General Relativity
General RelativityMA4448Transcribed by Debbie IpTaught by Dr Peter TaylorHilary Term 2012
Contents1 Introduction1.1 Newtonian Theory of Gravity . . . . . . . . . . . . . . . . . .1.2 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . .1.3 General Relativitiy (Guiding Principles) . . . . . . . . . . . .11232 Einstein Equations from an Action Principle2.1 Principle of Least Action . . . . . . . . . . . . . . . . .2.2 The Stress-Energy-Momentum Tensor . . . . . . . . . .2.3 Varying the Metric Inverse and the Metric Determinant2.4 The Einstein Hilbert Action . . . . . . . . . . . . . . .2.5 Einstein’s Field Equations . . . . . . . . . . . . . . . .2.6 Further Remarks on the Field Equations . . . . . . . .4455789.3 The Stress-Energy-Momentum Tensor103.1 Decomposition of the Stress-Energy-Momentum Tensor in anOrthonormal Tetrad . . . . . . . . . . . . . . . . . . . . . . . 103.2 Stress-Energy-Momentum Tensor for a Perfect Fluid and forDust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 The4.14.24.34.4Schwarzschild SolutionCanonical Form of a Spherically Symmetric Line-ElementThe Schwarzschild Solution . . . . . . . . . . . . . . . .Properties of the Schwarzschild Solution . . . . . . . . .4.3.1 Limiting Cases M 0, r . . . . . . . . . .4.3.2 The Coordinate Singularity at r 2M . . . . . .Birkhoff’s Theorem . . . . . . . . . . . . . . . . . . . . .5 Solar System Tests of GR5.1 The Gravitational Red-Shift . . . . . . . . .5.2 Planetary Motion and Perihelian Advance of5.2.1 Geodesic Equations . . . . . . . . . .5.2.2 Propagation Equation for θ(τ ) . . . .5.2.3 Newtonian Result . . . . . . . . . . .5.2.4 Shape of General Relativistic Orbit .5.3 Light Reflection . . . . . . . . . . . . . . . . . . . .Mercury. . . . . . . . . . . . . . . . . . . . .14141516161717.18182020212223246 Black Holes276.1 Radial In-falling Photons . . . . . . . . . . . . . . . . . . . . . 276.2 Radially In-falling Particles . . . . . . . . . . . . . . . . . . . 286.3 The Kruskal Extension of the Schwarzschild Manifold . . . . . 292
7 Cosmology7.1 The Cosmological Principle: . . . . . . . . . . . .7.2 Kinematics of the Continuum . . . . . . . . . . .7.2.1 Connecting Vectors . . . . . . . . . . . . .7.2.2 Shear, Stress and Expansion . . . . . . . .7.3 The Friedman-Robertson-Walker (FRW) Metric .7.3.1 Isotropy . . . . . . . . . . . . . . . . . . .7.3.2 Comoving Coordinates . . . . . . . . . . .7.3.3 The Spacetime Metric . . . . . . . . . . .7.4 Cosmological Red-Shift and Hubble’s Law . . . .7.5 Einstein’s Equations . . . . . . . . . . . . . . . .7.5.1 Matter Content of the Universe . . . . . .7.5.2 The Friedman and Raychauduri Equations7.6 Cosmological Models with Vanishing Λ . . . . . .7.6.1 Geometry of the Universe . . . . . . . . .7.6.2 Big Bang Singularities . . . . . . . . . . .7.6.3 The Fate of the Universe (Eschatology) . .8 The8.18.28.38.4.Linear ApproximationThe Einstein Equations in the Linear ApproximationGauge Transformations . . . . . . . . . . . . . . . . .The Newtonian Limit . . . . . . . . . . . . . . . . . .Gravitational Waves . . . . . . . . . . . . . . . . . .8.4.1 The Linearised Vacuum Solutions . . . . . . .8.4.2 Energy Transfer . . . . . . . . . . . . . . . . 9 Einstein-Maxwell Theory539.1 The Field Equations . . . . . . . . . . . . . . . . . . . . . . . 539.2 The Reissner-Nordstrom Solution . . . . . . . . . . . . . . . . 55
Suggested ReadingR. D’Inverno, Introducing Einstein’s Relativity ( - - - )B. Shutz, A First Course in General Realtivity (- )S.M.Carroll, Lecture Notes on GR, online (- )H. Stephani, General Relativity ( - - -)R.M. Wald, General Relativity (- )C.W.Misner, K.S. Throne, J.A.Wheeler, Gravitation (- )
1Introduction1.1Newtonian Theory of GravityFor a distribution of matter with density ρ(t, x, y, z), we have a gravitationalpotential ϕ.Field Equation: 2 ϕ 4πGρwhere 2 2 2 x2 y 2 z 2ϕ is given at points inside the matter distribution by solving Poisson’sequation with ρ given.Outside the matter distribution (i.e. in a vacuum) ρ is given by solving 2 2 ϕ 0Equations of MotionThe equations of motion of a test particle arexi xi (t), i 1, 2, 3 i.e.d 2 xi ϕ,idt2d2 x ϕ 2dt x2dy ϕ 2dt yd2 z ϕ 2dt zNewtonian theory can be written as a metric theory of gravity, thoughthe spacetime is complicated to describe geometrically. (Requires additionalstructures such as absolute time and simultaneous points forming a Euclidean3-manifold)E.g. We can rewrite the equations of motion in the geodesic equationform.1
Definexµ (t, x, y, z) ẋµ (1, ẋi )ẍµ (0, ẍi ) (0, ϕ,i )i.e.ẗ 0 and ẍi ϕ,i 0 ẍi ϕ,i ẋ0 ẋ0 0Compare with the geodesic equationλνd 2 xµµ dx dx 0 Γνλdt2dt dtWe can read off the Newtonian connectionΓi N 00 ϕ,i and Γµ N νλ 0 otherwise RN i0j0 ϕ,i and RN µνλρ 0 otherwise RN 00 2 ϕ and RN µν 0 otherwisei.e. Newtonian spacetime is curved.1.2Special RelativityDescribes non gravitational physics such as electrodynamics, standard modeletc.Discard the notion of absolute time, we introduce a 4D space continuum.To each event in spacetime, we assign the coordinates (t, x, y, z) and theinfinitesimal interval ds between the infinitesimally separated events satisfiesthe Minkowski line element.ds2 dt2 dx2 dy 2 dz 2 ηµν dxµ dxνwhereηµν diag ( 1, 1, 1, 1)The Minkowski line element is invariant under Lorentz transformations0xµ xµ2
where00Λµ µ Λν ν ηµ0 ν 0 ηµνand under arbitrary translationsxµ x µ d µwhere dµ are constants. Together, these define the Poincare group.Null cones of νµν describe light rays in a vacuum.Time-like geodesics of νµν describe force free motion of massive particles andZZZ1µν 12τ dτ ( ηµν dx dx ) (1 v 2 ) 2 dtis the proper time measured by a standard clock associated with the particlewhere the integral is taken along a time like path representing the particle’strajectory.Note Used ‘relativistic’ units c 1. In non-relativistic units 1Z v2 2τ 1 2dtc1.3General Relativitiy (Guiding Principles) All observers are equivalent. physics should be coordinate independent. (Principle of GeneralCovariance) tensor equations are the most natural mathematical framework. Should agree locally with SR Admit a class of preferred relatively accelerated world lines representingfree fall. Should admit a tensor related to the source of the gravitational field. Should explain observed solar system phenonema such as light deflection, perihelian advance of Mercury, time-delay etc.General relativity assumes spacetime is a pseudo-riemannian manifoldwith signature(- ).3
Null geodesics represent light rays.Timelike geodesics represent paths of freely falling particles.Locally, we can always choose a chart s.t. gµν ηµν , i.e. SR valid locally.The field equations areGµν κT µνwhere κ is a constant fixed by the Newtonian limit, Gµν is the EinsteinTensor, and T µν describes the source of the gravitational field.2Einstein Equations from an Action PrincipleWe first recall two important results:(i) Fundamental Lemma of Calculus of Variations: IfZ x2ϕ(x)η(x)dx 0x1where ϕ(x) is continuous, ad η(x) twice differentiable and vanishes on boundary η(x1 ) η(x2 ) 0, then ϕ(x) 0 on [x1 , x2 ](ii) Gauss Divergence Theorem:ZZµχµ dΣµ µ χ dΩ VVµwhere χ is a vector density of weight 1. An immediate corollary isZZ µ g µ X dΩ g X µ dΣµV Vwhere X µ is a vector field.2.1Principle of Least ActionWe start with an actionZLdΩS all spacewhere L is a Langrangian density of weight 1. We consider small variations in the metric tensor gµν gµν Sgµν which inducs a variation in theaction functional S S δS. We also assume the metric variations and itsderivatives vanish at infinity.4
The action principle impliesZLµν δgµν dΩ 0δS all spacewhere L2.2µν δLδgµνis a( 20 )tensor density of weight 1.The Stress-Energy-Momentum TensorIn General Relativity, we must allow for the definition of a tensor related tothe source of the gravitational field, i.e. the action has contributions comingfrom the matter fields and the gravitational fieldsZS Sµ Sg (Lµ Lg )dΩall spaceWe defineZδSµ all spacewhere we have definedδLµ1δgµν dΩ δgµν2Z g T µν δgµν dΩall space2 δLµT µν g δgµνwhich is the stress-energy-momentum tensor.2.3Varying the Metric Inverse and the Metric DeterminantIn what follows, we shall require δg µν in terms of δgµν . We note thatg µν gµν δλµ δg µν gνλ g µν gνλ 0Multiply by g λρ δg µν δνρ g λρ g µν δgνλ δg µρ g λρ g µν δgνλ(2.1) We also require δ g. We note that for any non-singular matrix aµν , withinverse aµν and determinant a. Each element has a cofactor given byAµν aνµ a5
Also, the determinant is obtained by expanding across any row.a A(µ)ν a(µ)ν(no sum over µ)which imples that a Aµν aνµ a aµνTherefore a a g, this gives a aµν aνµ a aµν aµν 11 ( g) ( g) 2 g211 ( g) 2 gg νµ gµν21 gg µν gµν2 1 δ( g) g g µν gµν2An immediate consequence of equation (2.2) is 1 ( g),λ g g µν gµν,λ2(2.2)(2.3)Example 2.3.1Show that equation (2.3) leads to µ ( g) 0 g is a scalar density of weight 1. The covariant derivative of a scalardensity of weight a is λ χ χ,λ ω Γµµλ χWe wish to show that λ ( g) ( g),λ g Γµµλ 0We know that0 µ gνλ gνλ,µ gνρ Γρλµ gλρ Γρνµ gνλ,µ gνρ Γρλµ gλρ Γρνµ6
Equation (2.3) implies: 1 ( g),λ gg µν gµν,λ2 1 g g µν (gµρ Γρνλ gνρ Γρµλ )2 1 g(δρν Γρνλ δρµ Γρµλ )2 1 g(Γννλ Γµµλ )2 g Γµµν λ ( g) 02.4 The Einstein Hilbert ActionWe now consider the contribution to the action coming from the gravitationalfield:ZSg Lg dΩspaceThe only scalar density of weight 1 involving the metric and its derivatives up to second order is gR. i.e. we take Lg κ 1 gR κ 1 g g µν RµνZ 1 δSg κ[δ( g g µν )Rµν g g µν δRµν ]δΩWe require our expression for δRµν schematically, we haveR Γ Γ ΓΓ ΓΓThusδR (δΓ) (δΓ) δΓΓ ΓδΓ δΓΓ ΓδΓδΓµνλis a well defined tensor (even though Γµνλ is not, since this involves thedifference of two connections) we can therefore convert partial derivatives tocovariant derivatives:µδRνλρ (δΓµνρ );λ (δΓµνλ );ρ7
The second term of the gravitational action is thereforeZ νρ µ 1κ g (g δΓνρ );µ (g νρ δΓµνµ );ρ dΩNow g νρ δΓµνρ and g νρ Γµνµ are vectors, so we may apply the corollary to thedivergence theorem to convert to a surface integralZ 1κ g(g νµ δΓρνµ g νρ δΓµνµ )dΣρ 0)The gravitational action reduces toZ 1δSg κδ( g g µν )Rµν dΩ 1 κ 1 κ2.5Z [δ( g)g µν Rµν gδg µν Rµν ]dΩ 1 λρµνµλ g g δgλρ g Rµν g g δgλρ Rµν dΩ2 Z 1 λρ 1λρ κg R R gδgλρ dΩ2Z 1 κ g Gλρ δgλρ dΩZ Einstein’s Field EquationsPutting the results together, we haveδS δSµ δSgZZ 1 µν 1 g T δgµν dΩ κ g Gµν δgµν dΩ2 Z 1 µν 1 µν gT κ Gδgµν dΩ 02Since we assume metric variations vanish at the boundary, we must have1 µνT κ 1 Gµν 02κ Gµν T µν28
In the weak field slow moving approximations, they reproduce Poisson’sequation only when16πGκ (D’Inverno)c4or, in natural units c g 1, κ 16π.Gµν δπT µν Recall the twice contracted Bianchi identitiesGµν;ν 0 T;νµν 02.6Further Remarks on the Field EquationsThe history of an isolated body in spacetime is a timelike world tube filledwith the world lines of the constituent particles. Inside the world tube, wehave T µν 6 0 , and we solve the non vacuum Einstein Field Equations.Gµν δπT µνOutside the world tube, T µν 0 and we solve the vacuum field equationsGµν 0 Rµν 0Agreement with Newtonian limit requires κ 16π. G 8πT µνThe world line of a particle xµ (s) with non zero mass is timelike. Taking sto be arc-length along the curve, we havedxµ dxν 1ds dsIf the particle is a test particle (doesn’t perturb the geometry of spacetime),then the world line is a timelike geodesic satisfyinggµνλνd 2 xµµ dx dx Γ 0νλds2ds dswhere s is now the proper time along the curve.We take the world line of massless particles to be null geodesicsνλd 2 xµµ dx dx Γ 0νλdr2dr drandgµνdxµ dxν 0dr dr9
33.1The Stress-Energy-Momentum TensorDecomposition of the Stress-Energy-Momentum Tensor in an Orthonormal TetradThe stress-energy-momentum tensor satisfies1 µνGand T;νµν 08πT µν is a symmetric 4x4 matrix and in general will have 4 mutually orthogonaleigenvectors; one timelike and three spacelike.Let uµ be the unit timelike eigenvector of T µν with eigenvalue ρ, i.e.T µν T µν uν ρ uµ , uµ uµ 1 ρ Gµλ uλWe take timelike worldlines tangent to uµ (i.e. the integral curves of uµ )to be the worldlines of the constituent particles of the matter distribution.We take ρ to be the proper density of the matter (density observed in therest frame of the constituent particle). uµ is the 4-velocity of a constituentparticle, and it describes the interval motion of the body.We further defineSµν ρuµ uν Tµν SµνSµν uν ρuµ (uν uν ) Tµν uν ρuµ ρuµ 0uν is a unit timelike eigenvector of Sµν with eigenvalue zero.Sµν has 6 independent components, and is called the stress tensor of thematter distribution.We now let {eµ(1) , eµ(2) , eµ(3) } {eµ(i) }3i 1 be the unit spacelike eigenvectorsof T µν with eigenvalues {p(i) }3i 1 , respectively.Tµν eν(i) p(i) e(i)µi 1, 2, 3 no sum over iMutual orthogonality impliesuµ e(i)µ uµ eµ(i) 0ThereforeSµν eν(i) ρuµ uν eν(i) Tµν eν(i) ρ(i) e(i)µ10i 1, 2, 3
Hence {eν(i) }3i 1 are the unit spacelike eigenvectors of Sµν with e-value {p(i) }3i 1 .These are called the 3 principle stresses in the matter distribution.For pressures p(i) 0.For tensions p(i) 0.So we have 4 mutually orthogonal eigenvectors satisfyinguµ uµ 1eµ(i) e(j)µ δ(i)(j)eµ(i) uµ 0We set uµ eµ(0) , then we haveeµ(0) e(0)µ 1eµ(1) e(1)µ δ(i)(j)eµ(0) e(1)µ 0 eµ(a) e(b)µ η(a)(b)(a,b 0,1,2,3)(parenthesis around indices to distinguish tetrad indices from spacetime indices) eµ(a) e(b)µ η(a)(B) diag( 1, 1, 1, 1) gµν eµ(0) eν(b) η(a)(b){eµ(a) }3a 0 is an orthonormal tetrad. η(a)(b) are the components of the metrictensor on this orthonormal tetrad.Any vector or tensor may be projected onto the tetrad from, for example,the components of the curvature tensor in the orthonormal tetrad areR(a)(b)(c)(d) Rµνλσeµ(a) eν(b) eλ(c) eσ(d)We can also write the metric components g µν in terms of {eµ(a) } followingfrom the orthonormality conditions:g µν η (a)(b) eµ(a) eν(b)Therefore, we can pass freely from tensor components to tetrad componentsand vice-versa.11
Note:T(0)(0) Tµν eµ(0) eν(0) Tµν uµ uν ρuν uν ρT(0)(i) Tµν eµ(0) eν(i) Tνµ uµ eν(i) 0(i 1,2,3)T(i)(j) Tµν eµ(i) eν(j) p(i) e(i)ν eν(j) p(i) δ(i)(j) T(a)(b) diag(ρ, p(1) , p(2) , p(3) )3.2gµν eµ(a) eν(b) η(a)(b)(a)g µν η (a)(b) eµ(a) eν(b)(b)Stress-Energy-Momentum Tensor for a Perfect Fluidand for DustWriting (b) out explicitly:g µν η (0)(b) eµ(0) eν(b) η (1)(b) eµ(1) eν(b) η (2)(b) eµ(2) eν(b) η (3)(b) eµ(3) eν(b) η (0)(0) eµ(0) eν(0) η (1)(1) eµ(1) eν(1) η (2)(2) eµ(2) eν(2) η (3)(3) eµ(3) eν(3)µ ν u u 3Xeµ(i) eν(i)i 1 3Xeµ(i) eν(i) g µν uµ uνi 1RecallSµν eν(i) p(i) e(i)µ 3XSµν eν(i) eλ(i) 3Xi 1LHS Sµνp(i) e(i)µ eλ(i)i 13Xeν(i) eλ(i) Sµν (g νλ uν uλ ) Sµλ 0i 1 Sµν 3Xp(i) eµ(i) eν(i)i 1For a perfect fluid, the stress is an isotropic pressure (no preferred direction)p(1) p(2) p(3) p12
Sµν p3Xeµ(i) eν(i)i 1 S µν p(g µν uµ uν )(stress tensor for a perfect fluid)where 4µ is the 4-velocity.By definition, we haveTµν ρuµ uν Sµν ρuµ uν p(gµν uµ uν )Tµν (ρ p)uµ uν pgµν(stress-energy-momentum tensor for a perfect fluid)Example 3.2.1Show that for incoherent matter with proper density ρ, that ρ changes alongintegral curves of uµ according toρ,µ uµ ρuµ;µ 0Further show that the world lines of the dust particles are timelike geodesics.We have T µν ρuµ uµ . The conservation equations areT;νµν 0 0 ν (ρuµ uν ) ρ,ν uµ uν ρ( ν uµ )uν ρuµ ν uν uµ (ρ,ν uν ρuν;ν ) ρuµ;ν uννρuν;ν ) 0 (ρ,ν u 1But uµ uµ;ν (uµ uµ );ν 02(1)ρuµ uµ;ν(as required)Sub this result back into (1) uµ;ν uν 0 Du u 0i.e. the integral curve of the dust particle parallel transports its own tangentvector geodesics.13
4The Schwarzschild Solution4.1Canonical Form of a Spherically Symmetric LineElementWe shall consider spherically symmetric solutions to Einstein’s vacuum fieldequations.Spherical symmetry implies that there exists a coordinate system (t, r, θ, ϕ)say, in which the line-element is invariant under the reflectionsθ θ0 π θϕ ϕ0 ϕi.e. no cross terms of the form drdθ, drdϕ, dθdϕ, dθdϕ, dtdθ, dtdϕ and thateach 2D submanifold defined by t const, r const, are the 2-spheres.dl2 a2 (dθ2 sin2 θdϕ2 )Therefore, the spherically symmetric line-element has the formds2 A(r, t)dt2 2B(r, t)dtdr C(t, r)dr2 D(t, r)(dθ2 sin2 θdϕ2 ) Changing the radial coordinate r r̃ D ds2 Ã(t, r̃)dt2 2B̃(t, r̃)dtdr̃ C̃(t, r̃)dr̃2 r̃2 (dθ2 sin2 θdϕ2 )Introduce a new time coordinate bydt̃ I(t, r̃)[ Ã(t, r̃)dt B̃(t, r̃)dr̃] dt̃2 I(t, r̃)[Ã2 dt2 2ÃB̃dtdr̃ B̃ 2 dr̃2 ]dt̃2B̃ 2 2 Ãdt 2B̃dtdr̃ dr̃I 2 ÃÃThe line-element now reads 2 dt2B2ds 2 C dr2 r2 (dθ2 sin2 θdϕ2 ) (dropped the tildes)I AA2Defining 2 new functions p p(t, r); q q(t, r) by1 ep ;2I AB2 C eqAOur canonical form of a spherically symmetric line-element readsds2 ep dt2 eq dr2 r2 (dθ2 sin2 θdϕ2 )14
4.2The Schwarzschild SolutionTo determine the functions p(t, r), q(t, r) we must solve the vacuum fieldequations Gµν 0. The non vanishing components of the Einstein tensor are 1 q11t qGt e 2 2(i)r r rre q q2(ii)Gt r t 1 p11Grr e q 2 2(iii)r r rr1Gθθ Gϕϕ e q21 e p2! 21 p q 1 q 1 p 1 p 2p 22 r r r r r r 2 r r! 2 2 q 1 q1 1 p 2 t2 t2 t tWe see that the Einstein equations give us 4 non trivial equations. However,they are not all independent. The twice contracted Bianchi identities Gµν;ν 0imply that vanishing of (i) (iii) implies vanishing of (iv). So we have 3independent equations 11 q1 qe 2 2 0(a)r r rr q 0(b) t 111 p 2 2 0(c)e qr r rrIt is immediately obvioius from (b) that q is a function of r only. i.e q q(r)and therefore, (a) becomes a simple ODE:e q e q rdq 1drd(re q ) 1dr re q r const Taking our constant of integration to be 2M (which we will interpret later)yields: 12M qe 1 r15
To optain p we note that adding (a) and (c) gives p q 0 r ri.e. p q f (t) ep e q ef (t) 2M 1 ef (t)rOur line element reads 1 2M2M2f (t) 2ds 1 e dt 1 dr2 r2 dθ2 sin2 θdϕ2rrFinally, we may eliminate f (t) by redefining our time coordinate by1e 2 f (t) dt dt0Z t10e 2 f (u) du t cwhich gives (after dropping primes) 12M2M22ds 1 dt 1 dr2 r2 (dθ2 sin2 θdϕ2 )rr(Schwarzschild Solution)4.34.3.1Properties of the Schwarzschild SolutionLimiting Cases M 0, r It is clear that by setting M 0 we retrieve the Minkowski metric in sphericalpolar coordinates. The parameter M represents the mass/energy and onemay interpret the Schwarzschild solution as the geometry due to a pointmass M at the origin.We further note that as r , we again retrieve the Minkowski metric. We did not impose asymptotic flatness! Spherically symmetric vacuumsolutions of Einstein’s equations are necessarily asymptotically flat.16
4.3.2The Coordinate Singularity at r 2MThe metric components of Gµν are singular at r 0, and r 2M (r 2GMinC2non natural units). The r 0 singularity is known as a curvature singularityand is irremovable. The r 2M singularity is a coordinate singularityand may be rmoved by an appropriate coordinate transformation (thoughr 2M still has important physical implications). To see this, we make thecoordinate transformation (t, r, θ, ϕ) (u, r, θ, ϕ) whereu t r 2M log(r 2M ) 1 2Mdr du dt 1 rIn these coordinates, the metric reads 12M2ds 1 du2 2dudr r2 (dθ2 sin2 θdϕ2 )rIn coordinates (u, r, θ, ϕ) the components of Gµν are non singular at r 2M ) 100 (1 2Mr 10 00 gµν 00 r20 00 0 r2 sin2 θWe also note that in the standard form of the Schwarzschild metric 2Mgtt 1 0, r 2Mr 0, r 2Mso that the signature of the metric is ( ) for r 2M . In this region,r takes on the character of a time coordinate and t a spatial coordinate. Wecall the region r 2M the exterior Schwarzschild geometry, and the region0 r 2M the interior Schwarzschild geometry.4.4Birkhoff ’s TheoremDefinition: Static space time: A space time is said to be static if there existsa coordinate system in which the metric components are time- independentand the metric is time reversal invariant, i.e. there exists a coordinate systemsuch that gµν,t 0, and there are no cross terms dtdxi (i 1, 2, 3)17
Note: The chart independent definition relies on the existence of a timelike killing vector that is hypersurface orthogonal.We note that the Schwarzschild solution is static, but we did not imposethis!Birkhoff ’s Theorem: A spherically symmetric vacuum solution in theexterior region is necessarily static.Corollary: For a spherically symmetric source in the region r a, wherea 2M , the exterior Schwarzschild solution is the unique solution.5Solar System Tests of GRIn order for GR to be considered a viable theory of gravitation, it oughtto be able to explain various phenomena in our solar system such as lightdeflectionWe model the gravitational field by the Schwarzschild solution with M M , the mass of the sun. We model the planets as text particles which moveon timelike geodesics of the Schwarzschild spcacetime. There are 3 classicaltests we shall consider:5.1The Gravitational Red-ShiftLet Co and C1 be the timelike world lines of an emitter and receiver of light,respectively. Let τ be the proper time along them. Let P0 and P1 be the nullworldline of a photon emitted at the even P0 on C0 and received at the eventP1 on C1 . Suppose in a short interval dτ0 of proper time on C0 , n photonsare emitted and these are received in an interval dτ1 of proper time on C1 .Then,ν0 frequency of emission no. of photons per unit timen dτ018
Similarly,ν1 frequency of receptionn dτ1 ν0 dτ0 ν1 dτ1ν0dτ1 ν1dτ0If λ0 , λ1 are the emitted and received wavelengths respectively, then11λ1 (c 1)λ0 ν0ν1λ1dτ1 λ0dτ0A signal is red shifted (loses energy) if λ1 λ0 or if dτ1 dτ0 .Suppose the emitter is at rest on the surface of the sun. Then the worldline C0 would be given byr a solar radiusθ θ0ϕ ϕ0On C0 : 2Mds 1 dt2ar 2Mdτ ds2 1 dta2 Similarly, on C1 :rdτ1 ForMa1 2Mdtrq1 2Mdτ1λ1r q λ0dτ01 2Ma small Mr smallq 2 ! 2 !1 2MMMMMrq 1 O1 Orraa1 2Ma 1 MM ar19
MMλ1 1 λ0arSince Ma Mr , we have λ1 λ0 . i.e. signals are red-shifted as they passthrough the gravitational field λ0 λ1λ1MM λ 1 z λ0λ0λ0ar 1 1r in standard units)(or GMc2aNote: This is not a Doppler shift since there is no relative motion betweenobservers.5.25.2.1Planetary Motion and Perihelian Advance of MercuryGeodesic EquationsWe treat planets as test particles moving among timelike geodesics of Schwarzschildspacetime. Line element 12M2M22ds 1 dt 1 dr2 r2 (dθ2 sin2 θdϕ2 )rr 12M 22ML gµν ẋ ẋ 1 ṫ 1 ṙ2 r2 (θ̇2 sin2 θϕ̇2 ) 1rr(5.1)dtṫ d(proper time) d L LE L µ 0µdτ ẋ xµ ν20
1 #2M 2 1 ṙr 2M 1 ṫ Er" # 1 # " 2 2M2M 2d2M 22M1 µ r :0 ṙ 2 ṫ 1 ṙ 2r(θ̇2 sin2 θϕ̇2 )2dτrrrr 1 MM2M2M 22222 r̈ 2 1 ṙ (2 2M )(θ̇ sin θϕ̇ ) 2 1 ṫ 0rrrrdµ θ :0 (2r2 θ̇) 2r2 sin θ cos θϕ̇2dτ2 θ̈ ṙθ2 sin θ cos θϕ̇2 (iv)rdµ ϕ :0 (2r2 sin2 θϕ̇)dτ r2 sin2 θϕ̇ hdµ t:dτ"5.2.2Propagation Equation for θ(τ )(iv) π2is a solution. Assume θ(0) π2 , θ̇(0) 0 θ̈(0) 0.Differentiating (iv) gives θ̇(0) 0. all derivatives of θ vanish.Consider τ τ1 0 close to τ 0, then1 .1θ(τ0 ) θ(0) θ̇(0)τ1 θ̈(0)τ12 θ (0)τ13 .23!1 .θ̇(τ1 ) θ̇(0) θ̈(0)τ1 θ (0)τ12 .2π θ(τ1 ) , θ̇(τ1 ) 02Therefore we have shown that assuming θ(0) π2 , θ̇(0) 0, then it remainstrue for some nearby point. By induction, it is true for all values of τ ,θ(τ ) π2 , θ̇(τ ) 0. only consider equatorial plane. We now have 2Mṫ E1 r21
ṙ2 Substitution: h h2r2 r2 ϕ̇ h 2M2M21 E 1 0rrr0rdr dudr dτdu dτdr du dϕ du dϕ dτ 2 rdu ϕ̇ dϕr0h du r0 dϕṙ 2 du2Mh2 22M u 2u 1 E2 1 0dϕr0r0r0 2du2M ur0 2M u3r2 u2 0 (E 2 1) dϕhh2r0u2r02(omitting 2M u3,r0we retrieve the Newtonian result)Differentiation gives the more familiar formM r0 3M u2d2 u u dϕ2h2r05.2.3(Relativistic Binet Equation)Newtonian ResultIgnoring2M u3r0and writing 2M ε 1r0 2 uN r2duNr2 u2N 0 (E 2 1) ε 2 0dϕhhThis can be solved exactly by writing the solution as uN u0 v., where u0is a constant chosen to eliminate the term linear in v. 2dvr2εr2 u20 2u0 v v 2 02 (E 2 1) 20 (u0 v)dϕhhu0 is chosen such that2u0 εr021 εr02 u 0h22 h222
dvdϕ 2 v2 r02 2εr02 u02(E 1) u k20h2h2 v(ϕ) k sin(ϕ ϕ0 ) uN u0 (1 e sin(ϕ ϕ0 ))with e uk (defines ellipse for 0 e 1) 23r2 (c2 1)du0 u2 0 h2 2 Mhur 2 Mr0u2dϕNewtonian result obtained by ignoring u3 term. Solved with AnsatzuN u0 v u0 (1 esin(ϕ ϕ0 ))ellipse with period 2π5.2.4Shape of General Relativistic OrbitAgain we take u u0 v where u0 is a constant chosen to eliminate the termlinear in v. This leads to requiring that u0 satisfies the quadratic3 u20 2u0 r02 0h2where 2M1 and we choose the solution that is closest to the Newtonianr0result. Then v satisfies 2dv(c2 1)M r0 u0 u20 v 2 r02 2 u30 3 u0 v 2 v 322dϕhhIgnoring the v 3 term and collecting constants 2dv v 2 (1 3 u0 ) k 2dϕwhich is easily solved, yieldingv ksinω(ϕ ϕ0 )ω23
where ω 2 1 3 u0 . i.e. the shape of the orbit as predicted by relativity isan ellipse with a periodicity2π3 2π(1 u0 )ω2the periolian advance is given by (in standard units) ϕ 3π u0 GM u0 6π 2cr0To approximate ur00 we use the fact that each orbit is approximately Newtonian and we know for an ellipsermax a(1 e) rmin a(1 e)where a is the semi-major axisr0r0 rmina(1 e)r0r0 (uN )min u0 (1 e) rmaxa(1 e)122u01 r0a(1 e2 ) a(1 e2 )a(1 e2 )(uN )max u0 (1 e) ϕ 6πGM e2 )c2 a(1For Mercury, this predicts a shift of 43” per century while the observed valueis 43”.1 0.55.3Light ReflectionWe consider photon paths in the Schwarzschild gravitational field. We describe the photons by null geodesicsẍµ Γµνλ ẋµ ẋλ 0andgµν ẋµ ẋν24
µwhere ẋµ dxis an affine parameter.dsAgain, without loss of generality, we take the photon path to be in theequatorial plane θ(s) π2 for all s. Our geodesic equations are 2M1 ṫ Err2 ϕ̇ hM2r̈ 2r 1 2M2M 2M221 1 ṫ 0ṙ (r 2M )ϕ̇ rrrThe 1st integral of the motion0 gµν ẋµ ẋν 12M 22M2 2 1 ṫ r ϕ̇ 1 ṙ2rrUsing the fact thatdrh drϕ̇ 2dϕr dϕand the conservation equations to simplify 1 2 2 12Mhdrh22M21 2 E 1 0rr4 dϕrrṙ Again, we take u r0r dudϕ 2 dudϕ 2 u2 E 2 r22M u3 20r0hor u2 u3 E 2 r02h2where 2M1r0Take u u0 u, and subbing into our equation and equating equal ordersof gives 2 r2hdu0 u20 02 where d (A)dϕdEand 2du0dϕ du1dϕ 2u0 u1 u30 025(B)
Equation (A) is easily solvedr0sin ϕdu0 takingϕ0 0Then subbing this into equation (B) du11 r2cos ϕ sin ϕu1 02 sin3 ϕ 0dϕ2dTry a solution of the formu1 A B sin ϕ C cos2 ϕ B sin ϕ(A C) sin3 ϕ(C B 0, A C, C 1 r02) 02 d21 r022 d21 r02(1 cos2 ϕ)2 d2r0 u sin ϕ (a cos2 ϕ)d2We require the total deflection in the asymptotic regions r (u 0).r , as ϕ ϕ1r , as ϕ π ϕ2subbing these into our equation gives u1 1 r2r0( ϕ1 ) 02 (1 1 O( 2 ))d2dr0 ϕ1 d21rr00 ϕ2 02 (1 1 O( 2 ))drdr0 ϕ2 d2r0 t4M ϕ ϕ1 ϕ2 dd0 ϕ 4GMc2 d(total deflection angle (in standard units))Take M M , d R , gives ϕ 1”.75. Observed in 1919 by Sir ArthurEddington during a solar eclipse.26
66.1Black HolesRadial In-falling PhotonsConsider an observer at rest relative to the source of the Schwarzschild gravitational field. The observer’s world line is r constant, θ constant, ϕ constant and 2M2dτ 1 dt2rwhere τ is proper time 1dτ2M 2 1 dtrFor r 2M , then along the observer is world linedτ 1 t τ (choosing τ (0) 0)dtTherefore, t corresponds to proper time measured by an observer at restat infinity. How does such an observer ‘see’ a radially in-falling photon asr 2M ?The world line of a radially in-falling photon satisfies 1 2M2M2dr 1 dt21 rr 1dt2M 1 drrwhere represents an outgoing photon and - represents an ingoing photon.Solving givest (r 2M log(r 2M ) C) u t (r 2M log(r 2M )) constant along radially null geodisics.Clearly, as r 2M, t . i.e. an observer at infinity will never ‘see’the photon cross the horizon (r 2M ), according to this observer, it takesan infinite amount of time to reach r 2M .dtNote: As r , we have dr 1 t r c.i.e. as r , ingoing and outgoing null rays are straight lines with angle 45 .27
6.2Radially In-falling ParticlesA radially in-falling particle will move on a timelike geodesic given by 2M1 ṫ Er 1 2M 22M 1 ṫ 1 ṙ2 1rrIf we consider a particle initially at rest at infinity E 1Then the geodesic equations give 1 12M2M 1 1 ṙ2 1rr dτdr 2r2M r 21 dτ dr2M(minus sign reflects the fact that the particle is ingoing.Integrating, we obtain33222(rτ 10 r )3(2M ) 2 where the particle is at r0 at τ τ0 . Now the proper time to reach thesingular
0) tensor density of weight 1. 2.2 The Stress-Energy-Momentum Tensor In General Relativity, we must allow for the de nition of a tensor related to the source of the gravitational eld, i.e. the action has contributions coming from the matter elds and the gravitational elds S S S g Z all space (L L g)d We de ne S Z all space L g g d 1 2 .
1 RELATIVITY I 1 1.1 Special Relativity 2 1.2 The Principle of Relativity 3 The Speed of Light 6 1.3 The Michelson–Morley Experiment 7 Details of the Michelson–Morley Experiment 8 1.4 Postulates of Special Relativity 10 1.5 Consequences of Special Relativity 13 Simultaneity and the Relativity
Sean Carroll, “Spacetime and Geometry” A straightforward and clear introduction to the subject. Bob Wald, “General Relativity” The go-to relativity book for relativists. Steven Weinberg, “Gravitation and Cosmology” The go-to relativity book for particle physicists. Misner, Thorne and Wheeler, “Gravitation”
Theory of Relativity. Einstein's General Theory of Relativity by Asghar Qadir. Einstein'sGeneralTheoryofRelativity ByAsgharQadir . Relativity: An Introduction to the Special Theory (World Scientific 1989) or equivalent, but do not have a sound background of Geometry. It can be used
Introduction Special Relativity General Relativity Curriculum Books The Geometry of Special Relativity Tevi
The theory of relativity is split into two parts: special and general. Albert Einstein came up with the spe-cial theory of relativity in 1905. It deals with objects mov-ing relative to one another, and with the way an observer's experience of space and time depends on how she is mov-ing. The central ideas of special relativity can be formu-
General Relativity and Newtonian Gravitation Theory "530-47773_Ch00_2P.tex" — 1/23/2012 — 17:18 — page ii 1 0 1 . by Robert Geroch (1984) Useful Optics, by Walter T. Welford . This manuscript began life as a set of lecture notes for a two-quarter (twenty-week) course on the foundations of general relativity that I taught at .
Topics in the foundations of general relativity and Newtonian gravitation theory / David Malament. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-226-50245-8 (hardcover : alkaline paper) ISBN-10: 0-226-50245-7 (hardcover : alkaline paper) 1. Relativity (Physics) 2. Gravitatio
Differential Topology in Relativity (Philadelphi3' Society for Industrial and Applied Mathe matics. 1972). More accessible than either is Robert .Geroch, "Space-Time Structure from a Global Viewpoint," in B. K. Sachs, ed., General Relativity and Cosmology (New York Academic Press, 1971). 3. A future end point need not be a point on the curve.
