality of both sides in the specifiedcoordinate system.

3.4 Differentiation of tangent vector

fields

3.4.1 Tangent tensor fields

(For the remainder of Chapter 3, except for Sec. 3.11, we shall assumethat the hypersurface Σ is either spacelike or timelike. In Sec. 3.11 weshall return to the case of null hypersurfaces.)

Once we are presented with a hypersurface Σ, it is a common situa-tion to have tensor fields Aαβ··· that are defined only on Σ and which arepurely tangent to the hypersurface. Such tensors admit the followingdecomposition:

Aαβ··· = Aab··· eαaeβ

b · · · , (3.4.1)

where eαa = ∂xα/∂ya are basis vectors on Σ. Equation (3.4.1) implies

that Aαβ···nα = Aαβ···nβ = · · · = 0, which confirms that Aαβ··· is tan-gent to the hypersurface. We note that an arbitrary tensor Tαβ··· canalways be projected down to the hypersurface, so that only its tangen-tial components survive. The quantity that effects the projection ishαβ ≡ habeα

aeαβ = gαβ − εnαnβ, and hα

µhβν · · ·T µν··· is evidently tangent

to the hypersurface.The projections

Aαβ··· eαaeβ

b · · · = Aab··· ≡ hamhbn · · ·Amn··· (3.4.2)

give the three-tensor Aab··· associated with the tangent tensor fieldAαβ···; latin indices are lowered and raised with hab and hab, respec-tively. Equations (3.4.1) and (3.4.2) show that one can easily go backand forth between a tangent tensor field Aαβ··· and its equivalent three-tensor Aab···. We emphasize that while Aab··· transforms as a tensorunder a transformation ya → ya′ of the coordinates intrinsic to Σ, it isa scalar under a transformation xα → xα′ of the spacetime coordinates.

3.4 Differentiation of tangent vector fields 71

3.4.2 Intrinsic covariant derivative

We wish to consider how tangent tensor fields are differentiated. Wewant to relate the covariant derivative of Aαβ··· (with respect to a con-nection that is compatible with the spacetime metric gαβ) to the covari-ant derivative of Aab···, defined in terms of a connection that is compat-ible with the induced metric hab. For simplicity, we shall restrict ourattention to the case of a tangent vector field Aα, such that

Aα = Aaeαa , Aαnα = 0, Aa = Aα eα

a .

Generalization to three-tensors of higher ranks will be obvious.We define the intrinsic covariant derivative of a three-vector Aa to

be the projection of Aα;β onto the hypersurface:

Aa|b ≡ Aα;β eαaeβ

b . (3.4.3)

We will show that Aa|b, as defined here, is nothing but the covariantderivative of Aa defined in the usual way in terms of a connection Γa

bc

that is compatible with hab.To get started, let us express the right-hand side of Eq. (3.4.3) as

Aα;βeαaeβ

b = (Aαeαa );βeβ

b − Aαeαa;βeβ

b

= Aa,βeβb − eaγ;βeβ

b Aceγc

=∂Aa

∂xβ

∂xβ

∂yb− eγ

c eaγ;βeβb Ac

= Aa,b − ΓcabAc,

where we have definedΓcab = eγ

c eaγ;βeβb . (3.4.4)

Equation (3.4.3) then reads

Aa|b = Aa,b − ΓcabAc, (3.4.5)

which is the familiar expression for the covariant derivative.The connection used here is the one defined by Eq. (3.4.4), and we

would like to show that it is compatible with the induced metric. Inother words, we would like to prove that Γcab, as defined by Eq. (3.4.4),can also be expressed as

Γcab =1

2(hca,b + hcb,a − hab,c). (3.4.6)

This could be done by directly working out the right-hand side ofEq. (3.4.4). It is easier, however, to show that the connection is suchthat hab|c ≡ hαβ;γ eα

aeβb eγ

c = 0. Indeed,

hαβ;γ eαaeβ

b eγc = (gαβ − εnαnβ);γ eα

aeβb eγ

c

= −ε(nα;γnβ + nαnβ;γ) eαaeβ

b eγc

= 0,

72 Hypersurfaces

because nα eαa = 0. Intrinsic covariant differentiation is therefore the

same operation as straightforward covariant differentiation of a three-tensor.

3.4.3 Extrinsic curvature

The quantities Aa|b = Aα;βeαaeβ

b are the tangential components of the

vector Aα;βeβ

b . The question we would like to investigate now is whetherthis vector possesses also a normal component.

To answer this we re-express Aα;βeβ

b as gαµA

µ;βeβ

b and decomposethe metric into its normal and tangential parts, as in Eq. (3.1.12). Thisgives

Aα;βeβ

b = (εnαnµ + hameαaemµ)Aµ

;βeβb

= ε(nµAµ;βeβ

b )nα + ham(Aµ;βeµmeβ

b )eαa ,

and we see that only the second term is tangent to the hypersurface.We now use Eq. (3.4.3) and the fact that Aµ is orthogonal to nµ:

Aα;βeβ

b = −ε(nµ;βAµeβb )nα + hamAm|be

αa

= Aa|b eα

a − εAa(nµ;βeµae

βb )nα.

At this point we introduce the three-tensor

Kab ≡ nα;β eαaeβ

b , (3.4.7)

called the extrinsic curvature, or second fundamental form, of the hy-persurface Σ. In terms of this, we have

Aα;βeβ

b = Aa|b eα

a − εAaKabnα, (3.4.8)

and we see that Aa|b gives the purely tangential part of the vector field,

while −εAaKab represents the normal component. This answers ourquestion: the normal component vanishes if and only if the extrinsiccurvature vanishes.

We note that if eαa is substituted in place of Aα, then Ac = δc

a andEqs. (3.4.5), (3.4.8) imply

eαa;βeβ

b = Γcab eα

c − εKabnα. (3.4.9)

This is known as the Gauss-Weingarten equation.The extrinsic curvature is a very important quantity; we will en-

counter it often in the rest of this book. We may prove that it is asymmetric tensor:

Kba = Kab. (3.4.10)

3.5 Gauss-Codazzi equations 73

The proof is based on the properties that (i) the vectors eαa and nα

are orthogonal, and (ii) the basis vectors are Lie transported along oneanother, so that eα

a;βeβb = eα

b;βeβa . We have

nα;βeαaeβ

b = −nαeαa;βeβ

b

= −nαeαb;βeβ

a

= nα;βeαb eβ

a ,

and Eq. (3.4.10) follows. The symmetry of the extrinsic curvature im-plies the relations

Kab = n(α;β)eαaeβ

b =1

2(£ngαβ)eα

aeβb , (3.4.11)

and the extrinsic curvature is therefore intimately related to the normalderivative of the metric tensor.

We also note the relation

K ≡ habKab = nα;α, (3.4.12)

which shows that K is equal to the expansion of a congruence ofgeodesics that intersect the hypersurface orthogonally (so that theirtangent vector is equal to nα on the hypersurface). From this result weconclude that the hypersurface is convex if K > 0 (the congruence isdiverging), or concave if K < 0 (the congruence is converging).

We see that while hab is concerned with the purely intrinsic aspectsof a hypersurface’s geometry, Kab is concerned with the extrinsic as-pects — the embedding of the hypersurface in the enveloping spacetimemanifold. Taken together, these tensors provide a virtually completecharacterization of the hypersurface.

3.5 Gauss-Codazzi equations

3.5.1 General form

We have introduced the induced metric hab and its associated intrinsiccovariant derivative. A purely intrinsic curvature tensor can now bedefined by the relation

Ac|ab − Ac

|ba = −RcdabA

d, (3.5.1)

which of course implies

Rcdab = Γc

db,a − Γcda,b + Γc

maΓmdb − Γc

mbΓmda. (3.5.2)

The question we now examine is whether this three-dimensional Rie-mann tensor can be expressed in terms of Rγ

δαβ — the four-dimensionalversion — evaluated on Σ.

74 Hypersurfaces

To answer this we start with the identity

(eαa;βeβ

b );γeγc = (Γd

abeαd − εKabn

α);γeγc

which follows immediately from Eq. (3.4.9). We first develop the left-hand side:

lhs = (eαa;βeβ

b );γeγc

= eαa;βγe

βb eγ

c + eαa;βeβ

b;γeγc

= eαa;βγe

βb eγ

c + eαa;β(Γd

bceβd − εKbcn

β)

= eαa;βγe

βb eγ

c + Γdbc(Γ

eade

αe − εKadn

α)− εKbceαa;βnβ.

Next we turn to the right-hand side:

rhs = (Γdabe

αd − εKabn

α);γeγc

= Γdab,ce

αd + Γd

abeαd;γe

γc − εKab,cn

α − εKabnα;γe

γc

= Γdab,ce

αd + Γd

ab(Γedce

αe − εKdcn

α)− εKab,cnα − εKabn

α;γe

γc .

We now equate the two sides and solve for eαa;βγe

βb eγ

c . Subtracting a

similar expression for eαa;γβeγ

c eβb gives −Rα

µβγeµae

βb eγ

c , the quantity weare interested in. After some algebra, we find

Rµαβγe

αaeβ

b eγc = Rm

abceµm + ε(Kab|c −Kac|b)n

µ + εKabnµ;γe

γc − εKacn

µ;βeβ

b .

Projecting along edµ gives

Rαβγδ eαaeβ

b eγc e

δd = Rabcd + ε(KadKbc −KacKbd), (3.5.3)

and this is the desired relation between Rabcd and the full Riemanntensor. Projecting instead along nµ gives

Rµαβγnµeα

aeβb eγ

c = Kab|c −Kac|b. (3.5.4)

Equations (3.5.3) and (3.5.4) are known as the Gauss-Codazzi equa-tions. They reveal that the spacetime curvature can be expressed interms of the intrinsic and extrinsic curvatures of a hypersurface.

3.5.2 Contracted form

The Gauss-Codazzi equations can also be written in contracted form,in terms of the Einstein tensor Gαβ = Rαβ − 1

2Rgαβ. The spacetime

Ricci tensor is given by

Rαβ = gµνRµανβ

= (εnµnν + hmneµmeν

n)Rµανβ

= εRµανβnµnν + hmnRµανβeµmeν

n,

3.5 Gauss-Codazzi equations 75

and the Ricci scalar is

R = gαβRαβ

= (εnαnβ + habeαaeβ

b )(εRµανβnµnν + hmnRµανβeµmeν

n)

= 2εhabRµανβnµeαanνeβ

b + habhmnRµανβeµmeα

aeνne

βb .

A little algebra then reveals the relations

−2εGαβnαnβ = 3R + ε(KabKab −K2) (3.5.5)

andGαβ eα

anβ = Kba|b −K,a. (3.5.6)

Here, 3R = habRmamb is the three-dimensional Ricci scalar. The impor-

tance of Eqs. (3.5.5) and (3.5.6) lies with the fact that they form partof the Einstein field equations on a hypersurface Σ; this observationwill be elaborated in the next section. We note that Gαβ eα

aeβb , the re-

maining components of the Einstein tensor, cannot be expressed solelyin terms of hab, Kab, and related quantities.

3.5.3 Ricci scalar

We now complete the computation of the four-dimensional Ricci scalar.Our starting point is the relation

R = 2εhabRµανβnµeαanνeβ

b + habhmnRµανβeµmeα

aeνneβ

b ,

which was derived previously. The first term is simplified by using thecompleteness relation (3.1.12) and the fact that Rµανβnµnαnνnβ = 0;it becomes 2εRαβnαnβ. Using the definition of the Riemann tensor, werewrite this as

Rαβnαnβ = −nα;αβnβ + nα

;βαnβ

= −(nα;αnβ);β + nα

;αnβ;β + (nα

;βnβ);α − nα;βnβ

;α.

In the second term of this last expression we recognize K2, where K =nα

;α is the trace of the extrinsic curvature. The fourth term, on theother hand, can be expressed as

nα;βnβ

;α = gβµgανnα;βnµ;ν

= (εnβnµ + hβµ)(εnαnν + hαν)nα;βnµ;ν

= (εnβnµ + hβµ)hανnα;βnµ;ν

= hβµhανnα;βnµ;ν

= hbmhannα;βeαaeβ

b nµ;νeµmeν

n

= hbmhanKabKmn

= KabKba

= KabKab.

76 Hypersurfaces

In the second line we have inserted the completeness relation (3.1.12)and used the notation hαβ = habeα

aeβb . In the third and fourth lines

we have used the fact that nαnα;β = 12(nαnα);β = 0. In the sixth line

we have substituted the definition (3.4.7) for the extrinsic curvature.Finally, in the last line we have used the fact that Kab is a symmetricthree-tensor.

The previous manipulations take care of the first term in our start-ing expression for the Ricci scalar. The second term is simplified bysubstituting the Gauss-Codazzi equation (3.5.3),

habhmnRµανβeµmeα

aeνne

βb = habhmn

[Rmanb + ε(KmbKan −KmnKab)

]

= 3R + ε(KabKab −K2).

Putting all this together, we arrive at

R = 3R + ε(K2 −KabKab) + 2ε(nα;βnβ − nαnβ

;β);α. (3.5.7)

This is the four-dimensional Ricci scalar evaluated on the hypersurfaceΣ. This result will be put to good use in Chapter 4.

3.6 Initial-value problem

3.6.1 Constraints

In Newtonian mechanics, a complete solution to the equations of motionrequires the specification of initial values for the position and velocityof each moving body. In field theories, a complete solution to the fieldequations requires the specification of the field and its time derivativeat one instant of time.

A similar statement can be made for general relativity. Because theEinstein field equations are second-order partial differential equations,we would expect that a complete solution should require the specifica-tion of gαβ and gαβ,t at one instant of time. While this is essentiallycorrect, it is desirable to convert this decidedly noncovariant statementinto something more geometrical.

The initial-value problem of general relativity starts with the se-lection of a spacelike hypersurface Σ which represents an “instant oftime”. This hypersurface can be chosen freely. On this hypersurfacewe put arbitrary coordinates ya.

The spacetime metric gαβ, when evaluated on Σ, has componentsthat characterize displacements away from the hypersurface. (For ex-ample, gtt is such a component if Σ is a surface of constant t.) Thesecomponents cannot be given meaning in terms of the geometric proper-ties of Σ alone. To provide meaningful initial values for the spacetimemetric, we must consider displacements within the hypersurface only.

3.6 Initial-value problem 77

In other words, the initial values for gαβ can only be the six components

of the induced metric hab = gαβ eαaeβ

b ; the remaining four componentsare arbitrary, and this reflects the complete freedom in choosing thespacetime coordinates xα.

Similarly, the initial values for the “time derivative” of the met-ric must be described by a three-tensor that carries information aboutthe derivative of the metric in the direction normal to the hypersur-face. Because Kab = 1

2(£ngαβ) eα

aeβb , the extrinsic curvature is clearly

an appropriate choice.The initial-value problem of general relativity therefore consists in

specifying two symmetric tensor fields, hab and Kab, on a spacelikehypersurface Σ. In the complete spacetime, hab is interpreted as theinduced metric on the hypersurface, while Kab is the extrinsic curvature.These tensors cannot be chosen freely: they must satisfy the constraintequations of general relativity. These are given by Eqs. (3.5.5) and(3.5.6), together with the Einstein field equations Gαβ = 8πTαβ:

3R + K2 −KabKab = 16πTαβnαnβ ≡ 16πρ (3.6.1)

andKb

a|b −K,a = 8πTαβeαanβ ≡ 8πja. (3.6.2)

The remaining components of the Einstein field equations provide evo-lution equations for hab and Kab; these will be considered in Chapter4.

3.6.2 Cosmological initial values

As an example, let us solve the constraint equations for a spatially flat,isotropic, and homogeneous cosmology. To satisfy these requirements,the three-metric must take the form

ds2 = a2(dx2 + dy2 + dz2),

where a is the scale factor, which is a constant on the hypersurface.Isotropy and homogeneity also imply ρ = constant, ja = 0, and

Kab =1

3Khab,

where K is a constant. The second constraint equation is thereforetrivially satisfied. The first one implies

16πρ = K2 −KabKab =2

3K2,

and this provides the complete solution to the initial-value problem.To recognize the physical meaning of this last equation, we use

the fact that in the complete spacetime, K = nα;α, where nα is the

78 Hypersurfaces

unit normal to surfaces of constant t. The full metric is given by theFriedmann-Robertson-Walker form

ds2 = −dt2 + a2(t)(dx2 + dy2 + dz2),

so that nα = −∂αt and K = 3a/a, where an overdot indicates differ-entiation with respect to t. The first constraint equation is thereforeequivalent to

3(a/a)2 = 8πρ,

which is one of the Friedmann equations governing the evolution of thescale factor.

3.6.3 Moment of time symmetry

We notice from the previous example that Kab = 0 when a = 0, that is,the extrinsic curvature vanishes when the scale factor passes througha turning point of its evolution. Because the dynamical history ofthe scale factor is time-symmetric about the time t = t0 at which theturning point occurs, we may call this time a moment of time symmetryin the evolution of the spacetime. Thus, Kab = 0 at this moment oftime symmetry.

Generalizing, we shall call any hypersurface Σ on which Kab = 0a moment of time symmetry in spacetime. Because Kab is essentiallythe “time derivative” of the metric, a moment of time symmetry corre-sponds to a turning point of the metric’s evolution, at which its “timederivative” vanishes. The dynamical history of the metric is then “time-symmetric” about Σ. From Eq. (3.6.2) we see that a moment of timesymmetry can occur only if ja = 0 on that hypersurface.

3.6.4 Stationary and static spacetimes

A spacetime is stationary if it admits a timelike Killing vector ξα. Thismeans that in a coordinate system (t, xa) in which ξα ∗

= δαt, the metric

does not depend on the time coordinate t: gαβ,t∗= 0 (see Sec. 1.5). For

example, a rotating star gives rise to a stationary spacetime if its massand angular velocity do not change with time.

A stationary spacetime is also static if the metric does not changeunder a time reversal, t → −t. For example, the spacetime of a rotatingstar is not static because a time reversal changes the direction of rota-tion. In the specified coordinate system, invariance of the metric undera time reversal implies gta

∗= 0. This, in turn, implies that the Killing

vector is proportional to a gradient: ξα∗= gtt∂αt. Thus, a spacetime is

static if the timelike Killing vector field is hypersurface orthogonal.We may show that if a spacetime is static, then Kab = 0 on those

hypersurfaces Σ that are orthogonal to the Killing vector; these hy-persurfaces therefore represent moments of time symmetry. If Σ is

3.6 Initial-value problem 79

orthogonal to ξα, then its unit normal must given by nα = µ ξα,where 1/µ2 = −ξαξα. This implies that nα;β = µ ξα;β + ξαµ,β, andn(α;β) = ξ(αµ,β) because ξα is a Killing vector. That Kab = 0 followsimmediately from Eq. (3.4.11) and the fact that ξα is orthogonal to eα

a .

3.6.5 Spherical space, moment of time symmetry

As a second example, we solve the constraint equations for a sphericallysymmetric spacetime at a moment of time symmetry. The three-metriccan be expressed as

ds2 = [1− 2m(r)/r]−1 dr2 + r2dΩ2,

for some function m(r); to enforce regularity of the metric at r = 0 wemust impose m(0) = 0. The Ricci scalar is given by 3R = 4m′/r2, witha prime denoting differentiation with respect to r. Because Kab = 0 ata moment of time symmetry, Eq. (3.6.1) implies 16πρ = 3R. Solvingfor m(r) gives

m(r) =

∫ r

0

4πr′2ρ(r′) dr′.

This states, loosely speaking, that m(r) is the mass-energy containedinside a sphere of radius r, at the selected moment of time symmetry.

3.6.6 Spherical space, empty and flat

We now solve the constraint equations for a spherically symmetric spaceempty of matter (so that ρ = 0 = ja). We assume that we can endowthis space with a flat metric, so that

hab dyadyb = dr2 + r2 dΩ2.

We also assume that the hypersurface does not represent a moment oftime symmetry. While the flat metric and Kab = 0 make a valid solutionto the constraints, this is a trivial configuration — a flat hypersurfacein a flat spacetime.

Let na = ∂ar be a unit vector that points radially outward on thehypersurface. The fact that Kab is a spherically symmetric tensor meansthat it can be expressed as

Kab = K1(r)nanb + K2(r)(hab − nanb),

with K1 representing the radial component of the extrinsic curvature,and K2 the angular components. In the usual spherical coordinates(r, θ, φ), we have Ka

b = diag(K1, K2, K2), which is the most generalexpression admissible under the assumption of spherical symmetry.

80 Hypersurfaces

Because the space is empty and flat, the first constraint equationreduces to K2 − KabKab = 0, an algebraic equation for K1 and K2.This gives us the condition (2K1 + K2)K2 = 0. Choosing K2 = 0would eventually return the trivial solution Kab = 0. We choose insteadK2 = −2K1 and re-express the extrinsic curvature as

Kab = K(r)(2

3hab − nanb

),

where K = −3K1 is the sole remaining function to be determined.To find K(r), we turn to the second constraint equation, Kb

a|b −K,a = 0, which becomes

1

3K,a + (Knb

|b + K,bnb)na + Kna|bn

b = 0.

With K,a = K ′na (with a prime denoting differentiation with respect tor), nb

|b = 2/r, and na|bnb = 0 (because the radial curves are geodesics

of the hypersurface), we arrive at 2rK ′ + 3K = 0. Integration yields

K(r) = K0(r/r0)3/2,

with K0 denoting the value of K at the arbitrary radius r0.We have found a nontrivial solution to the constraint equations for

a spherical space that is both empty and flat. The physical meaning ofthis configuration will be revealed in Sec. 3.13, Problem 1.

3.6.7 Conformally-flat space

A powerful technique for generating solutions to the constraint equa-tions consists of writing the three-metric as

hab = ψ4δab,

where ψ(ya) is a scalar field on the hypersurface. Such a metric is saidto be conformally related to the flat metric, and the space is said to beconformally flat. For this metric the Ricci scalar is 3R = −8ψ−5∇2ψ,and Eq. (3.6.1) takes the form of Poisson’s equation,

∇2ψ = −2πρeff ,

where

ρeff = ψ5[ρ +

1

16π

(KabKab −K2

)]

is an effective mass density on the hypersurface. At a moment of timesymmetry, this simplifies to ρeff = ψ5ρ, and one possible strategy forsolving the constraint is to specify ρeff , solve for ψ, and then see whatthis produces for the actually mass density ρ. If ρ = 0 at the moment of

3.7 Junction conditions and thin shells 81

time symmetry, then the constraint becomes Laplace’s equation ∇2ψ =0, and this admits many interesting solutions. A well-known exampleis Misner’s (1960) solution, which describes two black holes about toundergo a head-on collision. This initial data set has been vigourouslystudied by numerical relativists.

3.7 Junction conditions and thin shells

The following situation sometimes presents itself: A hypersurface Σpartitions spacetime into two regions + and − (Fig. 3.5). In + themetric is g+

αβ, and it is expressed in a system of coordinates xα+. In −

the metric is g−αβ, and it is expressed in coordinates xα−. We ask: what

conditions must be put on the metrics to ensure that + and − are joinedsmoothly at Σ, so that the union of g+

αβ and g−αβ forms a valid solutionto the Einstein field equations? To answer this question is not entirelystraightforward because in practical situations, the coordinate systemsxα± will often be different, and it may not be possible to compare the

metrics directly. To circumvent this difficulty we will endeavour toformulate junction conditions that involve only three-tensors on Σ. Inthis section we will assume that Σ is either timelike or spacelike; wewill return to the case of a null hypersurface in Sec. 3.11.

3.7.1 Notation and assumptions

We assume that the same coordinates ya can be installed on both sidesof the hypersurface, and we choose nα, the unit normal to Σ, to pointfrom − to +. We suppose that an overlapping coordinate system xα,distinct from xα

±, can be introduced in a neighbourhood of the hyper-surface. (This is for our short-term convenience; the final formulationof the junction conditions will not involve this coordinate system.) Weimagine Σ to be pierced by a congruence of geodesics that intersect itorthogonally. We take ` to denote proper distance (or proper time)along the geodesics, and we adjust the parameterization so that ` = 0when the geodesics cross the hypersurface; our convention is that ` is

Figure 3.5: Two regions of spacetime joined at a common boundary.

− : xα−

Σ : ya

+ : xα+

nα

82 Hypersurfaces

negative in − and positive in +. We can think of ` as a scalar field:The point P characterized by the coordinates xα is linked to Σ by amember of the congruence, and `(xα) is the proper distance (or propertime) from Σ to P along this geodesic. Our construction implies thatnα is equal to dxα/d` at the hypersurface, and that

nα = ε∂α`; (3.7.1)

we also have nαnα = ε.We will use the language of distributions. We introduce the Heavi-

side distribution Θ(`), equal to +1 if ` > 0, 0 if ` < 0, and indeterminateif ` = 0. We note the following properties:

Θ2(`) = Θ(`), Θ(`)Θ(−`) = 0,d

d`Θ(`) = δ(`),

where δ(`) is the Dirac distribution. We also note that the productΘ(`)δ(`) is not defined as a distribution.

The following notation will be useful:

[A] ≡ A(+)|Σ − A(−)|Σ,

where A is any tensorial quantity defined on both sides of the hyper-surface; [A] is therefore the jump of A across Σ. We note the relations

[nα] = [eαa ] = 0, (3.7.2)

where eαa = ∂xα/∂ya. The first follows from the relation nα = dxα/d`

and the continuity of both ` and xα across Σ; the second follows fromthe fact that the coordinates ya are the same on both sides of thehypersurface.

3.7.2 First junction condition

We begin by expressing the metric gαβ, in the coordinates xα, as adistribution-valued tensor:

gαβ = Θ(`) g+αβ + Θ(−`) g−αβ, (3.7.3)

where g±αβ is the metric in ± expressed in the coordinates xα. Wewant to know if the metric of Eq. (3.7.3) makes a valid distributionalsolution to the Einstein field equations. To decide, we must verifythat geometrical quantities constructed from gαβ, such as the Riemanntensor, are properly defined as distributions. We must then try toeliminate, or at least give an interpretation to, singular terms thatmight arise in these geometric quantities.

Differentiating Eq. (3.7.3) yields

gαβ,γ = Θ(`) g+αβ,γ + Θ(−`) g−αβ,γ + εδ(`)[gαβ]nγ,

3.7 Junction conditions and thin shells 83

where Eq. (3.7.1) was used. The last term is singular, and it causesproblems when we compute the Christoffel symbols, because it gener-ates terms proportional to Θ(`)δ(`). If the last term were allowed tosurvive, therefore, the connection would not be defined as a distribu-tion. To eliminate this term, we impose continuity of the metric acrossthe hypersurface: [gαβ] = 0. This statement holds in the coordinatesystem xα only. However, we can easily turn this into a coordinate-invariant statement: 0 = [gαβ]eα

aeβb = [gαβeα

aeβb ]; this last step follows

by virtue of Eq. (3.7.2). We have obtained

[hab] = 0, (3.7.4)

the statement that the induced metric must be the same on both sidesof Σ. This is clearly required if the hypersurface is to have a well-defined geometry. Equation (3.7.4) will be our first junction condition,and it is expressed independently of the coordinates xα or xα

±.

3.7.3 Riemann tensor

To find the second junction condition requires more work: we mustcalculate the distribution-valued Riemann tensor. Using the resultsobtained thus far, we have that the Christoffel symbols are

Γαβγ = Θ(`) Γ+α

βγ + Θ(−`) Γ−αβγ,

where Γ±αβγ are the Christoffel symbols constructed from g±αβ. A straight-

forward calculation then reveals

Γαβγ,δ = Θ(`) Γ+α

βγ,δ + Θ(−`) Γ−αβγ,δ + εδ(`)[Γα

βγ]nδ,

and from this follows the Riemann tensor:

Rαβγδ = Θ(`) R+α

βγδ + Θ(−`) R−αβγδ + δ(`)Aα

βγδ, (3.7.5)

whereAα

βγδ = ε([Γαβδ]nγ − [Γα

βγ]nδ). (3.7.6)

We see that the Riemann tensor is properly defined as a distribution,but the δ-function term represents a curvature singularity at Σ. Oursecond junction condition will seek to eliminate this term. Failing this,we will see that a physical interpretation can nevertheless be given tothe singularity. This is our next topic.

3.7.4 Surface stress-energy tensor

Although they are constructed from Christoffel symbols, the quanti-ties Aα

βγδ form a tensor, because the difference between two sets of

84 Hypersurfaces

Christoffel symbols is a tensorial quantity (see Sec. 1.2). We must nowfind an explicit expression for this tensor.

The fact that the metric is continuous across Σ in the coordinatesxα implies that its tangential derivatives must also be continuous. Thismeans that if gαβ,γ is to be discontinuous, the discontinuity must bedirected along the normal vector nα. There must therefore exist atensor field καβ such that

[gαβ,γ] = καβ nγ; (3.7.7)

this tensor is given explicitly by

καβ = ε[gαβ,γ]nγ. (3.7.8)

Equation (3.7.7) implies

[Γαβγ] =

1

2(κα

βnγ + καγnβ − κβγn

α),

and we obtain

Aαβγδ =

ε

2(κα

δnβnγ − καγnβnδ − κβδn

αnγ + κβγnαnδ).

This is the δ-function part of the Riemann tensor.Contracting over the first and third indices gives the δ-function part

of the Ricci tensor:

Aαβ ≡ Aµαµβ =

ε

2(κµαnµnβ + κµβnµnα − κnαnβ − εκαβ),

where κ ≡ καα. After an additional contraction we obtain the δ-function

part of the Ricci scalar,

A ≡ Aαα = ε(κµνn

µnν − εκ).

With this we form the δ-function part of the Einstein tensor, and afterusing the Einstein field equations, we obtain an expression for the stress-energy tensor:

Tαβ = Θ(`) T+αβ + Θ(−`) T−

αβ + δ(`)Sαβ, (3.7.9)

where 8πSαβ ≡ Aαβ− 12Agαβ. In Eq. (3.7.9), the first and second terms

represent the stress-energy tensors of regions + and −, respectively. Theδ-function term, on the other hand, comes with a clear interpretation:it is associated with the presence of a thin distribution of matter —a surface layer, or a thin shell — at Σ; this thin shell has a surfacestress-energy tensor given by Sαβ.

3.7 Junction conditions and thin shells 85

3.7.5 Second junction condition

Explicitly, the surface stress-energy tensor is given by

16πεSαβ = κµαnµnβ + κµβnµnα − κnαnβ − εκαβ − (κµνnµnν − εκ)gαβ.

From this we notice that Sαβ is tangent to the hypersurface: Sαβnβ = 0.It therefore admits the decomposition

Sαβ = Sabeαaeβ

b , (3.7.10)

where Sab = Sαβeαaeβ

b is a symmetric three-tensor. This is evaluated asfollows:

16πSab = −καβeαaeβ

b − ε(κµνnµnν − εκ)hab

= −καβeαaeβ

b − κµν(gµν − hmneµ

meνn)hab + κhab

= −καβeαaeβ

b + hmnκµνeµmeν

n hab.

On the other hand, we have

[nα;β] = −[Γγαβ]nγ

= −1

2(κγαnβ + κγβnα − καβnγ)n

γ

=1

2(εκαβ − κγαnβnγ − κγβnαnγ),

which allows us to write

[Kab] = [nα;β]eαaeβ

b =ε

2καβeα

aeβb .

Combining these results, we obtain

Sab = − ε

8π

([Kab]− [K]hab

), (3.7.11)

which relates the surface stress-energy tensor to the jump in extrinsiccurvature from one side of Σ to the other. The complete stress-energytensor of the surface layer is

TαβΣ = δ(`) Sabeα

aeβb . (3.7.12)

We conclude that a smooth transition across Σ requires [Kab] = 0 — theextrinsic curvature must be the same on both sides of the hypersurface.This requirement does more than just remove the δ-function term fromthe Einstein tensor: In Sec. 3.13, Problem 4 you will be asked to provethat [Kab] = 0 implies Aα

βγδ = 0, which means that the full Riemanntensor is then nonsingular at Σ.

The condition [Kab] = 0 is our second junction condition, and it isexpressed independently of the coordinates xα and xα

±. If this condition

86 Hypersurfaces

is violated, then the spacetime is singular at Σ, but the singularitycomes with a straightforward interpretation: a surface layer with stress-energy tensor Tαβ

Σ is present at the hypersurface.Notice that a minor miracle is at work here: When [Kab] 6= 0, only

the Ricci part of the Riemann tensor acquires a singularity, and it isthis part that can readily be associated with matter. The remainingpart of the Riemann tensor — the Weyl part — is smooth even whenthe extrinsic curvature is discontinuous.

3.7.6 Summary

The junction conditions for a smooth joining of two metrics at a hy-persurface Σ (assumed not to be null) are

[hab] = [Kab] = 0.

If the extrinsic curvature is not the same on both sides of Σ, then athin shell with surface stress-energy tensor

Sab = − ε

8π

([Kab]− [K]hab

)

is present at Σ. The complete stress-energy tensor of the surface layeris given by Eq. (3.7.12) in the overlapping coordinates xα. In the coor-dinate system xα

± used originally in ±, it is

TαβΣ = Sab

(∂xα±

∂ya

)(∂xβ±

∂yb

)δ(`).

This follows from Eq. (3.7.12) by a simple coordinate transformationfrom xα to xα

±; such a transformation leaves both ` and Sab invariant.This formulation of the junction conditions is due to Darmois (1927)

and Israel (1966). The thin-shell formalism is due to Lanczos (1922and 1924) and Israel (1966); an extension to null hypersurfaces will beconsidered in Sec. 3.11.

3.8 Oppenheimer-Snyder collapse

In 1939, J. Robert Oppenheimer and his student Hartland Snyder pub-lished the first solution to the Einstein field equations that describesthe process of gravitational collapse to a black hole. For simplicity, theymodeled the collapsing star as a spherical ball of pressureless matterwith a uniform density. (A perfect fluid with negligible pressure is usu-ally called dust.) The metric inside the dust is a Friedmann-Robertson-Walker (FRW) solution, while the metric outside is the Schwarzschild

3.8 Oppenheimer-Snyder collapse 87

solution (Fig. 3.6). The question considered here is whether these met-rics can be joined smoothly at their common boundary, the surface ofthe collapsing star.

The metric inside the collapsing dust (which occupies the region −)is given by

ds2− = −dτ 2 + a2(τ)(dχ2 + sin2 χdΩ2), (3.8.1)

where τ is proper time on comoving world lines (along which χ, θ, and φare all constant), and a(τ) is the scale factor. By virtue of the Einsteinfield equations, this satisfies

a2 + 1 =8π

3ρa2, (3.8.2)

where an overdot denotes differentiation with respect to τ . By virtue ofenergy-momentum conservation in the absence of pressure, the dust’smass density ρ satisfies

ρa3 = constant ≡ 3

8πamax, (3.8.3)

where amax is the maximum value of the scale factor. The solution toEqs. (3.8.2) and (3.8.3) has the parametric form

a(η) = 12amax(1 + cos η), τ(η) = 1

2amax(η + sin η);

the collapse begins at η = 0 when a = amax, and it ends at η = π whena = 0. The hypersurface Σ coincides with the surface of the collapsingstar, which is located at χ = χ0 in our comoving coordinates.

The metric outside the dust (in the region +) is given by

ds2+ = −f dt2 + f−1 dr2 + r2 dΩ2, f = 1− 2M/r, (3.8.4)

where M is the gravitational mass of the collapsing star. As seen fromthe outside, Σ is described by the parametric equations r = R(τ), t =

Figure 3.6: The Oppenheimer-Snyder spacetime.

Σ

− : FRW

+ : Schwarzschild

nα

uα

88 Hypersurfaces

T (τ), where τ is proper time for observers comoving with the surface.Clearly, this is the same τ that appears in the metric of Eq. (3.8.1).

It is convenient to choose ya = (τ, θ, φ) as coordinates on Σ. Thisimplies that eα

τ = uα, where uα is the four-velocity of an observercomoving with the surface of the collapsing star.

We now calculate the induced metric. As seen from −, the metricon Σ is

ds2Σ = −dτ 2 + a2(τ) sin2 χ0 dΩ2.

As seen from +, on the other hand,

ds2Σ = −(FT 2 − F−1R2) dτ 2 + R2(τ) dΩ2,

where F = 1 − 2M/R. Because the induced metric must be the sameon both sides of the hypersurface, we have

R(τ) = a(τ) sin χ0, F T 2 − F−1R2 = 1. (3.8.5)

The first equation determines R(τ), and the second equation can besolved for T :

FT =√

R2 + F ≡ β(R, R). (3.8.6)

This equation can be integrated for T (τ), and the motion of the bound-ary in + is completely determined.

The unit normal to Σ can be obtained from the relations nαuα = 0,nαnα = 1. As seen from −, uα

− ∂α = ∂τ and n−α dxα = a dχ; we havechosen nχ > 0 so that nα is directed toward +. As seen from +, uα

+ ∂α =

T ∂t + R ∂r and n+α dxα = −R dt+ T dr, with a consistent choice for the

sign.The extrinsic curvature is defined on either side of Σ by Kab =

nα;βeαaeβ

b . The nonvanishing components are Kττ = nα;βuαuβ = −nαuα;βuβ =

−aαnα (where aα is the acceleration of an observer comoving with thesurface), Kθθ = nθ;θ, and Kφφ = nφ;φ. A straightforward calculationreveals that as seen from −,

Kτ−τ = 0, Kθ

−θ = Kφ−φ = a−1 cot χ0. (3.8.7)

The first relation follows immediately from the fact that the comovingworld lines of a FRW spacetime are geodesics. As seen from +,

Kτ+τ = β/R, Kθ

+θ = Kφ+φ = β/R, (3.8.8)

where β(R, R) is defined by Eq. (3.8.6).To have a smooth transition at the surface of the collapsing star, we

demand that Kab be the same on both sides of the hypersurface. It istherefore necessary for uα

+ to satisfy the geodesic equation (aα+ = 0) in

+. It is easy to check that the geodesic equation implies R2 + F = E2,

3.9 Thin-shell collapse 89

where E = −ut is the (conserved) energy parameter of the comovingobserver. This relation implies β = E, and the fact that β is a constantenforces Kτ

+τ = 0, as required. On the other hand, [Kθθ] = 0 implies

cot χ0/a = β/R = E/(a sin χ0), or

β = E = cos χ0. (3.8.9)

We have found that the requirement for a smooth transition at Σ isthat the hypersurface be generated by geodesics of both − and +, andthat the parameters E and χ0 be related by Eq. (3.8.9). With the helpof Eqs. (3.8.2), (3.8.5), and (3.8.6), we may turn Eq. (3.8.9) into

M =4π

3ρR3, (3.8.10)

which equates the gravitational mass of the collapsing star to the prod-uct of its density and volume. This relation has an immediate intu-itive meaning, and it neatly summarizes the complete solution to theOppenheimer-Snyder problem.

3.9 Thin-shell collapse

As an application of the thin-shell formalism, we consider the gravita-tional collapse of a thin spherical shell. We assume that spacetime isflat inside the shell (in −). Outside (in +), the metric is necessarily aSchwarzschild solution (by virtue of the assumed spherical symmetry).We assume also that the shell is made of pressureless matter, in thesense that its surface stress-energy tensor is constrained to have theform

Sab = σ uaub, (3.9.1)

in which σ is the surface density and ua = dya/dτ the shell’s velocityfield. Our goal is to derive the shell’s equations of motion under thestated conditions.

Using the results derived in the preceding section, we have

Kτ±τ = β±/R,

Kθ±θ = Kφ

±φ = β±/R,

β+ =

√R2 + 1− 2M/R,

β− =√

R2 + 1,

where R(τ) is the shell’s radius, and M its gravitational mass. As wedid before, we use (τ, θ, φ) as coordinates on Σ. Equation (3.7.11) allows

90 Hypersurfaces

us to calculate the components of the surface stress-energy tensor, andwe find

−σ = Sττ =

β+ − β−4πR

, 0 = Sθθ =

β+ − β−8πR

+β+ − β−

8πR.

The second equation can be integrated immediately, giving (β+−β−)R =constant. Substituting this into the first equation yields 4πR2σ =−constant.

We have obtained

4πR2σ ≡ m = constant (3.9.2)

and β− − β+ = m/R. The first equation states that m, the shell’srest mass, stays constant during the evolution. Squaring the secondequation converts it to

M = m√

1 + R2 − m2

2R, (3.9.3)

which comes with a nice physical interpretation. The first term onthe right-hand side is the shell’s relativistic kinetic energy, includingrest mass. The second term is the shell’s binding energy, the workrequired to assemble the shell from its dispersed constituents. The sumof these is the total (conserved) energy, and this is equal to the shell’sgravitational mass M . Equation (3.9.3) provides a vivid illustration ofthe general statement that all forms of energy contribute to the totalgravitational mass of an isolated body.

Equations (3.9.2) and (3.9.3) are the shell’s equations of motion. Itis interesting to note that if M < m, then the motion exhibits a turningpoint at R = Rmax ≡ m2/[2(m−M)]: an expanding shell with M < mcannot escape its own gravitational pull.

3.10 Slowly rotating shell

Our next application of the thin-shell formalism is concerned with thespacetime of a slowly rotating, spherical shell. We take the exteriormetric to be the slow-rotation limit of the Kerr solution,

ds2+ = −f dt2 + f−1 dr2 + r2 dΩ2 − 4Ma

rsin2 θ dtdφ. (3.10.1)

Here, f = 1 − 2M/r, with M denoting the shell’s gravitational mass,and a = J/M ¿ M , where J is the shell’s angular momentum. Through-out this section we will work consistently to first order in a.

The metric of Eq. (3.10.1) is cut off at a radius r = R at which theshell is located. As viewed from the exterior, the shell’s induced metricis

ds2Σ = −(1− 2M/R) dt2 + R2 dΩ2 − 4Ma

Rsin2 θ dtdφ.

3.10 Slowly rotating shell 91

It is possible to remove the off-diagonal term by going to a rotatingframe of reference. We therefore introduce a new angular coordinate ψrelated to φ by

ψ = φ− Ωt, (3.10.2)

where Ω is the angular velocity of the new frame with respect to theinertial frame of Eq. (3.10.1). We anticipate that Ω will be propor-tional to a, and this allows us to approximate dφ2 by dψ2 + 2Ω dtdψ.Substituting this into ds2

Σ returns a diagonal metric if Ω is chosen tobe

Ω =2Ma

R3. (3.10.3)

With this, the induced metric becomes

hab dyadyb = −(1− 2M/R) dt2 + R2(dθ2 + sin2 θ dψ2). (3.10.4)

It is now clear that the shell has a spherical geometry. As Eq. (3.10.4)indicates, we will use the coordinates ya = (t, θ, ψ) on the shell.

We take spacetime to be flat inside the shell, and we write theMinkowski metric in the form

ds2− = −(1− 2M/R) dt2 + dρ2 + ρ2(dθ2 + sin2 θ dψ2), (3.10.5)

where ρ is a radial coordinate. This metric must be cut off at ρ = Rand matched to the exterior metric of Eq. (3.10.1). The shell’s intrinsicmetric, as computed from the interior, agrees with Eq. (3.10.4). Conti-nuity of the induced metric is therefore established, and we must nowturn to the extrinsic curvature.

We first compute the extrinsic curvature as seen from the shell’sexterior. In the metric of Eq. (3.10.1), the shell’s unit normal is nα =f−1/2∂αr. The parametric equations of the hypersurface are t = t,θ = θ, and φ = ψ + Ωt, and they have the generic form xα = xα(ya).These allow us to compute the tangent vectors eα

a = ∂xα/∂ya, and weobtain eα

t ∂α = ∂t + Ω∂φ, eαθ ∂α = ∂θ, and eα

ψ∂α = ∂φ. From all this wefind that the nonvanishing components of the extrinsic curvature are

Ktt =

M

R2√

1− 2M/R,

Ktψ = − 3Ma sin2 θ

R2√

1− 2M/R,

Kψt =

3Ma

R4

√1− 2M/R,

Kθθ =

1

R

√1− 2M/R = Kψ

ψ.

As now seen from the shell’s interior, the unit normal is nα = ∂αρ, andthe tangent vectors are eα

t ∂α = ∂t, eαθ ∂α = ∂θ, and eα

ψ∂α = ∂ψ. From

92 Hypersurfaces

this we find that Kθθ = 1/R = Kψ

ψ are the only two nonvanishingcomponents of the extrinsic curvature. This could have been obtaineddirectly by setting M = 0 in our previous results.

We have a clear discontinuity in the extrinsic curvature, and Eq. (3.7.11)allows us to calculate Sab, the shell’s surface stress-energy tensor. Aftera few lines of algebra, we obtain

Stt = − 1

4πR

(1−

√1− 2M/R

),

Stψ =

3Ma sin2 θ

8πR2√

1− 2M/R,

Sψt = − 3Ma

8πR4

√1− 2M/R,

Sθθ =

1−M/R−√

1− 2M/R

8πR√

1− 2M/R= Sψ

ψ.

These results, while giving us a complete description of the surfacestress-energy tensor, are not terribly illuminating. Can we make senseof this mess?

We will attempt to cast Sab in a perfect-fluid form,

Sab = σuaub + p(hab + uaub), (3.10.6)

in terms of a velocity field ua, a surface density σ, and a surface pressurep. How do we find these quantities? First we notice that Eq. (3.10.6)implies Sa

bub = −σua, which shows that ua is a normalized eigenvector

of the surface stress-energy tensor, with eigenvalue −σ. This gives usthree equations for three unknowns, the density and the two indepen-dent components of the velocity field. Once those have been obtained,the pressure is found by projecting Sab in the directions orthogonal toua. The rest is just a matter of algebra.

We can save ourselves some work if we recognize that the shell mustmove rigidly in the ψ direction, with a uniform angular velocity ω. Itsvelocity vector can then be expressed as

ua = γ(ta + ωψa), (3.10.7)

where ta = ∂ya/∂t and ψa = ∂ya/∂ψ are the Killing vectors of theinduced metric hab. In Eq. (3.10.7), ω = dψ/dt is the shell’s angularvelocity in the rotating frame of Eq. (3.10.2), and γ is determined bythe normalization condition, habu

aub = −1. We can simplify thingsfurther if we anticipate that ω will be proportional to a. For example,neglecting O(ω2) terms when normalizing ua gives

γ =1√

1− 2M/R. (3.10.8)

3.10 Slowly rotating shell 93

With these assumptions, we find that the eigenvalue equation pro-duces ω = −Sψ

t/(−Stt + Sψ

ψ) and σ = −Stt. After simplification, the

first equation becomes

ω =6Ma

R3

1− 2M/R

(1−√

1− 2M/R)(1 + 3√

1− 2M/R), (3.10.9)

and the second is

σ =1

4πR

(1−

√1− 2M/R

). (3.10.10)

We now have the surface density and the velocity field. The surfacepressure can easily be obtained by projecting Sab in the directions or-thogonal to ua: p = 1

2(hab + uaub)S

ab = 12(S + σ), where S = habS

ab.This gives p = Sθ

θ, and

p =1−M/R−

√1− 2M/R

8πR√

1− 2M/R. (3.10.11)

The shell’s surface stress-energy tensor can therefore be given an inter-pretation in terms of a perfect fluid of density σ, pressure p, and angularvelocity ω. When R is much larger than 2M , Eqs. (3.10.9)–(3.10.11)reduce to ω ' 3a/(2R2), σ ' M/(4πR2), and p ' M2/(16πR3), re-spectively.

The spacetime of a slowly rotating shell offers us a unique oppor-tunity to explore the rather strange relativistic effects associated withrotation. We will conclude this section with a short description of theseeffects.

The metric of Eq. (3.10.1) is the metric outside the shell, and it isexpressed in a coordinate system that goes easily into a Cartesian frameat infinity. This is the frame of the “fixed stars”, and it is this framewhich sets the standard of no rotation. The metric of Eq. (3.10.5), onthe other hand, is the metric inside the shell, and it is expressed in acoordinate system that is rotating with respect to the frame of the fixedstars. The transformation is given by Eq. (3.10.2), and it shows thatan observer at constant ψ moves with an angular velocity dφ/dt = Ω.Inertial observers inside the shell are therefore rotating with respectto the fixed stars, with an angular velocity Ωin ≡ Ω. According toEq. (3.10.3), this is

Ωin =2Ma

R3. (3.10.12)

This angular motion is induced by the rotation of the shell, and theeffect is known as the dragging of inertial frames. It was first discoveredin 1918 by Thirring and Lense.

The shell’s angular velocity ω, as computed in Eq. (3.10.9), is mea-sured in the rotating frame. As measured in the nonrotating frame,

94 Hypersurfaces

the shell’s angular velocity is Ωshell = dφ/dt = dψ/dt + Ω = ω + Ωin.According to Eqs. (3.10.9) and (3.10.12), this is

Ωshell =2Ma

R3

1 + 2√

1− 2M/R

(1−√

1− 2M/R)(1 + 3√

1− 2M/R). (3.10.13)

When R is much larger than 2M , Ωin/Ωshell ' 4M/(3R), and the inter-nal observers rotate at a small fraction of the shell’s angular velocity.As R approaches 2M , however, the ratio approaches unity, and theinternal observers find themselves corotating with the shell. This is arather striking manifestation of frame dragging. (The phrase “Mach’sprinciple” is often attached to this phenomenon.) This spacetime, ad-mittedly, is highly idealized, and we may wonder whether corotationcould ever occur in realistic situations. We will see in Chapter 5 thatthe answer is yes: a very similar phenomenon occurs in the vicinity ofa rotating black hole.

3.11 Null shells

We saw in Secs. 3.1 and 3.2 that the description of null hypersurfacesrequires some care and involves interesting subtleties, and we shouldnot be surprised to find that the same is true of the description of nullsurface layers. Our purpose here, in the last section of Chapter 3, is toface these subtleties and extend the formalism of thin shells, as devel-oped in Sec. 3.7, to the case of a null hypersurface. The presentationgiven here is adapted from Barrabes and Israel (1991).

3.11.1 Geometry

As in Sec. 3.7 we consider a hypersurface Σ that partitions spacetimeinto two regions ± in which the metric is g±αβ when expressed in co-ordinates xα

±. Here we assume that the hypersurface is null, and ourconvention is such that − is in the past of Σ, and + in its future. Weassume also that the hypersurface is singular, in the sense that theRiemann tensor possesses a δ-function singularity at Σ. We will char-acterize the Ricci part of this singular curvature tensor, and relate itto the surface stress-energy tensor of the shell.

(We note in passing that the Weyl part of the curvature tensor mayalso be singular at the hypersurface, and that this may happen evenin the absence of a singular Ricci tensor: the two types of singularityare entirely decoupled. This property is peculiar to null hypersurfaces:in the case of timelike or spacelike shells, the curvature singularity iscompletely supported by the Ricci tensor, and the Weyl tensor is alwayssmooth. In this section we shall be concerned only with the singularity

3.11 Null shells 95

in the Ricci tensor, and we shall have to leave unexplored the interestingphysical effects associated with a singular Weyl tensor.)

As in Sec. 3.1.2 we install coordinates

ya = (λ, θA)

on the hypersurface, and as in Sec. 3.7.1 we assume that these coordi-nates are the same of both sides of Σ. We take λ to be an arbitraryparameter on the null generators of the hypersurface, and we use θA

to label the generators. We shall see below that in general, it is notpossible to make λ an affine parameter on both sides of Σ.

In ±, Σ is described by the parametric relations xα±(ya), and we can

introduce tangent vectors eα±a = ∂xα

±/∂ya on each side of the hyper-surface. These are naturally separated into a null vector kα

± that istangent to the generators, and two spacelike vectors eα

±A that point inthe directions transverse to the generators. Explicitly,

kα =(∂xα

∂λ

)θA

= eαλ , eα

A =(∂xα

∂θA

)λ. (3.11.1)

(Here and below, in order to keep the notation simple, we refrain fromusing the “±” label in displayed equations; this should not cause anyconfusion.) By construction, these vectors satisfy

kαkα = 0 = kαeαA. (3.11.2)

On the other hand, the remaining inner products

σAB(λ, θC) ≡ gαβ eαAeβ

B (3.11.3)

do not vanish, and we assume that they are the same on both sides ofΣ:

[σAB] = 0. (3.11.4)

As in Sec. 3.1.3, we find that it is the two-tensor σAB which acts as ametric on Σ,

ds2Σ = σAB dθAdθB,

and the condition (3.11.4) ensures that the hypersurface possesses awell-defined intrinsic geometry.

As in Sec. 3.1.3 we complete the basis by adding an auxiliary nullvector Nα

± satisfying

NαNα = 0, Nαkα = −1, NαeαA = 0. (3.11.5)

This gives us the convenient expression

gαβ = −kαNβ −Nαkβ + σABeαAeβ

B (3.11.6)

96 Hypersurfaces

for the inverse metric on either side of Σ (in the coordinates xα±); σAB

is the inverse of σAB, and it is the same on both sides.To complete the geometric setup we must introduce a congruence

of geodesics that cross the hypersurface. In Sec. 3.7, in which Σ waseither timelike or spacelike, the congruence was selected by demandingthat the geodesics intersect the hypersurface orthogonally: the vectorfield uα

± tangent to the congruence was set equal (on Σ) to the normalvector nα

±. When the hypersurface is null, however, this requirementdoes not produce a unique congruence, because a vector orthogonal tokα can still possess an arbitrary component along kα.

We shall have to give up on the idea of adopting a unique congru-ence. An important aspect of our description of null shells is thereforethat it involves an arbitrary congruence of timelike geodesics intersect-ing Σ. This arbitrariness comes with the lightlike nature of the singularhypersurface, and it cannot be removed. It can, however, be physicallymotivated: The arbitrary vector field uα

± that enters our descriptionof null shells can be identified with the four-velocity of a family of ob-servers making measurements on the shell; since many different familiesof observers could be introduced to make such measurements, there isno reason to demand that the vector field be uniquely specified.

We therefore introduce a congruence of timelike geodesics γ thatarbitrarily intersect the hypersurface. The geodesics are parameterizedby proper time τ , which is adjusted so that τ = 0 when a geodesiccrosses Σ; thus, τ < 0 in −, and τ > 0 in +. The vector tangent to thegeodesics is

uα =dxα

dτ. (3.11.7)

To ensure that the congruence is smooth at the hypersurface, we de-mand that uα

± be “the same” on both sides of Σ. This means thatuαeα

a , the tangential projections of the vector field, must be equal whenevaluated on either side of the hypersurface:

[−uαkα] = 0 = [uαeαA]. (3.11.8)

If, for example, uα− is specified in −, then the three conditions (3.11.8)

are sufficient (together with the geodesic equation) to determine thethree independent components of uα

+ in +. We note that −uαNα, thetransverse projection of the vector field, is allowed to be discontinuousat Σ.

The proper-time parameter on the timelike geodesics can be thoughtof as a scalar field τ(xα

±) defined in a neighbourhood of Σ: Select a pointxα± off the hypersurface and locate the unique geodesic γ that connects

this point to Σ; the value of the scalar field at xα± is equal to the proper-

time parameter of this geodesic at that point. The hypersurface Σ canthen be defined by the statement

τ(xα) = 0,

3.11 Null shells 97

and the normal vector k±α will be proportional to the gradient of τ(xα±)

evaluated at Σ. It is easy to check that the expression

kα = −(−kµuµ)

∂τ

∂xα(3.11.9)

is compatible with Eq. (3.11.7). We recall that the factor −kµuµ in

Eq. (3.11.9) is continuous across Σ.

3.11.2 Surface stress-energy tensor

As in Sec. 3.7 we introduce an overlapping coordinate system xα, dis-tinct from xα

±, in a neighbourhood of the hypersurface; the final formu-lation of our null-shell formalism will be independent of these coordi-nates. We express the metric as a distribution-valued tensor:

gαβ = Θ(τ) g+αβ + Θ(−τ) g−αβ,

where g±αβ(xµ) is the metric in ±. We assume that in these coordinates,the metric is continuous at Σ: [gαβ] = 0; Eq. (3.11.4) is compatiblewith this requirement. We also have [kα] = [eα

A] = [Nα] = [uα] = 0.Differentiation of the metric proceeds as in Secs. 3.7.2 and 3.7.3, exceptthat we now write τ instead of `, and we use Eq. (3.11.9) to relate thegradient of τ to the null vector kα. We arrive at a Riemann tensor thatcontains a singular part given by

R αΣ βγδ = −(−kµu

µ)−1([Γαβδ]kγ − [Γα

βγ]kδ)δ(τ), (3.11.10)

where [Γαβγ] is the jump in the Christoffel symbols across Σ.

In order to make Eq. (3.11.10) more explicit we must characterizethe discontinuous behaviour of gαβ,γ. The condition [gαβ] = 0 guaran-tees that the tangential derivatives of the metric are continuous:

[gαβ,γ]kγ = 0 = [gαβ,γ] e

γC .

The only possible discontinuity is therefore in gαβ,γNγ, the transverse

derivative of the metric, and we conclude that there must exist a tensorfield γαβ such that

[gαβ,γ] = −γαβkγ. (3.11.11)

This tensor is given explicitly by γαβ = [gαβ,γ]Nγ, and it is now easy to

check that

[Γαβγ] = −1

2(γα

βkγ + γαγkβ − γβγk

α). (3.11.12)

Substituting this into Eq. (3.11.10) gives

R αΣ βγδ =

1

2(−kµu

µ)−1(γαδkβkγ − γβδk

αkγ − γαγkβkδ + γβγk

αkδ)δ(τ),

(3.11.13)

98 Hypersurfaces

and we see that kα and γαβ give a complete characterization of thesingular part of the Riemann tensor.

From Eq. (3.11.13) it is easy to form the singular part of the Einsteintensor, and the Einstein field equations then give us the singular partof the stress-energy tensor:

TαβΣ = (−kµu

µ)−1Sαβ δ(τ), (3.11.14)

where

Sαβ =1

16π(kαγβ

µkµ + kβγα

µkµ − γµ

µkαkβ − γµνk

µkνgαβ)

is the surface stress-energy tensor of the null shell — up to a factor−kµu

µ that depends on the choice of observers making measurementson the shell. Its expression can be simplified if we decompose Sαβ inthe basis (kα, eα

A, Nα). For this purpose we introduce the projections

γA ≡ γαβ eαAkβ, γAB ≡ γαβ eα

AeβB, (3.11.15)

and we use the completeness relation (3.11.6) to find that the vectorγα

µkµ admits the decomposition

γαµk

µ =1

2(γµ

µ − σABγAB)kα + (σABγB) eαA − (γµνk

µkν)Nα.

Substituting this into our previous expression for Sαβ and involvingonce more the completeness relation, we arrive at our final expressionfor the surface stress-energy tensor:

Sαβ = µkαkβ + jA(kαeβA + eα

Akβ) + p σABeαAeβ

B. (3.11.16)

Here,

µ ≡ − 1

16π(σABγAB)

can be interpreted as the shell’s surface density,

jA ≡ 1

16π(σABγB)

as a surface current, and

p ≡ − 1

16π(γαβkαkβ)

as a surface pressure.The surface stress-energy tensor of Eq. (3.11.16) is expressed in

the overlapping coordinates xα. As a matter of fact, the derivation ofEq. (3.11.16) relies heavily on these coordinates: the introduction of γαβ

rests on the fact that in these coordinates, gαβ is continuous at Σ, so

3.11 Null shells 99

that an eventual discontinuity of the metric derivative must be directedalong kα. In the next subsection we will remove the need to involvethe coordinates xα in practical applications of the null-shell formalism.For the time being we simply note that while Eq. (3.11.16) is indeedexpressed in the overlapping coordinates xα, it is a tensorial equationinvolving vectors (kα and eα

A) and scalars (µ, jA, and p). This equationcan therefore be expressed in any coordinate system; in particular, whenviewed from ±, the surface stress-energy tensor can be expressed in theoriginal coordinates xα

±.

3.11.3 Intrinsic formulation

In Sec. 3.7, the surface stress-energy tensor of a timelike or spacelikeshell was expressed in terms of intrinsic three-tensors — quantities thatcan be defined on the hypersurface only. The most important ingre-dients in this formulation were hab, the (continuous) induced metric,and [Kab], the discontinuity in the extrinsic curvature. We would liketo achieve something similar here, and remove the need to involve anoverlapping coordinate system xα to calculate the surface quantities µ,jA, and p.

We can expect that the intrinsic description of the surface stress-energy tensor of a null shell will involve σAB, the nonvanishing com-ponents of the induced metric. We might also expect that it shouldinvolve the jump in the extrinsic curvature of the null hypersurface,which would be defined by Kab = kα;β eα

aeβb = 1

2(£kgαβ) eα

aeβb . Not so.

The reason is that there is nothing “transverse” about this object: Inthe case of a timelike or spacelike hypersurface, the normal nα pointsaway from the surface, and £ngαβ truly represents the transverse deriva-tive of the metric; when the hypersurface is null, on the other hand, kα

is tangent to the surface, and £kgαβ is a tangential derivative. Thus, theextrinsic curvature is necessarily continuous when the hypersurface isnull, and it cannot be related to the tensor γαβ defined by Eq. (3.11.11).

There is, fortunately, an easy solution to this problem: we can intro-duce a transverse curvature Cab that properly represents the transversederivative of the metric. This shall be defined by Cab = 1

2(£Ngαβ) eα

aeβb =

12(Nα;β + Nβ;α)eα

aeβb , or

Cab = −Nα eαa;βeβ

b . (3.11.17)

To arrive at Eq. (3.11.17) we have used the fact that Nαeαa is a constant,

and the identity eαa;βeβ

b = eαb;βeβ

a , which states that each basis vector eαa

is Lie transported along any other basis vector; this property ensuresthat Cab, as defined by Eq. (3.11.17), is a symmetric three-tensor.

In the overlapping coordinates xα, the jump in the transverse cur-

100 Hypersurfaces

vature is given by

[Cab] = [Nα;β] eαaeα

b

= −[Γγαβ]Nγe

αaeα

b

=1

2γαβ eα

aeαb ,

where we have used Eq. (3.11.12) and the fact that kα is orthogonalto eα

a . We therefore have [Cλλ] = 12γαβkαkβ, [CAλ] = 1

2γαβeα

Akβ ≡ 12γA,

and [CAB] = 12γαβeα

AeβB ≡ 1

2γAB, where we have involved Eq. (3.11.15).

Finally, we find that the surface quantities can be expressed as

µ = − 1

8πσAB[CAB], jA =

1

8πσAB[CλA], p = − 1

8π[Cλλ].

(3.11.18)We have established that the shell’s surface quantities can all be relatedto the induced metric σAB and the discontinuity in the transverse cur-vature Cab. This completes the intrinsic formulation of our null-shellformalism.

3.11.4 Summary

A singular null hypersurface Σ possesses a surface stress-energy tensorcharacterized by tangent vectors kα

± and eα±A, as well as a surface density

µ, a surface current jA, and a surface pressure p. The surface quan-tities can all be related to a discontinuity in the surface’s transversecurvature,

Cab = −Nα eαa;β eβ

b ,

which is defined on either side of Σ in the appropriate coordinate systemxα±. The relations are

µ = − 1

8πσAB[CAB], jA =

1

8πσAB[CλA], p = − 1

8π[Cλλ].

The surface stress-energy tensor is given by

Sαβ = µkαkβ + jA(kαeβA + eα

Akβ) + p σABeαAeβ

B,

and the complete stress-energy tensor of the surface layer is

TαβΣ = (−kµu

µ)−1Sαβ δ(τ).

In this expression, the factor (−kµuµ)−1 is continuous at the shell, and

the vector uα± = dxα

±/dτ is tangent to an arbitrary congruence of time-like geodesics (which represents a family of observers making measure-ments on the shell). The presence of this factor implies that µ, jA, and

3.11 Null shells 101

p are not the physically-measured surface quantities; those are insteadgiven by

µphysical = (−kµuµ)−1µ, jA

physical = (−kµuµ)−1jA, pphysical = (−kµu

µ)−1p.

The arbitrariness associated with the choice of congruence is thus lim-ited to a single multiplicative factor; the “bare” quantities µ, jA, andp are independent of this choice.

3.11.5 Parameterization of the null generators

Our null-shell formalism is now complete, and it is ready to be involvedin applications. We will consider a few in the following subsections, butwe first return to a statement made earlier, that in general, λ cannot bean affine parameter on both sides of the hypersurface. We shall justifythis here, and also consider what happens to µ, jA, and p when theparameterization of the null generators is altered.

Whether or not λ is an affine parameter can be decided by com-puting κ±, the “acceleration” of the null vector kα

±. This is defined oneither side of the hypersurface by (Sec. 1.3)

kα;βkβ = κkα,

and λ will be an affine parameter on the ± side of Σ if κ± = 0. Accordingto Eq. (3.11.5), κ = −Nαkα

;βkβ = −Nαeαλ;βeβ

λ = Cλλ, where we havealso used Eqs. (3.11.2) and (3.11.17). Equation (3.11.18) then relatesthe discontinuity in the acceleration to the surface pressure:

[κ] = −8π p. (3.11.19)

We conclude that λ can be an affine parameter on both sides of Σ onlywhen the null shell has a vanishing surface pressure. When p 6= 0, λcan be chosen to be an affine parameter on one side of the hypersurface,but it will not be an affine parameter on the other side.

Additional insight into this matter can be gained from Raychaud-huri’s equation, which describes the transverse evolution of a congru-ence of null geodesics (Sec. 2.4). In Sec. 2.6, Problem 8, Raychaudhuri’sequation was written in terms of an arbitrary parameterization of thenull geodesics. When the congruence is hypersurface orthogonal, itreads

dθ

dλ+

1

2θ2 + σαβσαβ = κ θ − 8πTαβkαkβ,

where θ and σαβ are the expansion and shear of the congruence, re-spectively; the equation holds on either side of Σ. The left-hand sideof Raychaudhuri’s equation, because it depends only on the intrinsic

102 Hypersurfaces

geometry of the hypersurface, is guaranteed to be continuous across theshell. Continuity of the right-hand side therefore implies

[κ]θ = 8π[Tαβkαkβ]. (3.11.20)

This relation shows that [κ] 6= 0 (and p 6= 0) whenever the componentTαβkαkβ of the stress-energy tensor is discontinuous at the shell. Thus,we conclude that λ cannot be an affine parameter on both sides of Σwhen [Tαβkαkβ] 6= 0. (Notice that this conclusion breaks down whenθ = 0, that is, when the shell is stationary.)

Recalling (Sec. 2.4.8) that the expansion θ is equal to the fractionalrate of change of the congruence’s cross-sectional area, we find thatwith the help of Eq. (3.11.19), Eq. (3.11.20) can be expressed as

pd

dλdS + [Tαβkαkβ]dS = 0, (3.11.21)

where dS =√

σ d2θ is an element of cross-sectional area on the shell(Sec. 3.2.2). This equation has a simple interpretation: The first termrepresents the work done by the shell as it expands or contracts, whilethe second term is the energy absorbed by the shell from its surround-ings; Eq. (3.11.21) therefore states that all of the absorbed energy goesinto work.

Having established that λ cannot, in general, be an affine parameteron both sides of the hypersurface, let us now investigate how a changeof parameterization might affect the surface density µ, surface currentjA, and surface pressure p of the null shell. Because each generatorcan be reparameterized independently of any other generator, we mustconsider transformations of the form

λ → λ(λ, θA). (3.11.22)

The question before us is: How do µ, jA, and p change under such atransformation?

To answer this we need to work out how the transformation ofEq. (3.11.22) affects the vectors kα, eα

A, and Nα. We first note thatthe differential form of Eq. (3.11.22) is

dλ = eβ dλ + cA dθA, (3.11.23)

where

eβ ≡(∂λ

∂λ

)θA

, cA ≡( ∂λ

∂θA

)λ; (3.11.24)

both eβ and cA depend on ya = (λ, θA), but because they depend on theintrinsic coordinates only, we have that [eβ] = 0 = [cA]. A displacementwithin the hypersurface can then be described either by

dxα = kα dλ + eαA dθA,

3.11 Null shells 103

where kα = (∂xα/∂λ)θAand eα

A = (∂xα/∂θA)λ, or by

dxα = kα dλ + eαA dθA,

where kα = (∂xα/∂λ)θAand eα

A = (∂xα/∂θA)λ; these relations holdon either side of Σ, in the relevant coordinate system xα

±. UsingEq. (3.11.23), it is easy to see that the tangent vectors transform as

kα = e−β kα, eαA = eα

A − cAe−β kα (3.11.25)

under the reparameterization of Eq. (3.11.22). It may be checked thatthe new basis vectors satisfy the orthogonality relations (3.11.2), andthat the induced metric σAB is invariant under this transformation:σAB ≡ gαβ eα

AeβB = gαβeα

AeβB ≡ σAB. To preserve the relations (3.11.5)

we let the new auxiliary null vector be

Nα = eβ Nα +1

2(σABcAcB)e−β kα − (σABcB)eα

A. (3.11.26)

This transformation ensures that the completeness relation (3.11.6)takes the same form in the new basis.

It is a straightforward (but slightly tedious) task to compute howthe transverse curvature Cab changes under a reparameterization of thegenerators, and to then compute how the surface quantities transform.You will be asked to go through this calculation in Sec. 3.13, Problem8. The answer is that under the reparameterization of Eq. (3.11.22),the surface quantities transform as

µ = eβµ + 2cAjA + (σABcAcB)e−βp,

A = jA + (σABcB)e−βp (3.11.27)

p = e−βp.

These transformations, together with Eq. (3.11.25), imply that thesurface stress-energy tensor becomes Sαβ = e−βSαβ. We also have(−kµu

µ)−1 = eβ(−kµuµ), and these results reveal that the combination

(−kµuµ)−1Sαβ is invariant under the reparameterization. This, finally,

establishes the invariance of TαβΣ , the full stress-energy tensor of the

surface layer.As a final remark, we note that under the reparameterization of

Eq. (3.11.22), the physically-measured surface quantities transform as

µphysical = e2βµphysical + 2cAeβjAphysical + (σABcAcB)pphysical,

Aphysical = eβjA

physical + (σABcB)pphysical, (3.11.28)

pphysical = pphysical;

we see in particular that the physically-measured surface pressure is aninvariant.

104 Hypersurfaces

3.11.6 Imploding spherical shell

For our first application of the null-shell formalism, we take anotherlook at the gravitational collapse of a thin spherical shell, a problemthat was first considered in Sec. 3.9. Here we imagine that the collapseproceeds at the speed of light, and that the thin shell lies on a nullhypersurface Σ. We take spacetime to be flat inside the shell (in −),and write the metric there as

ds2− = −dt2− + dr2 + r2 dΩ2,

in terms of spatial coordinates (r, θ, φ) and a time coordinate t−. Themetric outside the shell (in +) is the Schwarzschild solution,

ds2+ = −f dt2+ + f−1 dr2 + r2 dΩ2,

expressed in the same spatial coordinates but in terms of a distincttime t+; here, f = 1− 2M/r and M designates the gravitational massof the collapsing shell.

As seen from −, the null hypersurface Σ is described by the equationt− + r ≡ v− = constant, which means that the induced metric onΣ is given by ds2

Σ = r2 dΩ2. As seem from +, on the other hand,the hypersurface is described by t+ + r∗(r) ≡ v+ = constant, wherer∗(r) =

∫f−1 dr = r + 2M ln(r/2M − 1), and this gives rise to the

same induced metric. From these considerations we see that it waspermissible to express the metrics of ± in terms of the same spatialcoordinates (r, θ, φ), but that t+ cannot be equal to t−. The inducedmetric on the shell is

σAB dθAdθB = λ2(dθ2 + sin2 θ dφ2),

where we have set θA = (θ, φ) and identified −r with the parameter λon the null generators of the hypersurface; we shall see that here, λ isan affine parameter on both sides of Σ.

As seen from −, the parametric equations xα− = xα

−(λ, θA) describingthe hypersurface have the explicit form t− = v−+λ, r = −λ, θ = θ, andφ = φ. These give us the basis vectors kα∂α = ∂t − ∂r, eα

θ ∂α = ∂θ, andeα

φ∂α = ∂φ, and the basis is completed by Nα dxα = −12(dt− dr). From

all this and Eq. (3.11.17) we find that the nonvanishing components ofthe transverse curvature are

C−AB =

1

2rσAB.

The fact that C−λλ = 0 confirms that λ ≡ −r is an affine parameter on

the − side of Σ.As seen from +, the parametric equations describing the hyper-

surface are t+ = v+ − r∗(−λ), r = −λ, θ = θ, and φ = φ. The

3.11 Null shells 105

basis vectors are kα∂α = f−1∂t − ∂r, eαθ ∂α = ∂θ, eα

φ∂α = ∂φ, and

Nα dxα = −12(f dt − dr). The nonvanishing components of the trans-

verse curvature are now

C+AB =

f

2rσAB.

The fact that C+λλ = 0 confirms that λ ≡ −r is an affine parameter on

the + side of Σ; λ is therefore an affine parameter on both sides.The angular components of the transverse curvature are discontinu-

ous across the shell: [CAB] = −(M/r2)σAB. According to Eq. (3.11.18),this means that the shell has a vanishing surface current jA and a van-ishing surface pressure p, but that its surface density is

µ =M

4πr2.

We have therefore obtained the very sensible result that the surfacedensity of a collapsing null shell is equal to its gravitational mass di-vided by its (ever decreasing) surface area. Notice that µphysical = µfor observers at rest in −. Because of the focusing action of the nullshell, however, these observers do not remain at rest after crossing overto the + side: A simple calculation, based on Eq. (3.11.8), reveals thatan observer at rest before crossing the shell will move according todr/dτ = −(E2 − f)1/2 after crossing the shell; the energy parameter Evaries from observer to observer, and is related by E = 1 − M/rΣ tothe radius rΣ at which a given observer crosses the hypersurface.

3.11.7 Accreting black hole

In our second application of the null-shell formalism, we consider anonrotating black hole of mass (M −m) which suddenly acquires ad-ditional material of mass m and angular momentum J ≡ aM . Wesuppose that the accretion process is virtually instantaneous, that thematerial falls in with the speed of light, and that J ¿ M2. We idealizethe accreting material as a singular stress-energy tensor supported ona null hypersurface Σ.

The spacetime in the future of Σ — in + — is that of a slowlyrotating black hole of mass M and (small) angular momentum aM .We write the metric in + as in Eq. (3.10.1),

ds2+ = −f dt2 + f−1 dr2 + r2 dΩ2 − 4Ma

rsin2 θ dtdφ,

where f = 1− 2M/r; this is the slow-rotation limit of the Kerr metric,and throughout this subsection we will work consistently to first orderin the small parameter a.

106 Hypersurfaces

As seen from +, the null hypersurface Σ is described by v ≡ t+r∗ =0, where r∗ =

∫f−1 dr = r + 2M ln(r/2M − 1); you may check that

in the slow-rotation limit, every surface v = constant is null. It followsthat the vector kα = gαβ(−∂βv) is normal to Σ and tangent to its nullgenerators. We have

kα∂α =1

f∂t − ∂r +

2Ma

r3f∂φ,

and from this expression we deduce four important properties of thegenerators. First, the generators are affinely parameterized by λ ≡ −r.Second, as measured by inertial observers at infinity, the generatorsmove with an (ever increasing) angular velocity

dφ

dt≡ Ωgenerators =

2Ma

r3.

Third, θ is constant on each generator. And fourth, integration ofdφ/(−dr) = 2Ma/(r3f) reveals that

ψ ≡ φ +a

r

(1 +

r

2Mln f

)

also is constant on the generators.We shall use ya = (λ ≡ −r, θ, ψ) as coordinates on Σ; as we have

just seen, these coordinates are well adapted to the generators, andthis property is required by the null-shell formalism. Rememberingthat dt = −dr/f and dφ = dψ − (2Ma/r3f) dr on Σ, we find that theinduced metric is

σAB dθAdθB = r2(dθ2 + sin2 θ dψ2),

and that the hypersurface is intrinsically spherical.The parametric description of Σ, as seen from +, is xα(−r, θ, ψ), and

from this we form the tangent vectors eαλ = kα, eα

θ = δαθ , and eα

ψ = δαφ .

The basis is completed by Nα dxα = 12(−f dt+ dr). From Eq. (3.11.17)

we obtain

C+λψ =

3Ma

r2sin2 θ, C+

AB =f

2rσAB

for the nonvanishing components of the transverse curvature.The spacetime in the past of Σ — in − — is that of a nonrotating

black hole of mass (M −m). Here we write the metric as

ds2− = −F dt 2 + F−1 dr2 + r2(dθ2 + sin2 θ dψ2),

in terms of a distinct time coordinate t and the angles θ and ψ; wealso have F ≡ 1 − 2(M − m)/r. This choice of angular coordinatesimplies that inertial observers within − corotate with the shell’s null

3.11 Null shells 107

generators; this is another manifestation of the dragging of inertialframes, a phenomenon first encountered in Sec. 3.10. As we shall seepresently, this choice of coordinates is dictated by continuity of theinduced metric at Σ.

The mathematical description of the hypersurface, as seen from −,is identical to its external description provided that we make the sub-stitutions t → t, φ → ψ, M → M −m, and a → 0. According to this,the induced metric on Σ is still given by ds2

Σ = r2(dθ2 + sin2 θ dψ2), asrequired. The basis vectors are now kα∂α = F−1∂t − ∂r, eα

θ ∂α = ∂θ,eα

ψ∂α = ∂ψ, and Nα dxα = 12(−F dt + dr). This gives us

C−AB =

F

2rσAB

for the nonvanishing components of the transverse curvature.The transverse curvature is discontinuous at Σ, and Eqs. (3.11.18)

allow us to compute the shell’s surface quantities. Because the gener-ators are affinely parameterized by −r on both sides of the shell, wehave that p = 0 — the shell has a vanishing surface pressure. On theother hand, its surface density is given by

µ =m

4πr2,

the ratio of the shell’s gravitational mass m to its (ever decreasing)surface area 4πr2. Thus far our results are virtually identical to thoseobtained in the preceding subsection. What is new in this context isthe presence of a surface current jA, whose sole component is

jψ =3Ma

8πr4.

This comes from the shell’s rotation, and the fact that the situation isnot entirely spherically symmetric.

To better understand the physical significance of the surface current,we express the shell’s surface stress-energy tensor,

Sαβ = µkαkβ + jψ(kαeβψ + eα

ψkβ),

in terms of the vector `α ≡ kα +(jψ/µ) eαψ. This vector is null (when we

appropriately discard terms of order a2 in the calculation of gαβ`α`β),and it has the components

`α∂α =1

f∂t − ∂r +

1

fΩfluid∂φ

in the coordinates xα = (t, r, θ, φ) used in +; we have set

Ωfluid ≡ 2Ma

r3+

3Ma

2mrf.

108 Hypersurfaces

The shell’s surface stress-energy tensor is now given by the simple ex-pression

Sαβ = µ `α`β,

which corresponds to a pressureless fluid of density µ moving with afour-velocity `α. We see that the fluid is moving along null curves (notgeodesics!) that do not coincide with the shell’s null generators, andthat the motion across generators is created by a mismatch betweenΩfluid, the fluid’s angular velocity, and Ωgenerators, the angular velocityof the generators. The mismatch is directly related to jA:

Ωrelative ≡ Ωfluid − Ωgenerators =jψ

µ=

3Ma

2mrf.

Notice that the fluid rotates faster than the generators, which sharetheir angular velocity with inertial observers within −; such a phe-nomenon was encountered before, in the context of the stationary ro-tating shell of Sec. 3.10. But notice also that Ωrelative decreases to zeroas r approaches 2M : the fluid ends up corotating with the generatorswhen the shell crosses the black-hole horizon.

3.11.8 Cosmological phase transition

In this third (and final) application of the formalism, we consider anintriguing (but entirely artificial) cosmological scenario according towhich the universe was initially expanding in two directions only, butwas then made to expand isotropically by a sudden explosive event.

The − region of spacetime is the one in which the universe is ex-panding in the x and y directions only. Its metric is

ds2− = −dt2 + a2(t)(dx2 + dy2) + dz2

−,

and the scale factor is assumed to be given by a(t) ∝ t1/2. The cos-mological fluid moves with a four-velocity uα = ∂xα/∂t, and it has adensity and (isotropic) pressure given by ρ− = p− = 1/(32πt2), respec-tively.

In the + region of spacetime, the universe expands uniformly in allthree directions. Here the metric is

ds2+ = −dt2 + a2(t)(dx2 + dy2 + dz2

+),

with the same scale factor a(t) as in −, and the cosmological fluid has adensity and pressure given by ρ+ = 3p+ = 3/(32πt2), respectively; thiscorresponds to a radiation-dominated universe.

The history of the explosive event that changes the metric fromg−αβ to g+

αβ traces a null hypersurface Σ in spacetime. This surfacemoves in the positive z± direction, and as we shall see, it supports a

3.11 Null shells 109

singular stress-energy tensor. The “agent” that alters the course of theuniverse’s expansion is therefore a null shell.

As seen from −, the hypersurface is described by t = z− + constant,and the vector kα∂α = ∂t + ∂z is tangent to the null generators, whichare parameterized by t. In fact, because kα

;βkβ = 0, we have that t isan affine parameter on this side of the hypersurface. The coordinatesx and y are constant on the generators, and we use them, togetherwith t, as intrinsic coordinates on Σ. We therefore have ya = (t, θA),θA = (x, y), and the shell’s induced metric is

σAB dθAdθB = a2(t)(dx2 + dy2).

The remaining basis vectors are eαx∂α = ∂x, eα

y ∂α = ∂y, and Nα dxα =−1

2(dt+dz−). The nonvanishing components of the transverse curvature

are

C−AB =

1

4tσAB.

We note that on the − side of Σ, the null generators have an expansiongiven by θ = kα

;α = 1/t, and that Tαβkαkβ = ρ− + p− = 1/(16πt2),where Tαβ is the stress-energy tensor of the cosmological fluid.

As seen from +, the description of the hypersurface is obtained byintegrating dt = a(t) dz+, and kα∂α = ∂t + a−1∂z is tangent to the nullgenerators. We note that t is not an affine parameter on this side ofthe hypersurface: we have that kα

;βkβ = (2t)−1kα. The remaining basis

vectors are eαx∂α = ∂x, eα

y ∂α = ∂y, Nα dxα = −12(dt + a dz+), and the

nonvanishing components of the transverse curvature are now

C+tt =

1

2t, C+

AB =1

4tσAB.

On this side of Σ, the generators have an expansion also given by θ =1/t (since continuity of θ is implied by continuity of the induced metric),and Tαβkαkβ = ρ+ + p+ = 1/(8πt2).

The fact that t is an affine parameter on one side of the hypersurfaceonly tells us that the shell must possess a surface pressure. In fact,continuity of CAB across the shell implies that p is the only nonvanishingsurface quantity. It is given by

p = − 1

16πt,

the negative sign indicating that this surface quantity would be betterdescribed as a tension, not a pressure. The shell’s surface stress-energytensor is Sαβ = p σAB eα

AeαB. If we select observers comoving with the

cosmological fluid as our preferred observers to make measurementson the shell, then −kαuα = 1 and the full stress-energy tensor of thesingular hypersurface is Tαβ

Σ = Sαβδ(t− tΣ), with tΣ denoting the time

110 Hypersurfaces

at which a given observer crosses the shell. We see that for theseobservers, −p is the physically-measured surface tension.

Finally, we note that the expressions −p = 1/(16πt), θ = 1/t,and [Tαβkαkβ] = 1/(16πt2) are compatible with the general relation−p θ = [Tαβkαkβ] derived in Sec. 3.11.5. This shows that the energyreleased by the shell as it expands is absorbed by the cosmologicalfluid, whose density increases by a factor of ρ+/ρ− = 3; this energy isprovided by the shell’s surface tension.

3.12 Bibliographical notes

During the preparation of this chapter I have relied on the followingreferences: Barrabes and Israel (1991); Barrabes and Hogan (1998); dela Cruz and Israel (1968); Israel (1966); Misner, Thorne, and Wheeler(1973); Musgrave and Lake (1997); and Wald (1984).

More specifically:Sections 3.1, 3.2, and 3.3 are based partially on unpublished lecture

notes by Werner Israel. Sections 3.4, 3.5, and 3.9 are based on Israel’spaper. Section 3.6 is based on Sec. 10.2 of Wald. Sections 3.7 and3.11 (as well as Problem 9 below) are based on Barrabes and Israel.Section 3.8 is based on Exercise 32.4 of Misner, Thorne, and Wheeler.Section 3.10 is based on de la Cruz and Israel. Finally, the examplesof Secs. 3.11.7 and 3.11.8 are adapted from Musgrave & Lake, andBarrabes & Hogan, respectively.

3.13 Problems

Warning: The results derived in Problem 1 are used in later portionsof this book.

1. We consider a hypersurface T = constant in Schwarzschild space-time, where

T = t + 4M

[√r/2M +

1

2ln

(√r/2M − 1√r/2M + 1

)].

We use (r, θ, φ) as coordinates on the hypersurface.

a) Calculate the unit normal nα, and find the parametric equa-tions describing the hypersurface.

b) Calculate the induced metric hab.

c) Calculate the extrinsic curvature Kab. Verify that your resultsagree with those of Sec. 3.6.6, and show that K is equalto the expansion of the geodesic congruence considered inSec. 2.3.7.

3.13 Problems 111

d) Prove that when it is expressed in terms of the coordinates(T, r, θ, φ), the Schwarzschild metric takes the form

ds2 = −dT 2 + (dr +√

2M/r dT )2 + r2 dΩ2.

This shows very clearly that the sections T = constant areintrinsically flat. [This coordinate system was discoveredindependently by Painleve (1921) and Gullstrand (1922).]

2. A four-dimensional hypersurface is embedded in a flat, five-dimensionalspacetime. We use coordinates zA in the five-dimensional world,and express the metric as

ds2 = ηAB dzAdzB = −(dz0)2 + (dz1)2 + (dz2)2 + (dz3)2 + (dz4)2;

we let uppercase latin indices run from 0 to 4. In the four-dimensional world we use coordinates xα = (t, χ, θ, φ). The hy-persurface is defined by parametric relations zA(xα). Explicitly,

z0 = a sinh(t/a), z1 = a cosh(t/a) cos χ, z2 = a cosh(t/a) sin χ cos θ,

z3 = a cosh(t/a) sin χ sin θ cos φ, z4 = a cosh(t/a) sin χ sin θ sin φ,

where a is a constant.

a) Compute the unit normal nA and the tangent vectors eAα =

∂zA/∂xα to the hypersurface.

b) Compute the induced metric gαβ. What is the physical signif-icance of this four-dimensional metric? Does it satisfy theEinstein field equations?

c) Compute the extrinsic curvature Kαβ. Use the Gauss-Codazziequations to prove that the induced Riemann tensor can beexpressed as

Rαβγδ =1

a2(gαγgβδ − gαδgβγ).

This implies that the four-dimensional hypersurface is a space-time of constant Ricci curvature.

3. In this problem we consider a spherically symmetric space at amoment of time symmetry. We write the three-metric as

ds2 = d`2 + r2(`) dΩ2,

where ` is proper distance from the centre.

112 Hypersurfaces

a) Show that in these coordinates, the mass function introducedin Sec. 3.6.5 is given by

m(r) =r

2

[1− (dr/d`)2

].

b) Solve the constraint equations for a uniform energy density ρon the hypersurface. Make sure to enforce the asymptoticcondition r(` → 0) → `, so that the three-metric is regularat the centre.

c) Prove that r(`) can be no larger than rmax =√

3/8πρ.

d) Prove that 2m(rmax) = rmax, and that m(rmax) is the maxi-mum value of the mass function.

e) What happens when ` → πrmax?

4. Prove the statement made toward the end of Sec. 3.7.5, that[Kab] = 0 is a sufficient condition for the regularity of the fullRiemann tensor at the hypersurface Σ.

5. Prove that the surface stress-energy tensor of a thin shell satisfiesthe conservation equation

Sab|b = −ε[ja],

where ja ≡ Tαβeαanβ. Interpret this equation physically. (Con-

sider the case where the shell is timelike.)

6. The metric

ds2 = −dt2 + d`2 + r2(`) dΩ2,

where r(`) = ` when 0 < ` < `0 and r(`) = 2`0− ` when `0 < ` <2`0, describes a spacetime with closed spatial sections. (What isthe volume of a hypersurface t = constant?) The spacetime isflat in both − (` < `0) and + (` > `0), but it contains a surfacelayer at ` = `0.

a) Calculate the surface stress-energy tensor of the thin shell.Express this in terms of a velocity field ua, a density σ, anda surface pressure p.

b) Consider a congruence of outgoing null geodesics in this space-time, with its tangent vector kα = −∂α(t− `). Calculate θ,the expansion of this congruence. Show that it abruptlychanges sign (from positive to negative) at ` = `0. The sur-face layer therefore produces a strong focusing of the nullgeodesics.

3.13 Problems 113

c) Use Raychaudhuri’s equation to prove that the discontinuity indθ/d` is precisely accounted for by the surface stress-energytensor.

7. Two Schwarzschild solutions, one with mass parameter m−, theother with mass parameter m+, are joined at a radius r = R(τ) bymeans of a spherical, massive thin shell. Here, τ denotes propertime for an observer comoving with the shell. It is assumed thatm− is the interior mass (m+ is the exterior mass), that m+ > m−,and that R(τ) > 2m+ for all values of τ . The shell’s surfacestress-energy tensor is given by

Sab = (σ + p)uaub + phab,

where ua = dya/dτ is the fluid’s velocity field, σ(τ) the surfacedensity, p(τ) the surface pressure, and hab the induced metric.

a) Derive, and interpret physically, the equation

d

dτ(σR2) + p

d

dτ(R2) = 0.

b) Find the values of σ and p which allow for a static configu-ration: R(τ) = R0 = constant. Verify that both σ and pare positive. [The stability of these static configurations wasexamined by Brady, Louko, and Poisson (1991).]

8. Derive the relations (3.11.26).

9. Let spacetime be partitioned into two regions ± with metrics

ds2± = −f± dv2 + 2 dvdr + r2 dΩ2.

We assume that the coordinate system (v, r, θ, φ) is common toboth − and +. (In each region we could introduce a conventionaltime coordinate t± defined by dt± = dv − dr/f±, but it is muchmore convenient to work with the original system.) In − we setf− = 1 − r0/r, so that the metric is a Schwarzschild solutionwith mass parameter M ≡ 1

2r0. In + we set f+ = 1− (r/r0)

2, sothat the metric is a de Sitter solution with cosmological constantΛ ≡ 3/r0

2. (This metric is a solution to the modified Einsteinfield equations, Gαβ + Λgαβ = 0.) The boundary Σ between thetwo regions is the null surface r = r0, the common horizon of theSchwarzschild and de Sitter spacetimes.

Using ya = (v, θ, φ) as coordinates on Σ, calculate the surfacequantities µ, jA, and p associated with the null shell. Explainwhether your results are compatible with the general relationp θ = [Tαβkαkβ] derived in Sec. 3.11.5.

114 Hypersurfaces

Chapter 4

Lagrangian andHamiltonian formulations

of general relativity

Variational principles play a central role in virtually all areas of physics,and general relativity is no exception. This chapter is devoted to a gen-eral discussion of the Lagrangian and Hamiltonian formulations of fieldtheories in curved spacetime, with a special focus on general relativity.

The Lagrangian formulation of a field theory (Sec. 4.1) begins withthe introduction of an action functional, which is usually defined as anintegral of a Lagrangian density over a finite region of spacetime. Aswe shall see, general relativity is peculiar in this respect, as its actioninvolves also an integration over ∂, the boundary of the region ; this isnecessary for the well-posedness of the variational principle. We will,in this chapter, provide a systematic treatment of the boundary termsin the gravitational action.

The Hamiltonian formulation of a field theory (Sec. 4.2) involves adecomposition of spacetime into space and time. Geometrically, thiscorresponds to a foliation of spacetime into nonintersecting spacelikehypersurfaces Σ. In this 3+1 decomposition, the spacetime metric gαβ

is decomposed into an induced metric hab, a shift vector Na, and a lapsescalar N ; while the induced metric is concerned with displacementswithin Σ, the lapse and shift are concerned with displacements awayfrom the hypersurface. The Hamiltonian is a functional of the fieldconfiguration and its conjugate momentum on Σ. In general relativity,the Hamiltonian is a functional of hab and its conjugate momentum pab,which is closely related to the extrinsic curvature of the hypersurface Σ;the lapse and shift are freely specifiable, and they do not appear in theHamiltonian as dynamical variables. The gravitational Hamiltonianinherits boundary terms from the action functional; those are definedon the two-surface S formed by the intersection of ∂ and Σ.

There is a close connection between the gravitational Hamiltonianand the total mass M and angular momentum J of an asymptotically-

115

116Lagrangian and Hamiltonian formulations of general

relativity

flat spacetime; this connection is explored in Sec. 4.3. We will see thatthe value of the gravitational Hamiltonian for a solution to the Einsteinfield equations depends only on the conditions at the two-dimensionalboundary S. When the spacetime is asymptotically flat and S is pushedto infinity, the Hamiltonian becomes M if the lapse and shift are cho-sen so as to correspond to an asymptotic time translation. For analternative choice of lapse and shift, corresponding to an asymptoticrotation about an axis, the Hamiltonian becomes J , the componentof the angular-momentum vector along this axis. These Hamiltoniandefinitions for mass and angular momentum form the starting point, inSec. 4.3, of a rather broad review of the different notions of mass andangular momentum in general relativity.

4.1 Lagrangian formulation

4.1.1 Mechanics

In the Lagrangian formulation of Newtonian mechanics, one is given aLagrangian L(q, q), a function of the generalized coordinate q and itsvelocity q ≡ dq/dt. One then forms an action functional S[q],

S[q] =

∫ t2

t1

L(q, q) dt, (4.1.1)

by integrating the Lagrangian over a selected path q(t). The paththat satisfies the equations of motion is the one about which S[q] isstationary: Under a variation δq(t) of this path, restricted by

δq(t1) = δq(t2) = 0 (4.1.2)

but otherwise arbitrary in the interval t1 < t < t2, the action does notvary, δS = 0.

The change in the action is given by

δS =

∫ t2

t1

δL dt

=

∫ t2

t1

(∂L

∂qδq +

∂L

∂qδq

)dt

=∂L

∂qδq

∣∣∣t2

t1+

∫ t2

t1

(∂L

∂q− d

dt

∂L

∂q

)δqdt,

where, in the last step, we have used δq = d(δq)/dt and integrated byparts. The boundary terms vanish by virtue of Eq. (4.1.2). Becausethe variation is arbitrary between t1 and t2,

δS = 0 ⇒ d

dt

∂L

∂q− ∂L

∂q= 0. (4.1.3)

4.1 Lagrangian formulation 117

This is the Euler-Lagrange equation for a one-dimensional mechanicalsystem. Generalization to higher dimensions is immediate.

4.1.2 Field theory

We now consider the dynamics of a field q(xα) in curved spacetime.Although this field could be of any type (scalar, vector, tensor, spinor),for simplicity we shall restrict our attention to the case of a scalar field.

In the Lagrangian formulation of a field theory, one is given anarbitrary region of the spacetime manifold, bounded by a closed hy-persurface ∂. One is also given a Lagrangian density (q, q,α), a scalarfunction of the field and its first derivatives. The action functional isthen

S[q] =

∫(q, q,α)

√−g d4x. (4.1.4)

Dynamical equations for q are obtained by introducing a variationδq(xα) that is arbitrary within but vanishes everywhere on ∂,

δq|∂ = 0, (4.1.5)

and by demanding that δS vanish if the variation is about the ac-tual path q(xα). Equation (4.1.5) is the field-theoretical counterpart toEq. (4.1.2).

Upon such a variation (we use the notation ′ ≡ ∂/∂q, α ≡ ∂/∂q,α),

δS =

∫(′δq + αδq,α)

√−g d4x

=

∫[′ δq + (αδq);α − α

;αδq]√−g d4x

=

∫(′ − α

;α)δq√−g d4x +

∮

∂

αδq dΣα,

where Gauss’ theorem (Sec. 3.3) was used in the last step. The surfaceintegral vanishes by virtue of Eq. (4.1.5), and because δq is arbitrarywithin , we obtain

δS = 0 ⇒ ∇α∂

∂q,α

− ∂

∂q= 0. (4.1.6)

This is the Euler-Lagrange equation for a single scalar field q. Gener-alization to a collection of fields is immediate, and the procedure canbe taken over to fields of arbitrary tensorial or spinorial types.

As a concrete example, let us consider a Klein-Gordon field ψ withLagrangian density

= −1

2

(gµνψ,µψ,ν + m2ψ2

).

118Lagrangian and Hamiltonian formulations of general

relativity

We have α = −gαβψ,β, α;α = −gαβψ;αβ, and ′ = −m2ψ. The Euler-

Lagrange equation becomes

gαβψ;αβ −m2ψ = 0,

which is the curved-spacetime version of the Klein-Gordon equation.

4.1.3 General relativity

The action functional for general relativity contains a contributionSG[g] from the gravitational field gαβ and a contribution SM [φ; g] fromthe matter fields, which we collectively denote φ.

The gravitational action contains a Hilbert term IH [g]/(16π), aboundary term IB[g]/(16π), and a nondynamical term I0/(16π) thataffects the numerical value of the action but not the equations of mo-tion. More explicitly,

SG[g] =1

16π

(IH [g] + IB[g]− I0

), (4.1.7)

where

IH [g] =

∫R√−g d4x, (4.1.8)

IB[g] = 2

∮

∂

εK|h|1/2 d3y, (4.1.9)

I0 = 2

∮

∂

εK0|h|1/2 d3y. (4.1.10)

Here, R is the Ricci scalar in , K is the trace of the extrinsic curvature of∂, ε is equal to +1 where ∂ is timelike and −1 where ∂ is spacelike (it isassumed that ∂ is nowhere null), and h is the determinant of the inducedmetric on ∂. Coordinates xα are used in , and coordinates ya are usedon ∂. The factor of (16π)−1 on the right-hand side of Eq. (4.1.7) will beseen to give rise to the factor of 8π in the Einstein field equations. Therole of IB[g] in the variational principle will be elucidated below. Thepresence of I0 in the action will also be explained, and this explanationwill come with a precise definition for the quantity K0.

The matter action is taken to be of the form

SM [φ; g] =

∫(φ, φ,α; gαβ)

√−g d4x, (4.1.11)

for some Lagrangian density . As Eq. (4.1.11) indicates, it is assumedthat only gαβ, and none of its derivatives, appears in the matter action.This assumption is made for simplicity, and it could easily be removed.

The complete action functional is therefore

S[g; φ] =

∫( R

16π+

)√−g d4x +1

8π

∮

∂

ε(K −K0)|h|1/2 d3y. (4.1.12)

4.1 Lagrangian formulation 119

The Einstein field equations, Gαβ = 8πTαβ, are recovered by varyingS[g, φ] with respect to gαβ. The variation is subjected to the condition

δgαβ|∂ = 0. (4.1.13)

This implies that hab = gαβ eαaeβ

b , the induced metric on ∂, is held fixedduring the variation.

4.1.4 Variation of the Hilbert term

It is convenient to use the variations δgαβ instead of δgαβ. These are ofcourse not independent: the relations gαµgµβ = δα

β imply

δgαβ = −gαµgβν δgµν . (4.1.14)

We recall (from Sec. 1.7) that the variation of the metric determinantis given by δ ln |g| = gαβδgαβ = −gαβδgαβ, which implies

δ√−g = −1

2

√−ggαβ δgαβ. (4.1.15)

We also recall (from Sec. 1.2) that although Γαβγ is not a tensor, the dif-

ference between two sets of Christoffel symbols is a tensor; the variationδΓα

βγ is therefore a tensor.We now proceed with the variation of the Hilbert term in the grav-

itational action:

δIH =

∫δ(gαβRαβ

√−g) d4x

=

∫(Rαβ

√−g δgαβ + gαβ√−g δRαβ + R δ

√−g) d4x

=

∫(Rαβ − 1

2Rgαβ

)δgαβ

√−g d4x +

∫gαβδRαβ

√−g d4x.

In the last step we have used Eq. (4.1.15). The first integral seemsto give us what we need for the left-hand side of the Einstein fieldequations, but we must still account for the second integral.

Let us work on this integral. We begin with δRαβ, which we calcu-late in a local Lorentz frame at a point P :

δRαβ∗= δ(Γµ

αβ,µ − Γµαµ,β)

∗= (δΓµ

αβ),µ − (δΓµαµ),β

∗= (δΓµ

αβ);µ − (δΓµαµ);β.

Here, covariant differentiation is defined with respect to the referencemetric gαβ, about which the variation is taken. We notice that the last

120Lagrangian and Hamiltonian formulations of general

relativity

expression is tensorial; it is therefore valid in any coordinate system.We have found

gαβδRαβ = δ−vµ;µ, δ−vµ = gαβδΓµ

αβ − gαµδΓβαβ. (4.1.16)

We use the “slash” notation δ−vµ to emphasize the fact that δ−vµ is not thevariation of some quantity vµ. Using Eq. (4.1.16), the second integralin δIH becomes

∫gαβδRαβ

√−g d4x =

∫δ−vµ

;µ

√−g d4x

=

∮

∂

δ−vµ dΣµ

=

∮

∂

ε δ−vµnµ|h|1/2 d3y,

where nµ is the unit normal to ∂ and ε ≡ nµnµ = ±1.We must now evaluate δ−vµnµ, keeping in mind that on ∂, δgαβ =

0 = δgαβ. Under these conditions,

δΓµαβ|∂ =

1

2gµν(δgνα,β + δgνβ,α − δgαβ,ν),

and substituting this into Eq. (4.1.16) yields δ−vµ = gαβ(δgµβ,α−δgαβ,µ),so that

nµδ−vµ|∂ = nµ(εnαnβ + hαβ)(δgµβ,α − δgαβ,µ)

= nµhαβ(δgµβ,α − δgαβ,µ).

In the first line we have substituted the completeness relations gαβ =εnαnβ + hαβ, where hαβ ≡ habeα

aeβb (see Sec. 3.1). To obtain the second

line we have multiplied nαnµ by the antisymmetric quantity within thebrackets. Now, because δgαβ vanishes everywhere on ∂, its tangentialderivatives must vanish also: δgαβ,γe

γc = 0. It follows that hαβδgµβ,α =

0, and we finally obtain

nµδ−vµ|∂ = −hαβδgαβ,µnµ. (4.1.17)

This is nonzero because the normal derivative of δgαβ is not requiredto vanish on the hypersurface.

Gathering the results, we obtain

δIH =

∫Gαβδgαβ

√−g d4x−∮

∂

εhαβδgαβ,µnµ|h|1/2 d3y. (4.1.18)

The boundary term in Eq. (4.1.18) will be canceled by the variation ofIB[g]: this is the reason for including a boundary term in the gravita-tional action. That a boundary term is needed is due to the fact thatR, the gravitational Lagrangian density, contains second derivatives ofthe metric tensor, a nontypical feature of field theories.

4.1 Lagrangian formulation 121

4.1.5 Variation of the boundary term

We now work on the variation of IB[g], as given by Eq. (4.1.9). Becausethe induced metric is fixed on ∂, the only quantity to be varied is K,the trace of the extrinsic curvature. We recall from Sec. 3.4 that

K = nα;α

= (εnαnβ + hαβ)nα;β

= hαβnα;β

= hαβ(nα,β − Γγαβnγ),

so that its variation is

δK = −hαβδΓγαβnγ

= −1

2hαβ(δgµα,β + δgµβ,α − δgαβ,µ)nµ

=1

2hαβδgαβ,µn

µ;

we have used the fact that the tangential derivatives of δgαβ vanish on∂. We have obtained

δIB =

∮

∂

εhαβδgαβ,µnµ|h|1/2 d3y, (4.1.19)

and we see that this indeed cancels out the second integral on the right-hand side of Eq. (4.1.18). Because δI0 ≡ 0, the complete variation ofthe gravitational action is

δSG =1

16π

∫Gαβ δgαβ

√−g d4x. (4.1.20)

This produces the correct left-hand side to the Einstein field equations.

4.1.6 Variation of the matter action

Variation of SM [φ; g], as given by Eq. (4.1.11), yields

δSM =

∫δ(√−g) d4x

=

∫( ∂

∂gαβδgαβ

√−g + δ√−g

)d4x

=

∫( ∂

∂gαβ− 1

2gαβ

)δgαβ

√−g d4x.

If we define the stress-energy tensor by

Tαβ ≡ −2∂

∂gαβ+ gαβ, (4.1.21)

122Lagrangian and Hamiltonian formulations of general

relativity

then

δSM = −1

2

∫Tαβ δgαβ

√−g d4x, (4.1.22)

and this produces the correct right-hand side to the Einstein field equa-tions.

We have obtained

δ(SG + SM) = 0 ⇒ Gαβ = 8πTαβ, (4.1.23)

because the variation δgαβ is arbitrary within . The Einstein fieldequations therefore follow from a variational principle, and the actionfunctional for the theory is given by Eq. (4.1.12).

To see that Eq. (4.1.21) gives a reasonable definition for the stress-energy tensor, let us consider once more a Klein-Gordon field ψ withLagrangian density

= −1

2

(gµνψ,µψ,ν + m2ψ2

).

It is easy to check that for this, Eq. (4.1.21) becomes

Tαβ = ψ,αψ,β − 1

2

(ψ,µψ,µ + m2ψ2

)gαβ.

This is the correct expression for the Klein-Gordon stress-energy tensor.You may look into the consistency of this result by checking that thestatement of energy-momentum conservation, T αβ

;β = 0, implies theKlein-Gordon equation.

4.1.7 Nondynamical term

What is the role of

I0 = 2

∮

∂

εK0|h|1/2 d3y

in the gravitational action? Because I0 depends only on the inducedmetric hab (through the factor |h|1/2 in the integrand), its variationwith respect to gαβ gives zero, and the presence of I0 cannot affect theequations of motion. Its purpose can only be to change the numericalvalue of the gravitational action.

Let us first assume that gαβ is a solution to the vacuum field equa-tions. Then R = 0 and the numerical value of the gravitational actionis

SG =1

8π

∮

∂

εK|h|1/2 d3y,

where we omit the subtraction term K0 for the time being. Let usevaluate this for flat spacetime. We choose ∂ to consist of two hyper-surfaces t = constant and a large three-cylinder at r = R (Fig. 4.1).

4.1 Lagrangian formulation 123

It is easy to check that K = 0 on the hypersurfaces of constant time.On the three-cylinder, the induced metric is ds2 = −dt2 + R2 dΩ2, sothat |h|1/2 = R2 sin θ. The unit normal is nα = ∂αr, so that ε = 1 andK = nα

;α = 2/R. We then have

∮

∂

εK|h|1/2 d3y = 8πR(t2 − t1),

and this diverges when R → ∞, that is, when the spatial boundary ispushed all the way to infinity. The gravitational action of flat space-time is therefore infinite, even when is bounded by two hypersurfaces ofconstant time. Because this problem does not go away when the space-time is curved, this would imply that the gravitational action is nota well-defined quantity for asymptotically-flat spacetimes. (Of course,this is not a problem if the spacetime manifold is compact.)

This problem is remedied by I0. Apart from a factor of 16π, thisterm is chosen to be equal to the gravitational action of flat spacetime,as regularized by the procedure used above. The difference IB − I0 isthen well defined in the limit R →∞, and there is no longer a difficultyin defining a gravitational action for asymptotically-flat spacetimes.(The subtraction term is irrelevant for compact manifolds.) In otherwords, the choice

K0 = extrinsic curvature of ∂ embedded in flat spacetime (4.1.24)

cures the divergence of the gravitational action, which is then well de-fined when the spacetime is asymptotically flat. In particular, SG ≡ 0for flat spacetime.

4.1.8 Bianchi identities

The Lagrangian formulation of general relativity provides us with anelegant derivation of the contracted Bianchi identities,

Gαβ;β = 0. (4.1.25)

Figure 4.1: The boundary of a region of flat spacetime.

t = t1

r = R

t = t2

124Lagrangian and Hamiltonian formulations of general

relativity

In this approach, Eq. (4.1.25) comes as a consequence of the invarianceof SG[g] under a change of coordinates in .

To prove this, it is sufficient to consider infinitesimal transforma-tions,

xα → x′α = xα + εα, (4.1.26)

where εα is an infinitesimal vector field, arbitrary within but con-strained to vanish on ∂. The variation of the metric under such atransformation is

δgαβ ≡ g′αβ(x)− gαβ(x)

= g′αβ(x′)− gαβ(x) + g′αβ(x)− g′αβ(x′)

=∂xµ

∂x′α∂xν

∂x′βgµν(x)− gαβ(x) + g′αβ(x)− g′αβ(x + ε)

= (δµα − εµ

,α)(δνβ − εν

,β)gµν(x)− gαβ(x)− gαβ,µ(x)εµ

= −εµ,αgµβ − εµ

,βgαµ − gαβ,µεµ

= −£εgαβ,

discarding all terms of the second order in εα. Using Eq. (4.1.14), wefind that the metric variation is

δgαβ = εα;β + εβ;α. (4.1.27)

Substituting this into Eq. (4.1.20), we find

8π δSG =

∫Gαβεα;β

√−g d4x

= −∫

Gαβ;βεα

√−g d4x +

∮

∂

Gαβεα dΣβ.

With εα arbitrary within but vanishing on ∂, the contracted Bianchiidentities follow from the requirement that δSG = 0 under the variationof Eq. (4.1.27).

4.2 Hamiltonian formulation

4.2.1 Mechanics

The Hamiltonian formulation of Newtonian mechanics begins with theintroduction of the canonical momentum p, defined by

p =∂L

∂q. (4.2.1)

It is assumed that this relation can be inverted to give q as a functionof p and q. The Hamiltonian is then

H(p, q) = p q − L. (4.2.2)

4.2 Hamiltonian formulation 125

Hamilton’s form of the equations of motion can be derived from a vari-ational principle. Here, the action is varied with respect to p and q in-dependently, with the restriction that δq must vanish at the endpoints.Thus,

δS =

∫ t2

t1

δ(p q −H) dt

=

∫ t2

t1

(p δq + q δp− ∂H

∂pδp− ∂H

∂qδq

)dt

= p δq∣∣∣t2

t1+

∫ t2

t1

[−

(p +

∂H

∂q

)δq +

(q − ∂H

∂p

)δp

]dt.

Because the variation is arbitrary between t1 and t2, but δq(t1) =δq(t2) = 0, we have

δS = 0 ⇒ p = −∂H

∂q, q =

∂H

∂p. (4.2.3)

These are Hamilton’s equations.

4.2.2 3 + 1 decomposition

The Hamiltonian formulation of a field theory is more involved. Here,the Hamiltonian H[p, q] is a functional of q, the field configuration, andp, the canonical momentum, on a spacelike hypersurface Σ. To expressthe action in terms of the Hamiltonian, it is necessary to foliate into afamily of spacelike hypersurfaces, one for each “instant of time”. Thisis the purpose of the 3 + 1 decomposition.

To effect this decomposition, we introduce a scalar field t(xα) suchthat t = constant describes a family of nonintersecting spacelike hy-persurfaces Σt. This “time function” is completely arbitrary; the onlyrequirements are that t be a single-valued function of xα, and nα ∝ ∂αt,the unit normal to the hypersurfaces, be a future-directed timelike vec-tor field.

On each of the hypersurfaces Σt we install coordinates ya. A priori,the coordinates on one hypersurface need not be related to the coordi-nates on another hypersurface. It is, however, convenient to introducea relationship, as follows (Fig. 4.2). Consider a congruence of curves γintersecting the hypersurfaces Σt. We do not assume that these curvesare geodesics, nor that they intersect the hypersurfaces orthogonally.We use t as a parameter on the curves, and the vector tα = dxα/dt istangent to the curves. It is easy to check that the relation

tα∂αt = 1 (4.2.4)

follows from the construction. A particular curve γP from the congru-ence defines a mapping from a point P on Σt to a point P ′ on Σt′ , and

126Lagrangian and Hamiltonian formulations of general

relativity

then to a point P ′′ on Σt′′ , and so on. To fix the coordinates of P ′ andP ′′, given ya(P ) on Σt, we simply demand ya(P ′′) = ya(P ′) = ya(P ).Thus, ya is held constant on each of the curves γ.

This construction defines a coordinate system (t, ya) in . Thereexists a transformation between this and the system xα originally inuse: xα = xα(t, ya). We have

tα =(∂xα

∂t

)ya

, (4.2.5)

and we define

eαa =

(∂xα

∂ya

)t

(4.2.6)

to be tangent vectors on Σt. These relations imply that in the coordi-nates (t, ya), tα

∗= δα

t and eαa∗= δα

a. We also have

£t eαa = 0, (4.2.7)

which holds in any coordinate system.We now introduce the unit normal to the hypersurfaces:

nα = −N∂αt, nαeαa = 0, (4.2.8)

where the scalar function N , called the lapse, ensures that nα is prop-erly normalized. Because the curves γ do not generally intersect Σt

orthogonally, tα is not parallel to nα. We may decompose tα in thebasis provided by the normal and tangent vectors (Fig. 4.3):

tα = Nnα + Naeαa ; (4.2.9)

the three-vector Na is called the shift. It is easy to check that Eq. (4.2.9)is compatible with Eq. (4.2.4).

Figure 4.2: Foliation of spacetime into spacelike hypersurfaces.

γP γQ

Σt

Σt′

Σt′′

P

P ′

P ′′

Q

Q′

Q′′

4.2 Hamiltonian formulation 127

We can use the coordinate transformation xα = xα(t, ya) to expressthe metric in the coordinates (t, ya). We start by writing

dxα = tα dt + eαa dya

= (N dt)nα + (dya + Na dt)eαa ,

which follows at once from Eqs. (4.2.5), (4.2.6), and (4.2.9). The lineelement is then given by ds2 = gαβdxαdxβ, or

ds2 = −N2dt2 + hab(dya + Na dt)(dyb + N b dt), (4.2.10)

where hab = gαβ eαaeβ

b is the induced metric on Σt.We may now express the metric determinant g in terms of h ≡

det[hab] and the lapse function. We recall that gtt = cofactor(gtt)/g =h/g, as follows from Eq. (4.2.10). But gtt = gαβt,αt,β = N−2gαβnαnβ =−N−2, where Eq. (4.2.8) was used. The desired expression is therefore

√−g = N√

h. (4.2.11)

Equations (4.2.9), (4.2.10), and (4.2.11) are the fundamental results ofthe 3 + 1 decomposition.

4.2.3 Field theory

We now return to the Hamiltonian formulation of a field theory. Forsimplicity, we will assume that the field is a scalar, but the procedurecan easily be extended to fields of other tensorial types. We begin bydefining the “time derivative” of q to be its Lie derivative along theflow vector tα,

q ≡ £t q. (4.2.12)

In the coordinates (t, ya), £tq∗= ∂q/∂t, and q reduces to the ordinary

time derivative. We also introduce the spatial derivatives, q,a ≡ q,αeαa .

The field’s Lagrangian density can then be expressed as (q, q, q,a).

Figure 4.3: Decomposition of tα into lapse and shift.

γ

Σt

Nnα tα

Naeαa

128Lagrangian and Hamiltonian formulations of general

relativity

Figure 4.4: The region , its boundary ∂, and their foliations.

St

Σt

Σt1

Σt2

The field’s canonical momentum p is defined by

p =∂

∂q

(√−g). (4.2.13)

It is assumed that this relation can be inverted to give q in terms of q,q,a, and p. The Hamiltonian density is then

(p, q, q,a) = p q −√−g . (4.2.14)

Because of the factors of√−g in Eqs. (4.2.13) and (4.2.14), the Hamil-

tonian density is not a scalar with respect to transformations ya → ya′ .We might introduce a scalarized version scalar, defined by =

√h scalar =√−g scalar/N , but such an object would turn out not to be as useful as

the original, nonscalar, Hamiltonian density. The Hamiltonian func-tional is defined by

H[p, q] =

∫

Σt

(p, q, q,a) d3y. (4.2.15)

The Hamiltonian functional is an ordinary (nonscalar) function of timet.

We consider a region of spacetime foliated by spacelike hypersur-faces Σt bounded by closed two-surfaces St (Fig. 4.4); itself is boundedby the hypersurfaces Σt1 , Σt2 , and , the union of all two-surfaces St.To obtain the Hamilton form of the field equations, we will vary theaction with respect to q and p, treating the variations δq and δp asindependent. We will demand that δq vanish on the boundaries Σt1 ,Σt2 , and .

The action functional is given by

S =

∫ t2

t1

dt

∫

Σt

(p q − ) d3y,

and variation yields

δS =

∫ t2

t1

dt

∫

Σt

(p δq + q δp− ∂

∂pδp− ∂

∂qδq − ∂

∂q,a

δq,a

)d3y.

4.2 Hamiltonian formulation 129

The first term may be integrated by parts:∫ t2

t1

dt

∫

Σt

p δq d3y =

∫ t2

t1

dtd

dt

∫

Σt

p δq d3y −∫ t2

t1

dt

∫

Σt

p δq d3y

=

∫

Σt2

p δq d3y −∫

Σt1

p δq d3y −∫ t2

t1

dt

∫

Σt

p δq d3y

= −∫ t2

t1

dt

∫

Σt

p δq d3y,

because δq = 0 on Σt1 and Σt2 . We treat the last term similarly:

−∫ t2

t1

dt

∫

Σt

∂

∂q,a

δq,a d3y = −∫ t2

t1

dt

∫

Σt

∂scalar

∂q,a

δq,a

√h d3y

= −∫ t2

t1

dt

∮

St

∂scalar

∂q,a

δq dSa

+

∫ t2

t1

dt

∫

Σt

(∂scalar

∂q,a

)|a

δq√

h d3y

=

∫ t2

t1

dt

∫

Σt

( ∂

∂q,a

),a

δq d3y.

In the second line we have used the three-dimensional version of Gauss’theorem, with dSa denoting the surface element on St. In the third linewe have used the divergence formula Aa

|a = h−1/2(h1/2Aa),a and thefact that δq vanishes on St.

Gathering the results, we have

δS =

∫ t2

t1

dt

∫

Σt

−

[p +

∂

∂q−

( ∂

∂q,a

),a

]δq +

[q − ∂

∂p

]δp

d3y,

and

δS = 0 ⇒ p = − ∂

∂q+

( ∂

∂q,a

),a, q =

∂

∂p. (4.2.16)

These are Hamilton’s equations for a scalar field q and its canonicalmomentum p.

As a concrete example, we consider once again a Klein-Gordon fieldψ with its Lagrangian density

= −1

2

(gµνψ,µψ,ν + m2ψ2

).

For simplicity, we choose our foliation to be such that Na = 0. Thisimplies gtt = 1/gtt, gta = 0, and gab = hab. Then = −1

2(gttψ2 +

habψ,aψ,b + m2ψ2), p = −√−g gttψ, and Eq. (4.2.14) gives

= − p2

2√−g gtt

+1

2

√−g (habψ,aψ,b + m2ψ2).

130Lagrangian and Hamiltonian formulations of general

relativity

The equations of motion are

ψ = − p√−g gtt, p = −√−g m2ψ + (

√−g habψ,b),a.

It is easy to check that from these follow the Klein-Gordon equation,gαβψ;αβ −m2ψ = 0, in the selected foliation.

4.2.4 Foliation of the boundary

Before tackling the case of the gravitational field, we need to provideadditional details regarding the foliation of , the timelike boundary of, by the two-surfaces St. (Refer back to Fig. 4.4.)

The closed two-surface St is the boundary of the spacelike hyper-surface Σt, on which we have coordinates ya, tangent vectors eα

a , andan induced metric hab. It is described by an equation of the formΦ(ya) = 0, or by parametric relations ya(θA), where θA are coordinateson St. We use ra to denote the unit normal to St, and we define anassociated four-vector rα by

rα = raeαa . (4.2.17)

This satisfies the relations rαrα = 1 and rαnα = 0, where nα is thenormal to Σt. The three-vectors ea

A = ∂ya/∂θA are tangent to St, sothat rae

aA = 0. This implies rαeα

A = 0, where

eαA ≡ eα

aeaA =

∂xα

∂θA. (4.2.18)

In this equation, it is understood that xα stands for the functionsxα(ya(θA)), where xα(ya) are the parametric relations defining Σt.

The induced metric on St is given by

ds2 = σAB dθAdθB, (4.2.19)

where σAB = hab eaAeb

B = (gαβ eαaeβ

b )eaAeb

B, or, using Eq. (4.2.18),

σAB = gαβ eαAeβ

B. (4.2.20)

Its inverse is denoted σAB. The three-dimensional completeness rela-tions, hab = rarb + σABea

AebB, are easily established (see Sec. 3.1). It

follows that the four-dimensional relations, gαβ = −nαnβ +habeαaeβ

b , canbe expressed as

gαβ = −nαnβ + rαrβ + σAB eαAeβ

B. (4.2.21)

This can be established directly by computing all inner products be-tween the vectors nα, rα, and eα

A.

4.2 Hamiltonian formulation 131

The extrinsic curvature of St embedded in Σt is defined by kAB =ra|bea

AebB (see Sec. 3.4), or

kAB = rα;β eαAeβ

B. (4.2.22)

We use k to denote its trace: k = σABkAB.A priori, the coordinates θA on a given two-surface (St′ , say) are

not related to the coordinates on another two-surface (St′′ , say). Tointroduce a relationship, we consider a congruence of curves β runningon , intersecting the two-surfaces St orthogonally, and therefore havingnα as their tangent vectors. We demand that if the curve βP intersectsSt′ at the point P ′ labelled by θA, then the same coordinates will des-ignate the point P ′′ at which βP intersects St′′ . Because θA does notvary along these curves, and because t can be chosen as a parameter,we have

nα = N−1(∂xα

∂t

)θA

, (4.2.23)

where the factor N−1 comes from Eq. (4.2.8) and the normalizationcondition nαnα = −1. The construction ensures that nα and eα

A areeverywhere orthogonal.

The hypersurface is foliated by the two-surfaces St. We put coor-dinates zi on , and introduce the tangent vectors eα

i = ∂xα/∂zi. Theinduced metric on is then given by

γij = gαβ eαi eβ

j . (4.2.24)

Its inverse is γij, and the completeness relations take the form

gαβ = rαrβ + γij eαi eβ

j . (4.2.25)

While the coordinates zi are a priori arbitrary, the choice zi = (t, θA)is clearly convenient. In these coordinates,

dxα =(∂xα

∂t

)θA

dt +(∂xα

∂θA

)tdθA

= Nnα dt + eαA dθA,

where Eqs. (4.2.18) and (4.2.23) were used. For displacements within ,the line element is

ds2 = gαβdxαdxβ

= gαβ(Nnα dt + eαA dθA)(Nnβ dt + eβ

B dθB)

= (gαβnαnβ)N2 dt2 + (gαβeαAeβ

B) dθAdθB,

where the relation nαeαA = 0 was used. We have obtained

γij dzidzj = −N2 dt2 + σAB dθAdθB. (4.2.26)

132Lagrangian and Hamiltonian formulations of general

relativity

This implies√−γ = N

√σ, where γ and σ are the determinants of γij

and σAB, respectively.

Finally, we let ij be the extrinsic curvature of embedded in space-time. This is given by

ij = rα;β eαi eβ

j , (4.2.27)

because rα, the unit normal to the two-surfaces St, is also normal to .We will use to denote its trace: = γij

ij.

Table 4.1 provides a list of the various geometric quantities intro-duced in this subsection.

4.2.5 Gravitational action

As a first step toward constructing the gravitational Hamiltonian, wemust subject the gravitational action SG to the 3 + 1 decompositiondescribed in Sec. 4.2.2. Our starting point is Eq. (4.1.12),

16πSG =

∫R√−g d4x + 2

∮

∂

εK|h|1/2 d3y,

where the subtraction term I0 is omitted for the time being; it willbe re-instated at the end of the calculation. Here, ∂ is the closedhypersurface bounding the four-dimensional region , ya are coordinateson ∂, hab is the induced metric, K is the trace of the extrinsic curvature,and ε = nαnα, where nα is the outward normal to ∂.

Throughout this section, the quantities nα, ya, hab, and Kab havereferred specifically to the spacelike hypersurfaces Σt, and we need tobe more careful with our notation. We have seen that ’s boundary isthe union of two spacelike hypersurfaces Σt1 and Σt2 with a timelikehypersurface (Fig. 4.4):

∂ = Σt2 ∪ (−Σt1) ∪ .

The minus sign in front of Σt1 serves to remind us that while the normalto ∂ must be directed outward, the normal to Σt1 is future-directed and

Table 4.1: Geometric quantities of Σt, St, and .

Surface Σt St

Unit normal nα rα rα

Coordinates ya θA zi

Tangent vectors eαa eα

A eαi

Induced metric hab σAB γij

Extrinsic curvature Kab kAB ij

4.2 Hamiltonian formulation 133

therefore points inward. With the notation introduced in the precedingsubsection, the gravitational action takes the form

16πSG =

∫R√−g d4x−2

∫

Σt2

K√

h d3y+2

∫

Σt1

K√

h d3y+2

∫√−γ d3z,

and the integration over Σt1 incorporates the extra minus sign justdiscussed.

The region is foliated by spacelike hypersurfaces Σt on which theRicci scalar is given by (Sec. 3.5.3)

R = 3R + KabKab −K2 − 2(nα;βnβ − nαnβ

;β);α,

where 3R is the Ricci scalar constructed from hab. Using Eq. (4.2.11),which we write as

√−g d4x = N√

h dtd3y, we have that∫

R√−g d4x =

∫ t2

t1

dt

∫

Σt

(3R + KabKab −K2)N√

h d3y

− 2

∮

∂

(nα;βnβ − nαnβ

;β) dΣα.

The new boundary term can be expressed as integrals over Σt1 , Σt2 , and. On Σt1 , dΣα = nα

√h d3y — this also incorporates an extra minus

sign — and

−2

∫

Σt1

(nα;βnβ−nαnβ

;β) dΣα = −2

∫

Σt1

nβ;β

√h d3y = −2

∫

Σt1

K√

h d3y.

We see that this term cancels out the other integral over Σt1 comingfrom the original boundary term in the gravitational action. The inte-grals over Σt2 cancel out also. There remains a contribution from , onwhich dΣα = rα

√−γ d3z, giving

−2

∫(nα

;βnβ−nαnβ;β) dΣα = −2

∫nα

;βnβrα

√−γ d3z = 2

∫rα;βnαnβ

√−γ d3z,

where we have used nαrα = 0.Gathering the results, we have

16πSG =

∫ t2

t1

dt

∫

Σt

(3R + KabKab −K2)N√

h d3y

+ 2

∫( + rα;βnαnβ)

√−γ d3z.

We now use the fact that is foliated by the closed two-surfaces St. Wesubstitute

√−γ d3z = N√

σ dtd2θ and express as

= γijij

= γij(rα;β eαi eβ

j )

= rα;β(γijeαi eβ

j )

= rα;β(gαβ − rαrβ),

134Lagrangian and Hamiltonian formulations of general

relativity

so that the integrand becomes

+ rα;βnαnβ = rα;β(gαβ − rαrβ + nαnβ)

= rα;β(σABeαAeβ

B)

= σAB(rα;βeαAeβ

B)

= σABkAB

= k.

We have used Eqs. (4.2.21) and (4.2.25) in these manipulations. Sub-stituting this into our previous expression for the gravitational action,we arrive at

SG =1

16π

∫ t2

t1

dt

∫

Σt

(3R + KabKab −K2

)N√

h d3y

+ 2

∮

St

(k − k0)N√

σ d2θ

. (4.2.28)

We have re-instated the subtraction term, by inserting k0 into the in-tegral over St. This is justified by the fact that for flat spacetime, theintegral over Σt vanishes, so that the sole contribution to SG comesfrom the boundary integral; the k0 term prevents this integral from di-verging in the limit St →∞, and it ensures that SG vanishes identicallyfor flat spacetime. Thus,

k0 = extrinsic curvature of St embedded in flat space.

The k0 term makes the gravitational action well defined for any asymptotically-flat spacetime. For compact spacetime manifolds, this term is irrele-vant.

The matter action should also be subjected to the 3+1 decomposi-tion. Because the procedure is straightforward, and because we woulddo well to keep things as simple as possible, we shall omit this stephere. In the rest of this section we will consider pure gravity only, andput the matter action to zero.

4.2.6 Gravitational Hamiltonian

To construct the Hamiltonian, we must express SG in terms of

hab ≡ £thab, (4.2.29)

where tα is the timelike vector field defined by Eq. (4.2.9). We calculatethis as follows. First we recall the definition of the induced metric andwrite

hab = £t(gαβeαaeβ

b ) = (£tgαβ)eαaeβ

b ,

4.2 Hamiltonian formulation 135

where we have used Eq. (4.2.7). Equation (4.2.9) implies that the Liederivative of the metric is given by

£tgαβ = tα;β + tβ;α

= (Nnα + Nα);β + (Nnβ + Nβ);α

= nαN,β + N,αnβ + N(nα;β + nβ;α) + Nα;β + Nβ;α,

where Nα = Naeαa . Finally, projecting this along eα

aeβb gives

hab = 2NKab + Na|b + Nb|a,

where we have used the definitions of extrinsic curvature and intrinsiccovariant differentiation found in Sec. 3.4.

We have obtained

Kab =1

2N

(hab −Na|b −Nb|a

). (4.2.30)

The gravitational action therefore depends on hab through the extrinsiccurvature. Notice that the action does not involve N nor Na, so thatmomenta conjugate to N and Na are not defined. This means thatunlike hab, the lapse and the shift are not dynamical variables. Thiswas to be expected: N and Na only serve to specify the foliation ofinto the spacelike hypersurfaces Σt; because this foliation is arbitrary,we are completely free in our choice of lapse and shift.

The momentum conjugate to hab is defined by

pab =∂

∂hab

(√−g G

), (4.2.31)

where G is the “volume part” of the gravitational Lagrangian. (The“boundary part” is independent of hab.) Because G is expressed interms of Kab, it is convenient to write Eq. (4.2.31) in the form

16π pab =∂Kmn

∂hab

∂

∂Kmn

(16π

√−g G

),

where16π

√−g G = [3R + (hachbd − habhcd)KabKcd]N√

h

follows from Eq. (4.2.28). Evaluating the partial derivatives gives

16π pab =√

h (Kab −Khab), (4.2.32)

and we see that the canonical momentum is closely related to the ex-trinsic curvature.

The “volume part” of the Hamiltonian density is

G = pab hab −√−g G. (4.2.33)

136Lagrangian and Hamiltonian formulations of general

relativity

Using our previous results, we have

16πG =√

h (Kab −Khab)(2NKab + Na|b + Nb|a)

− (3R + KabKab −K2)N√

h

= (KabKab −K2 − 3R)N√

h + 2(Kab −Khab)Na|b√

h

= (KabKab −K2 − 3R)N√

h + 2[(Kab −Khab)Na]|b√

h

− 2(Kab −Khab)|bNa

√h.

The full Hamiltonian is obtained by integrating G over Σt and addingthe boundary terms:

16πHG =

∫

Σt

16πG d3y − 2

∮

St

(k − k0)N√

σ d2θ

=

∫

Σt

[N(KabKab −K2 − 3R)− 2Na(K

ab −Khab)|b]√

h d3y

+ 2

∮

St

(Kab −Khab)Na dSb − 2

∮

St

(k − k0)N√

σ d2θ.

Writing dSb = rb

√σ d2θ, the gravitational Hamiltonian becomes

16πHG =

∫

Σt

[N(KabKab −K2 − 3R)− 2Na(K

ab −Khab)|b]√

h d3y

− 2

∮

St

[N(k − k0)−Na(K

ab −Khab)rb

]√σ d2θ. (4.2.34)

It is understood that here, Kab stands for the function of hab and pab

defined by Eq. (4.2.32); this is given explicitly by

√hKab = 16π

(pab − 1

2phab

), (4.2.35)

where p ≡ hab pab.

4.2.7 Variation of the Hamiltonian

The equations of motion for the gravitational field are obtained byvarying the action of Eq. (4.2.28) with respect to N , Na, hab, andpab, which are all treated as independent variables. The variation isrestricted by the conditions

δN = δNa = δhab = 0 on St, (4.2.36)

but there is no requirement that δpab vanish on the boundary. As a pre-liminary step toward calculating δSG, we shall now carry out the vari-ation of HG. The computations presented here are rather formidable;the punch line is delivered in Eq. (4.2.46).

4.2 Hamiltonian formulation 137

We begin with a variation with respect to both N and Na. TakingEq. (4.2.36) into account, Eq. (4.2.34) gives

16π δNHG =

∫

Σt

(−C δN − 2 Ca δNa)√

h d3y, (4.2.37)

where

C ≡ 3R + K2 −KabKab, Ca ≡ (K ba −Kδ b

a )|b. (4.2.38)

This was easy; the remaining variations will require considerably morelabour.

To carry out a variation with respect to hab or pab, we must expressHG in terms of these variables, instead of hab and Kab as was done inEq. (4.2.34). Using Eq. (4.2.35), a few steps of algebra give

16π HG = HΣ + HS, (4.2.39)

where

HΣ =

∫

Σt

[Nh−1/2(pabpab − 1

2p2)−Nh1/2 3R− 2Nah

1/2(h−1/2 pab)|b]d3y

(4.2.40)is the “bulk” term, while

HS = −2

∮

St

[N(k − k0)−Nah

−1/2 pabrb

]√σ d2θ (4.2.41)

is the “boundary” term. We have introduced the notation HSigma ≡16π HΣ, pab ≡ 16π pab, and so on; this usage was anticipated in Eqs. (4.2.38).

We first vary HG with respect to pab. From Eq. (4.2.40) we have

δpHΣ =

∫

Σt

Nh−1/2δp(pabpab− 1

2p2) d3y−2 δp

∫

Σt

Na(h−1/2 pab)|bh

1/2 d3y.

We substitute

δp(pabpab − 1

2p2) = 2(pab − 1

2p hab)δp

ab

inside the first integral, and we integrate the second by parts. Thisgives

δpHΣ =

∫

Σt

2[Nh−1/2(pab − 1

2p hab) + N(a|b)

]δpab d3y

− 2

∮

St

Nah−1/2δpabrb

√σ d2θ.

The boundary term is precisely equal to (minus) the variation of HS.We therefore have obtained

δpHG =

∫

Σt

Hab δpab d3y, (4.2.42)

138Lagrangian and Hamiltonian formulations of general

relativity

where

Hab = 2Nh−1/2(pab − 1

2p hab

)+ 2N(a|b). (4.2.43)

To vary HG with respect to hab is more labourious, and we will relyon computations already presented in Sec. 4.1. We begin with the bulkterm:

δhHΣ =

∫

Σt

[−Nh−1(pabpab − 1

2p2)δhh

1/2 + Nh−1/2δh(pabpab − 1

2p2)

−Nδh(h1/2 3R)

]d3y − 2δh

∮

St

Nah−1/2 pabrb

√σ d2θ

+ 2δh

∫

Σt

Na|b pab d3y,

in which the last term on the right-hand side of Eq. (4.2.40) was inte-grated by parts. The variation of the integral over St vanishes becausehab is fixed on the boundary. In the first term within the integral overΣt we substitute

δhh1/2 =

1

2h1/2habδhab,

while in the second term,

δh(pabpab − 1

2p2) = 2(pa

c pcb − 12p pab)δhab.

In the third term, we use the three-dimensional version of Eq. (4.1.16),

δhh1/2 3R = −h1/2Gabδhab + h1/2 δ−vc

|c,

where Gab = Rab − 12

3R hab is the three-dimensional Einstein tensor,and δ−vc = habδΓc

ab − hacδΓbab. Finally, in the last term we substitute

δhNa|b = N c|bδhac + hacN

dδΓcbd.

After a few steps of algebra, we obtain

δhHΣ =

∫

Σt

[−1

2Nh−1/2(pcdpcd − 1

2p2)hab + 2Nh−1/2(pa

c pbc − 12p pab)

+ Nh1/2Gab + 2pc(aNb)|c

]δhab d3y

+

∫

Σt

[−Nh1/2 δ−vc|c + 2pb

cNdδΓc

bd

]d3y.

We now leave the first integral alone, and set to work on the secondintegral, beginning with the first term. After integrating by parts,

−∫

Σt

Nh1/2δ−vc|c d3y =

∫

Σt

N,cδ−vch1/2 d3y −

∮

St

Nδ−vcrc

√σ d2θ

=

∫

Σt

N,cδ−vch1/2 d3y +

∮

St

Nhabδhab,crc√

σ d2θ,

4.2 Hamiltonian formulation 139

where the three-dimensional version of Eq. (4.1.17) was used. To ex-press the first integral in terms of δhab, we use the relation

δΓcab =

1

2hcd[(δhda)|b + (δhdb)|a − (δhab)|d],

which is easy to establish. (Note that the covariant derivative is definedwith respect to the reference metric hab, about which the variation istaken.) We have

δ−vc =1

2(habhcd − hachbd)[(δhda)|b + (δhdb)|a − (δhab)|d]

and then

N,cδ−vc =

1

2(habN ,d −N ,ahbd)[(δhda)|b + (δhdb)|a − (δhab)|d]

= −(habN ,d −N ,ahbd)(δhab)|d;

the second line follows by virtue of the antisymmetry in a and d of thefirst factor. After another integration by parts we obtain

−∫

Σt

Nh1/2δ−vc|c d3y =

∫

Σt

(habN|d

d −N |ab)δhabh1/2 d3y

+

∮

St

Nhabδhab,crc√

σ d2θ,

where we have used the fact that δhab vanishes on St. All this takescare of the first term inside the second integral for δhHΣ. We now turnto the second term inside the same integral. We have

∫

Σt

2pbcN

dδΓcbd d3y =

∫

Σt

pabNd[(δhab)|d + (δhad)|b − (δhbd)|a] d3y

=

∫

Σt

h−1/2 pabNd(δhab)|dh1/2 d3y

= −∫

Σt

(h−1/2 pabNd)|d δhabh1/2 d3y,

where we have integrated by parts and put δhab = 0 on St.Gathering the results, we find that the variation of the bulk term is

δhHΣ =

∫

Σt

Pabδhab d3y +

∮

St

Nhabδhab,crc√

σ d2θ,

where Pab will be written in full below. On the other hand, variationof the boundary term gives

δhHS = −2

∮

St

Nδk√

σ d2θ,

140Lagrangian and Hamiltonian formulations of general

relativity

and δk = 12habδhab,cr

c is the three-dimensional analogue of a resultpreviously derived in Sec. 4.1.5. Thus,

δhHS = −∮

St

Nhabδhab,crc√−σ d2θ,

and this cancels out the boundary integral in δhHΣ. The variation ofthe full Hamiltonian is therefore

δhHG =

∫

Σt

Pabδhab d3y, (4.2.44)

where

Pab = Nh1/2Gab − 12Nh−1/2(pcdpcd − 1

2p2)hab + 2Nh−1/2(pa

c pbc − 12p pab)

− h1/2(N |ab − habN |cc)− h1/2(h−1/2 pabN c)|c + 2pc(aN

b)|c. (4.2.45)

Here, as before, Gab = Rab − 12

3R hab is the three-dimensional Einsteintensor.

Combining Eqs. (4.2.37), (4.2.42), and (4.2.44), we find that thecomplete variation of the gravitational Hamiltonian, under the condi-tions of Eq. (4.2.36), is given by

δHG =

∫

Σt

(Pab δhab +Hab δpab − C δN − 2 Ca δNa

)d3y, (4.2.46)

where Pab ≡ Pab/(16π) is given by Eq. (4.2.45), Hab by Eq. (4.2.43),while C ≡ C/(16π) and Ca ≡ Ca/(16π) are given by Eq. (4.2.38).

4.2.8 Hamilton’s equations

The equations of motion are obtained by varying the gravitational ac-tion, expressed as

SG =

∫ t2

t1

dt[∫

Σt

pab hab d3y −HG

],

with respect to the independent variables N , Na, hab, and pab. Variationyields

δSG =

∫ t2

t1

dt[∫

Σt

(pab δhab + hab δpab) d3y − δHG

],

where δHG is given by Eq. (4.2.46). After integrating the first term byparts, we obtain

δSG =

∫ t2

t1

dt

∫

Σt

[(hab−Hab) δpab−(pab+Pab) δhab+C δN+2 Ca δNa

]d3y.

(4.2.47)

4.2 Hamiltonian formulation 141

Demanding that the action be stationary implies

hab = Hab, pab = −Pab, C = 0, Ca = 0. (4.2.48)

These are the vacuum Einstein field equations in Hamilton form. Thefirst two govern the evolution of the conjugate variables hab and pab; itis easy to check that hab = Hab just reproduces the relation betweenhab and pab implied by Eqs. (4.2.30) and (4.2.35). The last two arethe constraints equations first derived in Sec. 3.6; the relations C = 0and Ca = 0 are usually referred to as the Hamiltonian and momentumconstraints of general relativity, respectively.

The Hamiltonian formulation of general relativity suggests the fol-lowing strategy for solving the Einstein field equations. First, selecta foliation of spacetime into spacelike hypersurfaces by specifying thelapse N and the shift Na; the choice of foliation is completely arbitrary.Defining hab to be the induced metric on the spacelike hypersurfaces,the full spacetime metric is given by Eq. (4.2.10):

ds2 = −N2dt2 + hab(dya + Na dt)(dyb + N b dt). (4.2.49)

Next, choose initial values for the tensor fields hab and Kab, where Kab isthe extrinsic curvature of the spacelike hypersurfaces. This choice is notentirely arbitrary, because the constraint equations must be satisfied:the initial values must be solutions to

3R + K2 −KabKab = 0, (Kab −Khab)|b = 0, (4.2.50)

where 3R is the Ricci scalar associated with hab, and K = habKab.Finally, evolve these initial values using Hamilton’s equations, hab =Hab and pab = −Pab, which may be written in the form (Sec. 4.5,Problem 4)

hab = 2NKab + £Nhab (4.2.51)

and

Kab = N|ab −N(Rab + KKab − 2KcaKbc) + £NKab. (4.2.52)

In these equations, the Lie derivatives are directed along Na, the shiftvector. This formulation of the field equations, usually referred to astheir 3 + 1 decomposition, is the usual starting point of numerical rel-ativity.

4.2.9 Value of the Hamiltonian for solutions

We now return to Eq. (4.2.34) and ask: What is the value of the gravita-tional Hamiltonian when the fields hab and Kab satisfy the vacuum field

142Lagrangian and Hamiltonian formulations of general

relativity

equations (4.2.50)–(4.2.52)? The answer is that by virtue of the con-straint equations, only the boundary term contributes to the solution-valued Hamiltonian:

HsolutionG = − 1

8π

∮

St

[N(k − k0)−Na(K

ab −Khab)rb

]√σ d2θ. (4.2.53)

As was discussed previously, this boundary term is relevant only whenthe spacetime manifold is noncompact. For compact manifolds, Hsolution

G ≡0. The physical significance of Hsolution

G for asymptotically-flat space-times will be examined in the next section.

4.3 Mass and angular momentum

4.3.1 Hamiltonian definitions

It is natural to expect that the gravitational mass of an asymptotically-flat spacetime — its total energy — should be related to the value ofthe gravitational Hamiltonian for this spacetime. We will explore thisrelation in this section, and motivate another between the Hamiltonianand the spacetime’s total angular momentum.

The solution-valued Hamiltonian, HsolutionG given by Eq. (4.2.53),

depends on the asymptotic behaviour of the spacelike hypersurface Σt,and on the asymptotic behaviour of the lapse and shift. While the lapseand shift are always arbitrary, the fact that the spacetime is asymp-totically flat gives us a preferred behaviour for the hypersurfaces. Weshall demand that Σt asymptotically coincide with a surface of con-stant time in Minkowski spacetime: If (t, x, y, z) is a Lorentzian frameat infinity, then the asymptotic portion of Σt must coincide with a sur-face t = constant. In this portion of Σt, the (arbitrary) coordinatesya are related to the spatial Minkowski coordinates, and we have theasymptotic relations ya → ya(x, y, z); similarly, xα → xα(t, x, y, z).We note that t is proper time for an observer at rest in the asymp-totic region, and infer that this observer moves with a four-velocityuα = ∂xα/∂t. Because this vector is normalized and orthogonal to thesurfaces t = constant, it must coincide with the normal vector nα, andwe have another asymptotic relation, nα → ∂xα/∂t. Substituting thisinto Eq. (4.2.9) gives us

tα → N(∂xα

∂t

)ya

+ Na(∂xα

∂ya

)t,

an asymptotic relation for the flow vector. This shows that once theasymptotic behaviour of Σt has been specified, there is a one-to-onecorrespondence between a choice of lapse and shift and a choice of flowvector. The solution-valued Hamiltonian can then be regarded eitheras a function of N and Na, or as a function of tα.

4.3 Mass and angular momentum 143

We shall define M , the gravitational mass of an asymptotically-flatspacetime, to be the limit of Hsolution

G when St is a two-sphere at spatialinfinity, evaluated with the following choice of lapse and shift: N = 1and Na = 0. From Eq. (4.2.53),

M = − 1

8πlim

St→∞

∮

St

(k − k0)√

σ d2θ. (4.3.1)

Here, σAB is the metric on St, k = σABkAB is the extrinsic curvature ofSt embedded in Σt, and k0 is the extrinsic curvature of St embedded inflat space. The quantity defined by Eq. (4.3.1) is called the ADM massof the asymptotically-flat spacetime; the name refers to the seminalwork by Arnowitt, Deser, and Misner.

The choice N = 1, Na = 0 implies that asymptotically, tα →∂xα/∂t, so that the flow vector generates an asymptotic time trans-lation. The ADM mass is then just the gravitational Hamiltonianfor this choice of flow vector, and we have made a formal connec-tion between total energy and time translations. This connection isboth deep and compelling, and it can be adapted to give a definitionof total angular momentum. Indeed, the gravitational Hamiltonianshould provide a similar connection between angular momentum andasymptotic rotations, which are characterized by tα → φα ≡ ∂xα/∂φ,where φ is a rotation angle defined in the asymptotic region in termsof the Cartesian frame (x, y, z). This corresponds to the choice N = 0,Na = φa ≡ ∂ya/∂φ of lapse and shift.

We shall define J , the angular momentum of an asymptotically-flatspacetime, to be (minus) the limit of Hsolution

G when St is a two-sphere atspatial infinity, evaluated with N = 0 and Na = φa. From Eq. (4.2.53),

J = − 1

8πlim

St→∞

∮

St

(Kab −Khab)φarb√

σ d2θ. (4.3.2)

A minus sign was inserted to recover the usual right-hand rule for theangular momentum. Notice that this definition of angular momentumrefers to a specific choice of rotation axis, and φ is the angle aroundthis axis.

4.3.2 Mass and angular momentum for stationary,axially symmetric spacetimes

To show that these definitions are in fact reasonable, we shall calculateM and J for an asymptotically-flat spacetime that is both stationaryand axially symmetric. In the asymptotic region r À m, the metric ofsuch a spacetime can be expressed as

ds2 = −(1−2m

r

)dt2+

(1+

2m

r

)(dr2+r2 dΩ2)−4j sin2 θ

rdtdφ, (4.3.3)

144Lagrangian and Hamiltonian formulations of general

relativity

where m and j are the spacetime’s mass and angular-momentum pa-rameters, respectively. We will show that M = m and J = j, andconfirm that the Hamiltonian definitions are well founded. We notethat the validity of this metric in the asymptotic region could alwaysbe used to define mass and angular momentum. Our Hamiltonian def-initions are more powerful, however, because they do not involve aparticular coordinate system, and they stay meaningful even when thespacetime is not stationary nor axially symmetric.

We choose the hypersurfaces Σt to be surfaces of constant t, andnα = −(1−m/r)∂αt is the unit normal. (Throughout this calculationwe work consistently to first order in m/r.) The induced metric on Σt

is given by

habdyadyb =(1 +

2m

r

)(dr2 + r2 dΩ2).

The boundary St is the two-sphere r = R, and the limit R →∞ will betaken at the end of the calculation. Its unit normal is ra = (1+m/r)∂ar,and

σABdθAdθB =(1 +

2m

R

)R2 dΩ2

gives the two-metric on St.To evaluate M we must first calculate k. This is given by k = ra

|aand a brief calculation yields k = 2(1 − 2m/R)/R. To this we mustsubtract k0, the extrinsic curvature of a two-surface of identical intrinsicgeometry, but embedded in flat space. On this surface,

σ0ABdθAdθB = R′2 dΩ2,

where R′ ≡ R(1 + m/R), so that σ0AB = σAB. We have k0 = 2/R′ =

2(1 −m/R)/R, and simple algebra yields k − k0 = −2m/R2. On theother hand,

√σ d2θ = R2(1 + 2m/R) sin θ dθdφ, and substitution into

Eq. (4.3.1) yieldsM = m, (4.3.4)

the desired result.To evaluate J we must first calculate Kabφ

arb = Kφr(1 − m/r),

where Kab = nα;β eαaeβ

b . (The second term in the integrand, Khabφarb,

vanishes identically.) The relevant component of the extrinsic curvatureis Kφr = (1−m/r)Γt

φr. Using

gtt = −(1 +

2m

r

), gtφ = −2j

r3,

we find that Γtφr = −3j sin2 θ/r2, and this gives Kφr = −3j sin2 θ/R2.

Substituting this into Eq. (4.3.2) yields J = (3j/4)∫

sin3 θ dθ, or

J = j, (4.3.5)

the desired result.

4.3 Mass and angular momentum 145

4.3.3 Komar formulae

An appealing feature of the Hamiltonian definitions for mass and an-gular momentum is that they do not involve a specific choice of coordi-nates. Alternative definitions that share this property can be producedfor stationary and axially symmetric spacetimes. These are known asthe Komar formulae, and they are

M = − 1

8πlim

St→∞

∮

St

∇αξβ(t) dSαβ (4.3.6)

and

J =1

16πlim

St→∞

∮

St

∇αξβ(φ) dSαβ. (4.3.7)

Here, ξα(t) is the spacetime’s timelike Killing vector, and ξα

(φ) is the rota-tional Killing vector; they both satisfy Killing’s equation, ξα;β+ξβ;α = 0.The surface element is given by (Sec. 3.2)

dSαβ = −2n[αrβ]

√σ d2θ, (4.3.8)

where nα and rα are the timelike and spacelike normals to St, respec-tively.

To establish that these formulae do indeed give M = m and J = j,we must prove that for the spacetime of Eq. (4.3.3), the relations

−2∇αξβ(t)nαrβ = −2m/r2 = k − k0, ∇αξβ

(φ)nαrβ = Kabφarb

hold in the limit r →∞. We begin with the first relation:

−2∇αξβ(t)nαrβ = 2ξα

(t);βnαrβ

= 2Γαβγnαrβξγ

(t)

= −2(1− 2m/r)Γtrt

= −2m/r2,

as required. We have used Killing’s equation in the first line, andinserted nα = −(1 − m/r)∂αt, rα = raeα

a = (1 − m/r)∂xα/∂r, andΓt

rt = m(1 + 2m/r)/r2 in the following steps. To prove the secondrelation requires less work:

∇αξβ(φ)nαrβ = −ξα

(φ);βnαrβ

= ξα(φ)nα;βrβ

= nα;β(φa eαa )(rb eβ

b )

= (nα;β eαaeβ

b )φarb

= Kabφarb.

146Lagrangian and Hamiltonian formulations of general

relativity

Once again, Killing’s equation was used in the first line. The secondline follows from the fact that the Killing vector is orthogonal to nα. Inthe third line, the vectors ξα

(φ) and rβ were decomposed into the basis eαa .

Finally, the last line follows from the definition of the extrinsic curva-ture. These computations prove that the definitions of Eq. (4.3.6) and(4.3.7) do indeed imply M = m and J = j. We see that for stationaryand axially symmetric spacetimes, the Komar formulae are equivalentto our Hamiltonian definitions for mass and angular momentum.

The Komar formulae can be turned into hypersurface integrals byinvoking Stokes’ theorem (Sec. 3.3),∮

S

Bαβ dSαβ = 2

∫

Σ

Bαβ;β dΣα,

where Bαβ is any antisymmetric tensor field, and S is the two-dimensionalboundary of the hypersurface Σ. This is possible because when ξα isa Killing vector, the tensor Bαβ = ∇αξβ is necessarily antisymmetric.We have

Bαβ;β = (∇αξβ);β = −(∇βξα);β = −¤ξα,

where ¤ ≡ ∇α∇α. Using the fact that all Killing vectors satisfy ¤ξα =−Rα

βξβ (Sec. 1.13, Problem 9), we have established the identity∮

S

∇αξβ dSαβ = 2

∫

Σ

Rαβξβ dΣα.

Because the hypersurface Σ is spacelike, we have that dΣα = −nα

√h d3y.

Using the Einstein field equations, we then obtain∮

S

∇αξβ dSαβ = −16π

∫

Σ

(Tαβ − 1

2Tgαβ

)nαξβ

√h d3y.

Finally, combining this with Eqs. (4.3.6) and (4.3.7), we arrive at

M = 2

∫

Σ

(Tαβ − 1

2Tgαβ

)nαξβ

(t)

√h d3y (4.3.9)

and

J = −∫

Σ

(Tαβ − 1

2Tgαβ

)nαξβ

(φ)

√h d3y. (4.3.10)

In these equations, Σ stands for any spacelike hypersurface that extendsto spatial infinity. If Σ had two boundaries instead of just one, thenan additional contribution from the inner boundary would appear onthe right-hand side of Eqs. (4.3.9) and (4.3.10). Such a situation ariseswhen the stationary, axially symmetric spacetime contains a black hole(see Sec. 5.5.3).

It is a remarkable fact that M and J are defined fundamentally interms of integrals over a closed two-surface at infinity. These quantitiesshould therefore be thought of as properties of the asymptotic structureof spacetime. It is only in the case of stationary, axially symmetricspacetimes that M and J can be defined as hypersurface integrals.

4.3 Mass and angular momentum 147

4.3.4 Bondi-Sachs mass

The ADM mass was constructed in Sec. 4.3.1 by selecting a closed two-surface St ≡ S(t, r), integrating k−k0 over this surface, and then takingthe limit r →∞. Thus,

MADM(t) = − 1

8π

∮

S(t,r→∞)

(k − k0)√

σ d2θ. (4.3.11)

Here, S(t, r) denotes a surface of constant t and r which becomes around two-sphere of area 4πr2 as r → ∞. This limit, which is takenwhile keeping t fixed, is what defines “spatial infinity”.

There exists another way of reaching infinity, and to this new limit-ing procedure corresponds a distinct notion of mass. This is the Bondi-Sachs mass, which is obtained by taking S(t, r) to “null infinity” in-stead of spatial infinity. To define this we introduce the null coordinatesu = t− r (retarded time) and v = t+ r (advanced time). In these coor-dinates, a two-surface of constant t and r becomes a surface of constantu and v, which we denote S(u, v). Null infinity corresponds to the limitv →∞ keeping u fixed, and the Bondi-Sachs mass is defined by

MBS(u) = − 1

8π

∮

S(u,v→∞)

(k − k0)√

σ d2θ. (4.3.12)

The physical importance of the Bondi-Sachs mass comes from the factthat when an isolated object emits radiation (in the form, say, of elec-tromagnetic or gravitational waves), the rate of change of MBS(u) isdirectly related to the outward flux of radiated energy. If F denotesthis flux, then the Bondi-Sachs mass satisfies

dMBS

du= −

∮

S(u,v→∞)

F√

σ d2θ. (4.3.13)

Thus, the mass of a radiating object decreases as the radiation escapesto infinity. The proof of this statement is rather involved; it can befound in the original papers by Bondi, Sachs, and their collaborators.

4.3.5 Distinction between ADM and Bondi-Sachsmasses: Vaidya spacetime

For stationary spacetimes, the ADM and Bondi-Sachs masses are iden-tical: there is no distinction. For the dynamical spacetime of an isolatedbody emitting gravitational (or other types of) radiation, the two no-tions of mass are distinct. For such a system, the Bondi-Sachs massdecreases according to Eq. (4.3.13), while the ADM mass stays con-stant.

148Lagrangian and Hamiltonian formulations of general

relativity

The metric of a radiating spacetime is difficult to write down; usu-ally it is expressed as a messy expansion in powers of 1/r. We shallnot attempt to deal with these complications here. For the purpose ofillustrating the difference between the ADM and Bondi-Sachs masses,we shall instead adopt a simple spherically-symmetric model. Con-sider the Schwarzschild metric expressed in terms of the null coordi-nate u = t − r − 2M ln(r/2M − 1), and allow the mass parameter Mto become a function of retarded time: M → m(u). This new metricis given by

ds2 = −f du2 − 2 dudr + r2 dΩ2, f = 1− 2m(u)/r, (4.3.14)

and it is a good candidate to represent a radiating spacetime. To seeif it makes a sensible solution to the Einstein field equations, let usexamine the Einstein tensor, whose only nonvanishing component isGuu = −(2/r2)(dm/du). This means that the stress-energy tensor mustbe of the form

Tαβ = −dm/du

4πr2lαlβ, (4.3.15)

where lα = −∂αu is tangent to radial, outgoing null geodesics. Thisstress-energy tensor describes a pressureless fluid with energy densityρ = (−dm/du)/(4πr2) moving with a four-velocity lα. Such a fluidis usually referred to as null dust; it gives a good description of high-frequency radiation. It is easy to check that the form (function of u)/r2

for the energy density is dictated by energy-momentum conservation.You may also verify that all the standard energy conditions are satisfiedby Tαβ if dm/du < 0, that is, if m decreases with increasing retardedtime. We conclude that the metric of Eq. (4.3.14), called the outgoingVaidya metric, makes a physically reasonable solution to the Einsteinfield equations.

We wish to compute the ADM and Bondi-Sachs masses for theVaidya spacetime. The first step is to select a spacelike hypersurface Σbounded by a closed two-surface S; this hypersurface must asymptot-ically coincide with a surface t = constant of Minkowski spacetime. Asuitable choice is to let Σ be a surface of constant t ≡ u + r, for whichthe unit normal

nα = −(2− f)−1/2 ∂α(u + r)

is everywhere timelike. From Eq. (4.3.14) we obtain that the inducedmetric on Σ is

hab dyadyb = (2− f) dr2 + r2 dΩ2.

For S we choose the two-sphere r = R, where R is a constant muchlarger than the maximum value of 2m(u); eventually we will take thelimit R → ∞. Recall that spatial infinity corresponds to keeping t

4.3 Mass and angular momentum 149

fixed while taking the limit (which means that u → −∞), whereasnull infinity corresponds to keeping u fixed while taking the limit. Theunit normal on S is ra = (2 − f)1/2 ∂ar, and the induced metric isσAB dθAdθB = R2 dΩ2.

First we calculate

M(S) ≡ − 1

8π

∮

S

(k − k0)√

σ d2θ

for the bounded two-surface S; the two different limits to infinity willbe taken next. The extrinsic curvature of S embedded in Σ is calculatedas

k = ra|a =

2

R

[1 +

2m(u)

R

]−1/2

=2

R

[1− m(u)

R+ O(R−2)

],

and the extrinsic curvature of S embedded in flat space is k0 = 2/R.Subtracting, we have that k − k0 = −2m(u)/R2 + O(R−3), and inte-grating over S yields M(S) = m(u) + O(R−1).

We may now take the limit R →∞. As was mentioned previously,the ADM mass is obtained by keeping t = u + R fixed while taking thelimit. This gives

MADM(t) = m(−∞), (4.3.16)

and we see that the ADM mass is a constant, equal to the initial valueof the mass function. We may therefore say that MADM represents allthe mass initially present in the spacetime. (This interpretation is quitegeneral and not limited to this specific example.) For the Bondi-Sachsmass, we must keep u fixed while taking the limit. This gives

MBS(u) = m(u), (4.3.17)

and we see that the Bondi-Sachs mass is identified with the mass func-tion of the Vaidya spacetime. It decreases in response to the outflow ofradiation described by the stress-energy tensor of Eq. (4.3.15). Noticethat the field equation

dm

du= −4πr2Tuu = −4πr2(−T r

u) ≡ −4πr2F

is compatible with the general mass-loss formula, Eq. (4.3.13).It may appear paradoxical that the ADM mass of a dynamical space-

time should be a constant. This, however, is what should be expectedof a radiating spacetime (Fig. 4.5). The ADM mass represents all themass present on a spacelike hypersurface of constant t. This hyper-surface intersects the central object whose mass does decrease as aconsequence of radiation loss. But this does not mean that the ADMmass should decrease, because the hypersurface intersects also the ra-diation, and the ADM mass accounts for both forms of energy. The

150Lagrangian and Hamiltonian formulations of general

relativity

Figure 4.5: A radiating spacetime.

radiating mass

t = constant

u = constant

net result is a conserved quantity. On the other hand, the Bondi-Sachsmass represents all the mass present on a null hypersurface of constantu. Because this hypersurface fails to intersect any of the radiation thatwas emitted prior to the retarded time u, the net result is a quantitythat decreases with increasing retarded time.

4.3.6 Transfer of mass and angular momentum

We shall now derive expressions for the transfer of mass and angularmomentum across a hypersurface Σ in a stationary, axially symmetricspacetime.

To begin, consider the vector fields

εα = −T αβ ξβ

(t), `α = Tαβ ξβ

(φ), (4.3.18)

where T αβ is a test stress-energy tensor that does not influence thespacetime geometry. From the definition of the stress-energy tensor, εα

can be interpreted as an energy-density flux vector, while `α is inter-preted as an angular-momentum-density flux vector.

To see this clearly, consider the simple case of dust, a perfect fluidwith stress-energy tensor Tαβ = ρuαuβ, where ρ is the rest-mass den-sity, and uα the four-velocity. Energy-momentum conservation impliesthat uα satisfies the geodesic equation, and that jα = ρuα is a con-served vector: jα

;α = 0. This vector can be interpreted as the dust’smomentum density, or equivalently, as a rest-mass flux vector. Thenεα = Ejα and `α = Ljα, where E ≡ −uαξα

(t) is the conserved energy

per unit rest mass, and L ≡ uαξα(φ) the conserved angular momentum

per unit rest mass. (As we have indicated, both E and L are constantsof the motion.) These relations show quite clearly that εα represents aflux of energy density, while `α is a flux of angular-momentum density.

The vectors εα and `α are divergence-free. For example,

εα;α = −T αβ

;α ξ(t)β + Tαβ ξ(t)β;α = 0;

4.4 Bibliographical notes 151

the first term vanishes by virtue of energy-momentum conservation,and the second vanishes because ξ(t)β;α is an antisymmetric tensor field.This implies that the integral of εα or `α over a hypersurface ∂ enclosinga four-dimensional region is identically zero. For example,

∮

∂

εα dΣα = 0.

This equation states that the total transfer of energy across a closedhypersurface ∂ is zero. This is clearly a statement of conservation oftotal energy — or total mass.

The boundary ∂ can be partitioned into any number of pieces. Ifone such piece is the hypersurface Σ, then the integral of εα over Σrepresents the mass transferred across this piece of ∂. Thus,

∆M = −∫

Σ

Tαβ ξβ

(t) dΣα (4.3.19)

is the mass transferred across the hypersurface Σ, and similarly,

∆J =

∫

Σ

Tαβ ξβ

(φ) dΣα (4.3.20)

is the angular momentum transferred across Σ.For illustration, let us return to our previous example, and let us

choose Σ to be spacelike and orthogonal to the vector field uα. ThendΣα = −uα

√h d3y, and we find that ∆M =

∫Σ

Eρ√

h d3y and ∆J =∫Σ

Lρ√

h d3y. The first equation states that the transfer of energy across

Σ is the integral of Eρ, the energy density. The second equation comeswith a very similar interpretation.

4.4 Bibliographical notes

During the preparation of this chapter I have relied on the followingreferences: Arnowitt, Deser, and Misner (1962); Bondi, van der Burg,and Metzner (1962); Brown and York (1993), Brown, Lau, and York(1997); Carter (1979); Hawking and Horowitz (1996); Sachs (1962);Sudarsky and Wald (1992); and Wald (1984).

More specifically:An overview of the Lagrangian and Hamiltonian formulations of

general relativity is given in Appendix E of Wald. The Hamiltonianformulation was initiated by Arnowitt, Deser, and Misner, who alsointroduced the ADM mass. Early treatments of the Hamiltonian for-mulation often discarded the all-important boundary terms; carefultreatments are given in Sudarsky and Wald, Brown and York, andHawking and Horowitz. (Problem 7 below is based on this last paper.)

152Lagrangian and Hamiltonian formulations of general

relativity

The Hamiltonian definitions for mass and angular momentum are takenfrom Brown and York; the discussion of Sec. 4.2.4 is also based on theirpaper. Sections 4.3.3 and 4.3.5 are based on Carter’s Secs. 6.6.1 and6.6.2, respectively. The Bondi-Sachs mass was introduced by Bondiand his collaborators in an effort to put the notion of gravitational-wave energy on a firm footing. The definition given in Sec. 4.3.4 isdue to Brown, Lau, and York. I would like to point out that the firstoccurrence of the (k − k0) formula for the ADM mass can be found ina 1988 paper by Katz, Lynden-Bell, and Israel.

4.5 Problems

1. The Lagrangian density for the free electromagnetic field is

= − 1

16πF αβFαβ,

where Fαβ = Aβ;α−Aα;β is the Faraday tensor, expressed in termsof the vector potential Aα.

a) Derive the Maxwell field equations for vacuum, Fαβ;β = 0, on

the basis of this Lagrangian density.

b) Show that the stress-energy tensor for the electromagneticfield is given by

Tαβ =1

4π

(FαµF

µβ − 1

4gαβF µνFµν

).

2. The Lagrangian density for a point particle of mass m moving ona world line zα(λ) is given by

= −m

∫ √−gαβ zαzβ δ4(x, z) dλ,

where δ4(x, x′) is a four-dimensional, scalarized δ-function satis-fying ∫

δ4(x, x′)√−g d4x = 1

if x′ is within the domain of integration; we also have zα =dzα/dλ, and the parameterization of the world line is arbitrary.

a) Derive an expression for the stress-energy tensor of a pointparticle. To simplify this expression, set dλ = dτ (with τdenoting proper time on the world line) at the end of thecalculation.

4.5 Problems 153

b) Prove that when it is applied to a point particle, the state-ment Tαβ

;β = 0 gives rise to the geodesic equation for uα =dzα/dτ .

c) Explain whether the result of part b) constitutes a valid proofof the statement that the Einstein field equations predict themotion of a massive body to be geodesic.

3. Calculate the gravitational action SG for a region of Schwarzschildspacetime. Take to be bounded by the hypersurfaces Σt1 , Σt2 , ΣR,and Σρ, where Σt1 (Σt2) is the spacelike hypersurface describedby t = t1 (t = t2), and where ΣR (Σρ) is the three-cylinder atr = R (r = ρ). Here, 2M < ρ < R. At the end of the calculation,take the limits R →∞ and ρ → 2M .

4. Derive Eq. (4.2.52), the evolution equation for the extrinsic cur-vature. You may use pab = −Pab as a starting point, or proceedfrom scratch with the definition Kab = £t(nα;βeα

aeβb ). [Either way,

the calculation is tedious! You may want to consult York (1979).]

5. Recall that in Sec. 3.6.5 we introduced a mass function m(r) thatdetermines the three-metric of a spherically symmetric hypersur-face. Prove that

− 1

8π

∮

S(r)

(k − k0)√

σ d2θ = r(1−

√1− 2m/r

),

where S(r) is a two-surface of constant r. Use this to show thatm(∞) is the ADM mass of this hypersurface.

6. In this problem we explore some consequences of Eq. (4.3.9),which gives an expression for the ADM mass of a stationary space-time.

a) Prove that the right-hand side of Eq. (4.3.9) is independent ofthe choice of hypersurface Σ.

b) Show that if Tαβ is the stress-energy tensor of a static perfectfluid, then Eq. (4.3.9) reduces to

M =

∫

Σ

(ρ + 3p) eΦ√

h d3y,

where ρ is the mass density, p the pressure, and e2Φ ≡−gαβξα

(t)ξβ(t). [Hints: A perfect fluid is static if its four-

velocity uα is parallel to ξα(t). You may assume that the

spacetime also is static.]

154Lagrangian and Hamiltonian formulations of general

relativity

c) Specialize to spherical symmetry, and write the spacetimemetric as

ds2 = −e2Φ dt2 + (1− 2m/r)−1 dr2 + r2 dΩ2,

in which Φ and m are functions of r. Refer back to the resultof Problem 5 and deduce the identity

∫

Σ

ρ(1− 2m/r)1/2 dV =

∫

Σ

(ρ + 3p) eΦ dV,

where dV =√

h d3y is the natural volume element on thehypersurface Σ.

d) Specialize now to a weak-field situation, for which the metriccan be expressed as

ds2 = −(1 + 2Φ) dt2 + (1− 2Φ)(dx2 + dy2 + dz2);

the Newtonian potential Φ is a function of r =√

x2 + y2 + z2.Working consistently in the weak-field approximation, showthat the identity derived in part c) reduces to

1

8π

∫

Σ

|∇Φ|2 dV = 3

∫

Σ

p dV,

in which dV and all vectorial operations refer to the three-dimensional flat space of ordinary vector calculus. The left-hand side represents (minus) the total gravitational poten-tial energy of the system. For a monoatomic ideal gas inthermodynamic equilibrium, the right-hand side representstwice the total kinetic energy of the system. This equationis therefore a formulation of the virial theorem of Newto-nian gravitational physics. The identity of part c) can thenbe interpreted as the general-relativistic version of the virialtheorem.

7. The ADM mass is usually defined by

M =1

16π

∮

S→∞(Dbγab −Daγ)ra

√σ d2θ,

which is a very different expression from the one appearing inSec. 4.3.1. Here, S is the two-surface that encloses the spacelikehypersurface Σ. If hab is the metric on Σ in arbitrary coordinatesya, then γab ≡ hab − h0

ab, where h0ab is the metric of flat space

in the same coordinates. We also have γ ≡ γaa, and Da is the

covariant derivative associated with the flat metric h0ab, which is

4.5 Problems 155

used to raise and lower all indices. Finally, ra is the unit normalof the surface S, and

√σ d2θ is the surface element on S.

The purpose of this problem is to prove that this definition isequivalent to the one given in the text,

M = − 1

8π

∮

S→∞(k − k0)

√σ d2θ,

where k (k0) is the extrinsic curvature of S embedded in Σ (flatspace). You may proceed along the following lines:

Because both expressions are invariant under a coordinate trans-formation, we may use, in a neighbourhood of S, the coordinates(`, θA), where ` is proper distance off S in the direction orthogonalto S, and θA are the coordinates on S which are Lie transportedoff S along curves orthogonal to S. In these coordinates, themetric on Σ is given by

habdyadyb = d`2 + σAB(`) dθAdθB,

where σAB(`) (which also depends on θA) is such that σAB(0) =σAB, the induced metric on S. Similarly,

h0abdyadyb = d`2 + σ0

AB(`) dθAdθB.

Because the induced metrics must agree on S, we also have σ0AB(0) =

σAB. This implies that γab = 0 on S.

Using this information, show that both expressions for M reduceto the same form,

M = − 1

16π

∮

S→∞h0abγab,`

√σ d2θ.

This is sufficient to prove that the two expressions are indeedequivalent.

8. In this problem we study the transport of energy and angularmomentum by a scalar field ψ in flat spacetime. The metric isds2 = −dt2 + dr2 + r2 dΩ2, and the scalar field satisfies the waveequation gαβψ;αβ = −4πρ, where ρ is a specified source. It canbe shown that in the wave zone (where r is much larger than atypical wavelength of the radiation), the field is given by

ψ(t, r, θ, φ) =1

r

∞∑

`=0

∑

m=−`

a`m(u) Y`m(θ, φ) + O(r−2),

156Lagrangian and Hamiltonian formulations of general

relativity

where Y`m(θ, φ) are spherical harmonics; the amplitudes a`m areconstructed from ρ, and they are functions of retarded time u ≡t− r. The scalar field comes with a stress-energy tensor

Tαβ = ψ,αψ,β − 1

2(ψ,µψ,µ)gαβ,

and we are interested in the transfer of energy and angular mo-mentum across a null hypersurface Σ defined by v = constant,where v ≡ t + r is advanced time.

a) Show that dΣα = −r2kα dudΩ, where kα = −12∂αv and dΩ =

sin θ dθdφ, is a surface element on Σ.

b) Prove that for any test field producing a stress-energy tensorT αβ, the amount of energy crossing Σ per unit retarded timeis

dE

du=

∮

S

r2Tαβkαtβ dΩ,

where tα = ∂xα/∂t and S is a two-sphere of constant u andv. Prove also that the amount of angular momentum flowingacross Σ is given by

dJ

du= −

∮

S

r2Tαβkαφβ dΩ,

where φα = ∂xα/∂φ.

c) Show that for scalar radiation, the preceding expressions re-duce to

dE

du=

∞∑

`=0

∑

m=−`

|a`m(u)|2

and

dJ

du=

∞∑

`=0

∑

m=−`

im a`m(u)a∗`m(u)

in the limit v → ∞. Here, an overdot indicates differen-tiation with respect to u, and an asterisk denotes complexconjugation.

d) Suppose that the source producing the scalar radiation is inrigid rotation around the z axis, in the sense that the t andφ dependence of ρ resides entirely in the combination φ−Ωt,where Ω is a constant angular velocity. Prove that in thissituation, the field satisfies

ψ,α ξα = 0,

4.5 Problems 157

where ξα ≡ tα + Ωφα. Prove also that in the limit v → ∞,the transfers of energy and angular momentum are relatedby

dE

du= Ω

dJ

du.

This relation applies to any type of radiation emitted by asource in rigid rotation. It is valid also in curved spacetimes,provided that the spacetime is stationary, axially symmetric,and asymptotically flat.

158Lagrangian and Hamiltonian formulations of general

relativity

Chapter 5Black holes

The final chapter of this book is devoted to one of the most success-ful applications of general relativity, the mathematical theory of blackholes. In the first part of the chapter we explore three exact solu-tions to the Einstein field equations that describe black holes; those arethe Schwarzschild (Sec. 5.1), Reissner-Nordstrom (Sec. 5.2), and Kerr(Sec. 5.3) solutions. In Sec. 5.4 we move away from the specifics ofthose solutions and consider properties of black holes that can be for-mulated quite generally, without relying on the details of a particularmetric. In the final section of this chapter, Sec. 5.5, we present the fourfundamental laws of black-hole mechanics.

The most important feature of a black-hole spacetime is the eventhorizon, a null hypersurface which acts as a causal boundary betweentwo regions of the spacetime, the interior and exterior of the blackhole. Many physical quantities associated with the black hole, such asits mass, angular momentum, and surface area, are defined by integra-tion over the event horizon. The integration techniques introduced inChapter 3 will be put to direct use here, as well as the notions of massand angular momentum encountered in Chapter 4. And since the eventhorizon is generated by a congruence of null geodesics, the methods in-troduced in Chapter 2 will also be part of our discussion. So here it allcomes together in one final glorious moment!

5.1 Schwarzschild black hole

5.1.1 Birkhoff’s theorem

The Schwarzschild metric,

ds2 = −(1− 2M

r

)dt2 +

(1− 2M

r

)−1

dr2 + r2 dΩ2, (5.1.1)

is the unique solution to the Einstein field equations that describesthe vacuum spacetime outside a spherically symmetric body of massM . While this object could have a time-dependent mass distribution,

159

160 Black holes

the external spacetime is necessarily static, and its metric is given byEq. (5.1.1). This statement, known as Birkhoff’s theorem, implies thata spherical mass distribution cannot emit gravitational waves.

The proof of the theorem goes as follows. The metric of a sphericallysymmetric spacetime can always be cast in the form

ds2 = −e2ψf dt2 + f−1 dr2 + r2 dΩ2, (5.1.2)

involving the two arbitrary functions ψ(t, r) and f(t, r). It is convenientto also introduce a mass function m(t, r) defined by

f = 1− 2m

r. (5.1.3)

For the metric of Eq. (5.1.2), the Einstein field equations are

∂m

∂r= 4πr2(−T t

t),∂m

∂t= −4πr2(−T r

t),

(5.1.4)

∂ψ

∂r= 4πrf−1(−T t

t + T rr).

The first two equations motivate the name “mass” for the functionm(t, r), as −T t

t represents the density of mass-energy and −T rt its out-

ward radial flux; they imply that in vacuum, m(t, r) = M , a constant.The third gives ψ′ = 0, and ψ(t, r) can be set equal to zero withoutloss of generality. The Schwarzschild solution is thereby recovered.

5.1.2 Kruskal coordinates

The difficulties of the Schwarzschild metric at r = 2M are well known.While the spacetime is perfectly well behaved there, the coordinates(t, r) become singular at r = 2M — they are no longer in a one-to-one correspondence with spacetime events. This problem can becircumvented by introducing another coordinate system. The followingconstruction originates from the independent work of Kruskal (1960)and Szekeres (1960).

Consider a swarm of massless particles moving radially in the Schwarzschildspacetime — t and r vary, but not θ and φ. It is easy to check that ingo-ing particles move along curves v = constant, while outgoing particlesmove along curves u = constant, where

u = t− r∗, v = t + r∗,

(5.1.5)

r∗ =

∫dr

1− 2M/r= r + 2M ln

∣∣∣ r

2M− 1

∣∣∣.

5.1 Schwarzschild black hole 161

Figure 5.1: Spacetime diagram based on the (u, v) coordinates.

outgoing ray

ingoing ray

u v

In a spacetime diagram using v (advanced time) and u (retarded time)as oblique coordinates (both oriented at 45 degrees), the massless par-ticles propagate at 45 degrees, just as in flat spacetime (Fig. 5.1). Thenull coordinates (u, v) are therefore well suited to the description of(radial) null geodesics. In these coordinates, the Schwarzschild metrictakes the form

ds2 = −(1− 2M/r) dudv + r2 dΩ2. (5.1.6)

Here, r appears no longer as a coordinate, but as the function of uand v defined implicitly by r∗(r) = 1

2(v − u). In these coordinates, the

surface r = 2M appears at v − u = −∞, and it is still the locus of acoordinate singularity.

To see how this coordinate singularity might be eliminated, we focusour attention on a small neighbourhood of the surface r = 2M , in whichthe relation r∗(r) can be approximated by r∗ ' 2M ln |r/2M−1|. Thisimplies that r/2M ' 1 ± er∗/2M = 1 ± e(v−u)/4M , and f ' ±e(v−u)/4M .Here and below, the upper sign refers to the part of the neighbourhoodcorresponding to r > 2M , while the lower sign refers to r < 2M . Themetric (5.1.6) becomes

ds2 ' ∓(e−u/4M du)(ev/4M dv) + r2 dΩ2.

This expression motivates the introduction of a new set of null coordi-nates, U and V , defined by

U = ∓e−u/4M , V = ev/4M . (5.1.7)

It is now clear that when expressed in terms of these coordinates, themetric will be well behaved near r = 2M . Going back to the exactexpression (5.1.5) for r∗, we have that er∗/2M = e(v−u)/4M = ∓UV , or

er/2M( r

2M− 1

)= −UV, (5.1.8)

162 Black holes

which implicitly gives r as a function of UV . You may check that theSchwarzschild metric is now given by

ds2 = −32M3

re−r/2M dUdV + r2 dΩ2. (5.1.9)

This is manifestly regular at r = 2M . The coordinates U and V arecalled null Kruskal coordinates. In a Kruskal diagram (a map of the U -V plane; Fig. 5.2), outgoing light rays move along curves U = constant,while ingoing light rays move along curves V = constant.

In the Kruskal coordinates, a surface of constant r is described byan equation of the form UV = constant, which corresponds to a two-branch hyperbola in the U -V plane. For example, r = 2M becomesUV = 0, while r = 0 becomes UV = 1. There are two copies of eachsurface r = constant in a Kruskal diagram. For example, r = 2M canbe either U = 0 or V = 0. The Kruskal coordinates therefore revealthe existence of a much larger manifold than the portion covered by theoriginal Schwarzschild coordinates. In a Kruskal diagram, this portionis labeled I. The Kruskal coordinates do not only allow the continuationof the metric through r = 2M into region II, they also allow continua-tion into regions III and IV. These additional regions, however, existonly in the maximal extension of the Schwarzschild spacetime. If theblack hole is the result of gravitational collapse, then the Kruskal dia-gram must be cut off at a timelike boundary representing the surfaceof the collapsing object. Regions III and IV then effectively disappearbelow the surface of the collapsing star.

5.1.3 Eddington-Finkelstein coordinates

Because of the implicit nature of the relation between r and UV , theKruskal coordinates can be awkward to use in some computations. Infact, it is rarely necessary to employ coordinates that cover all four

Figure 5.2: Kruskal diagram.

U V

IIIIII

IV

r = 0

r = M

r = M

r = 0

r = 3M r = 3Mr=

2M

r=

2M

5.1 Schwarzschild black hole 163

regions of the Kruskal diagram, although it is often desirable to havecoordinates that are well behaved at r = 2M . In such situations, choos-ing v and r as coordinates, or u and r, does the trick. These coordinatesystems are called ingoing and outgoing Eddington-Finkelstein coordi-nates, respectively.

It is easy to check that in the ingoing coordinates, the Schwarzschildmetric takes the form

ds2 = −(1− 2M/r) dv2 + 2 dvdr + r2 dΩ2, (5.1.10)

while in the outgoing coordinates,

ds2 = −(1− 2M/r) du2 − 2 dudr + r2 dΩ2. (5.1.11)

It may also be verified that the (v, r) coordinates cover regions I andII of the Kruskal diagram, while u and r cover regions IV and I.

The Eddington-Finkelstein coordinates can also be used to con-struct spacetime diagrams (Fig. 5.3), but these do not have the propertythat both ingoing and outgoing null geodesics propagate at 45 degrees:While ingoing light rays move with dv = 0, that is, along coordinatelines that can be oriented at 45 degrees, the outgoing rays move withdv/dr = 2/(1− 2M/r), that is, with a varying slope.

5.1.4 Painleve-Gullstrand coordinates

Another useful set of coordinates for the Schwarzschild spacetime arethe Painleve-Gullstrand coordinates first considered in Sec. 3.13, Prob-lem 1. Here, as with the Eddington-Finkelstein coordinates, the spatialcoordinates (r, θ, φ) are the same as in the original form of the metric,Eq. (5.1.1), but the time coordinate is different: T is proper time asmeasured by a free-falling observer starting from rest at infinity andmoving radially inward.

Figure 5.3: Spacetime diagram based on the (v, r) coordinates.

r = 0 r = 2M

ingoing

r

outgoing

164 Black holes

The four-velocity of such an observer is given by uα ∂α = f−1 ∂t −√1− f ∂r, where f = 1 − 2M/r. From this we deduce that uα =

−∂αT , where the time function T is obtained by integrating dT = dt +f−1

√1− f dr ≡ dτ , where τ is proper time. (Integration is elementary,

and the result appears in Sec. 3.13, Problem 1.) After inserting thisexpression for dt into Eq. (5.1.1), we obtain the Painleve-Gullstrandform of the Schwarzschild metric:

ds2 = −dT 2 + (dr +√

2M/r dT )2 + r2 dΩ. (5.1.12)

The coordinates (T, r, θ, φ) give rise to a metric that is regular at r =2M , in correspondence with the fact that our free-falling observer doesnot consider this surface to be in any way special. Because this observeroriginates in region I of the spacetime (at r = ∞) and ends up in regionII (at r = 0), the new coordinates cover only these two regions of theKruskal diagram. By reversing the motion — letting dr become −dr inEq. (5.1.12) — an alternative coordinate system can be produced thatcovers regions VI and I instead.

From Eq. (5.1.12) we infer a rather striking property of the Painleve-Gullstrand coordinates: the hypersurfaces T = constant are all intrin-sically flat. This can be seen directly from the fact that the inducedmetric on any such hypersurface is given by ds2 = dr2 + r2 dΩ.

5.1.5 Penrose-Carter diagram

The double-null Kruskal coordinates make the causal structure of theSchwarzschild spacetime very clear, and this is their main advantage.Another useful set of double-null coordinates is obtained by applyingthe transformation

U = arctan U, V = arctan V. (5.1.13)

This rescaling of the null coordinates does not affect the appearanceof radial light rays, which still propagate at 45 degrees in a spacetime

Figure 5.4: Compactified coordinates for the Schwarzschild spacetime.

U V

U + V = −π/2

U + V = π/2

V = −π/2

U = π/2

U = −π/2

V = π/2

5.1 Schwarzschild black hole 165

Figure 5.5: Penrose-Carter diagram of the Schwarzschild spacetime.

r = 0

r = 0

i−

i0

i+

i−

i0

i+

−

+

−

+

r=

2M

r=

2M

diagram based on the new coordinates (Fig. 5.4). However, while therange of the initial coordinates was infinite (for example, −∞ < U <∞), it is finite for the new coordinates (for example, −π/2 < U <π/2). The entire spacetime is therefore mapped into a finite domain ofthe U -V plane. This compactification of the manifold introduces badcoordinate singularities at the boundaries of the new coordinate system,but these are of no concern when the purpose is simply to construct acompact map of the entire spacetime.

In the new coordinates, the surfaces r = 2M are located at U = 0and V = 0, and the singularities at r = 0, or UV = 1, are nowat U + V = ±π/2. The spacetime is also bounded by the surfacesU = ±π/2 and V = ±π/2. The four points (U , V ) = (±π

2,±π

2) are

singularities of the coordinate transformation: In the actual spacetime,the surfaces U = 0, U = ∞, and UV = 1 never meet.

It is useful to give names to the various boundaries of the compact-ified spacetime (Fig. 5.5). The surfaces U = π/2 and V = π/2 arecalled future null infinity, and are labeled + (pronounced “scri plus”).The diagram makes it clear that + contains the future endpoints ofall outgoing null geodesics (those along which r increases). Similarly,the surfaces U = −π/2 and V = −π/2 are called past null infinity,and are labeled −. These contain the past endpoints of all ingoing nullgeodesics (those along which r decreases). The points at which + and− meet are called spacelike infinity, and are labeled i0. These containthe endpoints of all spacelike geodesics. The points (U , V ) = (0, π

2)

and (U , V ) = (π2, 0) are called future timelike infinity, and are labeled

i+. These contain the future endpoints of all timelike geodesics thatdo not terminate at r = 0. Finally, the points (U , V ) = (0,−π

2) and

(U , V ) = (−π2, 0) are called past timelike infinity, and are labeled i−.

These contain the past endpoints of all timelike geodesics that do notoriginate at r = 0. Table 5.1 provides a summary of these definitions.

Compactified maps such as the one displayed in Fig. 5.5 are calledPenrose-Carter diagrams. They display, at a glance, the complete

166 Black holes

causal structure of the spacetime under consideration. They make avery useful tool in general relativity.

5.1.6 Event horizon

On a Kruskal diagram (Fig. 5.2), all radial light rays move along curvesU = constant or V = constant. The light cones are therefore orientedat 45 degrees, and timelike world lines, which lie within the light cones,move with a slope larger than unity. The one-way character of the sur-face r = 2M separating regions I and II of the Schwarzschild spacetimeis then quite clear: An observer crossing this surface can never retraceher steps, and cannot elude an encounter with the curvature singularityat r = 0. It is also clear that after crossing r = 2M , the observer canno longer send signals to the outside world, although she may continueto receive them. The surface r = 2M therefore prevents any exter-nal observer from detecting what goes on inside. In this context, itis called the black-hole’s event horizon. The region within the eventhorizon (region II) is called the black-hole region of the Schwarzschildspacetime.

The surface r = 2M that separates regions I and II must be distin-guished from the surface r = 2M that separates regions IV and I. Itis clear that the latter is an event horizon to all observers living insideregion IV (who cannot perceive what goes on in region I). It is alsoa one-way surface, because observers from the outside cannot cross it.To distinguish between the two surfaces r = 2M , it is usual to referto the first as a future horizon, and to the second as a past horizon.The region within the past horizon (region IV) is called the white-holeregion of the Schwarzschild spacetime.

5.1.7 Apparent horizon

Another important property of the surface r = 2M has to do with thebehaviour of outgoing light rays in a neighbourhood of this surface.Here, the term outgoing will refer specifically to those rays which moveon curves U = constant. This is potentially confusing, because the

Table 5.1: Boundaries of the compactified Schwarzschild spacetime.

Label Name Definition

+ Future null infinity v = ∞, u finite− Past null infinity u = −∞, v finitei0 Spatial infinity r = ∞, t finitei+ Future timelike infinity t = ∞, r finitei− Past timelike infinity t = −∞, r finite

5.1 Schwarzschild black hole 167

Figure 5.6: Trapped surfaces and apparent horizon of a spacelikehypersurface.

III

IIIIV

r=

2M

Σ

apparent horizon

trapped surface

radial coordinate r does not necessarily increase along those rays; infact, r increases only if U < 0 (outside the black hole), and it decreasesif U > 0 (inside the black hole). While the term “outgoing” shouldperhaps be reserved to designate rays along which r always increases,this choice of terminology is fairly standard. Similarly, we will usethe term ingoing to designate light rays which move on curves V =constant. If V > 0, then r decreases along the ingoing rays; if V < 0,r increases.

We will show that the expansion of a congruence of outgoing lightrays (as defined above) changes sign at r = 2M . (This should beobvious just from the fact that r increases along the geodesics thatare outside r = 2M , but decreases along geodesics that are inside.)Outgoing light rays have

kα = −∂αU (5.1.14)

as their (affinely parameterized) tangent vector, and their expansionis calculated as θ = kα

;α = |g|−1/2(|g|1/2kα),α. In Kruskal coordinates,

kV = |gUV |−1 is the only nonvanishing component of kα, and |g|1/2 =|gUV |r2 sin2 θ. This gives θ = 2r,V /r|gUV |, and using Eq. (5.1.8) and(5.1.9), we obtain

θ = kα;α = − U

2Mr. (5.1.15)

As was previously claimed, the expansion is positive for U < 0 (in thepast of r = 2M) and negative for U > 0 (in the future of r = 2M).The expansion therefore changes sign at r = 2M , and in this context,this surface is called an apparent horizon. (A similar calculation wouldreveal that for ingoing light rays, the expansion is negative everywherein regions I and II.)

To give a proper definition to the term “apparent horizon”, wemust first introduce the notion of a trapped surface (Fig. 5.6). LetΣ be a spacelike hypersurface. A trapped surface on Σ is a closed,two-dimensional surface S such that for both congruences (ingoing and

168 Black holes

outgoing) of future-directed null geodesics orthogonal to S, the expan-sion θ is negative everywhere on S. (It should be clear that each two-sphere U, V = constant in region II of the Kruskal diagram is a trappedsurface.) Let be the part of Σ that contains trapped surfaces; this isknown as the trapped region of Σ. The boundary of the trapped region,∂, is what is defined to be the apparent horizon of the spacelike hyper-surface Σ. (In Schwarzschild spacetime, this would be any two-sphereat r = 2M .) Notice that the apparent horizon is a marginally trappedsurface: For one congruence of null geodesics orthogonal to ∂, θ = 0.Notice also that the apparent horizon designates a specific two-surfaceS on a given hypersurface Σ. The apparent horizon can generally beextended toward the future (and past) of Σ, because hypersurfaces tothe future (and past) of Σ also contain apparent horizons. The unionof all these apparent horizons forms a three-dimensional surface calledthe trapping horizon of the spacetime. (In Schwarzschild spacetime,this would be the entire hypersurface r = 2M .) In the following wewill not distinguish between the two-dimensional apparent horizon andthe three-dimensional trapping horizon; we will refer to both as theapparent horizon. (This sloppiness of language is fairly standard.)

5.1.8 Distinction between event and apparent horizons:Vaidya spacetime

The event and apparent horizons of the Schwarzschild spacetime coin-cide, and it may not be clear why the two concepts need to be distin-guished. This coincidence, however, is a consequence of the fact thatthe spacetime is stationary; for more general black-hole spacetimes, theevent and apparent horizons are distinct hypersurfaces. To illustratethis we introduce a simple, non-stationary black-hole spacetime.

We express the Schwarzschild metric in terms of ingoing Eddington-Finkelstein coordinates,

ds2 = −f dv2 + 2 dvdr + r2 dΩ2, (5.1.16)

and we allow the mass function to depend on advanced time v:

f = 1− 2m(v)

r. (5.1.17)

This gives the ingoing Vaidya metric, a solution to the Einstein fieldequations with stress-energy tensor

Tαβ =dm/dv

4πr2lαlβ, (5.1.18)

where lα = −∂αv is tangent to ingoing null geodesics. This stress-energy tensor describes null dust, a pressureless fluid with energy den-sity (dm/dv)/(4πr2) and four-velocity lα. (A similar, outgoing Vaidya

5.1 Schwarzschild black hole 169

solution was considered in Sec. 4.3.5. A notable difference betweenthese solutions is that here, the mass function must increase for Tαβ tosatisfy the standard energy conditions.)

Consider the following situation. A black hole, initially of mass m1,is irradiated (with ingoing null dust) during a finite interval of advancedtime (between v1 and v2) so that its mass increases to m2. Such aspacetime is described by the Vaidya metric, with a mass function givenby

m(v) =

m1 v ≤ v1

m12(v) v1 < v < v2

m2 v ≥ v2

,

where m12(v) increases smoothly from m1 to m2. We would like to de-termine the physical significance of the surfaces r = 2m1, r = 2m12(v),and r = 2m2, and find the precise location of the event horizon.

It should be clear that r = 2m1 and r = 2m2 describe the apparenthorizon when v ≤ v1 and v ≥ v2, respectively. More generally, wewill show that the apparent horizon of the Vaidya spacetime is alwayslocated at r = 2m(v).

The null vector field kα dxα = −f dv + 2 dr is tangent to a con-gruence of outgoing null geodesics. It does not, however, satisfy thegeodesic equation in affine-parameter form: As a brief calculation re-veals, kα;βkβ = κ kα, where κ = 2m(v)/r2. To calculate the expansionof the outgoing null geodesics, we need to introduce an affine parameterλ∗ and a rescaled tangent vector kα

∗ = dxα/dλ∗. (The calculation canalso be handled via the results of Sec. 2.6, Problem 8.) As was shownin Sec. 1.3, the desired relation between these vectors is kα

∗ = e−Γkα,where dΓ/dλ = κ(λ), with λ denoting the original parameter. We have

θ = kα∗;α

= e−Γ(kα;α − Γ,αkα)

= e−Γ(kα

;α −dΓ

dλ

)

= e−Γ(kα;α − κ).

Here, the factor kα;α−κ is the congruence’s expansion when measured in

terms of the initial parameter λ — it is equal to (δA)−1d(δA)/dλ, whereδA is the congruence’s cross-sectional area. The factor e−Γ converts itto (δA)−1d(δA)/dλ∗, and this operation does not affect the sign of θ.A simple computation gives kα

;α = 2(r −m)/r2, and we arrive at

eΓθ =2

r2[r − 2m(v)].

So θ = 0 on the surface r = 2m(v), and we conclude that the apparenthorizon begins at r = 2m1 for v ≤ v1, follows r = 2m12(v) in theinterval v1 < v < v2, and remains at r = 2m2 for v ≥ v2.

170 Black holes

Figure 5.7: Black hole irradiated with ingoing null dust.

EH

AH

v=

v1

v=

v2

r = 0 r = 2m1r = 2m2

r

r = 0

r = 0

− −

++

v=

v1

v=

v2

EHAH

We may now show that while the apparent horizon is a null hyper-surface before v = v1 and after v = v2, it is spacelike in the intervalv1 ≤ v ≤ v2. This follows at once from the fact that if Φ ≡ r−2m(v) =0 describes the apparent horizon, then

gαβΦ,αΦ,β = −4dm

dv

is negative (so that the normal Φ,α is timelike) if dm/dv > 0 (so thatthe energy conditions are satisfied). We therefore see that the apparenthorizon is null when the spacetime is stationary, but that it is spacelikeotherwise.

Where is the event horizon? Clearly, it must coincide with thesurface r = 2m2 in the future of v = v2. But what is its extension tothe past of v = v2? Because the event horizon is defined as a causalboundary in spacetime, it must be a null hypersurface generated by nullgeodesics (more will be said on this in Sec. 5.4). The event horizon cantherefore not coincide with the apparent horizon in the past of v = v2.Instead, its location is determined by finding the outgoing null geodesicsof the Vaidya spacetime that connect smoothly with the generators ofthe surface r = 2m2. (See Fig. 5.7; a particular example is worked outin Sec. 5.7, Problem 2.)

It is clear that the generators of the event horizon have to be ex-panding in the past of v = v2 if they are to be stationary (in the sensethat θ = 0) in the future. Indeed, supposing that the null energycondition is satisfied (which will be true if dm/dv > 0), the focusingtheorem (Sec. 2.4) implies that the congruence formed by the null gen-erators of the event horizon will be focused by the infalling null dust;a zero expansion in the future of v = v2 guarantees a positive expan-sion in the past. The event horizon is therefore generated by those nullgeodesics that undergo just the right amount of focusing, so that afterencountering the last of the infalling matter, their expansion goes tozero.

5.1 Schwarzschild black hole 171

The event horizon coincides with the apparent horizon only in thefuture of v = v2. In the past, because the apparent horizon has aspacelike segment while the event horizon is everywhere null, the ap-parent horizon lies within the event horizon, that is, inside the blackhole (Fig. 5.7). As we shall see in Sec. 5.4, this observation is quitegeneral.

It is a remarkable property of the event horizon that the entirefuture history of the spacetime must be known before its position canbe determined: The black hole’s final state must be known before thehorizon’s null generators can be identified. This teleological propertyis not shared by the apparent horizon, whose location at any giventime (as represented by a spacelike hypersurface) depends only on theproperties of the spacetime at that time.

5.1.9 Killing horizon

The vector tα = ∂xα/∂t is a Killing vector of the Schwarzschild space-time. While this vector is timelike outside the black hole, it is null onthe event horizon, and it is spacelike inside:

gαβtαtβ = 1− 2M

r.

The surface r = 2M can therefore be called a Killing horizon, a hyper-surface on which the norm of a Killing vector goes to zero. In staticblack-hole spacetimes, the event, apparent, and Killing horizons all co-incide.

5.1.10 Bifurcation two-sphere

The point (U, V ) = (0, 0) in a Kruskal diagram, at which the pastand future horizons intersect, represents the bifurcation two-sphere ofthe Schwarzschild spacetime. This two-surface is characterized by thefact that the Killing vector tα = ∂xα/∂t vanishes there. To recognizethis, we need to work out the components of this vector in Kruskalcoordinates. From Eqs. (5.1.5) and (5.1.7) we get the relation et/2M =−V/U , and after using Eq. (5.1.8), we obtain

U2 = e(r−t)/2M( r

2M− 1

), V 2 = e(r+t)/2M

( r

2M− 1

).

Taking partial derivatives with respect to t, we arrive at

tU = − U

4M, tV =

V

4M. (5.1.19)

It follows immediately that tα = 0 at the bifurcation two-sphere. Itshould be noted that the bifurcation two-sphere exists only in the max-imally extended Schwarzschild spacetime. If the black hole is the result

172 Black holes

of gravitational collapse, then the bifurcation two-sphere is not part ofthe actual spacetime.

According to our previous calculation, tV is the only nonvanishingcomponent of the Killing vector on the future horizon. This impliesthat tα ∝ −∂αU at U = 0, and we have the important result thattα is tangent to the null generators of the event horizon. This was tobe expected from the fact that the event horizon of the Schwarzschildspacetime is also a Killing horizon.

5.2 Reissner-Nordstrom black hole

5.2.1 Derivation of the Reissner-Nordstrom solution

The Reissner-Nordstrom (RN) metric describes a static, sphericallysymmetric black hole of mass M possessing an electric charge Q. Webegin our discussion with a derivation of this solution to the Einstein-Maxwell equations.

We assume that the electromagnetic-field tensor Fαβ has no compo-nents along the θ and φ directions; this ensures that the field is purelyelectric when measured by stationary observers. Under this assump-tion, the only nonvanishing component is F tr. Maxwell’s equationsin vacuum are 0 = Fαβ

;β = |g|−1/2(|g|1/2F αβ),β. Using the metric of

Eq. (5.1.2), this implies (eψr2F tr)′ = 0, or

F tr = e−ψ Q

r2,

where Q is a constant of integration, to be interpreted as the black-holecharge. The stress-energy tensor for the electromagnetic field is

T αβ =

1

4π

(FαµFβµ − 1

4δα

βF µνFµν

),

and a few steps of algebra yield

Tαβ =

Q2

8πr4diag(−1,−1, 1, 1). (5.2.1)

The Einstein field equations (5.1.4) imply m′ = Q2/2r2, or m(r) =M − Q2/2r. The fact that T t

t = T rr implies ψ′ = 0, so that ψ can be

set to zero without loss of generality. The RN solution is therefore

ds2 = −(1− 2M

r+

Q2

r2

)dt2 +

(1− 2M

r+

Q2

r2

)−1

dr2 + r2 dΩ2, (5.2.2)

with an electromagnetic-field tensor whose only nonvanishing compo-nent is

F tr =Q

r2. (5.2.3)

5.2 Reissner-Nordstrom black hole 173

Here, M is total (ADM) mass of the spacetime, and Q is the blackhole’s electric charge.

To see that Q is indeed the charge, consider a nonsingular chargedistribution on a spacelike hypersurface Σ, described by a current den-sity jα. An appropriate definition for total charge is Q =

∫Σ

jα dΣα, or

Q = (4π)−1∫Σ

Fαβ;β dΣα after using Maxwell’s equations. Using Stokes’

theorem (Sec. 3.3.3), we rewrite this as an integral over a closed two-surface S bounding the charge distribution. This yields Gauss’ law,

Q =1

8π

∮

S

Fαβ dSαβ. (5.2.4)

The advantage of this expression for the total charge is that it is appli-cable even when the charge distribution is singular, which is the casein the present application. Also, this definition of total charge is in thesame spirit as the previously encountered definitions for total mass andangular momentum (Sec. 4.3). Substituting Eq. (5.2.3) and evaluatingfor a two-sphere of constant t and r confirms that the Q appearing inEq. (5.2.3) is indeed the black hole’s total charge.

5.2.2 Kruskal coordinates

The function f(r) = 1− 2M/r + Q2/r2 has zeroes at r = r±, where

r± = M ±√

M2 −Q2. (5.2.5)

The roots are both real, and the RN spacetime truly contains a blackhole, if |Q| ≤ M . The special case of a black hole with |Q| = M isreferred to as an extreme RN black hole. If |Q| > M , then the RNsolution describes a naked singularity at r = 0.

As for the Schwarzschild metric, the coordinates (t, r) are singularat the outer horizon (r = r+), and new coordinates must be introducedto extend the metric across this surface. This can be done with the helpof Kruskal coordinates. As we shall see, however, these coordinates failto be regular at the inner horizon (r = r−): Another coordinate trans-formation is required to extend the metric beyond this surface. Thus,Kruskal coordinates are specific to a given horizon, and a single coordi-nate patch is not sufficient to cover the entire RN manifold (Fig. 5.8).As we shall see below, the outer horizon in an event horizon for the RNspacetime, and the inner horizon is an apparent horizon.

Let us first take care of the extension across the outer horizon. Weexpress the RN metric in the form

ds2 = −f dt2 + f−1 dr2 + r2 dΩ2,

where f = 1 − 2M/r + Q2/r2. Near r = r+ this function can beapproximated by

f(r) ' 2κ+(r − r+),

174 Black holes

Figure 5.8: Kruskal patches for the Reisser-Nordstrom spacetime.

r = r1

r = r1

U+ V+

r=

r+

r=

r+

r = r1

r = r1

U− V−

r=

r+

r=

r+

r = 0 r = 0

r = r1

r = r1

U+ V+

r=

r+

r=

r+

where κ+ ≡ 12f ′(r+). It follows that near r = r+,

r∗ ≡∫

dr

f' 1

2κ+

ln|κ+(r − r+)|.

Introducing the null coordinates u = t− r∗ and v = t + r∗, the surfacer = r+ appears at v− u = −∞, and we define the Kruskal coordinatesU+ and V+ by

U+ = ∓e−κ+u, V+ = eκ+v. (5.2.6)

Here, the upper sign refers to r > r+ and the lower sign refers to r < r+.It is easy to check that f ' −2U+V+ near r = r+, so that the metricbecomes

ds2 ' − 2

κ+2dU+dV+ + r+

2 dΩ2.

This shows that when expressed in the coordinates (U+, V+), the metricis well behaved at the outer horizon. On the other hand, an exactintegration for r∗(r) would reveal that r∗ → +∞ at the inner horizon,which is then located at v − u = ∞, or U+V+ = ∞; the Kruskalcoordinates are singular at the inner horizon.

The coordinates (U+, V+) should be used only in the interval r1 <r < ∞, where r1 > r− is some cutoff radius. Inside r = r1, anothercoordinate system must be introduced. One such system is (t, r), inwhich the metric takes the standard form of Eq. (5.2.2). It is importantto understand that this new coordinate patch, which covers the portionof the RN spacetime corresponding to the interval r− < r < r+, isdistinct from the original patch covering the region r > r+. And indeed,because f is now negative, the new t must be interpreted as a spacelikecoordinate (because gtt > 0), while r must be interpreted as a timelikecoordinate (because grr < 0).

There still remains the issue of extending the spacetime beyondr = r−, where the new (t, r) coordinates fail. We want to constructa new set of Kruskal coordinates, U− and V−, adapted to the innerhorizon. Retracing the same steps as before, we have that near r = r−,the function f can be approximated by

f(r) ' −2κ−(r − r−),

5.2 Reissner-Nordstrom black hole 175

where κ− = 12|f ′(r−)|. It follows that

r∗ ' − 1

2κ−ln|κ−(r − r−)|.

With u = t− r∗ and v = t + r∗, the surface r = r− appears at v − u =+∞, and we define the new Kruskal coordinates by

U− = ∓eκ−u, V− = −e−κ−v. (5.2.7)

Here, the upper sign refers to r > r− and the lower sign refers to r < r−.Then f ' −2U−V− and the metric becomes

ds2 ' − 2

κ−2dU−dV− + r−2 dΩ2.

This is manifestly regular across r = r−. The new Kruskal coordinates,however, are singular at r = r+.

What happens now on the other side of the inner horizon? Themost noticeable feature is that the singularity at r = 0 appears as atimelike surface — this is markedly different from what happens insidea Schwarzschild black hole, where the singularity is spacelike. Becausef > 0 when r < r−, r re-acquires its interpretation as a spacelikecoordinate; any surface r = constant < r− is therefore a timelike hy-persurface, and this includes the singularity. Because it is timelike, thesingularity can be avoided by observers moving within the black hole.This is a striking new phenomenon, and we should think about thisvery carefully.

Consider the motion of a typical observer inside a RN black hole(Fig. 5.8). Before crossing the inner horizon (but after going across theouter horizon), r is a timelike coordinate and the motion necessarilyproceeds with r decreasing. After crossing r = r−, however, r becomesspacelike, and both types of motion (r decreasing or increasing) becomepossible. Our observer may therefore decide to reverse course, and ifshe does, she will avoid r = 0 altogether. Her motion inside the innerhorizon will then proceed with r increasing, and she will cross, oncemore, the surface r = r−. This, however, is another copy of the innerhorizon, distinct from the one encountered previously. (Recall thatthere are two copies of each surface r = constant in a Kruskal diagram.)After entering this new r > r− region, our observer notices that r hasonce again become timelike, and finds that reversing course is no longerpossible: her motion must proceed with r increasing, and this brings herin the vicinity of another surface r = r+. Because there is no reason forspacetime to just stop there, yet another Kruskal patch (U+, V+) mustbe introduced to extend the RN metric beyond this horizon. The newKruskal coordinates take over where the old patch (U−, V−) leaves off,at the spacelike hypersurface r = r1.

176 Black holes

The ultimate conclusion to these considerations is that our observereventually emerges out of the black hole, through another copy of theouter horizon, into a new asymptotically-flat universe. Her trip may notend there: Our observer could now decide to enter the RN black holethat resides in this new universe, and this entire cycle would repeat! Ittherefore appears that the RN metric describes more than just a singleblack hole. Indeed, it describes an infinite lattice of asymptotically-flatuniverses connected by black-hole tunnels.

Such a fantastic spacetime structure is best represented with aPenrose-Carter diagram (Fig. 5.9). This diagram makes it clear thatregion bounded by the surfaces r = r+ and r = r− contains trappedsurfaces: both ingoing and outgoing light rays originating from thisregion converge toward the singularity. The outer and inner horizonsare therefore apparent horizons, but only the outer horizon is an eventhorizon.

5.2.3 Radial observers in Reissner-Nordstrom spacetime.

The discovery of black-hole tunnels is so bizarre that it should be backedup by a solid calculation. Here we consider the geodesic motion of afree-falling observer in the RN spacetime. It is assumed that the motionproceeds entirely in the radial direction, and that initially, it is directedinward.

We will first work with the (v, r) coordinates, in which the metric

Figure 5.9: Penrose-Carter diagram of the Reisser-Nordstrom space-time.

r = 0 r = 0

r = 0 r = 0

− −

− −

+ +

r−

r+

r−

5.2 Reissner-Nordstrom black hole 177

takes the formds2 = −f dv2 + 2 dvdr + r2 dΩ2, (5.2.8)

where f = 1 − 2M/r + Q2/r2. The observer’s four-velocity is uα ∂α =v ∂v + r ∂r, where an overdot denotes differentiation with respect toproper time τ . The quantity E = −uαtα = −uv, the observer’s energyper unit mass, is a constant of the motion. In terms of v and r, this isgiven by E = fv − r. On the other hand, the normalization conditionuαuα = −1 gives fv2 − 2vr = 1, and these equations imply

rin = −(E2 − f)1/2, vin =E − (E2 − f)1/2

f, (5.2.9)

where the sign in front of the square root was chosen appropriately foran ingoing observer.

The equation for r can also be written in the form

r2 + f = E2, (5.2.10)

which comes with a nice interpretation as an energy equation (Fig. 5.10).Its message is clear: After crossing the outer and inner horizons, theobserver reaches a turning point (r = 0) at a radius rmin < r− suchthat f(rmin) = E2. The motion, which initially was inward, turns out-ward, and the observer eventually emerges out of the black hole, into anew external universe. During the outward portion of the motion, theobserver’s four-velocity is given by

rout = +(E2 − f)1/2, vout =E + (E2 − f)1/2

f, (5.2.11)

with the opposite sign in front of the square root.Let us examine the behaviour of v as the observer traverses a hori-

zon. When the motion is inward, we have that v ' (2E)−1 in the limit

Figure 5.10: Effective potential for radial motion in Reisser-Nordstrom spacetime.

r = 0 r = r− r = r+

r

f(r)E2

178 Black holes

Figure 5.11: Eddington-Finkelstein patches for the Reisser-Nordstromspacetime.

r = 0

r+

r−

−

+

v= −∞

v=∞

r = 0r−

r+

−

+

u=−∞

u=∞

f → 0. This means that v stays finite during the first crossings of theouter and inner horizons. When the motion is outward, v ' 2E/f inthe limit f → 0, and this means that the coordinates (v, r) become sin-gular during the second crossing of the inner horizon. The observer’smotion cannot be followed beyond this point, unless new coordinatesare introduced.

Let us therefore switch to the coordinates (u, r), in which the RNmetric takes the form

ds2 = −f du2 − 2 dudr + r2 dΩ2. (5.2.12)

In these coordinates, and during the outward portion of the motion(after the bounce at r = rmin), the four-velocity is given by

rout = +(E2 − f)1/2, uout =E − (E2 − f)1/2

f. (5.2.13)

We have that u ' (2E)−1 when f → 0, which shows that u stays finiteduring the second crossings of the inner and outer horizons.

We see that a large portion of the RN spacetime is covered bythe two coordinate patches employed here (Fig. 5.11). This includestwo asymptotically-flat regions connected by a black-hole tunnel thatcontains two copies of the outer horizon, and two copies of the innerhorizon. The complete spacetime is obtained by tessellation, using thepatches (v, r) and (u, r) as tiles; this gives rise to the diagram of Fig. 5.9.Because the completed spacetime contains an infinite number of black-hole tunnels, an infinite number of coordinate patches is required forits description.

The presence of black-hole tunnels in the RN spacetime is now wellestablished. These tunnels, of course, have a lot to do with the oc-currence of a turning point in the motion of our free-falling observer.

5.2 Reissner-Nordstrom black hole 179

This is a rather striking feature of the RN spacetime. While turningpoints are a familiar feature of Newtonian mechanics, in this contextthey are always associated with the presence of an angular-momentumterm in the effective potential: the centrifugal force is repulsive, andit prevents an observer from ever reaching the centre at r = 0. This,however, cannot explain what is happening here, because the motionwas restricted from the start to be purely radial — there is no angu-lar momentum present to produce a repulsive force. The gravitationalfield alone must be responsible for the repulsion, and we are forced toconclude that inside the inner horizon, the gravitational force becomesrepulsive! It is this repulsive gravity that ultimately is responsible forthe black-hole tunnels.

Such a surprising conclusion can perhaps be understood better ifwe recall our previous expression for the mass function:

m(r) = M − Q2

2r. (5.2.14)

This relation shows that m(r) becomes negative if r is sufficiently small,and clearly, this negative mass will produce a repulsive gravitationalforce. How can we explain this behaviour for the mass function? Werecall that m(r) represents the mass inside a sphere of radius r. Ingeneral, this will be smaller than the total mass M ≡ m(∞), be-cause a sphere of finite radius r excludes a certain amount — equalto Q2/(2r) — of electrostatic energy. If the radius is sufficiently small,then Q2/(2r) > M , and m(r) < 0. You may check that this alwaysoccurs within the inner horizon.

The conclusion that the RN spacetime contains black-hole tunnelsis firm. Should we then feel confident that a trip inside a chargedblack hole will lead us to a new universe? This answer is no. Thereason is that the existence of such tunnels depends very sensitivelyon the assumed symmetries of the RN spacetime, namely, staticity andspherical symmetry. These symmetries would not be exact in a realisticblack hole, and slight perturbations have a dramatic effect on the hole’sinternal structure. The tunnels are therefore unstable, and they do notappear in realistic situations. (More will be said on this in Sec. 5.7,Problem 3.)

5.2.4 Surface gravity

In Sec. 5.2.2, a quantity κ+ ≡ 12f ′(r+) was introduced during the con-

struction of Kruskal coordinates adapted to the outer horizon. We shallname this quantity the surface gravity of the black hole, and henceforthdenote it simply by κ. As we shall see in Sec. 5.5, the surface gravityprovides an important characterization of black holes, and it plays a

180 Black holes

key role in the laws of black-hole mechanics. For the RN black hole, itis given explicitly by

κ =r+ − r−

2r+2

=

√M2 −Q2

r+2

, (5.2.15)

where we have used Eq. (5.2.5). Notice that κ = 0 for an extreme RNblack hole. On the other hand,

κ =1

4M(5.2.16)

for a Schwarzschild black hole.The name “surface gravity” deserves a justification. Consider, in a

static and spherically symmetric spacetime with metric

ds2 = −f dt2 + f−1 dr2 + r2 dΩ2, (5.2.17)

a particle of unit mass held in place at a radius r. (Here, f is notnecessarily restricted to have the RN form, although this will be the caseof interest.) The four-velocity of the stationary particle is uα = f−1/2tα,and its acceleration is aα = uα

;βuβ. The only nonvanishing component

is ar = 12f ′, and its magnitude is

a(r) ≡ (gαβaαaβ)1/2 =1

2f−1/2f ′(r). (5.2.18)

This is the force required to hold the particle at r if the force is appliedlocally, at the particle’s position. This, not surprisingly, diverges in thelimit r → r+. But suppose instead that the particle is held in place byan observer at infinity, by means of an infinitely long, massless string.What is a∞(r), the force applied by this observer?

To answer this we consider the following thought experiment. Letthe observer at infinity raise the string by a small proper distance δs,thereby doing an amount δW∞ = a∞δs of work. At the particle’sposition, the displacement is also δs, but the work done is δW = a δs.(You may justify this statement by working in a local Lorentz frame atr.) Suppose now that the work δW is converted into radiation that isthen collected at infinity. The received energy is redshifted by a factorf 1/2, so that δE∞ = f 1/2a δs. But energy conservation demands thatthe energy extracted be equal to the energy put in, so that δE∞ = δW∞.This implies

a∞(r) = f 1/2a(r) =1

2f ′(r). (5.2.19)

This is the force applied by the observer at infinity. This quantity iswell behaved in the limit r → r+, and it is appropriate to call a∞(r+)the surface gravity of the black hole. Thus,

κ ≡ a∞(r+) =1

2f ′(r+). (5.2.20)

5.2 Reissner-Nordstrom black hole 181

The surface gravity is therefore the force required of an observer atinfinity to hold a particle (of unit mass) in place at the event horizon.

The surface gravity can also be defined in terms of the Killing vectortα. We have seen in Sec. 5.1.9 that the event horizon of a static space-time is also a Killing horizon, so that tα is tangent to the horizon’s nullgenerators. Because tα is orthogonal to itself on the horizon, it is alsonormal to the horizon. But Φ ≡ −tαtα = 0 on the horizon, and sincethe normal vector is proportional to Φ,α, there must exist a function κsuch that

(−tµtµ);α = 2κtα (5.2.21)

on the horizon. A brief calculation confirms that this κ is the surfacegravity: Using the coordinates (v, r), we have that tα ∂α = ∂v andtα dxα = dr on the horizon; with Φ = −gvv = f we obtain Φ,α = f ′∂αr,which is just Eq. (5.2.21) with κ = 1

2f ′(r+). This calculation reveals

also that the horizon’s null generators are parameterized by v, so thattα = dxα/dv.

Because tα is tangent to the horizon’s null generators, it must sat-isfy the geodesic equation at r = r+. This comes as an immediateconsequence of Eq. (5.2.21) and Killing’s equation: On the horizon,

tα;βtβ = κtα, (5.2.22)

and we see that v is not an affine parameter on the generators. Anaffine parameter λ can be obtained by integrating the equation dλ/dv =eκv (Sec. 1.3). This gives λ = V/κ, where V ≡ eκv is one of theKruskal coordinates adapted to the event horizon — it was denoted V+

in Sec. 5.2.2. It follows that on the horizon, the null vector

kα = V −1tα (5.2.23)

satisfies the geodesic equation in affine-parameter form.It is also possible to obtain an explicit formula for κ. Because the

congruence of null generators is necessarily hypersurface orthogonal,Frobenius’ theorem (Sec. 2.4.3) guarantees that the relation

t[α;βtγ] = 0

holds on the event horizon. Using Killing’s equation, this implies

tα;βtγ + tγ;αtβ + tβ;γtα = 0,

and contracting with tα;β yields

tα;βtα;βtγ = −tγ;αtα;βtβ + tβ;γtβ;αtα

= −κ tγ;αtα + κ tβ;γtβ

= −2κ2tγ.

182 Black holes

We have obtained

κ2 = −1

2tα;βtα;β, (5.2.24)

in which it is understood that the right-hand side is evaluated at r =r+. Equations (5.2.21), (5.2.22), and (5.2.24) can all be regarded asfundamental definitions of the surface gravity; these are of course allequivalent.

5.3 Kerr black hole

5.3.1 The Kerr metric

A solution to the Einstein field equations describing a rotating blackhole was discovered by Roy Kerr in 1963. (There is also a solution tothe Einstein-Maxwell equations that describes a charged, rotating blackhole. It is known as the Kerr-Newman solution, and it is described inSec. 5.7, Problem 8.) As we shall see, the Kerr metric can be written in anumber of different ways. In the standard Boyer-Lindquist coordinates,it is given by

ds2 = −(1− 2Mr

ρ2

)dt2 − 4Mar sin2 θ

ρ2dtdφ +

Σ

ρ2sin2 θ dφ2 +

ρ2

∆dr2 + ρ2 dθ2

(5.3.1)

= −ρ2∆

Σdt2 +

Σ

ρ2sin2 θ(dφ− ω dt)2 +

ρ2

∆dr2 + ρ2 dθ2,

where

ρ2 = r2 + a2 cos2 θ, ∆ = r2 − 2Mr + a2,

(5.3.2)

Σ = (r2 + a2)2 − a2∆ sin2 θ, ω ≡ − gtφ

gφφ

=2Mar

Σ.

The Kerr metric is stationary and axially symmetric; it therefore ad-mits the Killing vectors tα = ∂xα/∂t and φα = ∂xα/∂φ. It is alsoasymptotically flat. The Komar formulae (Sec. 4.3) confirm that Mis the spacetime’s ADM mass, and show that J ≡ aM is the angularmomentum (so that a is the ratio of angular momentum to mass).

The components of the inverse metric are

gtt = − Σ

ρ2∆, gtφ = −2Mar

ρ2∆, gφφ =

∆− a2 sin2 θ

ρ2∆ sin2 θ,

(5.3.3)

grr =∆

ρ2, gθθ =

1

ρ2.

5.3 Kerr black hole 183

The metric and its inverse have singularities at ∆ = 0 and ρ2 = 0. Todistinguish between coordinate and curvature singularities, we examinethe squared Riemann tensor of the Kerr spacetime:

RαβγδRαβγδ =48M2(r2 − a2 cos2 θ)(ρ4 − 16a2r2 cos2 θ)

ρ12. (5.3.4)

This reveals that the singularity of the metric at ∆ = 0 is just a coordi-nate singularity, but that the Kerr spacetime is truly singular at ρ2 = 0.The nature of the curvature singularity will be clarified in Sec. 5.3.8.

Various properties of the Kerr spacetime will be examined in the fol-lowing subsections. To facilitate this discussion we will introduce threefamilies of observers: zero-angular-momentum observers (ZAMOs), staticobservers, and stationary observers.

5.3.2 Dragging of inertial frames: ZAMOs

ZAMOs are freely moving observers with zero angular momentum: ifuα is the four-velocity, then L ≡ uαφα = 0. This implies gφtt+gφφφ = 0,where an overdot indicates differentiation with respect to proper timeτ . Using Eqs. (5.3.1), this translates to

Ω ≡ dφ

dt= ω, (5.3.5)

and we see that ZAMOs possess an angular velocity equal to ω =−gtφ/gφφ. This angular velocity increases as the observer approachesthe black hole, and it goes in the same direction as the hole’s own rota-tion — the ZAMOs rotate with the black hole. This striking propertyof the Kerr black hole, which in fact is shared by all rotating bodies,is called the dragging of inertial frames (see Sec. 3.10). At large dis-tances from the black hole, ω ' 2J/r3, and the dragging disappearscompletely at infinity.

5.3.3 Static limit: static observers

We now consider static observers in the Kerr spacetime. Such observershave a four-velocity proportional to the Killing vector tα:

uα = γtα, (5.3.6)

where the factor γ ≡ (−gαβtαtβ)−1/2 ensures that the four-velocity isproperly normalized. Because these observers must be held in placeby an external agent (a rocket engine, for example), the motion is notgeodesic.

Static observers cannot exist everywhere in the Kerr spacetime.This can be seen from the fact that tα is not everywhere timelike,

184 Black holes

but becomes null when γ−2 = −gtt = 0; when this occurs, Eq. (5.3.6)breaks down. The static limit is therefore described by gtt = 0 or, afterusing Eqs. (5.3.1) and (5.3.2), r2 − 2Mr + a2 cos2 θ = 0. Solving for rreveals that the static limit is located at r = rsl(θ), where

rsl(θ) = M +√

M2 − a2 cos2 θ. (5.3.7)

Thus, observers cannot remain static when r ≤ rsl(θ), even if an arbi-trarily large force is applied. Instead, the dragging of inertial framescompels them to rotate with the black hole. As we shall see, the staticlimit does not coincide with the hole’s event horizon. The finite regionbetween the horizon and the static limit is called the ergosphere of theKerr spacetime (Fig. 5.12).

5.3.4 Event horizon: stationary observers

We now consider observers moving in the φ direction with an arbitrary,but uniform, angular velocity dφ/dt = Ω. Because such observers donot perceive any time variation in the black hole’s gravitational field,they are called stationary observers. They move with a four-velocity

uα = γ(tα + Ωφα), (5.3.8)

where tα + Ωφα is a Killing vector for the Kerr spacetime, and γ a newnormalization factor given by

γ−2 = −gαβ(tα + Ωφα)(tβ + Ωφβ)

= −gtt − 2Ωgtφ − Ω2gφφ

= −gφφ(Ω2 − 2ωΩ + gtt/gφφ),

where ω = −gtφ/gtt.Stationary observers cannot exist everywhere in the Kerr spacetime:

the vector tα + Ωφα must be timelike, and this fails to be true whenγ−2 is nonpositive. It is easy to check that the condition γ−2 > 0 givesrise to the following requirement on the angular velocity:

Ω− < Ω < Ω+, (5.3.9)

where Ω± = ω±√ω2 − gtt/gφφ. After some algebra, using Eqs. (5.3.1)

and (5.3.2), this reduces to

Ω± = ω ± ∆1/2ρ2

Σ sin θ. (5.3.10)

A stationary observer with Ω = 0 is a static observer, and we alreadyknow that static observers exist only outside the static limit. It musttherefore be true that Ω− changes sign at r = rsl(θ). This is confirmed

5.3 Kerr black hole 185

Figure 5.12: Static limit and event horizon of the Kerr spacetime.ro

tati

onax

is

horizon

static limit

θ

by a few lines of algebra, using Eqs. (5.3.7) and (5.3.10). As r decreasesfurther from rsl(θ), Ω− increases while Ω+ decreases. Eventually wearrive at the situation Ω− = Ω+, which implies Ω = ω; at this pointthe stationary observer is forced to move around the black hole with anangular velocity equal to ω. This occurs when ∆ = 0, or r2−2Mr+a2 =0, and the largest solution is r = r+, where

r+ = M +√

M2 − a2. (5.3.11)

Notice that the roots of ∆ = 0 are real if and only if a ≤ M , or J ≤ M2:there is an upper limit on the angular momentum of a black hole. Kerrblack holes with a = M are said to be extremal. For a > M , the Kerrmetric describes a naked singularity.

The vector tα+Ωφα becomes null at r = r+, and stationary observerscannot exist inside this surface, which we identify with the black hole’sevent horizon (Fig. 5.12). The quantity

ΩH ≡ ω(r+) =a

r+2 + a2

(5.3.12)

is then interpreted as the angular velocity of the black hole. Stationaryobservers just outside the horizon have an angular velocity equal to ΩH

— they are in a state of corotation with the black hole.

To confirm that r = r+ is truly the event horizon, we use the prop-erty that in a stationary spacetime, the event horizon is also an appar-ent horizon — a surface of zero expansion for a congruence of outgoingnull geodesics orthogonal to the surface. The event horizon must there-fore be a null, stationary surface. Now, the normal to any stationarysurface must be proportional to ∂αr, and such a surface will be null ifgαβ(∂αr)(∂βr) = grr = 0. Using Eq. (5.3.3), this implies

∆ ≡ r2 − 2Mr + a2 = 0. (5.3.13)

186 Black holes

The largest solution, r = r+, designates the event horizon. The otherroot,

r− = M −√

M2 − a2, (5.3.14)

describes the black hole’s inner apparent horizon, which is analogousto the inner horizon of the Reissner-Nordstrom black hole.

We have found that the vector

ξα ≡ tα + ΩHφα (5.3.15)

is null at the event horizon. It is tangent to the horizon’s null gen-erators, which wrap around the horizon with an angular velocity ΩH .Because it is a linear combination of two Killing vectors, ξα is also aKilling vector, and the event horizon of the Kerr spacetime is a Killinghorizon. Notice an important difference between stationary and staticblack holes: For a static black hole, tα becomes null at the event hori-zon; for a stationary black hole, tα is null at the static limit, and ξα

becomes null at the event horizon.

5.3.5 The Penrose process

The fact that tα is spacelike in the ergosphere — the region r+ < r <rsl(θ) — implies that the (conserved) energy E = −pαtα of a particlewith four-momentum pα can be of either sign. Particles with negativeenergy can therefore exist in the ergosphere, but they would never beable to escape from this region. (Note that E < 0 refers to the energythat would be measured at infinity if the particle could be broughtthere. Any local measurement of the particle’s energy inside the staticlimit would return a positive value.)

It is easy to elaborate a scenario in which negative-energy par-ticles created in the ergosphere are used to extract positive energyfrom a Kerr black hole. Imagine that a particle of energy E1 > 0comes from infinity and enters the ergosphere. There, it decays intotwo new particles, one with energy −E2 < 0, the other with energyE3 = E1 +E2 > E1. While the negative-energy particle remains withinthe static limit, the positive-energy particle escapes to infinity, whereits energy is extracted. Because E3 is larger than the energy of the ini-tial particle, the black hole must have given off some of its own energy.This is the Penrose process, by which some of the energy of a rotatingblack hole can be extracted.

The Penrose process is self-limiting: only a fraction of the hole’stotal energy can be tapped. Suppose that in order to exploit the Pen-rose process, a rotating black hole is made to absorb a particle of energyE = −pαtα < 0 and angular momentum L = pαφα. Because the Killingvector ξα = tα + ΩHφα is timelike just outside the event horizon, thecombination E − ΩHL must be positive; otherwise the particle would

5.3 Kerr black hole 187

not be able to penetrate the horizon. Thus L < E/ΩH , and L must benegative if E < 0. The black hole will therefore lose angular momentumduring the Penrose process. Eventually the hole’s angular momentumwill go to zero, the ergosphere will disappear, and the Penrose processwill stop. We might say that only the hole’s rotational energy can beextracted by the Penrose process.

Note that in a process by which a black hole absorbs a particle ofenergy E (of either sign) and angular momentum L, its parameterschange by amounts δM = E and δJ = L. Since E − ΩHL must bepositive, we have

δM − ΩH δJ > 0.

As we shall see, this inequality is a direct consequence of the first andsecond laws of black-hole mechanics.

5.3.6 Principal null congruences

The Boyer-Lindquist coordinates, like the Schwarzschild coordinates,are singular at the event horizon: While a trip down to the event horizonrequires a finite proper time, the interval of coordinate time t is infinite.Moreover, because the angular velocity dφ/dt stays finite at the horizon,φ also increases by an infinite amount. We therefore need anothercoordinate system to extend the Kerr metric beyond the event horizon.It is advantageous to tailor these new coordinates to the behaviour ofnull geodesics. The two congruences considered here (which are knownas the principal null congruences of the Kerr spacetime) are especiallysimple to deal with; we will use them to construct new coordinates forthe Kerr metric.

It is a remarkable feature of the Kerr metric that the equationsfor geodesic motion can be expressed in a decoupled, first-order form.These equations involve three constants of the motion: the energy pa-rameter E, the angular-momentum parameter L, and the “Carter con-stant” . (This last constant appears because of the existence of a Killingtensor. This is explained in Sec. 5.7, Problem 4, which also provides aderivation of the geodesic equations.) For null geodesics, the equationsare

ρ2 t = −a(aE sin2 θ − L) + (r2 + a2)P/∆,

ρ2 r = ±√

R,

ρ2 θ = ±√

Θ,

ρ2 φ = −(aE − L/ sin2 θ) + aP/∆,

in which an overdot indicates differentiation with respect to the affineparameter λ, and

P = E(r2 + a2)− aL,

188 Black holes

R = P 2 −∆[(L− aE)2 +

],

Θ = + cos2 θ(a2E2 − L2/ sin2 θ).

We simplify these equations by making the following choices:

L = aE sin2 θ, = −(L− aE)2 = −(aE cos2 θ)2.

It is easy to check that these imply Θ = 0, so that our geodesics movewith a constant value of θ. We also have P = E ρ2 and R = (E ρ2)2,which gives

t = E(r2 + a2)/∆, r = ±E, θ = 0, φ = aE/∆.

The constant E can be absorbed into the affine parameter λ. We obtainan ingoing congruence by choosing the negative sign for r, and we shalluse lα to denote its tangent vector field:

lα ∂α =r2 + a2

∆∂t − ∂r +

a

∆∂φ. (5.3.16)

Choosing instead the positive sign gives an outgoing congruence, with

kα ∂α =r2 + a2

∆∂t + ∂r +

a

∆∂φ (5.3.17)

as its tangent vector field.To give the simplest description of the ingoing congruence, we in-

troduce new coordinates v and ψ defined by

v = t + r∗, ψ = φ + r], (5.3.18)

where

r∗ =

∫r2 + a2

∆dr

= r +Mr+√M2 − a2

ln∣∣∣ r

r+

− 1∣∣∣− Mr−√

M2 − a2ln

∣∣∣ r

r−− 1

∣∣∣(5.3.19)

and

r] =

∫a

∆dr =

a

2√

M2 − a2ln

∣∣∣r − r+

r − r−

∣∣∣. (5.3.20)

It is easy to check that in these coordinates, lr = −1 is the only nonva-nishing component of the tangent vector. This means that v and ψ (aswell as θ) are constant on each of the ingoing null geodesics, and that−r is the affine parameter.

The simplest description of the outgoing congruence is provided bythe coordinates (u, r, θ, χ), where

u = t− r∗, χ = φ− r]. (5.3.21)

5.3 Kerr black hole 189

In these coordinates, kr = +1 is the only nonvanishing component ofthe tangent vector. This shows that u and χ (as well as θ) are constantalong the outgoing null geodesics, and that r is the affine parameter.

The Kerr metric can be expressed in either one of these new coor-dinate systems. While the coordinates (v, r, θ, ψ) are well behaved onthe future horizon but singular on the past horizon, the coordinates(u, r, θ, χ) are well behaved on the past horizon but singular on thefuture horizon. For example, a straightforward computation revealsthat after a transformation to the ingoing coordinates, the Kerr metricbecomes

ds2 = −(1− 2Mr

ρ2

)dv2 + 2 dvdr − 2a sin2 θ drdψ

− 4Mar sin2 θ

ρ2dvdψ +

Σ

ρ2sin2 θ dψ2 + ρ2 dθ2. (5.3.22)

These coordinates produce an extension of the Kerr metric across thefuture horizon. Several coordinate patches, both ingoing and outgoing,are required to cover the entire Kerr spacetime, whose causal structureis very similar to that of the Reissner-Nordstrom spacetime. We shallreturn to this point in Sec. 5.3.9.

5.3.7 Kerr-Schild coordinates

Another useful set of coordinates for the Kerr metric is (t′, x, y, z), thepseudo-Lorentzian Kerr-Schild coordinates in terms of which the metrictakes a particularly interesting form. These are constructed as follows.

We start with Eq. (5.3.22) and separate out the terms that areproportional to M . After some algebra, we obtain

ds2 = −dv2 + 2 dvdr − 2a sin2 θ drdψ + (r2 + a2) sin2 θ dψ2 + ρ2 dθ2

+2Mr

ρ2(dv − a sin2 θ dψ)2.

The terms that do not involve M have a simple interpretation: they givethe metric of flat spacetime in a rather strange coordinate system. Therest of the line element can be written neatly in terms of lα: Recallingthat lr = −1 is the only nonvanishing component of lα, we find that

−lα dxα = dv − a sin2 θ dψ,

and the line element becomes

ds2 = (ds2)flat +2Mr

ρ2(lα dxα)2. (5.3.23)

The Kerr metric can therefore be expressed as

gαβ = ηαβ +2Mr

ρ2lαlβ, (5.3.24)

190 Black holes

where ηαβ is the metric of flat spacetime in the coordinates (v, r, θ, ψ).Equation (5.3.24) gives us the Kerr metric in a rather attractive

form. Any metric that can be written as gαβ = ηαβ + Hlαlβ, where His a scalar function and lα a null vector field, is known as a Kerr-Schildmetric. It is by adopting such an expression that Kerr discovered his so-lution in 1963. (Some general aspects of the Kerr-Schild decompositionare worked out in Sec. 5.7, Problem 5.)

The next order of business is to find the coordinate transformationthat brings ηαβ to the standard Minkowski form. The answer is

x + iy = (r + ia) sin θ eiψ, z = r cos θ, t′ = v − r. (5.3.25)

Going through the necessary algebra does indeed reveal that in thesecoordinates,

(ds2)flat = −dt′2 + dx2 + dy2 + dz2. (5.3.26)

It is easy to work out the components of lα in this coordinate system.Because the null geodesics move with constant values of v, θ, and ψ,we have that x + iy = − sin θ eiψ, z = − cos θ, and t′ = 1, where wehave used r = −1. Lowering the indices is a trivial matter (see Sec. 5.7,Problem 5), and expressing the right-hand sides in terms of the newcoordinates gives

−lα dxα = dt′ +rx + ay

r2 + a2dx +

ry − ax

r2 + a2dy +

z

rdz. (5.3.27)

The quantity r must now be expressed in terms of x, y, and z. Startingwith x2 + y2 = (r2 + a2) sin2 θ, it is easy to show that

r4 − (x2 + y2 + z2 − a2)r2 − a2z2 = 0, (5.3.28)

which may be solved for r(x, y, z). Equations (5.3.23), (5.3.26)–(5.3.28)give the explicit form of the Kerr metric in the Kerr-Schild coordinates.

5.3.8 The nature of the singularity

We have seen that the Kerr spacetime possesses a curvature singularityat

ρ2 ≡ r2 + a2 cos2 θ = 0.

According to this equation, the singularity occurs only in the equatorialplane (θ = π/2), at r = 0. The Kerr-Schild coordinates can help usmake sense of this statement. The relations x2 + y2 = (r2 + a2) sin2 θand z = r cos θ indicate that the “point” r = 0 corresponds in fact tothe entire disk x2 + y2 ≤ a2 in the plane z = 0. The points interior tothe disk correspond to angles such that sin2 θ < 1. The boundary,

x2 + y2 = a2,

5.3 Kerr black hole 191

corresponds to the equatorial plane, and this is where the Kerr metricis singular. The curvature singularity of the Kerr spacetime is thereforeshaped like a ring. This singularity can be avoided: Observers at r =0 can stay away from the equatorial plane, and they never have toencounter the singularity; such observers end up going through the ring.

5.3.9 Maximal extension of the Kerr spacetime

We have already constructed coordinate systems that allow the contin-uation of the Kerr metric across the event horizon. We now completethe discussion and show how the spacetime can also be extended be-yond the inner horizon. For simplicity, we shall work with the sectionof the Kerr spacetime obtained by setting θ = 0. This is the rotationaxis, and because the Kerr metric is not spherically symmetric, thisdoes represent a loss of generality.

Going back to Eq. (5.3.1) and the original Boyer-Lindquist coordi-nates, we find that when θ = 0, the Kerr metric reduces to

ds2 = −(1− 2Mr

r2 + a2

)dt2 +

r2 + a2

∆dr2

= − ∆

r2 + a2

(dt− r2 + a2

∆dr

)(dt +

r2 + a2

∆dr

),

ords2 = −f dudv, (5.3.29)

where u = t− r∗ and v = t + r∗ are the coordinates of Sec. 5.3.6. Here,

f =∆

r2 + a2=

(r − r+)(r − r−)

r2 + a2, (5.3.30)

and r± = M ± √M2 − a2 denote the positions of the outer and innerhorizons, respectively. The metric of Eq. (5.3.29) is extremely simple,and the construction of Kruskal coordinates for the θ = 0 section ofthe Kerr spacetime proceeds just as for the Reissner-Nordstrom (RN)black hole (Sec. 5.2.2).

We first consider the continuation of the metric across the eventhorizon. Near r = r+, Eq. (5.3.30) can be approximated by

f ' 2κ+(r − r+),

where κ+ = 12f ′(r+). It follows that

r∗ =

∫dr

f' 1

2κ+

ln|κ+(r − r+)|

and f ' ±2 e2κ+r∗ = ±2 eκ+(v−u); the upper sign refers to r > r+ andthe lower sign to r < r+. Introducing the new coordinates

U+ = ∓e−κ+u, V+ = eκ+v, (5.3.31)

192 Black holes

Figure 5.13: Kruskal patches for the Kerr spacetime.

r = r1

r = r1

U+ V+

r=

r+

r=

r+

r = r1

r = r1

U− V−

r=

r+

r=

r+

r = 0 r = 0

we find that near r = r+, the Kerr metric admits the manifestly regularform ds2 ' −2κ−2

+ dU+dV+.Just as for the RN spacetime, the coordinates U+ and V+ are sin-

gular at the inner horizon, and another coordinate patch is requiredto extend the Kerr metric beyond this horizon (Fig. 5.13). The pro-cedure is now familiar. Near r = r− we approximate Eq. (5.3.30) byf ' −2κ−(r − r−), where κ− = 1

2|f ′(r−)|, so that f ' ∓2 e−2κ−r∗ =

∓2 eκ−(u−v). The appropriate coordinate transformation is now

U− = ∓eκ−u, V+ = −e−κ−v, (5.3.32)

and the metric becomes ds2 ' −2κ−2− dU−dV−.

Just as for the RN spacetime, another copy of the outer horizonpresents itself in the future of the inner horizon, and another Kruskalpatch is required to extend the spacetime beyond this new horizon.This continues ad nauseam, and we see that the maximally extendedKerr spacetime represents an infinite succession of asymptotically-flatuniverses connected by black-hole tunnels. There is more, however. Itis easy to check that in a spacetime diagram based on the (U−, V−)coordinates, the surface r = 0 is represented by U−V− = −1. This is atimelike surface, and on the rotation axis, this surface is nonsingular.The Kerr spacetime can therefore be extended beyond r = 0, intoa region in which r adopts negative values. This new region has noanalogue in the RN spacetime; it contains no horizons, and it becomesflat in the limit r → −∞. Observers in this region interpret the Kerrmetric as describing the gravitational field of a (naked) ring singularity.You should be able to convince yourself that this singularity has anegative mass.

The maximally extended Kerr spacetime can be represented by aPenrose-Carter diagram (Fig. 5.14). The resulting causal structure isextremely complex. It should be kept in mind, however, that the in-terior of a Kerr black hole is subject to the same instability as that of

5.3 Kerr black hole 193

Figure 5.14: Penrose-Carter diagram of the Kerr spacetime.

r = 0 r = 0

r = 0 r = 0

r−

r+

r−

r+

r = −∞ r = −∞

r = −∞ r = −∞

r = −∞ r = −∞

r = −∞ r = −∞

− −

− −

+ +

a RN black hole (see Sec. 5.2.3 and Sec. 5.7, Problem 3). The tunnelsto other universes, and the regions of negative r, are not present insidephysically realistic black holes.

5.3.10 Surface gravity

As was pointed out in Sec. 5.3.4, the vector

ξα = tα + ΩHφα, (5.3.33)

where ΩH is given by Eq. (5.3.12), is null at the event horizon, and is infact tangent to the horizon’s null generators. From the same argumentsas those presented in Sec. 5.2.4, the black hole’s surface gravity κ canbe defined by

(−ξβξβ);α = 2κξα, (5.3.34)

or by

ξα;βξβ = κξα, (5.3.35)

or finally, by

κ2 = −1

2ξα;βξα;β. (5.3.36)

These definitions are all equivalent.Let us use Eq. (5.3.34) to calculate the surface gravity. The norm

of ξα is given by

ξβξβ =Σ sin2 θ

ρ2(ΩH − ω)2 − ρ2∆

Σ,

194 Black holes

and differentiation yields

(−ξβξβ);α =ρ2

Σ∆,α

on the horizon, at which ω = ΩH and ∆ = 0. We have that ∆,α =2(r+−M) ∂αr and ξα = (1− aΩH sin2 θ) ∂αr on the horizon, and a fewlines of algebra reveal that the surface gravity is

κ =r+ −M

r+2 + a2

=

√M2 − a2

r+2 + a2

. (5.3.37)

Notice that this is the same quantity that was denoted κ+ in Sec. 5.3.9.Notice also that κ = 0 for an extreme Kerr black hole. And finally,notice that in the general case, κ does not depend on θ — the sur-face gravity is uniform on the event horizon. We shall return to thisremarkable fact in Sec. 5.5.1.

5.3.11 Bifurcation two-sphere

In the coordinates (v, r, θ, ψ) which are regular on the event horizon,ξα ∂α = ∂v + ΩH ∂ψ. This shows that the horizon’s null generators areparameterized by the advanced-time coordinate v, but as Eq. (5.3.35)reveals, v is not affine. An affine parameter λ is obtained by integrating

dλ

dv= eκv,

so that κλ = eκv ≡ V . It follows that on the horizon, the vectorkα = V −1ξα satisfies the geodesic equation in affine-parameter form:kα

;βkβ = 0. [This vector is not equal to the kα introduced in Sec. 5.3.6.

It is easy to check that these vectors are related by kαnew = 1

2∆kα

old/(r2+

a2), where the right-hand side is to be evaluated on the horizon.] Ifκ 6= 0 and the event horizon is geodesically complete (in the sense thatthe null generators can be extended arbitrarily far into the past), therelation

ξα = V kα (5.3.38)

implies that ξα = 0 at V = 0. This defines a closed two-surface calledthe bifurcation two-sphere of the Kerr spacetime. The conditions aresometimes violated: The event horizon of a black hole formed by grav-itational collapse is not geodesically complete, because the horizon wasnecessarily formed in the finite past; and as we have seen, the surfacegravity of an extreme Kerr black hole (for which M = a) vanishes. Ineither one of these situations, the bifurcation two-sphere does not exist.

5.3 Kerr black hole 195

5.3.12 Smarr’s formula

There exists a simple algebraic relation between the black-hole massM , its angular momentum J ≡ Ma, and its surface area A. This isdefined by

A =

∮√σ d2θ, (5.3.39)

where is a two-dimensional cross section of the event horizon, describedby v = constant, r = r+, 0 ≤ θ ≤ π, and 0 ≤ ψ < 2π. From Eq. (5.3.22)we find that the induced metric is given by

σAB dθAdθB = ρ2 dθ2 +Σ

ρ2sin2 θ dψ2,

so that√

σ d2θ =√

Σ sin θ dθdψ = (r+2 + a2) sin θ dθdψ. Integration

yieldsA = 4π(r+

2 + a2). (5.3.40)

The algebraic relation, which was discovered by Larry Smarr in 1973,reads

M = 2 ΩHJ +κA

4π, (5.3.41)

where ΩH is the hole’s angular velocity and κ its surface gravity. Smarr’sformula is established by straightforward algebra: Substituting Eqs. (5.3.12),(5.3.37), and (5.3.40) into the right-hand side of Eq. (5.3.41) revealsthat it is indeed equal to M . We will generalize Smarr’s formula, andpresent an alternative derivation, in Sec. 5.5.2.

5.3.13 Variation law

It is clear that the surface area of a black hole is a function of its massand angular momentum: A = A(M, J). Suppose that a black hole ofmass M and angular momentum J is perturbed so that its parametersevolve to M + δM and J + δJ . (For example, the black hole mightabsorb a particle, as was considered in Sec. 5.3.5.) How does the areachange? There exists a simple formula relating δA to the changes inmass and angular momentum. It is

κ

8πδA = δM − ΩH δJ. (5.3.42)

To derive this we start with Eq. (5.3.40), which immediately implies

δA

8π= r+ δr+ + a δa.

But the horizon radius r+ depends on M and a; the defining relationis r+

2 − 2Mr+ + a2 = 0, and this gives us

(r+ −M) δr+ = r+ δM − a δa.

196 Black holes

This result can be substituted into the preceding expression for δA.The final step is to relate a to the hole’s angular momentum J ; wehave that a = J/M , and this implies M δa = δJ − a δM . Combiningthese results, we arrive at Eq. (5.3.42) after involving Eqs. (5.3.12) and(5.3.37).

In Sec. 5.3.5 we found that the right-hand side of Eq. (5.3.42) mustbe positive. What we have, therefore, is the statement that the surfacearea of a Kerr black hole always increases during a process by whichit absorbs a particle. This is a restricted version of the second law ofblack-hole mechanics, to which we shall return in Sec. 5.5.4.

5.4 General properties of black holes

The Kerr family of solutions to the Einstein field equations plays anextremely important role in the description of black holes, but thisdoes not mean that all black holes are Kerr black holes. For example,a black hole accreting matter is not stationary, and a stationary holeis not a Kerr black hole if it is tidally distorted by nearby masses. Inthis section we consider those properties of black holes that are quitegeneral, and not specific to any particular solution to the Einstein fieldequations.

5.4.1 General black holes

A spacetime containing a black hole possesses two distinct regions,the interior and exterior of the black hole; they are distinguished bythe property that all external observers are causally disconnected fromevents occurring inside. Physically speaking, this corresponds to thefact that once she has entered the black hole, an observer can no longersend signals to the outside world.

These fundamental notions can be cast in mathematical terms. Con-sider an event p and the set of all events that can be reached from pby future-directed curves, either timelike or null (Fig. 5.15). This setis denoted J+(p), and is called the causal future of p. A similar defini-tion can be given for its causal past, J−(p). These definitions can beextended to whole sets of events: If S is such a set, then J+(S) is theunion of the causal futures of all the events p contained in S; a similardefinition can be given for J−(S).

Loosely speaking, a spacetime contains a black hole if there existnull geodesics that never reach future null infinity, denoted +. Theseoriginate from the black-hole interior, a region characterized by thevery fact that all future-directed curves starting from it fail to reach+. Thus, events lying within the black-hole interior cannot be in thecausal past of +. The black-hole region B of the spacetime manifold is

5.4 General properties of black holes 197

therefore the set of all events p that do not belong to the causal pastof future null infinity:

B = − J−(+). (5.4.1)

The event horizon H is then defined to be the boundary of the black-hole region:

H = ∂B = ∂(J−(+)). (5.4.2)

The two-dimensional surface obtained by intersecting the event horizonwith a spacelike hypersurface Σ is denoted ; it is called a cross sectionof the horizon.

Because the event horizon is a causal boundary, it must be a null hy-persurface. Penrose (1968) was able to establish that the event horizonis a null hypersurface generated by null geodesics that have no futureend points. This means that: (i) when followed into the past, a gener-ator may, but does not have to, leave the horizon; (ii) once a generatorhas entered the horizon, it cannot leave; (iii) two generators can neverintersect, except possibly when they both enter the horizon; and fi-nally, (iv) through every point on the event horizon, except for those atwhich new generators enter, there passes one and only one generator. Itshould be clear that the entry points into the event horizon are causticsof the congruence of null generators (Fig. 5.16).

The black-hole region typically contains trapped surfaces, closedtwo-surfaces S with the property that for both ingoing and outgoingcongruences of null geodesics orthogonal to S, the expansion is neg-ative everywhere on S. (Exceptions are the extreme cases of Kerr,Kerr-Newman, or Reissner-Nordstrom black holes, which do not con-tain any trapped surfaces.) The three-dimensional boundary of theregion of spacetime that contains trapped surfaces — the trapped re-gion — is the trapping horizon, and its two-dimensional intersectionwith a spacelike hypersurface Σ is called an apparent horizon. Theapparent horizon is therefore a marginally trapped surface — a closedtwo-surface on which one of the congruences has a zero expansion. Theapparent horizon of a stationary black hole typically coincides with the

Figure 5.15: Causal future and past of an event p.

J−(p)

p

J+(p)

198 Black holes

event horizon. In dynamical situations, however, the apparent hori-zon always lies within the black-hole region (Fig. 5.16), unless the nullenergy condition is violated. (Refer back to Sec. 5.1.8 for a specificexample.)

The presence of trapped surfaces inside a black hole unequivocallyannounces the formation of a singularity; this is the content of the beau-tiful singularity theorems of Penrose (1965) and Hawking and Penrose(1970). The theorems rely on some form of energy condition (null forPenrose’s original formulation, strong for the Hawking-Penrose exten-sion) and require additional technical assumptions. The nature of the“singularity” predicted by the theorems is rather vague: The singu-larity is revealed by the presence inside the black hole of at least oneincomplete timelike or null geodesic, but the physical reason for in-completeness is not identified. In all known examples satisfying theconditions of the theorems, however, the black hole contains a curva-ture singularity at which the Riemann tensor diverges.

5.4.2 Stationary black holes

It was established by Hawking in 1972 that if a black hole is stationary,then it must be either static or axially symmetric. This means that thespacetime of a (stationary) rotating hole is necessarily axially symmet-ric, and that it must admit two Killing vectors, tα and φα. Hawkingwas also able to show that a linear combination of these vectors,

ξα = tα + ΩHφα, (5.4.3)

is null at the event horizon. Here, ΩH is the hole’s angular velocity,which vanishes if the spacetime is nonrotating (and therefore static).

Figure 5.16: Event and apparent horizons of a black-hole spacetime.

caustic

surface

BH

EH

AH

EH + AH

5.4 General properties of black holes 199

Thus, the event horizon is a Killing horizon, and ξα is tangent to thehorizon’s null generators: ξα = dxα/dv, with v denoting the (non-affine) parameter on the geodesics. The hole’s surface gravity κ is thendefined by the relation

ξα;βξβ = κξα, (5.4.4)

which holds on the horizon. We will prove in Sec. 5.5.2 that κ is constantalong the horizon’s null generators. (Indeed, it is uniform over the entirehorizon.) This means that we can replace v by an affine parameterλ = V/κ, where V = eκv (Sec. 5.3.10). Then

kα ≡ V −1ξα (5.4.5)

satisfies the geodesic equation in affine-parameter form. It follows thatif κ 6= 0 and the horizon is geodesically complete (in the sense that itsgenerators never leave the horizon when followed into the past), thenthere exists a two-surface, called the bifurcation two-sphere, on whichξα = 0.

The properties of stationary black holes listed here were all encoun-tered before, during our description of the Kerr solution. It should beappreciated, however, that Eqs. (5.4.3)–(5.4.5) hold by virtue of thesole fact that the black hole is stationary; these results do not dependon the specific details of a particular metric.

The observation that a stationary black hole must be axially sym-metric if it is rotating seems puzzling. After all, it should be possible toplace a nonsymmetrical distribution of matter outside the hole, and letit tidally distort the event horizon in a nonsymmetrical manner. This,presumably, would produce a black hole that is still stationary and ro-tating, but not axially symmetric. Hawking and Hartle (1972) haveshown that this, in fact, is false! The reason is that such a distributionof matter would impart a torque on the black hole, which would forceit to spin down to a nonrotating (and static) configuration. Thus, sucha situation would not leave the black hole stationary.

Additional properties of stationary black holes can be inferred fromRaychaudhuri’s equation,

dθ

dλ= −1

2θ2 − σαβσαβ −Rαβkαkβ, (5.4.6)

in which we have put ωαβ = 0 to reflect the fact that the congruence ofnull generators is necessarily hypersurface orthogonal. The event hori-zon will be stationary if θ and dθ/dλ are both zero. Using the Einsteinfield equations and the null energy condition, Eq. (5.4.6) implies thatthe stress-energy tensor must satisfy

Tαβξαξβ = 0 (5.4.7)

200 Black holes

on the horizon. This means that matter cannot be flowing across theevent horizon; if it were, the generators would get focused and the blackhole would not be stationary. Raychaudhuri’s equation also implies

σαβ = 0; (5.4.8)

the null generators of the event horizon have a vanishing shear tensor.

5.4.3 Stationary black holes in vacuum

In the absence of any matter in their exterior, stationary black holesadmit an extremely simple description.

If the black hole is static, then it must be spherically symmetric, andit can only be described by the Schwarzschild solution. This beautifuluniqueness theorem, the first of its kind, was established by WernerIsrael in 1967. It implies that in the absence of angular momentum,complete gravitational collapse must result in a Schwarzschild blackhole. This seems puzzling, because the statement is true irrespectiveof the initial shape of the progenitor, which might have been stronglynonspherical. The mechanism by which a nonspherical star shakes offits higher multipole moments during gravitational collapse was eluci-dated by Richard Price in 1972: These multipole moments are simplyradiated away, either out to infinity or into the black hole. After theradiation has faded away, the hole settles down to its final, sphericalstate.

If the black hole is axially symmetric, then it must be a Kerr blackhole. This extension of Israel’s uniqueness theorem was established byBrandon Carter (1971) and D.C. Robinson (1975).

The black-hole uniqueness theorems can be generalized to includesituations in which the black hole carries an electric charge. If the blackhole is static, then it must be a Reissner-Nordstrom black hole (Israel,1968). If it is axially symmetric, then it must be a Kerr-Newman blackhole (Mazur, 1982; Bunting, unpublished).

We see that a black hole in isolation can be described, uniquely andcompletely, by just three parameters: its mass, angular momentum, andcharge. No other parameter is required, and this remarkable propertyis at the origin of John A. Wheeler’s famous phrase, “a black hole hasno hair”. Chandrasekhar (1987) was well justified to write:

Black holes are macroscopic objects with masses varying from afew solar masses to millions of solar masses. To the extentthat they may be considered as stationary and isolated, tothat extent, they are all, every single one of them, describedexactly by the Kerr solution. This is the only instance wehave of an exact description of a macroscopic object. Macro-scopic objects, as we see them all around us, are governed

5.5 The laws of black-hole mechanics 201

by a variety of forces, derived from a variety of approxi-mations to a variety of physical theories. In contrast, theonly elements in the construction of black holes are our ba-sic concepts of space and time. They are, thus, almost bydefinition, the most perfect macroscopic objects there arein the universe. And since the general theory of relativityprovides a single unique two-parameter family of solutionsfor their descriptions, they are the simplest objects as well.

5.5 The laws of black-hole mechanics

In 1973, Jim Bardeen, Brandon Carter, and Stephen Hawking formu-lated a set of four laws governing the behaviour of black holes. Theselaws of black-hole mechanics bear a striking resemblance to the fourlaws of thermodynamics. While this analogy was at first perceived tobe purely formal and coincidental, it soon became clear that black holesdo indeed behave as thermodynamic systems. The crucial step in thisrealization was Hawking’s remarkable discovery of 1974 that quantumprocesses allow a black hole to emit a thermal flux of particles. It isthus possible for a black hole to be in thermal equilibrium with otherthermodynamic systems. The laws of black-hole mechanics, therefore,are nothing but a description of the thermodynamics of black holes.

5.5.1 Preliminaries

We begin our discussion of the four laws by collecting a few impor-tant results from preceding chapters; these will form the bulk of themathematical framework required for the derivations.

Let ya = (v, θA) be coordinates on the event horizon. The advanced-time coordinate v is a non-affine parameter on the horizon’s null gen-erators, and θA spans the two-dimensional space transverse to the gen-erators. The vectors

ξα =(∂xα

∂v

)θA

, eαA =

(∂xα

∂θA

)v

(5.5.1)

are tangent to the horizon; they satisfy ξαeαA = 0 = £ξe

αA, and ξα =

tα +ΩHφα is a Killing vector. We complete the basis by introducing anauxiliary null vector Nα, normalized by Nαξα = −1. This basis givesus the completeness relations (Sec. 3.1)

gαβ = −ξαNβ −Nαξβ + σABeαAeβ

B,

where σAB is the inverse of σAB = gαβ eαAeβ

B. The determinant of thetransverse two-metric will be denoted σ.

202 Black holes

The vectorial surface element on the event horizon can be expressedas (Sec. 3.2)

dΣα = −ξα dS dv, (5.5.2)

where dS =√

σ d2θ. The two-dimensional surface element on a crosssection v = constant is

dSαβ = 2ξ[αNβ] dS. (5.5.3)

We shall denote such a cross section by .Finally, we will need Raychaudhuri’s equation for the congruence of

null generators, expressed in a form that does not require the parameterto be affine. This was worked out in Sec. 2.6, Problem 8, and the answeris

dθ

dv= κ θ − 1

2θ2 − σαβσαβ − 8πTαβξαξβ; (5.5.4)

the last term would normally involve the Ricci tensor, but we haveused the Einstein field equations to write it in terms of the stress-energy tensor. We recall that θ is the fractional rate of change of thecongruence’s cross-sectional area: θ = (dS)−1d(dS)/dv.

5.5.2 Zeroth law

The zeroth law of black-hole mechanics states that the surface gravityof a stationary black hole is uniform over the entire event horizon. Wesaw in Sec. 5.3.10 that this statement is indeed true for the specific caseof a Kerr black hole, but the scope of the zeroth law is much wider: theblack hole need not be isolated, and its metric need not be the Kerrmetric.

To prove that κ is uniform on the event horizon, we need to establishthat (i) κ is constant along the horizon’s null generators, and (ii) κ doesnot vary from generator to generator. We will prove both statementsin turn, starting with

κ2 = −1

2ξα;βξα;β (5.5.5)

as our definition for the surface gravity. (We saw in Sec. 5.2.4 that thisrelation is equivalent to ξα

;βξβ = κξα.) We shall need the identity

ξα;µν = Rαµνβξβ, (5.5.6)

which is satisfied by any Killing vector ξα. (This was derived in Sec. 1.13,Problem 9.)

We differentiate Eq. (5.5.5) in the directions tangent to the horizon.(Because κ is defined only on the event horizon, its normal derivativedoes not exist.) Using Eq. (5.5.6), we obtain

2κκ,α = −ξµ;νRµναβξβ. (5.5.7)

5.5 The laws of black-hole mechanics 203

The fact that κ is constant along the generators follows immediatelyfrom this:

κ,αξα = 0. (5.5.8)

We must now examine how κ changes in the transverse directions.Equation (5.5.7) implies

2κκ,α eαA = −ξµ;νRµναβ eα

Aξβ,

and we would like to show that the right-hand side is zero. Let usfirst assume that the event horizon is geodesically complete, so thatit contains a bifurcation two-sphere, at which ξα = 0. Then the lastequation implies that κ,αeα

A = 0 at the bifurcation two-sphere. Becauseκ,αeα

A is constant on the null generators (Sec. 5.7, Problem 6), we havethat κ,αeα

A = 0 on every cross section v = constant of the event hori-zon. This shows that the value of κ does not change from generatorto generator, and we conclude that κ is uniform over the entire eventhorizon.

It is easy to see that the property κ,αeαA = 0 must be independent of

the existence of a bifurcation two-sphere. Consider two stationary blackholes, identical in every respect in the future of v = 0 (say), but differentin the past, so that only one of them possesses a bifurcation two-sphere.(We imagine that the first black hole has existed forever, and that thesecond black hole was formed prior to v = 0 by gravitational collapse;the second black hole is stationary only for v > 0.) Our proof thatκ,αeα

A = 0 on all cross sections v = constant of the event horizon appliesto the first black hole. But since the spacetimes are identical for v > 0,the property κ,αeα

A = 0 must apply also to the second black hole. Thus,the zeroth law is established for all stationary black holes, whether ornot they are geodesically complete.

It is clear that the relation ξµ;νRµναβ eαAξβ = 0 must hold every-

where on a stationary event horizon, but it is surprisingly difficult toprove this. In their original discussion, Bardeen, Carter, and Hawk-ing establish this identity by using the Einstein field equations and thedominant energy condition (Sec. 5.7, Problem 7). This restriction waslifted in a 1996 paper by Racz and Wald.

5.5.3 Generalized Smarr formula

Before moving on to the first law, we generalize Smarr’s formula (Sec. 5.3.12)that relates the black-hole mass M to its angular momentum J , angu-lar velocity ΩH , surface gravity κ, and surface area A. In the presentcontext, the black hole is stationary and axially symmetric, but it isnot assumed to be a Kerr black hole.

204 Black holes

Figure 5.17: A spacelike hypersurface in a black-hole spacetime.

−

+

SΣ

Our starting point is the Komar expressions for total mass andangular momentum (Sec. 4.3.3):

M = − 1

8π

∮

S

∇αtβ dSαβ, J =1

16π

∮

S

∇αφβ dSαβ,

where the integrations are over a closed two-surface at infinity. Weconsider a spacelike hypersurface Σ extending from the event horizonto spatial infinity (Fig. 5.17). Its inner boundary is , a two-dimensionalcross section of the event horizon, and its outer boundary is S. Us-ing Gauss’ theorem, as was done in Sec. 4.3.3 (but without the innerboundary), we find that M and J can be expressed as

M = MH + 2

∫

Σ

(Tαβ − 1

2Tgαβ

)nαtβ

√h d3y (5.5.9)

and

J = JH −∫

Σ

(Tαβ − 1

2Tgαβ

)nαφβ

√h d3y, (5.5.10)

where MH and JH are the black-hole mass and angular momentum,respectively. They are given by surface integrals over :

MH = − 1

8π

∮∇αtβ dSαβ (5.5.11)

and

JH =1

16π

∮∇αφβ dSαβ, (5.5.12)

where dSαβ is the surface element of Eq. (5.5.3). The interpretation ofEqs. (5.5.9) and (5.5.10) is clear: the total mass M (angular momentumJ) is given by a contribution MH (JH) from the black hole, plus acontribution from the matter distribution outside the hole. If the blackhole is in vacuum, then M = MH and J = JH .

5.5 The laws of black-hole mechanics 205

Smarr’s formula emerges after a few simple steps. Using Eqs. (5.5.3),(5.5.11) and (5.5.12), we have

MH − 2 ΩHJH = − 1

8π

∮∇α(tβ + ΩHφβ) dSαβ

= − 1

8π

∮∇αξβ dSαβ

= − 1

4π

∮ξβ;αξαNβ dS

= − 1

4π

∮κξβNβ dS

=κ

4π

∮dS,

where we have used the relation ξαNα = −1 and the fact that κ isconstant over . The last integration gives the horizon’s surface area,and we arrive at

MH = 2 ΩHJH +κA

4π, (5.5.13)

the generalized Smarr formula.

5.5.4 First law

We consider a quasi-static process during which a stationary black holeof mass M , angular momentum J , and surface area A is taken to a newstationary black hole with parameters M + δM , J + δJ , and A + δA.The first law of black-hole mechanics states that the changes in mass,angular momentum, and surface area are related by

δM =κ

8πδA + ΩH δJ. (5.5.14)

If the initial and final black holes are in vacuum, then they are Kerrblack holes by virtue of the uniqueness theorems, and a derivation ofEq. (5.5.14) was already presented in Sec. 5.3.13. That derivation,however, relied heavily on the details of the Kerr metric. We shallnow present a derivation that is quite insensitive to those details. Inparticular, we shall not assume that the black hole is in vacuum.

We suppose that a black hole, initially in a stationary state, isperturbed by a small quantity of matter described by the (infinitesimal)stress-energy tensor Tαβ. As a result, the mass and angular momentumof the black hole increase by amounts (Sec. 4.3.4)

δM = −∫

H

Tαβ tβ dΣα (5.5.15)

and

δJ =

∫

H

T αβ φβ dΣα, (5.5.16)

206 Black holes

where the integrations are over the entire event horizon. We will beworking to first order in the perturbation Tαβ, keeping tα, φα, and dΣα

at their unperturbed values. We assume that at the end of the process,the black hole is returned to another stationary state.

Substituting the surface element of Eq. (5.5.2) into Eqs. (5.5.15)and (5.5.16), we find

δM − ΩHδJ =

∫

H

Tαβ(tβ + ΩHφβ)ξα dS dv

=

∫dv

∮Tαβξαξβ dS.

To work out the integral, we turn to Raychaudhuri’s equation, Eq. (5.5.4).Because θ and σαβ are quantities of the first order in Tαβ, it is appro-priate to neglect the quadratic terms, so that we have

dθ

dv= κ θ − 8πTαβξαξβ,

and

δM − ΩHδJ = − 1

8π

∫dv

∮(dθ

dv− κ θ

)dS

= − 1

8π

∮θ dS

∣∣∣∞

−∞+

κ

8π

∫dv

∮θ dS.

Because the black hole is stationary both before and after the perturba-tion, θ(v = ±∞) = 0, and the boundary terms vanish. Using the factthat θ is the fractional rate of change of the congruence’s cross-sectionalarea, we obtain

δM − ΩHδJ =κ

8π

∫dv

∮( 1

dS

d

dvdS

)dS

=κ

8π

∮dS

∣∣∣∞

−∞

=κ

8πδA,

where δA is the change in the black hole’s surface area. This is Eq. (5.5.14),the statement of the first law of black-hole mechanics.

5.5.5 Second law

The second law of black-hole mechanics states that if the null energycondition is satisfied, then the surface area of a black hole can never de-crease: δA ≥ 0. This area theorem was established by Stephen Hawkingin 1971.

5.5 The laws of black-hole mechanics 207

Glossing over various technical details, the area theorem follows di-rectly from the focusing theorem (Sec. 2.4.5) and Penrose’s observationthat the event horizon is generated by null geodesics with no future endpoints. This statement means that the generators of the event horizoncan never run into caustics. (A generator can enter the horizon at acaustic point, but once in H it will never again meet another caustic.)The focusing theorem then implies that θ, the expansion of the congru-ence of null generators, must be positive, or zero, everywhere on theevent horizon. To see this, suppose that θ < 0 for some of the genera-tors. The focusing theorem then guarantees that these generators willconverge into a caustic, at which θ = −∞. We have a contradiction,and we must conclude that θ ≥ 0 everywhere on the event horizon.This implies that the horizon’s surface area will not decrease, which isjust the statement of the area theorem. (The fact that new generatorscan enter the event horizon contributes even further to the growth ofits area.)

5.5.6 Third law

The third law of black-hole mechanics states that if the stress-energytensor is bounded and satisfies the weak energy condition, then thesurface gravity of a black hole cannot be reduced to zero within a finiteadvanced time. A precise formulation of this law was given by WernerIsrael in 1986.

We have seen that a black hole of zero surface gravity is an ex-treme black hole. (Recall that a Kerr black hole is extremal if a = M ;for a Reissner-Nordstrom black hole, the condition is |Q| = M .) Anequivalent statement of the third law is therefore that under the statedconditions on the stress-energy tensor, it is impossible for a black holeto become extremal within a finite advanced time.

The proof of the third law is rather involved, and we will not attemptto go through it here. Instead of presenting a proof, we will illustratethe fact that the third law is essentially a consequence of the weakenergy condition.

For the purpose of this discussion, we need a black-hole spacetimewhich is sufficiently dynamical that it has the potential of becomingextremal at a finite advanced time v. A simple choice is the chargedgeneralization of the ingoing Vaidya spacetime, whose metric is givenby

ds2 = −f dv2 + 2 dvdr + r2 dΩ2, (5.5.17)

with

f = 1− 2m(v)

r+

q2(v)

r2. (5.5.18)

This metric describes a black hole whose mass m and charge q changewith time because of irradiation by charged null dust, a fictitious form

208 Black holes

of matter. This interpretation is confirmed by inspection of the stress-energy tensor,

Tαβ = T αβdust + Tαβ

em , (5.5.19)

where

Tαβdust = ρ lαlβ, ρ =

1

4πr2

∂

∂v

(m− q2

2r

)(5.5.20)

is the contribution from the null dust (lα = −∂αv is a null vector), and

T αem β = P diag(−1,−1, 1, 1), P =

q2

8πr4(5.5.21)

is the contribution from the electromagnetic field. The spacetime ofEqs. (5.5.17) and (5.5.18) will produce a violation of the third law ifm(v0) = q(v0) for some advanced time v0 < ∞.

An essential aspect of this discussion is the weak energy condition(Sec. 2.1), which states that the energy density measured by an observerwith four-velocity uα = dxα/dτ will always be positive:

Tαβuαuβ > 0.

Here, Tαβ is the stress-energy tensor of Eq. (5.5.19). If our observeris restricted to move in the radial direction only, then Tαβuαuβ =ρ(dv/dτ)2 + P . Because dv/dτ can be arbitrarily large, the weak en-ergy condition requires ρ > 0. In particular, ρ must be positive at theapparent horizon, r = r+(v), where r+ = m + (m2 − q2)1/2. This givesus the following condition:

4πr+3ρ(r+) = mm− qq + (m2 − q2)1/2 m > 0, (5.5.22)

where an overdot indicates differentiation with respect to v.Let us imagine a situation in which the black hole becomes extremal

at a finite advanced time v0. This means that ∆(v0) = 0, where ∆(v) ≡m(v) − q(v). Because the black hole was not extremal before v = v0,we have that ∆(v) > 0 for v < v0, and ∆(v) must be decreasing as vapproaches v0. However, Eq. (5.5.22) implies

m(v0) ∆(v0) > 0,

according to which ∆(v) must be increasing. We have a contradiction,and we conclude that the weak energy condition prevents the black holefrom ever becoming extremal at a finite advanced time.

5.5.7 Black-hole thermodynamics

The four laws of black-hole mechanics bear a striking resemblance tothe laws of thermodynamics, with κ playing the role of temperature,A that of entropy, and M that of internal energy. Hawking’s discovery

5.6 Bibliographical notes 209

that quantum processes give rise to a thermal flux of particles fromblack holes implies they do indeed behave as thermodynamic systems.Black holes have a well-defined temperature, which as a matter of factis proportional to the hole’s surface gravity:

T =~2π

κ. (5.5.23)

The zeroth law is therefore a special case of the corresponding law ofthermodynamics, which states that a system in thermal equilibrium hasa uniform temperature. The first law, when recognized as a special caseof the corresponding law of thermodynamics, implies that the black-hole entropy must be given by

S =1

4~A. (5.5.24)

The second law is therefore also a special case of the corresponding lawof thermodynamics, which states that the entropy of an isolated systemcan never decrease. In this regard it should be noted that Hawkingradiation actually causes the black-hole area to decrease, in violationof the area theorem. (The radiation’s stress-energy tensor does notsatisfy the null energy condition.) However, the process of black-holeevaporation does not violate the generalized second law, which statesthat the total entropy, the sum of radiation and black-hole entropies,does not decrease.

The fact that black holes behave as thermodynamic systems revealsa deep connection between such disparate fields as gravitation, quan-tum mechanics, and thermodynamics. This connection is still poorlyunderstood today.

5.6 Bibliographical notes

During the preparation of this chapter I have relied on the followingreferences: Bardeen, Carter, and Hawking (1973); Carter (1979); Chan-drasekhar (1983); Hayward (1994); Israel (1986a); Israel (1986b); Sul-livan and Israel (1980); Misner, Thorne, and Wheeler (1973); Wald(1984); and Wald (1992).

More specifically:The term “trapping horizon”, used in Secs. 5.1.7 and 5.4.1, was in-

troduced by Sean Hayward in his 1994 paper. The various definitionsfor the surface gravity (Secs. 5.2.4, 5.3.10, 5.4.2, and 5.5.2) are takenfrom Sec. 12.5 of Wald (1984). The presentation of the Kerr black holeis based on Secs. 33.1–5 of Misner, Thorne, and Wheeler, and Secs. 57and 58 of Chandrasekhar. The definitions for black-hole region andevent horizon are taken from Sec. 12.1 of Wald (1984); trapped sur-faces and apparent horizons are defined in Wald’s Sec. 9.5 and 12.2,

210 Black holes

respectively. Penrose’s theorem on the structure of the event horizon(Sec. 5.4.1) is very nicely discussed in Sec. 34.4 of Misner, Thorne,and Wheeler. Section 9.5 of Wald (1984) provides a thorough discus-sion of the singularity theorems. The general properties of stationaryblack holes (Sec. 5.4.2) are discussed in Sec. 12.3 of Wald (1984) andSec. 6.3.1 of Carter. An overview of the uniqueness theorems of black-hole spacetimes (Sec. 5.4.3) can be found in Sec. 12.3 of Wald (1984)and Sec. 6.7 of Carter. In Sec. 5.5, the proofs of the zeroth and firstlaws are taken from Wald’s 1992 Erice lectures. The generalized Smarrformula is derived in Sec. 6.6.1 of Carter. The discussion of the secondlaw is adapted from Sec. 6.1.2 of Carter. The final form of the thirdlaw was given in Israel (1986b); my discussion is based on Sullivan andIsrael. Finally, in the problems below, the material on the Majumdar-Papapetrou solution is taken from Sec. 113 of Chandrasekhar, the de-scription of null-dust collapse is adapted from Israel (1986a), and thealternative derivation of the zeroth law is based on Bardeen, Carter,and Hawking.

5.7 Problems

1. The metric of an extreme (Q = ±M) Reissner-Nordstrom blackhole is given by

ds2 = −(1− M

r

)2

dt2 +(1− M

r

)−2

dr2 + r2 dΩ2.

a) Find an appropriate set of Kruskal coordinates for this space-time.

b) Show that the region r ≤ M does not contain trapped sur-faces.

c) Sketch a Penrose-Carter diagram for this spacetime.

d) Find a coordinate transformation that brings the metric tothe form

ds2 = −(1 +

M

r

)−2

dt2 +(1 +

M

r

)2

(dx2 + dy2 + dz2),

where r2 = x2 + y2 + z2. Show that in these coordinates,the electromagnetic field tensor can be generated from thevector potential Aα dxα = ∓(1 + M/r)−1 dt, in which theupper (lower) sign gives rise to a positive (negative) electriccharge.

e) Show that the metric

ds2 = −Φ−2 dt2 + Φ2(dx2 + dy2 + dz2)

5.7 Problems 211

and the vector potential Aα dxα = ∓Φ−1 dt produce an ex-act solution to the Einstein-Maxwell equations provided thatΦ(x) satisfies Laplace’s equation ∇2Φ = 0. Here, ∇2 isthe usual Laplacian operator of three-dimensional flat space,and x = (x, y, z). This metric is known as the Majumdar-Papapetrou solution. Prove that if the spacetime is asymp-totically flat, then the total charge Q and the ADM mass Mare related by Q = ±M . Finally, find an expression for Φ(x)that corresponds to a collection of N black holes situated atarbitrary positions xn (n = 1, 2, · · · , N).

2. A black hole is formed by the gravitational collapse of null dust.During the collapse, the metric is given by an ingoing Vaidyasolution with mass function m(v) = v/16. Spacetime is assumedto be flat before the collapse (v < 0), and after the collapse(v > v0), the metric is given by a Schwarzschild solution withmass m0 = m(v0) = v0/16. We want to study various propertiesof this spacetime.

a) Show that in the interval 0 < v < v0, outgoing light rays aredescribed by the parametric equations

r(λ) = c λ e−λ, v(λ) = 4c (1 + λ) e−λ,

where c is a constant. Show that v = 4r also describes anoutgoing light ray. Plot a few of these curves in the (v, r)plane, using both positive and negative values of c. Plot alsothe position of the apparent horizon.

b) Find the parametric equations that describe the event horizon.

c) Prove that the curvature singularity at r = 0 is naked, inthe sense that it is visible to observers at large distances.Prove also that at the moment it is visible, the singularityis massless. [It is generally true that the central singularityof a spherical collapse must be massless if it is naked. Thiswas established by Lake (1992).]

3. In this problem we have a closer look at the instability of black-hole tunnels, a topic that was mentioned briefly in Secs. 5.2.3 and5.3.9. We will see that the instability is caused by the pathologicalbehaviour of the ingoing branch (v = ∞) of the inner horizon(r = r−). For reasons that will become clear, we shall call thisthe Cauchy horizon of the black-hole spacetime. For simplicity,we shall restrict our attention to the Reissner-Nordstrom (RN)spacetime. [The physics of the Cauchy-horizon instability wasthis author’s Ph.D. topic; see Poisson and Israel (1990). The

212 Black holes

book by Burko and Ori (1997) presents a rather complete reviewof this fascinating part of black-hole physics.]

a) Consider an event P located anywhere in the future of theCauchy horizon. Argue that the conditions at P are notuniquely determined by initial data placed on a spacelikehypersurface Σ located outside the black hole. Then arguethat the Cauchy horizon is the boundary of the region ofspacetime for which the evolution of this data is unique.(This region is called the domain of dependence of Σ, andwe say that the Cauchy problem of general relativity is wellposed in this region. The Cauchy horizon is the place atwhich the evolution ceases to be uniquely determined by theinitial data; the Cauchy problem breaks down. In effect, thepredictive power of the theory is lost at the Cauchy horizon.)

b) Consider a test null fluid in RN spacetime, with stress-energytensor Tαβ = ρ lαlβ, where ρ is the energy density andlα = −∂αv the four-velocity. The fluid moves parallel to theCauchy horizon, along ingoing null geodesics. Prove that ρmust be of the form

ρ =L(v)

4πr2,

where L(v) is an arbitrary function of advanced time v. Showthat if a finite quantity of energy is to enter the black hole,then L → 0 as v →∞. (How fast must L vanish?) Typically,radiative fields outside black holes decay in time accordingto an inverse power law (Price 1972). We shall therefore takeL(v) ∼ v−p as v →∞, with p larger than, say, 2.

c) Consider now a free-falling observer inside the RN black hole.This observer moves in the outward radial direction, encoun-ters the null dust, and measures its energy density to beTαβ uαuβ, where uα is the observer’s four velocity. Showthat as the observer crosses the Cauchy horizon,

Tαβ uαuβ =E2

4πr−2L(v) e2κ−v,

where E = −uαtα and κ− was defined in Sec. 5.2.2. Con-clude that the measured energy density diverges at the Cauchyhorizon, even though the total amount of energy entering thehole is finite. This is the pathology of the Cauchy horizon,which ultimately is responsible for the instability of black-hole tunnels.

4. The equations governing geodesic motion in the Kerr spacetimewere given without justification in Sec. 5.3.6. Here we provide a

5.7 Problems 213

derivation, which is valid both for timelike and null geodesics.[The general form of the geodesic equations can be found inSec. 33.5 of Misner, Thorne, and Wheeler (1973).]

a) By definition, a Killing tensor field ξαβ is one which satisfiesthe equation ξ(αβ;γ) = 0. Show that if ξαβ is a Killing tensorand uα satisfies the geodesic equation (uα

;βuβ = 0), then

ξαβ uαuβ is a constant of the motion.

b) Verify thatξαβ = ∆ k(αlβ) + r2gαβ

is a Killing tensor of the Kerr spacetime. Here, kα and lα

are the null vectors defined in Sec. 5.3.6.

b) Write the relations E = −tαuα and L = φαuα explicitly interms of uα = (t, r, θ, φ). Then invert these relations toobtain the equations for t and φ. [Hint: Make sure to involvethe inverse metric.]

c) The Carter constant is defined by

ξαβ uαuβ = + (L− aE)2.

By working out the left-hand side, derive the equation for r.[Hint: Express kα and lα in terms of the Killing vectors, andthen ξαβ uαuβ in terms of E, L, and r2.]

d) Finally, use the normalization condition gαβuαuβ = −ζ (whereζ = 1 for timelike geodesics and ζ = 0 for null geodesics) toobtain the equation for θ.

5. Let lα be a null, geodesic vector field in flat spacetime. With thisvector and an arbitrary scalar function H we construct a newmetric tensor gαβ:

gαβ = ηαβ + Hlαlβ,

where ηαβ is the Minkowski metric and lα = ηαβlβ. Such a metricis called a Kerr-Schild metric.

a) Show that lα is null with respect to both metrics.

b) Show that gαβ = ηαβ −Hlαlβ is the inverse metric.

c) Prove that lα = gαβlβ and lα = gαβlβ. Thus, indices on thenull vector can be lowered and raised with either metric.

d) Calculate the Christoffel symbols for gαβ. Show that theysatisfy the relations

lµΓµαβ = −1

2Hlαlβ, lµΓα

µβ =1

2Hlαlβ,

where H ≡ H,µlµ.

214 Black holes

e) Prove that lα;βlβ = 0. Thus, lα is a geodesic vector field inboth metrics.

f) Prove that the component Rαβlαlβ of the Ricci tensor vanishesfor any choice of function H.

6. Complete the discussion of the zeroth law by proving that κ,α eαA

is constant along the null generators of a stationary event horizon.

7. In this problem we provide an alternative derivation of the zerothlaw of black-hole mechanics. This derivation is based on the origi-nal presentation by Bardeen, Carter, and Hawking (1973); it usesthe Einstein field equations and the dominant energy condition.

a) We have seen that the vector ξα is tangent to the null genera-tors of the event horizon. It satisfies the following properties:(i) ξα is null on the horizon; (ii) ξα

;βξβ = κξα on the hori-zon; (iii) ξα is a Killing vector; and (vi) the congruence ofnull generators has zero expansion, shear, and rotation. Usethese facts to infer

ξα;β = (κNα + cAeAα)ξβ − ξα(κNβ + cBeBβ),

where cA ≡ σABξα;βNαeβB. This relation holds on the horizon

only.

b) Prove that the gradient of the surface gravity (in the directionstangent to the horizon) is given by

κ,α = −RαβγδξβNγξδ − (σABcAcB)ξα.

This immediately implies that κ is constant on each genera-tor: κ,αξα = 0.

c) Show that the result of part b) also implies

κ,αeαA = −Rαβ eα

Aξβ − σBCRαβγδ eαAeβ

BeγCξδ.

d) The quantities BAB ≡ ξα;β eαAeβ

B and their tangential deriva-tives must all vanish on the horizon. Use this observation toderive

Rαβγδ eαAeβ

BeγCξδ = 0.

This relations holds on the horizon only.

e) Collecting the results of parts c) and d), use the Einstein fieldequations to write

κ,αeαA = 8πjαeα

A,

where jα = −T αβξβ represents a flux of momentum across

the horizon.

5.7 Problems 215

f) The dominant energy condition states that jα should be eithertimelike or null, and future directed. Use this, together withthe stationary condition Tαβξαξβ = 0, to prove that jα mustbe parallel to ξα. Under these conditions, therefore,

κ,αeαA = 0,

and the zeroth law is established.

8. The unique solution to the Einstein-Maxwell equations describingan isolated black hole of mass M , angular momentum J ≡ aM ,and electric charge Q is known as the Kerr-Newman solution; itwas discovered by Newman et al. in 1965. The Kerr-Newmanmetric can be expressed as

ds2 = −ρ2∆

Σdt2 +

Σ

ρ2sin2 θ(dφ− ω dt)2 +

ρ2

∆dr2 + ρ2 dθ2,

where ρ2 = r2 + a2 cos2 θ, ∆ = r2 − 2Mr + a2 + Q2, Σ = (r2 +a2)2 − a2∆ sin2 θ, and ω = a(r2 + a2 −∆)/Σ. The metric comeswith a vector potential

Aα dxα = −Qr

ρ2(dt− a sin2 θ dφ).

When Q = 0, Aα = 0 and this reduces to the Kerr solution.

a) Find expressions for r+, the radius of the event horizon, andΩH , the angular velocity of the black hole.

b) Prove that the vector field

lα ∂α =r2 + a2

∆∂t − ∂r +

a

∆∂φ

is tangent to a congruence of ingoing null geodesics. Provealso that

v ≡ t +

∫r2 + a2

∆dr

and

ψ ≡ φ +

∫a

∆dr

are constant on each member of the congruence.

c) Show that in the coordinates (v, r, θ, ψ), the Kerr-Newmanmetric takes the form

ds2 = −∆− a2 sin2 θ

ρ2dv2 + 2 dvdr − 2a

r2 + a2 −∆

ρ2sin2 θ dvdψ

− 2a sin2 θ drdψ +Σ

ρ2sin2 θ dψ2 + ρ2 dθ2.

Find an expression for Aα is this coordinate system.

216 Black holes

d) Show that the vectors

ξα ∂α = ∂v + ΩH ∂ψ, eαθ ∂α = ∂θ, eα

ψ ∂α = ∂ψ,

and

Nα ∂α = − a2 sin2 θ

2(r+2 + a2 cos2 θ)

∂v − r+2 + a2

r+2 + a2 cos2 θ

∂r

− a

2(r+2 + a2)

2r+2 + a2(1 + cos2 θ)

r+2 + a2 cos2 θ

∂ψ

form a good basis on the event horizon. In particular, provethat they give rise to the completeness relations gαβ = −ξαNβ−Nαξβ+σAB eα

AeβB, where σAB is the inverse of σAB ≡ gαβ eα

AeβB.

e) Prove that the surface gravity of a Kerr-Newman black holeis given by

κ =r+ −M

r+2 + a2

.

Prove also that the hole’s surface area is

A = 4π(r+2 + a2).

f) Compute the black-hole mass MH and the black-hole angularmomentum JH of a Kerr-Newman black hole. (These quan-tities are defined in Sec. 5.5.3.) Make sure that your resultsare compatible with the following expressions:

MH =r+

2 + a2

2r+

[1− Q2

ar+

arctan(a/r+)

]

and

JH = ar+

2 + a2

2r+

1 +

Q2

2a2

[1− r+

2 + a2

ar+

arctan(a/r+)]

.

Verify that these expressions satisfy the generalized Smarrformula.

g) Derive the following alternative version of Smarr’s formula:

M = 2ΩHJ +κA

4π+ ΦHQ,

where

ΦH ≡ −Aαξα∣∣∣r=r+

=r+Q

r+2 + a2

is the electrostatic potential at the horizon.

5.7 Problems 217

h) Consider a quasi-static process during which a stationary blackhole of mass M , angular momentum J , and electric chargeQ is taken to a new stationary black hole with parametersM + δM , J + δJ , and Q + δQ. Prove that during such atransformation, the hole’s surface area A will change by anamount δA given by

δM =κ

8πδA + ΩH δJ + ΦH δQ.

This is the first law of black-hole mechanics for charged,rotating black holes.

9. Consider a quasi-static process during which the surface area of ablack hole changes. (By quasi-static we mean that dA/dv is verysmall.) Derive the Hawking-Hartle formula,

dA

dv=

8π

κ

∮( 1

8πσαβσαβ + Tαβξαξβ

)dS,

in which ξα = dxα/dv is tangent to the null generators of theevent horizon, and σαβ is their shear tensor. The second termwithin the integral represents the effect of accreting matter on thesurface area. The first term represents the effect of gravitationalradiation flowing across the horizon.

218 Black holes

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