Black Holes

Tom Charnock

Contents

1 Why Study Black Holes? 2

2 Relativity 42.1 The Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Christoffel Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Einstein’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Schwarzschild Geometry 93.1 Exterior Schwarzschild Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Eddington-Finkelstein Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Kruskal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Killing Vectors 13

5 Singularities 15

6 Structure of a Star 166.1 Perfect Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Static Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.2.1 Constant Density Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.3 Birkhoff’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.4 Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.5 Star Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.6 Non-Radial Null Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.7 Spacetime Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6.7.1 1+1 Minkowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1

Chapter 1

Why Study Black Holes?

Black holes exist. Probably. Although they are not directly observable they are believed to be at thecentre of galaxies. The object at the centre of the Milky Way has a mass of approximately 106M, whichis contained in a relatively small radius. This is inferred from the orbits of stars about the galactic centre.Some galaxies have black holes up the mass of ∼ 1010M. There are also individual (collapsed stars)stellar mass dark objects seen as part of a double star.

ys+

Star Dark Partner

X-RaysAccretedMatter

There are ∼ 10 of these systems documented where the dark partners are believed to be black holes. Oneof these systems is Cygnus X-1.

In terms of cosmology then one of the dark matter candidates is the black hole, although this is not themost popular model currently. There may also be mini-black holes which existed in the early universeand influenced the expansion. The problem with this is that there is no known mechanism for creatingblack holes at this period and there has been no observations of them.

Black holes began as a theoretical prediction of General Relativity in 1916 by Schwarzschild. In 1965then the term black hole arose and a proper understanding of them was initiated.

A black hole is an extreme example of curved spacetime which has interesting mathematical properties.One example of this is the singularity. Inside a Schwarzschild black hole then the contraction of theRiemann curvature tensor is RabcdR

abcd → ∞. The tidal forces grow to infinite strengths and so thedescription of spacetime breaks down. Particle physics and other well known models break down at thesingularity and so it is known that Einsteins’ equations are no longer good near the singularity. This canbe rectified by replacing Einsteins’ classical theory of gravity by a quantum theory of gravity.

There is no current quantum theory of gravity. If one was designed then it would need to recover generalrelativity in the correct limit and would also find Hawking Temperature.

Hawking Prediction 1974

Quantum matter is placed on a collapsing star spacetime. At late times then the black hole radiatesthermally. For the Schwarzschild black hole, with mass M , then:

T =~c3

8πMkB

2

This can be seen to be a quantum effect since ~ is present. This is not a quantum gravity but showsthat quantum matter feels the effect of the curvature. For M = M then T ≈ 10−7K which is totallyunobservable for a solar mass black hole. Say if M = 1014kg then T ≈ 109K. This would be visible, butthis mass would be equivalent to a kilometre cubed vessel full of sea water, or a small mountain. Theseare very small and so would be hard to observe and have not been so far.

The hint that Hawkings’ prediction is correct because black holes, which now have a temperature, canbe brought into discussions of thermodynamics. Without Hawking radiation then the second law ofthermodynamics could be violated since entropy would reduce due to black holes.

Hawking radiation arises due to pair creation on the horizon of a black hole so that one of the particlefalls into the black hole and the other escapes to infinity. The particle which falls in can be thought of asbringing with it an energy debt from the Heisenberg uncertainty principle and the black holes pays thedebt by dissipating some of its mass. This dissipation is slow, with the lifetime of the black hole being:

Black Hole Lifetime =(1010years

)( M

1012kg

)3

So for most reasonable sized black holes then this is orders of magnitudes longer than the length of theuniverse. The 1010 years arises from the type of quantum matter being radiated.

3

Chapter 2

Relativity

2.1 The Metric

Spacetime is possibly curved. For Minkowski space then the line element is the distance measure and isgiven by:

ds2 = −c2dt2 + dx2 + dy2 + dz2

c = 1 can be set by choice of natural units. In curved spacetime this is generalised to:

ds2 = gab(x)dxadxb

gab are the components of the metric tensor and is symmetric so gab = gba. This means that it is alwaysdiagonisable at a point in spacetime. It is also non-degenerate so that it has an inverse gab such thatgabgbc = δac . The eigenvalues of gab are (-+++). dxadxb is the symmetric product of differential forms:

gabdxadxb =

1

2gab(dx

a ⊗ dxb + dxb ⊗ dxa)

= gabdxa ⊗ dxb

The dimension of the line element is [ds2] = L2. There is some choice in whether the dimension is inthe coordinate, say dr2 or c2dt2 or in the metric, say the r2 component in front of dϑ2 in a line elementwritten in spherical polars.

Vectors are given by va and one-forms can be obtained using the metric tensor:

va = gabvb and va = gabvb

The vectors and one-forms can be classified by the value of their square. If vava > 0 then the vector va

or one-form va is called spacelike. vava = 0 indicates a null vector or one-form and vava < 0 has vectorsor one-forms called timelike.

2.1.1 Christoffel Connections

Γabc =1

2gad(gbd,c + gcd,b − gbc,d)

These connections are torsion-free and so are symmetric on the lower indices. This is used to define acovariant derivative:

Scalar ∇af = ∂af

Vector ∇avb = ∂avb + Γbacv

c

One-Form ∇avb = ∂avb − Γcabvc

4

With the Christoffel connection then ∇cgab = 0 and ∇cgab = 0. Another property is that:

∇c(gabvb) = (∇cgab)vb + gab∇cvb

= gab(∂cvb + Γbcdv

d)

= ∂cva + gabgbdΓbcdvb

= ∂cva − Γbacvb

= ∇cva

2.2 Curves

x2

x1

Xa(λ)

λ = 1λ = 2

λ = 3

λ is a parameter which increase monotonically along the curve. A tangent vector is given by:

ua =dxa

dλ

This tangent vector can be classified as timelike, spacelike or null everywhere and defines timelike, space-like or null curves. The proper time is defined for timelike curves by:

τ =

∫ √−uauadλ

This is the natural normalisation of the parameter along the curve and describes the worldline of amassive particle. For spacelike curves this parameter is:

` =

∫ √uauadλ

This is the proper length and describes the worldline of imaginary mass particles. Null curves have:∫ √−uauadλ = 0 =

∫ √uauadλ

There is no notion of proper time or length. This describes the trajectory of massless particles.

2.3 Geodesics

The action integral for the distance can be written down as:

S =1

2

∫dλgab(x)

dxa

dλ

dxb

dλ

5

The geodesic is the stationary curve of S. There is no square root which has some advantages, such asthe ability to not have to know whether the integral is valid because of the classification of the curve sinceit is always valid. The Euler-Lagrange equations can be solved to get the equations of motion. Thesecan always be rearranged into the form:

d2xc

dλ2+ Γcab

dxa

dλ

dxb

dλ= 0

This is a covariant equation since it can be written as:

ud∇duc = 0

This explains that the tangent vector to the curve is parallel transported along the curve.

λ is an affine parameter. This is true because if λ→ aλ+ b where a and b are constant then the geodesicequation remains unchanged.

On a geodesic then:

d

dλ(uau

a) =dxb

dλ

d

dxb(uau

a)

= ub∇b(uaua)

= 2uaub(∇bua)

= 2ua(0)

= 0

This means that uaua is constant so that a timelike geodesic stays timelike and similarly for the spacelikeand null geodesics. Because this is true then for timelike geodesics then λ = τ can be chosen arbitrarily,and for spacelike geodesics λ = `. For null geodesics then affine parameter needs to be used. Geodesicsare the trajectories of test of free particles.

2.4 Kinematics

Suppose Xa(λ) in a timelike curve then λ = τ can be chosen and uaua = −1 where ua is the four-velocity.The four-acceleration is obtained by:

Aa = ub∇bua

To see what this means then consider:

d

dτ(uaua) =

d

dτ(−1)

= 0

And also:

d

dτ(uaua) = 2ub∇bua

= 2uaAa

The contraction of a timelike vector with the acceleration is zero so they are orthogonal. This meansthat the acceleration must be spacelike or null. The scalar proper acceleration is:

A =√AaAa

6

2.5 Einstein’s Equations

The Riemann tensor is Rabcd. It is defined using parallel transport. Contracting this tensor gives theRicci tensor, Rabad = Rbd which is symmetric and from which the Ricci scalar can be obtained, gabRab =Raa = R. The Einstein tensor is then given by:

Gab = Rab −1

2Rgab

This satisfies ∇aGab = 0. Einstein’s equations are then given by:

Gab + Λgab = 8πGTab

Here Λ is a cosmological constant.

2.6 Hypersurfaces

Hypersurfaces, Σ, are created by a constant function f whose exterior derivative is ∂af 6= 0 which meansthat there are no intersections or cusps. ∂af = na is the normal one-form. The normal vector is obtainedfrom the metric.

For a Minkowski spacetime the 1 + 1 dimensional plane is given by ds2 = −dt2 +dx2 and is the hypersur-faces are created by f = −αt+ x where α is a constant then the normal one-form is na = ∂af = (−α, 1)and so the normal vector is na = (α, 1):

t

x

α > 1

0 < α < 1

na = va

α = 1

va

na

vana

7

When 0 < α < 1 then nana = −α2 + 1 > 0 so na is spacelike and so Σ is timelike. As expected the

tangent vector and the normal vector are orthogonal, vana = 0, eventhough the angle does not look likea right angle on the graph. This is because the geometry is Lorentzian and not Euclidean. Including theother two coordinates it can be seen that the signature of the induced metric is (−+ +).

When α > 1 then nana = −α2 < 0 and so na is timelike and hence σ is spacelike. The induced metric

has signature (+ + +).

When α = 1 then nana = −α2 + 1 = 0 and so na is null and the hypersurface is also called null. The

normal vector is also the tangent vector and so the tangent vector is (0 + +) and as such the metric onΣ is degenerate.

8

Chapter 3

Schwarzschild Geometry

The Schwarzschild line element is:

ds2 = −(

1− 2M

r

)dt2 +

dr2

1− 2M/r+ r2

(dϑ2 + sin2 ϑdφ2

)The Schwarzschild geometry is the only static matter-Λ free solution of the Einstein equation, Gab = 0,which is non-flat, Rabcd 6= 0. M is the mass parameter and has dimensions of length. It can be equatedto the physical mass by:

M =G

c2Mphys

If M → 0 then Minkowski space is recovered.

3.1 Exterior Schwarzschild Spacetime

The Schwarzschild spacetime has M > 0 and r > 2M and when r → ∞ then 2M/r → 0 so sphericalMinkowski spacetime metric is recovered. This means that it is called asymptotically flat or asymptoticallyMinkowski. Schwarzschild spacetime is also spherically symmetric and since gtt < 0 and ∂tgab = 0 thenthe metric is time-independent.

There is a pseudo-singularity when r → 2M since the formula, as written here, does not define a metricsince the r term diverges. The light cones of this metric are:

t

r2M

OutgoingRadial NullGeodesics

IngoingRadial NullGeodesics

9

If there is a geodesic which begins radially, ϑ = φ = 0 then it stays radial. Radial null geodesics have:

0 = −(

1− 2M

r

)dt2 +

dr2

1− 2M/r

These can be solved by integrating:

t = ±∫

dr

1− 2M/r

The lightcones get narrower the closer to the horizon they get. At r = 2M then the metric becomesdegenerate since the dt term disappears.

3.1.1 Eddington-Finkelstein Coordinates

This can be rectified by changing to the coordinate:

w = t+ 2M( r

2M− 1)

Finding dt in this coordinates allows the line element to be written as:

ds2 = −(

1− 2M

r

)dw2 +

4M

rdwdr +

(1 +

2M

r

)dr2 + r2dΩ2

Radial null geodesics have ds2 = dΩ2 = 0 and so there are two families of geodesics. The ingoing radialnull rays are described by dr + dw = 0 and the outgoing ones are:(

1− 2M

r

)dr −

(1− 2M

r

)dw = 0

The lightcone diagram is given by:

w

r

2M

OutgoingRadial NullGeodesics

IngoingRadial NullGeodesics

v

The ingoing lightcones actually carry on straight through r = 2M . This is because the metric is regularsince the components are all alright for r > 0 and the signature remains (− + ++) via continuity. Theexterior Schwarzschild spacetime has been extended into the region 0 < r < 2M by the Eddington-Finkelstein extension. There is also a radial null ray which stays at r = 2M .

The Schwarzschild coordinates have time inversion symmetry such that t → −t is an isometry. TheEddington-Finkelstein coordinates do not have time inversion symmetry and instead have two different

10

metrics to describe travelling into the future and travelling into the past. The ingoing coordinates givenby w can be written in terms of v = w + r so that:

v = t+ r + 2M ln( r

2M− 1)

Under t→ −t then the outgoing Eddington-Finkelstein coordinates are obtained and given in terms of:

u = t− r − 2M ln( r

2M− 1)

The lightcone diagram for these coordinates is:

?

r

2M

OutgoingRadial NullGeodesics

IngoingRadial NullGeodesics

3.1.2 Kruskal Coordinates

Another extension is the Kruskal-Szekeres-Frousdal extension, more often referred to as the Kruskalextension. Beginning with the ingoing and outgoing Eddington-Finkelstein then algebraically rescalingthem gives:

U = −e−u/4M and V = ev/4M

When r > 2M then U < 0 and V > 0. The whole metric is then given by:

ds2 = −32M2

re−r/2MdUdV + r2dΩ2

This r = r(U, V ) is a function of U and V . This can be found using:

−UV =( r

2M− 1)er/2M

This is not an elementary function but is almost elementary and good enough to do calculations with.There is still a r−1 term in the metric and so there is a singularity when r = 0. −UV is valid as long asit is greater than −1 or indeed UV < 1.

Radial null rays in Kruskal coordinates are given by dUdV = 0 which gives either U =constant orV =constant.

11

U V

r = constant

t = constant

r = 2M

r = 2M

t = constant

UV = 1

UV = 1

I: Original Exterior, r > 2M The Schwarzschild time coordinate is given by t = 2M ln(−V/U) andsurfaces of constant t are spacelike and to t is a timelike coordinate. The surfaces of constant r aretimelike and so the Schwarzschild r coordinate is spacelike.

II: Black Hole Interior, 0 < r < 2M This is the region described by the ingoing Eddington-Finkelstein coordinates. An exterior Schwarzschild-like time component t can be defined t = 2M ln(V/U)and is used to describe a metric:

ds2 = −(

1− 2M

r

)dt2 +

dr2

1− 2M/r+ r2dΩ2

Since 0 < r < 2M then it can be seen just from the metric that r has become the timelike componentand so surfaces of constant r are spacelike and t has become the spacelike component and so surfaces ofconstant t are timelike.

III: Second Black Hole Exterior, r > 2M Since the metric has (U, V ) → (V,U) as an isometrythen there is spatial inversion symmetry.

IV: White Hole Interior, 0 < r < 2M This is the region described by the outgoing Eddington-Finkelstein coordinates.

The t(t) coordinates are highly singular at the origin which is why the singularity at r = 2M occurs inthe Schwarzschild coordinates.

12

Chapter 4

Killing Vectors

Minkowski spacetime has ds2 = −dt2 +dx2 + · · · and is invariant under translations of the form x = x′+awhere a is a constant. This means that the metric has space translation symmetry or isometry. It alsohas time translation isometry t = t′ + b where b is a constant. These isometries are obvious from justplugging in these new values into the metric.

Killing vectors are a way of seeing these isometries in a coordinate free description. Self-isometry can bedescribed as:

Φ∗g = g

A one parameter group of isometries is given by Φt = etξ where ξ is a vector field. This family ofisometries have Φ∗g = g ∀t. This can also be denoted by Killings equation using the Lie derivative:

Lξg = ξc∂cgab + gac∂bξc + gcb∂aξ

c

= 0

Or alternatively as:∇aξb +∇bξa = 0

For Minkowski space then the Killing vector which translates across the x direction is ξ = ∂x and fortime translation then ξ = ∂t. Boosts are written as ξ = x∂t + t∂x. Suppose there are coordinates(x1, x2, x3, x4) such that ∂x1gab = 0 then ξ = k∂x1 is Killing, where k is a constant. This could bewritten as ξa = (1, 0, 0, 0).

Now suppose Xa(λ) is a geodesic and ξ is Killing and ua = Xa:

d

dλ(ξau

a) = uc∂c(ξaua)

= uc∇c(ξaua)

= uc∇c(ξaua)

= uauc(∇cξa) + ξauc(∇cua)

=1

2uauc(∇cξa) +

1

2ucua(∇aξc) + (0 from geodesic equation)

=uauc

2(∇cξa +∇aξc)

= 0

This means that ξaua =constant.

If there is a two dimensional manifold with a vector field then surfaces which are everywhere orthogonal

13

to the vector field can be constructed. In more than two dimensions then this is not generally true. Aone-form va is hypersurface orthogonal if it is orthogonal to surfaces of f =constant for some f :

v ∧ dv = 0

In less mathematical (differential geometry) language then this can be written as:

v[a∂bvc] =(va∂bvc + vb∂cva + vc∂avb−va∂cvb − vc∂bva − vb∂avc

)= 0

For the Christoffel connections then this is equivalent to:

v[a∇bvc] = 0

Stationary spacetime has a timelike Killing vector. There exists coordinates (t, x1, x2, x3) such that∂tgab = 0 and gtt < 0, but gti may be non-zero.

Static spacetime has a hypersurface-orthogonal Killing vector and has no cross terms in the time partof the metric, gti = 0. This means that there is time reversal invariance.

14

Chapter 5

Singularities

The two dimensional Euclidean plane ds2 = dx2 + dy2 is not singular because infinite distances can betravelled in any direction and no boundary will be reached. The two-sphere S2 in R3 is also not singularbecause any direction can be traversed to an infinite distance (even though steps may be retraced) andan edge will never be found. A bumped sphere again has no way to fall off the surface and so is notsingular.

In Lorentzian signatures such as Minkowski space then positive and negative infinite proper distances canbe reached as can proper times. The problem arises with null geodesics. This has no concept of properdistance or proper time but still a positive or negative infinite affine parameter can be reached. Sinceproper time and proper distance are affine then it can be seen that Minkowski spacetime is non-singularfor all affine parameter.

A complete geodesic is one where −∞ < λ < ∞. A geodesically complete spacetime is one where inte-grating along a geodesic can be done anywhere in the spacetime. Conversely, a geodesically incompletespacetime cannot be integrate along a geodesic continuously in the spacetime. If the integral falls of thespacetime then it is called geodesically incomplete. Geodesically incomplete spacetime can be artificialsay for R2 − 0, 0 and so this is not considered singular.

Singular spacetime must be geodesically incomplete and the spacetime cannot be extended into a geodesi-cally complete one.

Schwarzschild spacetime is singular. This can be seen be cause the Curvatuve Scalar RabcdRabcd → ∞

at r = 0. For a cone then Rabcd = 0 but this space is still singular so the curvature scalar does not haveto become infinite. Suppose the metric is ds2 = −dt2 + t2dφ2 with t > 0 and φ = φ + 2π. This is thesame as a FRW universe and so is an expanding cosmology. The singularity occurs at t = 0 which isidentified to the big bang. The Riemann curvature tensor is Rabcd = 0. This is also an example of asingular spacetime.

15

Chapter 6

Structure of a Star

6.1 Perfect Fluid

Particles travel along worldlines and have timelike fluid four-velocity ua which is the tangent vector tothe worldline normalised such that uaua = −1. Einsteins equation is:

Gab = 8πGTab

Tab is the energy momentum tensor. It is used to describe a fluid with pressure p and density %. In thecase of a perfect fluid then the energy momentum tensor is Tab = (%+ p)uaub + pgab. Using local inertialcoordinates then ua can be made to be wholey in the time direction, ua = (1, 0, 0, 0) and so:

Tab =

% · · ·· p · ·· · p ·· · · p

The time component of the tensor is the density which is what it is needed to be. The pressure is spatiallyisotropic and so physically there is no internal friction.

6.2 Static Star

A spherically symmetric static line element can be written as:

ds2 = −e2Φ(r)dt2 + e2Λ(r)dr2 + r2dΩ2

It can seen to be spherically symmetric since the unit S2 line element is present and there are no otherangular terms. It is static since there is a timelike killing vector, ∂t, which is hypersurface-orthogonalto surfaces of t =constant. Using the metric then the Einstein tensor can be found. When a fluidtravels along with normalised velocity ua = (e−Φ(r), 0, 0, 0) then assuming a static fluid then then energymomentum tensor can be found. Solutions to the Einstein equations can then be analysed. There arethree independent equations, the 00 part, the rr part and the ΩΩ part. Looking at the metric then it isconvenient to rewrite Λ(r) in terms of a now function, m(r), called the mass function:

e2Λ(r) =1

1− 2m(r)/r

The 00 part can now be written:dm(r)

dr= 4πGr2%

The rr part is also simplified to:

Φ′ =m(r) + 4πG%r3

r(r − 2m(r))

The ΩΩ part is still complicated but since ∇aGab = 0, which implies that ∇aTab = 0, then it followsthat:

(%+ p)Φ′ = −dpdr

16

If these three equations are satisfied then the ΩΩ equation is implied to be satisfied also. Outside thestar then there is no matter so % = 0 and p = 0 and when the differential equations are solved thenSchwarzschild is recovered.

When inside the star then % > 0, p ≥ 0 and a stability condition is also included:

dp

d%> 0

The Φ′ from the Einstein equations can be eliminated to leave:

dp

dr= − (%+ p)(m(r) + 4πGpr3)

r(r − 2m(r)

An equation of state now needs to be introduced to describe the stiffness of the matter. There are twoequations for two unknowns, m(r) and p(r), since p(r) determines %(r). The solutions are of the form:

p

pc

1

rS

rS r

rrS

rS

r

r

m(r)

e2φ(r)

M = m(rS)

Solid

Gas

%

%c

The mass can be found by integrating the dm(r)/dr equation:

M = G

∫ rS

0

%(4πr2)dr

In Newtonian gravity then the mass is given by:

mass =

∫%dV

=

∫ rS

0

%(4πr2)dr

This looks similar to the equation obtained from the differential equation but for the relativistic case:∫ rS

0

%(4πr2)dr 6=∫%dV

17

This is because when looking at the metric then the thickness is given by:

(d (thickness))2

=dr2

1− 2m(r)/r> dr2

This means that dV = 4πr2dr/√

1− 2m(r)/r and so:∫%dV >

∫ rS

0

%(4πr2)dr

This means that the mass is less than expected on Newtonian grounds. Using E = mc2 then the star canbe said to have less energy than expected. This is because building the star releases energy.

6.2.1 Constant Density Star

Take an equation of state % = constant > 0. This is not very physical but is solvable:

M

rS= −1

2

[1−

(1 + (%c/%)

1 + 3(%c/%)

)2]

%c/%

M/rS

4/9

rS > (2 + 1/4)M since the curve is under the curve. This means that the surface of the star is alwaysoutside the Schwarzschild radius. This was first found by Schwarzschild. If the star is squeezed then thedensity becomes arbitrarily large, but this is not physical. This is alright in this case since the equationof state is just a toy one.

6.3 Birkhoff’s Theorem

Every spherically symmetric solution to Gab is one, or part, of:

• Kruskal (M > 0)

• Minkowski (M = 0)

• M < 0 Schwarzschild (M < 0)

The spacetime outside a spherical star is Kruskal, even when the star is dynamical, i.e. pulsates or col-lapses. There are no spherically symmetric gravitational waves. The outcome of a spherically symmetricstar collapse must be part of Kruskal.

18

6.4 Redshift

τva

τua

p

q

γ = Xa(λ)

γ′ = Xa′(λ)= X(λ) + fa(λ)

A general redshift formula can be found using a vector field to move one geodesic to another close by:

(Xafa)p = (Xafa)q

Here (fa)p = uadτ and (fa)q = vadτ . This means the redshift can be written as:

dτ

dτ=

νobs

νsource

=(Xava)q

(Xaua)p

6.5 Star Collapse

U V

Surface of a Star

Observer

r = 2M

r = 2M

Geodesic

r = 0

r = 0

Radial Null Raysτ0

At tobs → ∞ then τ = τ0 −De−tobs/4M . No matter how large tobs gets it appears to the observer thatthe surface of the star never crosses the Schwarzschild radius.

dτ

dtobs=

D

4Me−tobs/4M

This is the redshift, and so it can be seen that it is exponential. The time scale for a scalar mass is∼ 4M ≈ 105s.

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6.6 Non-Radial Null Rays

ds2 = −F (r)dt2 +dr2

F (r)+ r2dΩ2

Unlike radial geodesics the non-radial ones cannot have dΩ = 0. The angular part can be simplifiedthough using symmetries. The coordinates can be chosen such that ϑ = π/2 so sin2 ϑ = 1 and dϑ = 0.This leaves the line element as:

ds2 = −F (r)dt2 +dr2

F (r)+ r2dφ2

The Lagrangian can then be written as:

L =1

2

[−F (r)t2 +

r2

F (r)+ c2φ2

]The constants of motion are the angular momentum, r2φ = ` and a conservation of energy-like term isF (r)t = ε. The three different types of geodesics can be considered:

ds2 =

−dτ2 timelike k = 1

0 null k = 0dτ2 spacelike k = 1

uaua = −k

This means that the geodesic for the radial part can be written as:

r2 +W (r) = ε2

Where W (r) = (k + `2/r2)F (r). For null geodesics in Schwarzschild then F (r) = 1− 2M/r and k = 0.

U V

r = 2M

r = 2M

r = 0

r = 0

2

2

3c

3a

1a1b3b

4r

W (r)

2

3b

1b

4

3c 3a

1a

ε = 0

0 < ε2 < `2/27M2

ε2 = `2/27M2

ε2 > `2/27M2

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6.7 Spacetime Diagrams

6.7.1 1+1 Minkowski

T

U V

X

V = constant

U = constant

TimelikeGeodesic

SpacelikeGeodesic

I − I −

I + I +

i−

i+

i0 i0

qp

To analyse what happens at infinity in 1+1 Minkowski then firstly the line element can be changed fromds2 = −dT 2 + dX2 to ds2 = −dV dU by coordinate transformation U = T − X and V = T + X. Therange of all these coordinates is −∞ < T,X,U, V <∞. Now taking U = tan p and V = tan q then it canbe seen that the range of p and q is −π/2 < p, q, π/2 and the metric can be written as:

ds2 = − dpdq

cos2 p cos2 q

A null geodesic with U =constant (left going) has a constant p and a null geodesic with V =constant(right going) has a constant q. A timelike geodesic must cross all the (right going forwards in time) nullgeodesics and so the trajectory always begins at i− and ends at i+ and likewise for a spacelike geodesicgo between i0’s.

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