notes on special and general relativity -...
TRANSCRIPT
Notes on Special and General Relativity
Andrew Forrester January 28, 2009
Contents
1 Questions and Ideas 2
2 The Big Picture 2
3 Notation 33.1 Ambiguous Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 Theoretical Summary 54.1 Fundamental Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 Developing the Special Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5 Basic Quantities and Terms 65.1 Physical Quantities and Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65.2 Mathematical Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6 Holors and Invariants 11
7 Holor Calculus and Differential Geometry 14
8 More Differential Geometry 178.1 Maps between Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.3 Parallel Propagation, Killing Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.4 Noncoordinate Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.5 Conformal Stuff... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
9 Lagrangian Formalism 17
10 Special Relativity 18
11 Equations 1811.1 Major Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1811.2 Minor Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
12 The Speed of Gravity, Etc. 20
13 Other Relativity Theories 20
14 Open Questions and Mysteries 20
15 Applications 20
16 Common Errors 2016.1 Mistaking Similar Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
16.1.1 Applying this notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2116.2 Silly Mistake(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
17 Useful Programs 21
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1 Questions and Ideas
• (Carroll [1], p.39: Why does δ(∂µΦi) = ∂µ(δΦi)?)• (Can we say that our relativistic velocity always has the same “pseudo” magnitude?)• Prove that the Lorentz transformation makes the coordinates “scissor”.• Are there particular parametrizations that are necessary to describe paths of massless particles (which travel at
the speed of light)? (affine parameters?)
Project Ideas
• Connection− What is it?− How (many ways) is it defined? Why?−
• Spacetime Inflation• Gyroscope precession
2 The Big Picture
Subtopics
The topic is most broadly divided into the following categories:
KinematicsStaticsDynamics
Domains
• Relativistic versus Non-relativistic• Relativity versus Quantum Mechanics• Newtonian Gravitation (Time-independent gravitational field, slow moving test particles)• Weak Linear Gravitation (Weak time-dependent gravitational field, no restriciton on test particle motion)• ...
2
3 Notation
Notation in These Notes
p = ~p 3-vector (usually representing a geometric object such as momentum)(I use the notation ~p in my handwriting and p in my electronic documents.)
pi the 3-vector p in a particular coordinate system (three numbers that transform together as aspatial vector)
pi =⟨p1, p2, p3
⟩ipi = {pi} the ith component (or i-component) of the 3-vector p
p2 = p ·p = pipi = (p1)2 + (p2)2 + (p3)2
p 4-vector (usually representing a geometric object such as 4-momentum or energy-momentum)pµ the 4-vector p in a particular coordinate system (four numbers that transform together as a 4-
vector)pµ = {pµ} the µ-component of the 4-vector p (one number)
Saying “µth component” could be misleading, since the counting starts at zero.p2 = p · p = pµpµ = −(p0)2 + (p1)2 + (p2)2 + (p3)2
bµ 4-tuple of 4-vectors (is this a 4-vector?)p = pµeµ = p0e0 + p1e1 + p2e2 + p3e3
We use the “spacelike convention” of metric signature (−1, 1, 1, 1), meaning that the flat, Minkowski spacetimemetric is η = diag(−1, 1, 1, 1), that is,
ηµν =
−1 0 0 0
0 1 0 00 0 1 00 0 0 1
µν
and we therefore also use 1
∂2 = ∂µ∂µ = −∂0
2 + ∂12 + ∂2
2 + ∂32 = − 1
c2 ∂t2 + ∂x
2 + ∂y2 + ∂z
2 = −∂02 +∇2.
We letKe ≡
14πε0
and Km ≡µ0
4π.
We use h to denote some form of contraction of the tensor h, for example, taking the “pseudo-trace” of hµν :h = h µ
µ = gµρhρµ; or the following contraction: hβ εγ = h β αε
α γ = gαδh β εα γδ . We will thus put bars over the
Riemann tensor symbol (R) to denote the Ricci tensor (R) and Ricci scalar ( ¯R) and over the Einstein tensor symbol(G) to denote the Einstein scalar (G), showing that they are contractions. We will use GN for Newton’s gravitationalconstant (except when accompanied by a mass M):
G = Gµµ 6= G 6= GN
GM ≡ GNM
Also, we will use G and Gi to denote the gravito-electric field. With these notations, the only ambiguity is betweenthe Einstein tensor G (when writing it without super- or subscripts) and the magnitude of the gravito-electric field|G| = G (which I think does not turn up in this document), so long as they are not multiplied by a mass M .
Overline notation: to distinguish between tensorial and non-tensorial indices. . . for example,
Rτµσν = gτρRρµσν
Rτµσν = −RτµνσRτµσν = −RµτσνRτµσν = Rσµτν
Rτµσν +Rτσνµ +Rτνµσ = 0; Rτ [µσν] = 01Sometimes ∂2 is written as �, in analogy to ∆, and sometimes it’s written as �2, in analogy to ∇2, usually in physics settings. I
prefer the �2 notation because it reveals its squared nature (while the box itself reveals its four-variable nature).
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or to distinguish between the “full object” tensor and the “parts” or merates
3.1 Ambiguous Notation
p This character is ambiguous because it could be the magnitude of p or it could be the “pseudo-magnitude” of p.
p?=√
p ·p
p?= √p · p
Both of these uses seem to be common. If either of these notations is used, it should be clearlyand loudly defined.
p2 If p is used, then one may wonder whether this is the 2-component of pµ (or pi) or the square of p.p2 ?= p · e2
p2 ?= p ·p
p2 ?= p · pIf p is used, then a preferrable notation for the square of p would be
(p)2
to distinguish it from the 2-component of p (or pi).Perhaps I should use the MTW notation for tensors: bold sans serif.
xµ =
x0
x1
x2
x3
µ
=
ctxyz
µ
OR���
�����
�����XXXXXXXXXXXXX
xµ =
x0
x1
x2
x3
µ
=
ctxyz
µ
x =
x0
x1
x2
x3
=
ctxyz
OR x =
x0
x1
x2
x3
=
ctxyz
OR x =
x0
x1
x2
x3
=
ctxyz
(With this notation, beware of the ambiguity between x or x the four-vector and x the component of the three-vector~x. In handwriting, you can’t distinguish between x and x, unless you use x...)
x =
x0
x1
x2
x3
=
ctxyz
Notation from Theory of Holors:
{xµ} =
x0 = ct if µ = 0x1 = x if µ = 1x2 = y if µ = 2x3 = z if µ = 3
Possible improvement:
xµ =
x0 = ct if µ = 0x1 = x if µ = 1x2 = y if µ = 2x3 = z if µ = 3
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Abbreviations in These Notes
• GR = General Relativity• SR = Special Relativity• EM = electromagnetic or electromagnetism
4 Theoretical Summary
4.1 Fundamental Principles
• General Equivalence?• Principle of General Covariance
“There are to be no preferred coordinates, but also, if we have two different spacetimes, representing two physicallydistinct gravitational fields, then there is to be no naturally preferred pointwise identification between the two –so we cannot say which particular spacetime point of one is to be regarded as the same point as some particularspacetime point of the other! This philosophical issue will concern us later, regarding how Einstein’s theoryrelates to the principles of quantum mechanics. It is for this reason, most particularly, that the tensor formalismis central to Einstein’s theory.” – Penrose [4] pg 459
• Postulate of Equivalence (from Goldstein): for Galilean transformationsrequires that physical laws must be phrased in an identical manner for all uniformly moving systems, i.e., becovariant when subjected to a Galilean transfomation.Measurements made entirely within a given system must be incapable of distinguishing that system from allothers moving uniformly with respect to it.
• Equivalence Principles (from Wikipedia)
− Weak Equivalence (universality of free fall):The trajectory of a falling test body depends only on its initial position and velocity, and is independent ofits composition.orAll bodies at the same spacetime point in a given gravitational field will undergo the same acceleration.(Are these the same?)The principle does not apply to large bodies, which might experience tidal forces, or heavy bodies, whosepresence will substantially change the gravitational field around them. This form of the equivalence principleis closest to Einstein’s original statement: in fact, his statements imply this one.
− Einstien Equivalence: The result of a local non-gravitational experiment in an inertial frame of referenceis independent of the velocity or location in the universe of the experiment.This is a kind of Copernican extension of Einstein’s original formulation, which requires that suitable framesof reference all over the universe behave identically. It is an extension of the postulates of special relativity inthat it requires that dimensionless physical values such as the fine-structure constant and electron-to-protonmass ratio be constant. Many physicists believe that any Lorentz invariant theory that satisfies the weakequivalence principle also satisfies the Einstein equivalence principle.
− Strong Equivalence: The results of any local experiment, gravitational or not, in an inertial frame ofreference are independent of where and when in the universe it is conducted.This is the only form of the equivalence principle that applies to self-gravitating objects (such as stars), whichhave substantial internal gravitational interactions. It requires that the gravitational constant be the sameeverywhere in the universe and is incompatible with a fifth force. It is much more restrictive than the Einsteinequivalence principle. General relativity is the only known theory of gravity compatible with this form of theequivalence principle.
− “we [. . . ] assume the complete physical equivalence of a gravitational field and a corresponding accelerationof the reference system.” (Einstein 1907)
− . . . the inertial mass in Newton’s second law, F = ma, mysteriously equals the gravitational mass in Newton’s
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law of universal gravitation. Under the equivalence principle, this mystery is solved because gravity is anacceleration from inertial motion caused by the mechanical resistance of the Earth’s surface. These considera-tions suggest the following corollary to the equivalence principle, which Einstein formulated precisely in 1911:“Whenever an observer detects the local presence of a force that acts on all objects in direct proportion tothe inertial mass of each object, that observer is in an accelerated frame of reference.”
− Einstein also referred to two reference frames, K and K ′. K is a uniform gravitational field, whereas K ′ hasno gravitational field but is uniformly accelerated such that objects in the two frames experience identicalforces:“We arrive at a very satisfactory interpretation of this law of experience, if we assume that the systems K andK ′ are physically exactly equivalent, that is, if we assume that we may just as well regard the system K asbeing in a space free from gravitational fields, if we then regard K as uniformly accelerated. This assumptionof exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system ofreference, just as the usual theory of relativity forbids us to talk of the absolute velocity of a system; and itmakes the equal falling of all bodies in a gravitational field seem a matter of course.” (Einstein 1911)
− This observation was the start of a process that culminated in general relativity. Einstein suggested that itshould be elevated to the status of a general principle when constructing his theory of relativity:“As long as we restrict ourselves to purely mechanical processes in the realm where Newton’s mechanics holdssway, we are certain of the equivalence of the systems K and K ′. But this view of ours will not have any deepersignificance unless the systems K and K ′ are equivalent with respect to all physical processes, that is, unlessthe laws of nature with respect to K are in entire agreement with those with respect to K ′. By assuming thisto be so, we arrive at a principle which, if it is really true, has great heuristic importance. For by theoreticalconsideration of processes which take place relatively to a system of reference with uniform acceleration, weobtain information as to the career of processes in a homogeneous gravitational field.” (Einstein 1911)
• “Covariance” of Physical Laws• Geodesics / Spacetime manifold curvature• Principle of Maximal Aging (or Stationary/Critical Proper Time) for geodesics
4.2 Developing the Special Theory of Relativity
Improve/correct this: (using Goldstein, 7-1)• Newton’s Laws (and various formulations of mechanics), which are Galilean-invariant, work• Maxwell’s Equations, which are Lorentz-invariant, work
There are many derivations of the Lorentz transformation. . . list (and learn) some of them.• . . .• Speed of light• . . .• It is postulated that the known physical laws can be improved by generalizing them somehow into Lorentz-
invariant forms• This postulate is verified by experiment
5 Basic Quantities and Terms
5.1 Physical Quantities and Terms
• Frames -− Inertial Frame− Lorentz Frame− Mathematical Frame (basis of vectors or forms? coordinate frame? local trivialization?)− Proper Frame (Rest frame, Self Frame)− “geodesic frame”, where ∂µgνρ = 0, so {ρµν} = 0 but ∂σ{ρµν} 6= 0
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• Spacetime -
• Minkowski Space -
• Spacetime Position and Coordinates - a.k.a. Spacetime Event or simply Event
• Proper Time - Self timedtdτ
=1√
1− β2≡ γ
β = v/c is called (the “relativistic beta”)? and γ = γβ is called the Lorentz factor.
• Spacetime Diagram -− Worldline -− Light Cone -
Past - the lower half-coneFuture - the upper half-coneElsewhere - the outside of the light cone
• Spacetime Interval - ... also Proper IntervalFlat spacetime, straight line path:
(∆s)2 = −(c∆t)2 + (∆x)2 + (∆y)2 + (∆z)2
= ηµν(∆x)µ(∆x)ν
(∆τ)2 = −(∆s)2
= −ηµν(∆x)µ(∆x)ν
Infinitesimal Spacetime Interval - ... or Line Element
ds2 = ηµνdxµdxν
Metric:g = gµνdxµdxν or g = gµν(dxµ ⊗ dxν)
Any spacetime, path: (since it’s unclear what∫ √
ηµνdxµdxν is supposed to mean, we parametrize the pathusing the dimensionless parameter λ, xµ(λ), and write the following...)
∆s =∫ √
ηµνdxµ
dλdxν
dλ dλ
∆τ =∫ √
−ηµν dxµ
dλdxν
dλ dλ
• Separation - (in flat spacetime only?)
Property Separation Type Light Cone Position
(∆s)2 > 0 Space-like “Outside” the light cone(∆s)2 = 0 Light-like or Null On the light cone(∆s)2 < 0 Time-like “Inside” the light cone
• Dust
• Perfect Fluid - In cosmology, the matter filling hte universe is typically modelled as a perfect fluid.
Tµν = (ρ+ p)UµUν + pgµν
where ρ is the energy density, p is the pressure, and Uµ is the four-velocity of the fluid.
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• Action - S, Lagrangian L, Lagrangian (Density) L
S =∫
d4xL(Φi, ∂µΦi)
L =∫
d3xL(Φi, ∂µΦi)
S =∫
dt L
• Energy− “Inertial” energy: E2 = p2c2 +m2c4; E = γmc2
− Slow-moving limit: rest energy Er = mc2; kinetic energy Ek = 12mv
2
• Time-Position 4-Vector (Position Four-Vector, Events)Four-Position
xµ ≡ (ct,x)µ
• Speed-Velocity 4-Vector (Velocity Four-Vector)Four-Velocity
uµ ≡ dxµ
dτ= γ(c,v)µ = (γc,u)µ
v ≡ dxdt
u ≡ dxdτ
= γv
where the path of the particle (or particle(s)/object) through some coordinate system is parametrized by theparticle’s proper time τ : xµ = xµ(τ)
• Energy-Momentum 4-VectorFour-Momentum
pµ =dxµ
dλ= (E/c,p)
p ≡ dxdλ
For massive particles:
pµ =dxµ
dλ= m
dxµ
dτ= muµ = γ(mc,pN)µ = (γmc,p)µ
λ = τ/m p = mdxdτ
= γpN pN = mv = mdxdt
- What are the limitations on λ for massless particles?
• Number-Flux 4-VectorFour-Number-Flux-Density
Nµ = nuµ
• Charge-Current 4-VectorFour-Current
Jµ = ρuµ
where ρ is the proper charge density (charge density in the rest frame of the (local?) charge distribution)
• Work/Power-Force 4-VectorFour-Force
fµ = md2xµ
dτ2=
dpµ
dτ- Lorentz force
fµ = quλFλµ
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• Frequency-Wave 4-Vector (Frequency-Wave-Vector Four-Vector, Wave Four-Vector)Four-Wave-Vector
Relativistic Doppler shift
• Stress Tensor - Energy-Momentum-Stress TensorSpin tensorWarning: In solid state physics and fluid mechanics, the stress tensor is defined to be the spatial componentsof the stress-energy tensor in the comoving frame of reference. In other words, the stress energy tensor inengineering differs from the stress energy tensor here by a momentum convective term.
• EM Four-Potential
Aµ =⟨
Φc, Ax, Ay, Ay
⟩µ• EM Field-Strength Tensor
Field-Strength Tensor (and Auxiliary)
Fαβ =
0BBB@0 −Ex −Ey −EzEx 0 −Bz ByEy Bz 0 −BxEz −By Bx 0
1CCCAαβ
≡ (E,B)αβ Gαβ =
0BBB@0 −Dx −Dy −DzDx 0 −Hz HyDy Hz 0 −HxDz −Hy Hx 0
1CCCAαβ
≡ (D,H)αβ
Ei = ∂iA0 − ∂0Ai Bi = −εijk(∂iAj − ∂kAj)
(check signs). . . with two covariant indices
Fαβ = gαγFγδgδβ =
0BBB@0 Ex Ey Ez
−Ex 0 −Bz By−Ey Bz 0 −Bx−Ez −By Bx 0
1CCCAαβ
≡ (E,B)αβ
. . . and polarization-magnetization tensor(P,−M)
Dual Field-Strength Tensor
eFαβ = 12εαβγδFγδ =
0BBB@0 −Bx −By −BzBx 0 Ez −EyBy −Ez 0 ExBz Ey −Ex 0
1CCCAαβ
5.2 Mathematical Terms
• Vector Space - (“Vector” here does not mean a physical vector, it means a mathematical vector. We may betalking about a four-vector space or a tensor space, etc.)
• Dimension -
• Inner Product - (pseudo-inner-product) Norm, Metric (“distance” function), Inverse Metric
• Poincare Transformation - ... Lorentz Transformation− Translation− Rotation− Boosting - Boost Parameter or Rapidity ζ: β = tanh ζ, γv = cosh ζ, γvβ = sinh ζ
− Proper/Improper - Proper Lorentz transformations are connected continuously to the identity transforma-tion and so have det Λ = 1. Improper Lorentz transformations may have det Λ = 1 or det Λ = −1 (a factrelated to the indefiniteness of the metric). Both Λ = η (space inversion) and Λ = −I (space-time inversion)are improper. If Λ is proper, −Λ is improper.
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− Orthogonal/Non-orthogonal− Orthochronous/Non-orthochronous
Group Structure: Lorentz Group (Homogeneous Lorentz Group), Poincare Group (Inhomogeneous LorentzGroup)
x′ = Λx
xµ′
= Λµ′
µ xµ
dxµ′
= Λµ′
µdxµ
Λ =
γv −γvβx 0 0−γvβx γv 0 0
0 0 1 00 0 0 1
=
cosh ζv − sinh ζv 0 0− sinh ζv cosh ζv 0 0
0 0 1 00 0 0 1
xµ′
:
x0′
= γv(x0 − βxx1) = x0 cosh ζv − x1 sinh ζvx1′
= γv(x1 − βxx0) = x1 cosh ζv − x0 sinh ζvx2′
= x2
x3′= x3
(See Lorentz Transformation paper and/or Jackson)
• Tangent Space -
• Tangent Bundle -The tangent space of a point consists of the possible directional derivatives at that point, and has the samedimension n as the manifold does. The collection of all tangent spaces can in turn be made into a manifold,the tangent bundle, whose dimension is 2n.
• Cotangent Bundle -Elements of the cotangent space can be thought of as infinitesimal displacements: if f is a differentiable functionwe can define at each point p a cotangent vector dfp which sends a tangent vector Xp to the derivative of fassociated with Xp. However, not every covector field can be expressed this way.
• Bundles− Tangent, Cotangent− Tensor Bundle - The tensor bundle is the direct sum of all tensor products of the tangent bundle and the
cotangent bundle. Each element of the bundle is a tensor field, which can act as a multilinear operator onvector fields, or on other tensor fields.The tensor bundle cannot be a differentiable manifold, since it is infinite dimensional. It is however analgebra over the ring of scalar functions.
− Fiber Bundle− Jet Bundle
A connection is a tensor on the jet bundle.
• Field -
• Dual Space -
• One-Forms - Dual Vector (Dual Tensor, e.g. dual field-strength tensor)“Four-Position” (Four-Position with index lowered) xµPartial Derivatives (Four-Gradiant) ∂µFour-Potential ((Four)-Vector Potential) Aµ
• Partial Derivative - Four-Derivative? Not a tensor (tensorial transformation) in general, right? (We’llhave to create the connection to make it tensorial.)
∂µ ≡∂
∂xµ
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∂µ = 〈∂0, ∂1, ∂2, ∂3〉µ =⟨
1c∂∂t ,
∂∂x ,
∂∂y ,
∂∂z
⟩µ
∂µ =⟨∂0, ∂1, ∂2, ∂3
⟩µ
=⟨−1c∂∂t ,
∂∂x ,
∂∂y ,
∂∂z
⟩µ
“Notice that, unlike the partial derivative, it makes sense to raise an index on the covariant derivative, due tometric compatibility.” gµλ∇λ = ∇µ but gµλ∂λ 6= ∂µ? (What about ηµλ∂λ = ∂µ?)
• Gradient - Four-Gradient (Derivative on a scalar field that yields a “vector” field...)
∂if = 〈∂1f, ∂2f, ∂3f〉i = (∇f)i (∇if) ?
∂µf = 〈∂0f, ∂1f, ∂2f, ∂3f〉i
• Divergence - Four-Divergence (Derivative on a “vector” field that yields a scalar field...)
∇ · x = ∂ixi = ∂1x
1 + ∂2x2 + ∂3x
3
∂ ·x = ∂µxµ = ∂0x
0 + ∂1x1 + ∂2x
2 + ∂3x3
• Curl - (Four-Curl?) (Goldstein [3] p.582: “Clearly, Fµν is a sort of four-dimensional curl of the vector Aµ”:Fµν = ∂µAν − ∂νAµ)
• d’Alembertian - (Four-Laplacian or as Jackson says (pg 555) “invariant four-dimensional Laplacian”; Andwhat about a “vector” four-Laplacian?) 2
�2 ≡ ∂2 = ∂ · ∂ = ηµν∂µ∂ν = −c2∂t2 +∇2
6 Holors and Invariants
• Holors, scalars, 4-vectors, TensorsKinds of “vectors” or “invariant vectors”
• Relativistic Invariant - Scalars, TensorsSpeed of LightSpacetime IntervalPlane-Wave Phase(pseudo-speed): square root of u2 = uµuµ = −γ2c2 + v2 = (−c2 + v2)/(1− v2/c2) = −c2(Laue’s scalar? T µ
µ )
• Four-Scalar (No, scalars don’t have four components)Four-Tensor? (versus Cartesian tensor, spherical tensor)
• CovarianceVarious meanings: invariance of the form of an equation or law; (co)varying in tandem with another varyingquantity in such a way as to keep a third quantity (that depends on the other two) constant;
• Vectors(shall we call any object that transforms as the space-time coordinate does under Lorentz transformationsa 4-vector? and shall we call any object that transforms as the space-time coordinate does under generalcoordinate transformations a “general vector”?)
− mathematical vector (an element of a vector space)− vector under some transformation group− vector under some physical transformation group
(transforms the same as some “physical” object such as spacetime coordinates. . . or velocity?)physical or geometric vector
2See the Notation section (footnotes) for comments on the Box notation that appear here.
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∗ spatial vector (rotations)pseudo vector, axial vector (parity, i.e. spatial inversion)
∗ four-vector (Lorentz transformations)∗ general vector (spacetime diffeomorphisms: general coordinate transformations)
• Four-Vector -Contravariant VectorCovariant VectorDoes this image help explain the meaning of contravariant and covariant?
• TensorsTensor Space, Tensor Product, Tensor FieldTensor products: p⊗ q(�, �) = 〈�,p〉 〈�, q〉(“invariant spatial tensor” versus tensor made from a tensor product versus holor)Kronecker Tensor δµν- δµν , δαβγδεη ?- δijab ≡ δiaδ
jb , δ
ijkabc ≡ δiaδ
jbδkc , and so forth.
Minkowski Metric ηµνMetric (Tensor) gµν : g(x,v) = gµν(x) dxµ(x,v) dxν(x,v)Electromagnetic Field Strenth Tensor Fµν
Stress Tensor (Stress-Energy-Momentum Tensor) Tµν (energy density, pressure, stress, strain?)Projection (Tensor) Operator Pσν = δσν + 1
c2uσuν
Einstein Tensor GµνLevi-Civita Tensor ε
ε = εµ1···µn dxµ1 ⊗ · · · ⊗ dxµn
=1n!εµ1···µn dxµ1 ∧ · · · ∧ dxµn
=1n!
√∣∣gµν∣∣ εµ1···µn dxµ1 ∧ · · · ∧ dxµn
=√∣∣gµν∣∣ dx0 ∧ · · · ∧ dxn−1
≡√∣∣gµν∣∣ dnx
So dnx ≡ dx0 ∧ · · · ∧ dxn−1 is a tensor density field (of weight 1?)Levi-Civita Symbol ε (a tensor density field of weight 1)
εijk : ε123 = 1, ε even permutation = 1, ε odd permutation = −1εµνρσ : ε0123 = 1, ε even permutation = 1, ε odd permutation = −1
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εijk = εijaδka
= εiab(δjkab − δjkba)
= εabc(δijkabc − δijkacb + δijkcab − δ
ijkcba + δijkbca − δ
ijkbac)
εµνρσ = εµνραδσα
= εµναβ(δρσαβ − δρσβα)
= εµαβγ(δνρσαβγ − δνρσαγβ + δνρσγαβ − δ
νρσγβα + δνρσβγα − δ
νρσβαγ)
= εαβγδ(δµνρσαβγδ − δµνρσαβδγ + δµνρσαδβγ − δ
µνρσαδγβ + δµνρσαγδβ − δ
µνρσαγβδ
+ δµνρσγαβδ − δµνρσγαδβ + δµνρσγδαβ − δ
µνρσγδβα + δµνρσγβδα − δ
µνρσγβαδ
+ δµνρσβγαδ − δµνρσβγδα + δµνρσβδγα − δ
µνρσβδαγ + δµνρσβαδγ − δ
µνρσβαγδ
+ δµνρσδβαγ − δµνρσδβγα + δµνρσδγβα − δ
µνρσδγαβ + δµνρσδαγβ − δ
µνρσδαβγ)
Might this notation ever be useful?:
εµνρσTνρσ = εµνρσ(Tνρσ − Tνσρ + Tσνρ − Tσρν + Tρσν − Tρνσ)
εµνρσTνρσ = εµνρσ(Tνρσ − Tνσρ + Tσνρ − Tσρν + Tρσν − Tρνσ)
εµAνAρAσ = εµαβγ(δAνAρAσαβγ − δAνAρAσαγβ + δA
νAρAσγαβ − δA
νAρAσγβα + δA
νAρAσβγα − δA
νAρAσβαγ)
εµHHνρσ = εµαβγ(δHHνρσαβγ − δHHνρσαγβ + δHH
νρσγαβ − δ
HHνρσγβα + δHH
νρσβγα − δ
HHνρσβαγ)
Alternators, or Generalized Kronecker Deltas (from Holors [5] book)
δijkl =
∣∣∣∣∣ δik δilδjk δjl
∣∣∣∣∣δijklmn =
∣∣∣∣∣∣∣δil δim δinδjl δjm δjnδkl δkm δkn
∣∣∣∣∣∣∣δr1r2···rms1s2···sm =
∣∣∣∣∣∣∣∣∣∣δr1s1 δr1s2 · · · δr1smδr2s1 δr2s2 · · · δr2sm...
.... . .
...δrms1 δrms2 · · · δrmsm
∣∣∣∣∣∣∣∣∣∣Signature: 3D Lorentz signature (+,+,−), Majorana (−,+,+)Space with Minkowski-Lorentz signature: (ds)2 = (dx)2 + (dy)2 − (dz)2
Space with Minkowski-Majorana signature: (ds)2 = −(dx)2 − (dy)2 + (dz)2
Landau-Lifshitz convention (terminology from D’Hoker)West coast convention (Bjorken-Drell) (terminology from D’Hoker)Bjorken metric: diag(1,-1,-1,-1) (terminology from Rajpoot)Pauli metric: diag(-1,1,1,1) (terminology from Rajpoot)
• Index GymnasticsSummation Conventions -Dummy Indices - ∑
i
εijkεilm = δjlδkm − δjmδkl εijkεilm = δjklm − δjkml∑
ij
εijkεijl = 2δkl εijkεijl = 2δkl
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Full Object versus Elements: something like this...
TαβγδSαβεδκ ≡
∑αβδ
TαβγδSαβεδκ = Rγκε
TαβγδSαβεδκ ≡
∑β
TαβγδSαβεδκ = Bγκε (α, δ)
ContractionRaising and Lowering IndicesSymmetrization
T(µ1µ2···µn)ρσ =
1n!
(Tµ1µ2···µnρσ + sum over permutations of indices µ1 · · ·µn)
T(µνρ)σ =16
(Tµνρσ + Tµρνσ + Tρµνσ + Tρνµσ + Tνρµσ + Tνµρσ)
Antisymmetrization
T[µ1µ2···µn]ρσ =
1n!
(Tµ1µ2···µnρσ + alternating sum over permutations of indices µ1 · · ·µn)
T[µνρ]σ =16
(Tµνρσ − Tµρνσ + Tρµνσ − Tρνµσ + Tνρµσ − Tνµρσ)
Decomposition into symmetric and antisymmetric partsTracePartial DerivativesProjection (Tensor) Operator Pσν = δσν + 1
c2uσuν
7 Holor Calculus and Differential Geometry
• Covariant Derivative - transforms like a tensor, etc...
∇T
The covariant derivative is defined to have the following properties:
1. Linearity: ∇(aT + bS) = a∇T + b∇S2. Leibniz (product) rule: ∇(T ⊗ S) = (∇T )⊗ S + T ⊗ (∇S)
3. Commutes with contractions: ∇µ(Tλλρ) = (∇T ) λµ λρ
4. Reduces to the partial derivative on scalars: ∇µφ = ∂µφ
(∇µ)αβ = ∂µ δαβ + Γαµβ
∇µV ν = ∂µVν + ΓνµλV
λ or (∇µV )ν =(∂µδ
νλ + Γνµλ
)V λ
∇µων = ∂µων − Γλµνωλ or (∇µω)ν =(∂µδ
λν − Γλµν
)ωλ
∇σTµ1µ2···µkν1ν2···νl = ∂σT
µ1µ2···µkν1ν2···νl
+ Γµ1σλT
λµ2···µkν1ν2···νl + Γµ2
σλTµ1λ···µk
ν1ν2···νl + · · ·−Γλσν1T
µ1µ2···µkλν2···νl − Γλσν1T
µ1µ2···µkν1λ···νl − · · ·
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• Connections -A connection is a tensor on the jet bundle. (Wikipedia: Differentiable Manifold)Christoffel Connection (Christoffel Symbol(s))Spin Connections (Cartan Structure Equations)(Gauge Connections?) (structure group, a Lie group; gauge transformations, gauge theories)− Kinds∗ Christoffel symbols of the first kind∗ Christoffel symbols of the second kind
− Torsion-free: Γλµν = Γλ(µν)
− Metric-compatible: ∇σgµν = 0Nice properties of a metric-compatible connection:∗ ∇λεµνρσ = 0∗ ∇λgµν = 0∗ gµλ∇σV λ = ∇σ(gµλV λ) = ∇σVµ
(Does “the metric is compatible with the connection” make sense?)
Connection induced by a metric gµν (Christoffel Levi-Civita Connection):
Γλµν = 12gλρ(∂µgνρ + ∂νgρµ − ∂ρgµν)
Torsion tensor: Tλµν = Γλµν − Γλνµ = 2Γλ[µν]
• Geodesic Equation
Ddλ
(dxµ
dλ
)= 0 ⇔ d2xµ
dλ2+ Γµρσ
dxρ
dλdxσ
dλ= 0
For time-like particle paths:pµ∇µpµ = 0
dpµ
dλ+ Γµρσp
ρpσ = 0
• Covariant Directional DerivativeDdλ
=dxµ
dλ∇µ(
Ddλ
)αβ
=dxµ
dλ(∇µ)αβ =
dxµ
dλ
(∂µ δ
αβ + Γαµβ
)=
dxµ
dλ
(∂∂xµ δ
αβ + Γαµβ
)=
ddλ
δαβ +dxµ
dλΓαµβ
• Parallel Transport - Geodesic Equation of Parallel Transport(DdλT
)µ1µ2···µk
ν1ν2···νl=
Ddλ
Tµ1µ2···µkν1ν2···νl = 0
Ddλ
V µ =dxµ
dλ∇µV µ =
ddλ
V µ + Γµσρ
Ddλ
V µ =dxν(λ)
dλ∇ν V µ
(x(λ)
)= 0
For a spacetime geodesic, the parallel transport of a vector must be valid when the parameter is the propertime λ = τ , thus any valid parametrization must be by a parameter that is related to the proper time by anaffine transformation. Any valid parameter is therefore called an affine parameter.Other parametrizations will yield a sort of “non-inertial frame” equation (my terminology) with a fictitiousforce f(α)
d2xµ
dα2+ Γµρσ
dxρ
dαdxσ
dα= f(α)
dxµ
dα
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Keeping an affine parametrization but adding non-gravitational forces yields a sort of “general Newton’s secondlaw” (my terminology):
d2xµ
dλ2+ Γµρσ
dxρ
dλdxσ
dλ=
q
mFµν
dxν
dλ
More ideas:
− Exponential map− Geodesisically incomplete manifolds
• Riemann Tensor - Curvature Tensor
Rρσµν = ∂µΓρνσ − ∂νΓρµσ + ΓρµλΓλνσ − ΓρνλΓλµσ
Rρµσν = ∂σΓρνµ − ∂νΓρσµ + ΓρσλΓλνµ − ΓρνλΓλσµNumber of independent components in n dimensions:
112n2(n2 − 1)
In 4-D, it has 20 independent components, “precisely the 20 degrees of freedom in the second derivatives ofthe metric that we could not set to zero by a clever choice of coordinates when we first discussed the locallyinertial coordinates in Chapter 2.”
Rρσµν = −Rρσνµ
Rτµσν = gτρRρµσν
Rτµσν = −RτµνσRτµσν = −RµτσνRτµσν = Rσµτν
Rτµσν +Rτσνµ +Rτνµσ = 0; Rτ [µσν] = 0
• Ricci Tensor -Rµν = Rλµλν
Rµν = Rνµ
• Ricci Scalar - Curvature ScalarR = Rµµ = gµνRνµ
• Weyl Tensor - Conformal Tensor - invariant under conformal transformations...basically the Riemanntensor with all of its contractions removed. In n dimensions (where n ≥ 3), it is
Cρσµν = Rρσµν −2
(n− 2)
(gρ[µRν]σ − gσ[µRν]ρ
)+
2(n− 1)(n− 2)
gρ[µgν]σR
“This messy formula is designed so that all possible contractions of Cρσµν vanish, wihle it retains the symmetriesof the Riemann tensor:”
Cρσµν = C[ρσ][µν]
Cρσµν = Cµνρσ
Cρ[σµν] = 0
• Einstein Tensor -Gµν = Rµν − 1
2Rgµν
Gµν = Gνµ due to the symmetry of the Ricci tensor and the metric“In 4D, the Einstein tensor can be thought of as a trace-reversed version of the Ricci tensor.” (Carroll pg 131)Einstein scalar?:
G = Gµµ
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• Bianchi Identity -∇[µRρσ]µν = 0
For a general connection there would be additional terms involving the torsion tensor. It implies (after con-tracting twice on (3.139))
∇µRρµ = 12∇ρR
(3.150)The twice-contracted Bianchi identity is equivalent to
∇µGµν = 0
From http://www.mathpages.com/home/kmath528/kmath528.htm . . .“The field equations of general relativity provide a good illustration of how non-linear laws imply constraints(e.g., the Bianchi identities) on the allowable initial conditions, so we are not free to specify a system at anarbitrary point in the phase space. This type of constraint applies to the Lorentz-Dirac equation as well, sinceit too is non-linear.”
8 More Differential Geometry
8.1 Maps between Manifolds
8.2 Stokes’ Theorem
8.3 Parallel Propagation, Killing Vectors
8.4 Noncoordinate Bases
8.5 Conformal Stuff...
9 Lagrangian Formalism
S =∫L(Φi,∇µΦi) dnx
where dnx and L are densities (tensor densities?) and their product is a tensor
L =√− |g| L
where L is a tensor (scalar).Euler-Lagrange equations:
∂ L∂Φ−∇µ
(∂ L
∂(∇µΦ)
)= 0
Hilbert-Einstein Action
SH =∫ √
− |g|Rdnx
S =1
16πGNSH + SM
where SM is the action for matter.A definition of the stress tensor:
Tµν ≡ −21√− |g|
δSM
δgµν
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Palatini Formalism
S[g,Γ] =∫d4x
√− |g| gµνRµν(Γ)
10 Special Relativity
Principles
• Equivalence• Velocity of Light
Other Things
Flat Spacetime:gµν(xσ) = ηµν for all four-positions xσ
Length Contraction:L(v) = γvL0
Time Dialation:(∆t)(v) = (∆t)0/γv?
∆t =∫ τ2
τ1
dτ√1− β2(τ)
=∫ τ2
τ1
γ(τ) dτ
Relativistic Doppler Shift:Some equation here
transverse relativistic doppler shiftRelativistic addition of velocity
u′ =dx′
dt′=cγv(dx1 − βxdx0)γv(dx0 − βxdx1)
Newton’s LawF = γm(13 + γ2vvT)a
where 13 is the 3×3 identity matrix and vT is the velocity row-vector ( http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html )
Anything else in particular? (From Jackson, say?)
11 Equations
11.1 Major Equations
• Conservation of Energy and Momentum
∂µTµν = 0ν
• Maxwell’s Equations∂νF
µν = Jµ
εµνρσ∂µFνρ = 0σ or ∂ν Fµν = 0µ
or something to that effect. Or, in other notation, the second equation is
∂[µFνλ] = 0(µ, ν, λ)
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∂[µFνλ] =16
(∂µFνλ − ∂µFλν + ∂λFµν − ∂λFνµ + ∂νFλµ − ∂νFµλ)
=16
(∂µFνλ + ∂µFνλ + ∂λFµν + ∂λFµν + ∂νFλµ + ∂νFλµ)
=13
(∂µFνλ + ∂λFµν + ∂νFλµ)
= 0(µ, ν, λ)
Four-PotentialFµν = ∂µAν − ∂νAµ
Gauge TransformationAµ → Aµ + ∂µλ(x)
Aµ′ =(Aµ + ∂µλ(x)
)δµµ′???
• Einstein’s Equation
Rµν −12Rgµν =
8πGc4
Tµν
Rµν = 8πG(Tµν − 1
2Tgµν)
In free space (no matter)Rµν = 0
With cosmological constant
Rµν −12Rgµν + (factor?)Λgµν = 8πGTµν
(Add a const to a Lagrangian and usually that doesn’t matter, but in gravity it does since even “constants”interact with gravity/matter. Explain that.)The cosmological constant is important but tiny.
• Geodesic EquationDdλ
(dxµ
dλ
)= 0 ⇔ d2xµ
dλ2+ Γµρσ
dxρ
dλdxσ
dλ= 0
For time-like particle paths:pµ∇µpµ = 0
dpµ
dλ+ Γµρσp
ρpσ = 0
• Euler-Lagrange Equations for a field theory in flat spacetime
δS
δΦi=
∂L∂Φi
− ∂µ
(∂L
∂(∂µΦi)
)= 0
• Klein-Gordon Equation(�−m2)φ = 0
11.2 Minor Equations
Tµνdust = pµNν = mnuµuν = ρuµuν
Tµνperfect fluid = (ρ+ p)uµuν + pηµν
Tµνscalar field theory = ηµληνσ∂λφ∂σφ− ηµν[
12ηλσ∂λφ∂σφ+ V (φ)
]TµνEM = FµλF νλ −
14ηµνFλσFλσ
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12 The Speed of Gravity, Etc.
13 Other Relativity Theories
• Weyl’s theory• Einstein-Cartan theory• Supergravity• String theory – 26D• Active Research
− String theory – 10D− Loop gravity
14 Open Questions and Mysteries
15 Applications
• GPS
− GR− SR− Sagnac effect
16 Common Errors
16.1 Mistaking Similar Quantities
There are several quantities of the same type that must be correctly distinguished:• Velocities
− Newtonian velocity: v
− Relativistic velocity: u = γv
− 4-velocity: uµ = γ(c,v)µ = (γc,u)µ
• Velocity magnitudes (speeds)
− Newtonian speed: v = ‖v‖− Relativistic speed: u = ‖u‖− “Relativistic pseudo-speed”:
√u2 =
√uµuµ =
√−c2 = ic or =
√c2 = c (depending on signature)
∗ Note that we will use the notation u2 ≡ uµuµ and u2 = u ·u(u2 6= u ·u, even though u ·u = ‖u‖2 and ‖u‖ = u)
∗ In quantum field theory (and perhaps in these notes somewhere) u may also refer to the 4-velocity inexpressions such as. . .
• Momenta
− Generalized, canonical, or conjugate momentum− Mechanical momentum: pm = mv (also, “Newtonian momentum” and “kinetic momentum”)− Relativistic momentum: p = γpm
− 4-momentum: pµ = (E/c,p)µ
For massive particles pµ = (E/c,p)µ = γ(mc,pm)µ = (γmc,p)µ
• Momenta magnitudes
20
− magnitude of generalized momentum− magnitude of mewtonian momentum: pm = mv
− magnitude of relativistic momentum: p = γpm
− “pseudo-magnitude” of 4-momentum:√p2 =
√pµpµ =
√−E2/c2 + p2
(=√−m2c2
)∗ Note that we will use the notation p2 ≡ pµpµ and p2 = p ·p
(p2 6= p ·p, even though p ·p = ‖p‖2 and ‖p‖ = p)∗ In quantum field theory (and perhaps in these notes somewhere) p may also refer to the 4-momentum in
expressions such as p ·x = ηµνpµxν = −p0x0 + p ·x.
• Mass
− Invariant mass: m− Relativistic mass: γm− Relation to point-particle energies (E = total energy; P = potential energy; K = kinetic energy):
P = mc2 + Premaining
γmc2 = mc2 +K
E = P +K = Premaining + γmc2
16.1.1 Applying this notation
• Relativistic Energy
− Given that p2 is the square of the 4-momentum and p is the relativistic momentumand given that E = γmc2
E2 6= p2c2 +m2c4 E2 = p2c2 +m2c4
We also have E2 = (p2 − p2)c2
and p2 = −m2c2
16.2 Silly Mistake(s)
• Confusing the forms of the velocity addition formula and gamma:
− (putting a plus/minus sign in the addition formula when you mean minus/plus:)− This one is hard to catch once it’s been written (with an equals sign):
γv 6=1√
1 + v2/c2
I made the error of equating the two when trying to prove. . .
17 Useful Programs
Programs (that use second party computer algebra systems)• GRTensorM, Cartan (both use Mathematica)• GRTensor II (Maple)
I haven’t used them (yet), though.
21
References
[1] Sean M. Carroll: Spacetime and Geometry: An Introduction to General Relativity, Addison-Wesley (2004)
[2] Charles W. Misner, Kip S. Thorne, John Archibald Wheeler: Gravitation, W. H. Freeman and Company (1973)
[3] Herbert Goldstien: Classical Mechanics, Second Edition, Addison-Wesley (1980)
[4] Roger Penrose: The Road to Reality: A Complete Guide to the Laws of the Universe, Alfred A. Knopf, a divisionof Random House (2004)
[5] Parry Moon, Domina E. Spencer: Theory of Holors: A generalization of tensors, Cambridge University Press(1986)
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