Lecture Notes On General Relativity - Kevin Zhou

61. Preliminaries From a particle physics perspective, tests of the equivalence principle are most useful forconstraining additional, non-gravitational long-range forces, which would generically violatethe WEP. Specific tests of the SEP are also useful. For instance, if one unluckily had a longrange force that coupled equally to protons and neutrons, there would be little apparent WEPviolation, since protons and neutrons have almost equal mass, but there would be significantSEP violation. It is also useful to do tests at a variety of scales (i.e. tabletop to astronomical),because the force could have a finite range µ, which would be more apparent at distances 1/µ. Deviations from the equivalence principles, particularly the SEP, could of course also be explained by modifying the structure of gravity itself. However, at some level “adding new forces”and “modifying gravity” aren’t all that different, because in most cases they both boil down toadding new fields and couplings to the Lagrangian. To date, all experimental tests are consistent with all equivalence principles, though specifictests of the SEP are less stringent.Example. The EEP predicts gravitational redshift. Consider two rockets in space a distance zapart, with acceleration a and velocities v c. The trailing rocket emits a photon of wavelengthλ0 , which reaches the other rocket after time t z/c. The receiving rocket has picked up anadditional velocity v a t az/c, so that the photon is redshifted by λaz v 2. λ0ccAbove we worked to first order in the velocities to avoid extra special relativistic effects.By the EEP, the same thing should happen if a photon is instead emitted upward a distance zin a uniform gravitational field, giving λgz 2.λ0cIn general relativity, the metric in a weak time-independent gravitational potential φ isc2 dτ 2 (1 2φ/c2 )c2 dt2 (1 2φ/c2 ) dr2 ,φ/c2 1Now consider two pulses of light sent between points A and B separated by a time t. Since thegravitational field is time-independent, the paths taken by the pulses are identical, so they alsoarrive separated in coordinate time t. Converting to proper time, τA2 (1 2φA /c2 ) δt2 ,so that a redshift of τB2 (1 2φB /c2 ) δt2 λ Φ 2λ0cis observed, where we are working to lowest order in φ/c2 . This is in agreement with the EEP result.This trick of using the separation time for pulses also lets us avoid the geodesic equation.1.3Physical DifferencesWe quickly review what changes when moving from special relativity (SR) to general relativity(GR). Note that when we refer to SR, we are referring to inertial frames in Minkowski space withCartesian coordinates.

7 1. Preliminaries In SR, we considered vectors as ‘free’, with a base point that could be moved. In GR, this mustbe performed by parallel transport, and the result generally is not unique. In SR, we considered spacetime events as vectors xµ . This is only possible because we identifiedspacetime itself with the tangent space at the origin, which we can’t do in GR. In SR, inertial frames were defined over all spacetime. In GR, they can’t due to tidal effects.In fact, we usually can’t define a global system of coordinates at all, as spacetime is a generalmanifold which may require multiple charts.R The time measured by a moving clock is still τ ds2 . Suppose a particle located at the origin of some coordinate system has momentum pµ . Ifthis coordinate system locally corresponds to the frame of an observer at the origin, they stillmeasure the energy of the particle to be p0 , and so on. In SR, the Levi–Civita symbole µνρσ sign(µνρσ)is a pseudotensor: it transforms properly under connected Lorentz transformations and picksup an extra sign from T and P . In GR, the Levi–Civita symbol is not a tensor at all. In SR, the partial derivative takes (r, s) tensors to (r, s 1) tensors. In GR, we instead mustuse the covariant derivative. (This also holds when we broaden the frames allowed in SR, aswe get a nontrivial connection.)Example. If two expressions agree in some frame, then they must agree in all frames. This allowsus to find general results with very little work. For example, the energy of a particle in some frameis p0 , and the velocity of an observer in its own frame is uµ (1, 0, 0, 0). Therefore E pµ uµ inthis frame, and hence it is true in all frames.As another example, consider the Lie derivative LV W where V 0 . ThenLV W ( 0 W i ) i .The right-hand side happens to be equal to the commutator [V, W ] in these coordinates, so ingeneral the Lie derivative is the commutator.Example. Newtonian gravity in index notation. The equation of motion for a particle is ẍi gi .However, in the spirit of the EEP, we note that this acceleration can always be set to zero in afalling frame, so instead we focus on tidal effects. Consider two particles separated by δx. Thenδ ẍi δxj j gi O(δx2 ).To simplify this we define the tidal tensorEij j gi ,δ ẍi Eij δxj 0.Since the gravitational field in Newtonian theory is curl-free, there exists a potential φ so thatgi i φ

81. Preliminarieswhich implies Eij Eji . In addition, since mixed partials commute, we haveEi[j,k] 0.Finally, the field is sourced by matter by Poisson’s equation, i i φ Eii 4πGρ.Similar equations appear in general relativity, where the tidal tensor corresponds to the Riemanntensor, and our identities for it correspond to the symmetries of the Riemann tensor and the Bianchiidentity. The potential roughly corresponds to the metric, and Poisson’s equation corresponds tothe Einstein field equation.1.4ManifoldsWe review the basics of differential geometry, considering structures that can be defined withoutusing a metric. We begin with the tangent space. Consider an n-dimensional manifold M . A scalar field f : M R is smooth if F f φ 1is smooth for all charts φ. For example, coordinate functions themselves are smooth becausetransition functions are smooth by the definition of a smooth manifold. A smooth curve is a smooth function λ : I M where I is an open interval in R. If f : M R is smooth, then f λ : I R is smooth, and in particular it has a derivative. Wethus define the tangent vector to λ at p as the map df (λ(t))Xp (f ) dtt 0where λ(0) p. Then Xp is a linear map on smooth functions and a derivation,Xp (f g) Xp (f )g(p) f (p)Xp (g).The set of tangent vectors of p forms a vector space in the usual way. We may also write the tangent vector in components, by notingf λ (f φ 1 ) (φ λ)which gives Xp (f ) F (x) xµ x φ(p)dxµ (λ(t))dt t 0where we used the chain rule. Now we check that the tangent space Tp is an n-dimensional vector space.– First, we check that it is closed under linear combinations. Note that if λ(t) and κ(t) givevectors Xp and Yp , thenν(t) φ 1 (α(φ(λ(t)) φ(p)) β(φ(k(t)) φ(p)) φ(p))has tangent vector αXp βYp , as desired.

91. Preliminaries– Next, we check that Tp has dimension n. The expression above shows that any Xp (f ) canbe written as a linear combination of F (x)/ xµ , so the vectors / xµ are a complete set;they correspond to paths that only change the coordinate xµ .– The vectors / xµ are independent because if αµ µ F 0 for all F , then choosing F xνgives αν 0. Thus they are a basis, giving the result.The basis / xµ depends on the coordinate chart. It is defined inside the entire patch andforms a coordinate basis in the patch. A general vector is written X X µ eµ for a basis eµ . To see how coordinate bases change under coordinate change, let φ and φ0 give coordinates xand x0 . Then formally,( µ )(f ) (f φ 1 ) ((f φ0 1 ) (φ0 φ 1 )) xµ xµwhere µ is an abstract vector. We can now use the chain rule, giving F F x0ν , xµ x0ν xµ µ x0ν 0 . xµ νMore casually, we can heuristically derive this result by writing f (x0 (x)) f x0 x x0 xwhere we implicitly identified a few quantities. By a very similar argument,X 0µ X ν x0µ. xνThis leaves vectors X µ µ invariant since the transformation factors cancel.Next, we define covectors. The dual space V of a vector space V is the set of linear maps V R. Given a basis eµ of Vthere is a dual basis f µ of V defined by f µ (eν ) δνµ . There is no natural isomorphism between V and V , though there is one between V and V by ‘shuffling parentheses’. Define the cotangent space Tp (M ) as the dual space of the tangent space. Given a smoothfunction f , we may define a covector df , called the gradient of f , by(df )p (X) X(f )p .In particular, dxµ is the dual basis to µ . Writing a covector as ω ωµ dxµ , we have the transformation lawsdxµ xµdx0ν , x0νAgain, these follow by ‘lining up the derivatives’.ωµ0 xνων . x0µ

10 1. PreliminariesNote. We use Greek indices in equations that are only true in a particular coordinate system, andLatin indices in equations that are always true. For example, for a vector X, we can write X µ δ0µin some coordinate system, but generally df (X) dfa X a . Equations in Latin should be interpretedas ‘component-free’, with the indices only indicating where the parentheses go.Finally, we introduce tensors. A tensor of type (r, s) at p is a multilinear map which takes r covectors and s vectors to R.For example, covectors are tensors of type (0, 1) and vectors are tensors of type (1, 0). Also,defining δ(ω, X) ω(X), δ is a (1, 1) tensor. Choosing a basis of vectors {eµ } with dual basis {f µ }, the components of a tensor areT µ1 .µrν1 .νs T (f µ1 , . . . , eν1 , . . .).For example, the components of δ areδ µν δ(f µ , eν ) f µ (eν ) δνµ .The set of tensors at p is a vector space with dimension nr s . Now we consider how tensor components change under a general change of coordinates,0000f µ Aµ ν f ν .The same arguments as before tell us thateµ0 (A 1 )ν µ0 eν ,X µ Aµ ν X ν ,ηµ0 (A 1 )ν µ0 ην .Plugging these results in, a tensor transforms as, e.g.0 000T µ ν ρ0 Aµ σ Aν τ (A 1 )λρ0 T στλ .00In the special case of a coordinate transformation, Aµ ν xµ / xν . In the even more specialcase of a Lorentz transformation, A is Λ. Given an (r, s) tensor, we can construct an (r 1, s 1) tensor by contracting two indices. Thisis done by plugging in a basis and dual basis,T (f µ , eµ , . . .) S(. . .).This is basis independent because the left-hand side transforms as T AA 1 T T . We can also construct tensors by the tensor product. For example,(S T )(ω, X) S(ω)T (X)with the same pattern holding for arbitrary tensors. The components simply multiply.

11 1. Preliminaries Finally, we may symmetrize and antisymmetrize tensors. For example, given a (0, 2) tensor T ,its symmetric and antisymmetric parts areS(X, Y ) T (X, Y ) T (Y, X),2A(X, Y ) T (X, Y ) T (Y, X)2which, in index notation, readsTµν Tνµ,2Sµν Aµν Tµν Tνµ.2We will also use vertical bars to denote exclusion from (anti)symmetrization. For example,T(µ νρ σ) Tµνρσ Tσνρµ.2The most useful property is that contractions of symmetric and antisymmetric tensors vanish. Similarly we may define vector and tensor fields on M . A vector field X is smooth if X(f ) is asmooth function for all smooth f , with other definitions similar.Next, we review some geometric objects derived from vector fields. Given a vector field X, an integral curve of X through p is a curve through p whose tangent atevery point is X. Taking coordinates, this means thatdxµ (t) X µ (x(t)),dtxµ (0) xµ p .Note that X µ (x(t)) means X µ evaluated at the point x(t), not acting on anything. If f is a function satisfying X(f ) 0, then f is conserved on integral curves. Flow along integral curves generates a one-parameter group of diffeomorphisms φt : M M byflowing along the integral curves for time t. Conversely, φt gives a vector field by differentiationat t 0. The commutator of two vector fields is[X, Y ](f ) X(Y (f )) Y (X(f )).It turns out to be a vector field, since the second derivatives cancel, with components[X, Y ]µ X ν ν Y µ Y ν ν X µ .The commutator operation turns the set of vector fields on M into a Lie algebra, whosecorresponding Lie group is the set of diffeomorphisms of M . Note that we can define the components of a tensor with respect to any set of vector fields {eµ }that form a basis at every point. By Frobenius’ theorem, we have [eµ , eν ] 0 if and only if theeµ are a coordinate basis, i.e. eµ µ . Most of the time we’ll work in a coordinate basis, butwe’ll try to point out what extra terms appear outside such a basis.Note. More examples of Lie algebras.

12 1. Preliminaries An explicit basis for the Lie algebra diff(R) isXα xα 1 x ,α Z,[Xα , Xβ ] (α β)Xα β . There exist only two two-dimensional Lie algebras,[X, Y ] 0 and [X, Y ] Y.The latter is the Lie algebra of affine transformations of the line. The Euclidean group E(2) acts on M R2 by rotations and translations, ax R(θ)x .bThen E(2) is a three-dimensional Lie group, parametrized by θ, a, and b. We can assign avector field to every infinitesimal transformation (alternatively, every one-parameter subgroupgives a one-parameter family of diffeomorphisms), givingXa x ,Xb y ,Xθ x y y x .which form a basis for e(2) with[Xa , Xb ] 0,[Xa , Xθ ] Xb ,[Xb , Xθ ] Xa .More generally, the set of Killing vectors will form a Lie algebra.Note. With index notation, we simultaneously speak about tensors and their components; however,this leads to ambiguity, especially when working with covariant derivatives, and is a bit inelegantto mathematicians because we always need to specify a coordinate system. If necessary, we will useabstract indices, which only mean the former. For instance, X a fa is simply a shorthand for X(f )and does not indicate a coordinate system; the “abstract index” a does not take numeric values.

13 2. Riemannian Geometry2Riemannian Geometry2.1The Metric The metric tensor gµν is a nondegenerate symmetric (0, 2) tensor. Since the metric is nondegenerate, g gµν 6 0. Then there exists an inverse metric, a (2, 0) tensor satisfyingg µν gνσ gσλ g λµ δσµ .For example, the trace of g is g µν gµν δµµ 4, in any signature. Unlike in special relativity, index placement now matters (it cannot be restored at the end ofthe calculation) because the metric has a nontrivial derivative. For example, since λ (g µν gνσ ) ( λ g µν )gνσ g µν ( λ gνσ ) 0we conclude that λ g µν g µσ g νρ λ gσρ .The minus sign is the same one as in (1/f )0 f 0 /f 2 . The metric is extremely useful: we use it to raise and lower indices, and compute path lengthsand proper times, giving geodesics. It determines causality and locally inertial frames. It is thegeneralization of both the Newtonian dot product and the Newtonian gravitational potential. The metric is in canonical form if it is diagonal, with p and q elements equal to 1 and 1respectively. Sylvester’s theorem states that this can always be done at any given point, withp and q unique. By continuity, the signature (p, q) is the same throughout the manifold. If q 0, the metric is called Euclidean/Riemannian, and if q 1 (as in relativity) the metricis called Lorentzian/pseudo-Riemannian; the canonical form is the Minkowski metric. The metric takes two vectors and gives a number, so it may be written asds2 gµν dxµ dxν .Here, ds2 is the metric tensor in component-free form, and dxµ dxν is a tensor product. Sincethe metric is symmetric, we use symmetrized tensor products so that dxdy dydx. The length of a spacelike curve isZ ps g(V, V ) dtwhere V µ (t) dxµ (t)/dt. Similarly, the proper time along a timelike curve isZ pτ g(V, V ) dt.Note that t is not time, but just an arbitrary parameter. If we parametrize by proper time,then V µ dxµ /dτ is the four-velocity, giving g(V, V ) 1 just as in special relativity.

14 2. Riemannian GeometryNote. Parameter counting for coordinate transformations. Let xµ (p) xµ̂ (p) 0, and expandgµ̂ν̂ xµ xνgµν xµ̂ xν̂in a Taylor series in xµ̂ about p. Expanding both sides to second order in the xµ̂ , dropping constantsand indices and evaluating all derivatives at p, we haveˆˆ(ĝ) ( ĝ)x̂ ( ˆ ĝ)x̂x̂ x x x 2 x x x ˆg g g x̂ x̂ x̂ x̂ x̂ x̂ x̂ x̂ x 3 x 2x 2x x 2 x ˆ x x ˆ ˆ g g g g x̂x̂ x̂ x̂ x̂ x̂ x̂ x̂ x̂ x̂ x̂ x̂ x̂ x̂ x̂Now let’s consider this expression order-by-order in x̂. At zeroth order, we get the transformation law at p. There are 16 parameters in the matrix xµ / xµ̂ , but the metric only has 10, since it’s symmetric. Therefore we can always bring themetric into canonical (i.e. Minkowski) form at a point, and the extra 6 degrees of freedom arethe Lorentz transformations. At first order, we have 40 numbers on the left-hand side, from 4 derivatives of 10 metriccomponents. On the right-hand side, the ( x/ x̂)2 term gives nothing since it was used up atzeroth order, but 2 xµ /( xµ̂1 xµ̂2 ) has 40 parameters, since the second derivative is symmetric.ˆ to zero.Then we have just enough freedom to set ĝ At second order, we have 100 numbers on the left-hand side, since ˆ ˆ is symmetric. On theright-hand side, we can only set 3 x/ x̂ x̂ x̂, which has 4 choices in the numerator and 20 inthe denominator, so we’re short by 20 degrees of freedom. These tell us about the curvature ofthe manifold; we will see later the Riemann tensor has 20 independent components.Note. As motivated above, at any point p, there should exist a coordinate system xµ withgµν (p) ηµν , σ gµν (p) 0.Such coordinates are called locally inertial coordinates, or Riemann normal coordinates, and theassociated basis vectors constitute a local Lorentz frame. These frames are associated with freelyfalling observers, as they see no effects of gravity besides tidal effects, which only appear to secondorder. Later, we will construct such coordinates using geodesics.Locally inertial coordinates are useful for extracting general expressions. While a calculationin curved spacetime may be difficult, we can always go into locally inertial coordinates at a pointand simplify using g η and g 0. As long as we phrase our final answers in terms of tensorialquantities, they must hold in all coordinate systems.Note. We can always choose a basis at every point so that the metric is in canonical form at everypoint. However, such a set of bases generally does not mesh together to form a coordinate system.Example. In a more elementary treatment, we think of dxµ as an infinitesimal displacement andds as an infinitesimal length. For example, consider the metricds2 dt2 t2q dx2 .

15 2. Riemannian GeometryWe want to find the null paths xµ (λ) followed by light. The tangent vector isV dxµ µ .dλThen the null paths must satisfyds2 (V, V ) dt2 (V, V ) t2q dx2 (V, V ) 0.Now, working very explicitly, we have22 dt (V, V ) [dt(V )] dtdλ 2 ,0 dtdλ 22q tdxdλ 2.We now have an ordinary differential equation, which simplifies todx t q

Carroll, Spacetime and Geometry. The canonical \friendly" general relativity book. Has either the advantage or disadvantage of moving most of the math to appendices, allowing the main text to be casual and conversational, including discussions of philosophical topics such as the meaning of the equivalence principle. Wald, General .