Problem Set #3: 3.1, 3.3, 3.5, 3.13, 3.15 (Due Wednesday, Nov. 11th) Midterm-exam: Friday, November 13th

Transcription

1 Chapter 3 Curvature Problem Set #3: 3.1, 3.3, 3.5, 3.13, 3.15 (Due Wednesday, Nov. 11th Midterm-exam: Friday, November 13th 3.1 Covariant derivative In the previous chapter we have shown that the partial derivative of a nonscalar tensor is not a tensor (see (2.34. It does not transform as a tensor but one might wonder if there is a way to define another derivative operator which would transform as a tensor and would reduce to the partial derivative in Minkowski space (note that exterior derivative does transform as a tensor, but does not reduce to partial derivative in the limit of flat space. The desired derivative operator (called covariant derivative and denote by can be constructed by by enforcing certain properties such as linearity, i.e. and product rule, i.e. (T + S = T + S, (3.1 (T S = T S + T S. (3.2 One can show (see Wlad s book that if the operator obeys product rule than it can be written as a partial derivative plus a linear correction whose coefficients are called the connection coefficients, µ V ν = µ V ν +Γ ν µλv λ. (3.3 Then the transformation properties of Γ ν µλ (which does not have to and will not transform as a tensor can be determined by demanding that µ V ν transforms as a (1, 1 tensor, i.e. µ V ν = xµ x µ x ν x ν µv ν. (3.4 29

2 CHAPTER 3. CURVATURE 30 By substituting (3.3 in (3.4 we obtain x µ x µ ( ( x ν x x µ x V ν λ +Γ ν ν µ λ x V λ λ µ V ν +Γ ν µ λ V λ = xµ x ν ( µ V ν +Γ ν x µ x µλv λ ν = xµ x ν x µ x ν x µ x ν x µ x µ x V ν x λ +Γ ν ν µ λ x V λ = xµ x ν λ x µ x ν Γν µλ V λ x µ V ν + xµ x µ x ν x ν Γν µλ V λ Γ ν µ λ V λ = xλ x λ x µ x µ x ν x ν Γν µλv λ xλ x λ x µ x µ x ν (3.5 x µ x V λ. λ we get a transformation law for the con- Since this must be true for all V λ nection coefficients Γ ν µ λ = xλ x λ x µ x µ x ν x ν Γν µλ xλ x λ x µ x µ 2 x ν x µ x λ. (3.6 Similarly we can show that a different set of coefficients Γ λ µν should be used to define a covariant derivative of a one form, µ ω ν = µ ω ν + Γ λ µν ω λ (3.7 where Γ λ µν although so far unrelated does transforms exactly as Γλ µν in (3.6. To establish a relation between Γ λ µν and Γ λ µν we demand that the covariant derivative reduces to partial derivative for scalars µ φ = µ φ. (3.8 Thus µ ( ωλ V λ = µ ( ωλ V λ ( µ ω λ V λ + ( µ V λ ω λ = ( µ ω λ V λ + ( µ V λ ω λ ( µ ω λ V λ + ( µ V λ ω λ = ( µ ω λ V λ + Γ σ µλ ω σv λ + ( µ V λ ω λ +Γ σ µλ ω σv λ Γ σ µλ ω σv λ = Γ σ µλ ω σv λ. (3.9 Since ω σ and V λ are arbitrary Γ σ µλ = Γ σ µλ. (3.10 and therefore µ ω ν = µ ω ν Γ λ µνω λ. (3.11

3 CHAPTER 3. CURVATURE 31 The covariant derivatives of an arbitrary rank tensor are given by σ T µ1...µ k µk...ν l = σ T µ1...µ k µk...ν l +Γ µ1 σλ T λµ2...µ k +... Γ λ σν 1 T µ1...µ k... (3.12 In general relativity the 4 3 = 64 independent connection coefficients of Γ λ µν are uniquely specified by the metric g µν. This is accomplished by demanding that the connection coefficient is torsion-free, i.e. and metric-compatible, i.e. Γ λ µν =Γλ (µν, (3.13 ρ g µν =0. (3.14 The torsion-free implies, for example, that the antisymmetrized covariant derivative is also the exterior derivative, i.e. [µ ω ν] = [µ ω ν] Γ λ [µν] ω λ = [µ ω ν]. The metric-compatibility implies a number of nice properties. First of all the covariant derivative of inverse metric also vanishes, i.e. ρ g µν = 0 (3.15 and thus the raising an lowering operators commute with covariant derivative ρ V µ = ρ (g µν V ν =g µν ρ V ν. (3.16 Moreover the torsion-free and metric-compatible properties of the connection single out a unique connection known as the Christoffel (or Levi-Civita connection of Christoffel symbol. The formula for the Christoffel symbol can be derived by from 0= ρ g µν = ρ g µν Γ λ ρµ g λν Γ λ ρν g µλ (3.17 0= µ g νρ = µ g νρ Γ λ µν g λρ Γ λ µρ g νλ (3.18 0= ν g ρµ = ν g ρµ Γ λ νρ g λµ Γ λ νµ g ρλ (3.19 by subtracting (3.18 and(3.19 from(3.17 and using (3.13, ρ g µν µ g νρ ν g µρ = ( Γ λ ρµ g λν Γ λ µρ g ( νλ + Γ λ ρν g µλ Γ λ νρ g ( λµ Γ λ µν g λρ +Γ λ νµ g (3.20 ρλ ρ g µν µ g νρ ν g µρ = 2Γ λ µν g λρ (3.21

4 CHAPTER 3. CURVATURE 32 or Γ λ µν = 1 2 gλρ ( µ g νρ + ν g µρ ρ g µν. (3.22 In Minkowski space described by Cartesian all of the Christoffel symbols vanish, but this does not have to be the case in curvilinear coordinates. As an example consider a two dimensional Euclidean space described by polar coordinates with metric ds 2 =(dr 2 + r 2 (dθ 2. (3.23 The non-vanishing components of the inverse metric are and for example the connection coefficient but Γ r rr = 1 2 grρ ( r g rρ + r g ρr ρ g rr = g rr = 1 (3.24 g θθ = r 2 (3.25 = 1 2 grr (2 r g rρ ρ g rr grθ (2 r g rρ ρ g rr = = 1 2 1(2 rg rr r g rr (2 rg rσ σ g rr = 0 (3.26 Γ r θθ = 1 2 grρ ( θ g θρ + θ g ρθ ρ g θθ = = 1 2 grr (2 θ g θr r g rr grθ (2 θ g θθ θ g θθ = = 1 2 1(2 θg θr r g rr (2 θg θρ ρ g θθ = = 1 2 rg rr = r. (3.27 It is a straightforward exercise to find all other coefficients, Γ r θr = Γ r rθ = 0 (3.28 Γ θ rr = 0 (3.29 Γ θ rθ = Γ θ θr = 1 r (3.30 Γ θ θθ = 0. (3.31 Just like one can make the connection coefficients to be non-zero in flat space it is possible to make the connection coefficients to vanish at some point curved space but not everywhere.

5 CHAPTER 3. CURVATURE 33 Since the covariant derivatives of a vector is and (one can also show µ V µ = µ V µ +Γ µ µλ V λ (3.32 Γ µ µν = 1 g ν g (3.33 then we obtain a useful expression µ V µ = 1 ( µ g V µ. (3.34 g 3.2 Parallel transport Up until now we were not able to compare different tangent vectors at different points since they where not elements of the same vector space. With the help of the connection coefficients we can continuously move the tangent vectors (or higher rank tensors from on point to another (or parallel transport, but the resulting vector will usually depend on the path along which it was moved. (Think about a parallel transport of a vector on the surface of a sphere to convince yourself that the parallel transported vector would depend on the path. This is a generic property of curved spaces which is why it makes no sense to ask what is a relative velocity of two particles in two distinct points. In fact interpreting the cosmological expansion ofspace by galaxies receding away at a speed defined by the redshift is incorrect and can lead to paradoxes involving superluminal velocities. Of course in cosmology nothing is receding away, but the metric between galaxies changes which causes the light between the object to change the wavelength (i.e. to redshift. In flat space the parallel transport of a tensor along a parametrized curve x µ (λ is given by the requirement ( D T µ 1...µ kν1...νl = dxµ µt µ1...µ k = 0 (3.35 which is generalized to curved spaces as ( D T µ 1...µ kν1...νl = dxµ µt µ1...µ k =0. (3.36

6 CHAPTER 3. CURVATURE 34 This is the parallel transport equation which, for example, take the following form for vectors D V µ = d V µ +Γ µ dx σ σρ V ρ =0. (3.37 It follows, for example, that the inner product of two parallel transported vectors is preserved, i.e. D (g µνv µ W ν = D (g µν V µ W ν D + g µν (V µ W ν + g µν V µ D (VWν =0. (3.38 Next we will obtain a formal solution of the parallel transport equation (3.37. Our task is to find the so-called parallel propagator matrix P µ ρ(λ 0,λ along trajectory γ(λ suchthat If we define a transition matrix V µ (λ =P µ ρ (λ, λ 0V ρ (λ 0. (3.39 A µ ρ (λ dx σ Γµ σρ then (3.37 can be written as a Schrodinger equation By substituting (3.39 into (3.41 we get (3.40 d V µ = A µ ρ V ρ. (3.41 d [ P µ ρ (λ, λ 0 V ρ (λ 0 ] = [ A µ σ P σ ρ (λ, λ 0 V ρ (λ 0 ] (3.42 d P µ ρ (λ, λ 0 = A µ σ P σ ρ (λ, λ 0. (3.43 By integrating both side we get P µ ρ (λ, λ 0=δ µ ρ + λ which can be solved by iteration P µ ρ (λ, λ 0=δ µ ρ + λ or in matrix notation P (λ, λ 0 =1+ λ 0 A µ ρ (η 1dη 1 + λ λ 0 A(η 1 dη 1 + λ 0 A µ σ (ηp σ ρ (η, λ 0dη (3.44 λ η1 λ 0 λ η2 λ 0 λ 0 A µ σ (η 1A σ ρ (η 2dη 1 dη (3.45 λ 0 A(η 2 A(η 1 dη 1 dη (3.46

7 CHAPTER 3. CURVATURE 35 which can be simplified using a path-ordered product of matrices, P[A(η n A(η n 1...A(η 1 ], as 1 λ P (λ, λ 0 =1+ P[A(η n...a(η 1 ]dη 1 dη 2... (3.47 n! λ 0 n=1 which is the series expansion of an exponential, ( λ P (λ, λ 0 =Pexp A(ηdη. (3.48 λ 0 Turning back to the components notation gives us ( λ P µ ν(λ, λ 0 =Pexp Γ µ dx σ σν λ 0 dη dη. (3.49 In quantum field theory the same formula is known as Dyson s formula which is due to the fact that (3.41 is mathematically equivalent to the Schrodinger equation. The parallel transport transformation around a closed loop is called the holonomy of the connection around the loop. For the metric-compatible connections the group of holonomy transformations at a point is a Lorentz group. Note that the knowledge of the holonomy group at each point is sufficient to determine the metric and thus might question what is more fundamental: holonomies of all loops or metric at all point. In the canonical quantum gravity the metric is treated as fundamental, when in the loop quantum gravity the holonomies are more fundamental. 3.3 Geodesics Geodesic connecting two points can be defined as a path which parallel transports its own tangent vector, i.e. D dx µ = d2 x µ dx ρ dx σ 2 +Γµ ρσ =0. (3.50 This is the geodesic equation which is obtained by substituting the tangent vector dxµ into (3.37. For the metric-compatible connection coefficients the geodesic is also the path of the largest proper time (there is no smallest proper time since one can always construct a path of zero length made out of null segments defined as τ[x] = g µν dx µ dx ν (3.51

8 CHAPTER 3. CURVATURE 36 that can be obtained using variational derivative, i.e. By Taylor expanding the metric δτ[x] =0. (3.52 δx µ we can vary the proper time we get δτ δx σ = = = = = g µν (x + δx g µν (x+δx σ σ g µν (x (3.53 δτ[x + δx] δτ[x] = δx σ dx g µ dx ν µν (g µν + δx ρ ρ g µν d(xµ +δx µ d(x ν +δx ν δx σ dx g µ dx ν µν dx g µ dx ν µν ( δx ρ dx ρ g µ dx ν µν g µν dδxµ dx ν g µν dxµ δx σ [ dx g µ dx ν µν 1 1+ ( dx g µ µν dx g µ dx ν µν ( gµν dx µ [ ( 1 dx µ gµν 2 dx ν dx ν 1 ( dx δxρ ρ g µ dx ν µν 2g µν dδxµ δx σ 1 ( δx ρ dx ρ g µ µν δx σ dx ν 1/2 ( 1 2 δxρ dx ρ g µ dx ν µν g µν dδxµ By changing the integration variable from λ to proper time dx = dτ ( g µ µν dx ν dx ν 2g µν dδxµ ] dx ν dδx ν ] dx ν δx σ (3.54 dx ν 1/2 (3.55 we get δτ δx σ = = = ( 1 2 dτ δxρ dx ρ g µ dx ν µν g dτ dτ µν dδxµ dτ δx ( σ 1 2 dτ dx ρg µ dx ν µν dτ dτ δxρ + d dτ δx σ dτ ( 12 dx µ dx ν σg µν dτ dτ + d dτ ( gµν dx ν dx ν dτ dτ δx µ dx (g ν σν dτ (3.56

9 CHAPTER 3. CURVATURE 37 or by demanding that δτ/δx σ vanishes for all variations 1 2 dx µ dx ν σg µν dτ dτ + d dx (g ν σν dτ dτ = dx µ dx ν σg µν dτ dτ + dx µ dx ν µg σν dτ dτ + g d 2 x ν σν dτ 2 = 0 d 2 x ν g σν dτ ( σg µν + µ g σν + ν g µσ dxµ dx ν dτ dτ = 0 d 2 x ρ dτ gρσ ( σ g µν + µ g σν + ν g µσ dxµ dx ν dτ dτ = 0. (3.57 The last equation is equivalent to the geodesic equation (3.50 for the Christoffel connection, but may different for more general connections. The equation describes the motion of unaccelarated particle independent on its mass, but it is also straightforward to include forces. For example, if the particle has a charge q and mass m, then the geodesic equation would be given by d 2 x µ dτ 2 dx ρ dx σ +Γµ ρσ dτ dτ = q m F µ dx ν ν dτ. (3.58 Note that the geodesics equation (3.50 is valid not only for proper time, but for any affine parameter defined as linear function of proper time, i.e. λ = aτ + b (3.59 and for non-linear parametrizations (3.50 would be modified d 2 x µ dα 2 dx ρ dx σ +Γµ ρσ dα dα = f(αdxµ dα (3.60 where f(α is some function which depends on the parametrization. Conversely if (3.60 is satisfied along some curve, then one can always find an affine parameter λ for which (3.50 is satisfied. In addition to parallel transport the geodesics x µ (λ passing through some point p can be used to map point from the tangent space T p to a neighborhood of p. Such mapping is called the exponential map exp p : T p M (3.61 which is defined as where exp p (k µ =x ν (λ = 1 (3.62 k µ = dxµ (λ =0 (3.63

10 CHAPTER 3. CURVATURE 38 is the tangent vector at point p where we have set λ =0. Ofcoursethemap is well defined and invertible on a subset of T p sufficiently close to k µ =0, and so (3.61 should not be taken literally. The range of the map can fail to be all of the manifold given that there can be points not connected bya geodesic, and the range of the map can fail to be all of the tangent space if the manifold is geodesically incomplete (has singularities or boundaries. 3.4 Riemann tensor The role of the Riemann (or curvature tensor is to be able to represent the features of the connection coefficients which would manifest the presence of curvature. Such manifestations include parallel lines remain parallel or parallel transports around closed loops change vectors. The change of parallel transported vectors depends on the loop, and for a local description of the curvature it would be more useful to consider parallel transport around infinitesimal loops. We can guess what kind of object the Riemann tensor R should be by considering a closed loop defined by two vectors A µ and B µ. Then a transformation around the loop should produce a vectorial change δv µ to a vector V µ such that δv ρ R ρ σµν V σ A µ B ν. (3.64 Thus we anticipate the Riemann tensor to be of rank (1,3. It should also be antisymmetric, i.e. R ρ σµν = Rρ σνµ, (3.65 so that the parallel transport in opposite direction comes with a negative sign. To obtain the exact expression for the Riemann tensor in terms of connection coefficients we consider instead the action of a commutator of two covariant derivatives on the vector, i.e. [ µ, ν ]V ρ. (3.66 Indeed, the covariant derivative of a vector in the direction of a parallel transport is zero by definition, and thus the covariant derivative measure by how much the vector changes compared to what it would have been if parallel transported. Then the commutation of two covariant derivatives (when contracted with A µ and B ν measuresthechangestoavectorv µ if it was first parallel transformed one way (first along A µ and then along B µ compared to the other way (first along B µ and then along A µ. The

11 CHAPTER 3. CURVATURE 39 commutator of a vector is easy to express in terms of connection coefficients [ µ, ν ]V ρ = 2 [µ ν] V ρ =2 ( [µ ν] V ρ Γ σ[µν] σv ρ +Γ ρ[µ σ ν]v σ = ( ( = 2 [µ ν] V ρ + [µ Γ ρ ν]σ V σ +Γ ρ [ν σ µ]v σ + 2 ( Γ σ [µν] σv ρ +Γ σ [µν] Γρ λσ V λ +2 (Γ ρ[µ σ ν]v σ +Γ ρ[µ σ Γσλ ν] V λ = ( = 2 [µ Γ ρ ν]σ +Γρ [µ λ Γλ σ ν] V σ 2Γ σ [µν] σ V ρ. (3.67 The second term is a torsion tensor, i.e. σ Tµν = 2Γ σ [µν] (3.68 which vanishes for Christoffel connections, and the first term defines the Riemann tensor, ( R ρ σµν =2 [µ Γ ρ ν]σ +Γρ [µ λ Γλ σ ν] = µ Γ ρ νσ ν Γ ρ µσ +Γ ρ µλ Γλ σν Γ ρ νλ Γλ σµ. (3.69 One can check that it is a legitimate (1,3 tensor, although it is not immediately clear that it is the same tensor as in (3.64 (see Wald for details. It is also straightforward to determine the action of the commutator [ ρ, σ ]on a tensor of arbitrary rank, [ ρ, σ ]X µ1...µ k = 2Γ λ [ρσ] λx µ1...µ k +R µ1 λρσ Xλµ2...µ k... R λ ν 1ρσ Xµ1...µ k λν2...ν l... (3.70 Sometimes it useful to express the relevant quantities in a coordinate independent way. Then one can think of the (1,2 torsion tensor as amap from two vectors to a third vector, T (X, Y = X Y Y X [X, Y ] (3.71 and the (1,3 Riemann tensor as a map from three vectors to a fourth vector, R(X, Y Z = X Y Z X Y Z [X,Y ] Z (3.72 where the commutator of two vectors is and the directional covariant derivative is [X, Y ] µ X λ λ Y µ Y λ λ X µ (3.73 X X µ µ. (3.74

12 CHAPTER 3. CURVATURE 40 Note that the third term in (3.72 vanishes when X and Y are coordinate basis since [ µ, ν ]=0. (3.75 In the case of metric compatible connection the Riemann tensor vanishes if the metric is constant everywhere on the manifold, g µν = const σ g µν = 0 Γ ρ µν = 0 σ Γ ρ µν = 0 One can also show that the opposite is true, i.e. Consider normal coordinates, i.e. R ρ σµν = 0. (3.76 R ρ σµν =0 g µν =const. (3.77 g µν = η µν (3.78 at some point p and with basis vectors ê (µ whose dot product is g σρ (pê σ (µ (pêρ (ν (p =η µν. (3.79 This set of basis vectors can be parallel transported to some other point q. Since the Riemann tensor is zero the result of the transport does not depend on the trajectory and the dot product remains unchanged, g σρ (qê σ (µ(qê ρ (ν (q =η µν. (3.80 Moreover the commutator of the parallel transported basis vectors vanishes for torsion-free connections, [ê (µ, ê (ν ]= ê(µ ê (ν ê(ν ê (µ T (ê (µ, ê (ν =0. (3.81 Then according to the Forbenius s Theorem one can always find coordinates x µ such that ê (µ = (3.82 x µ in which the metric would have the form of η µν.

13 CHAPTER 3. CURVATURE 41 Not all of the components of the Riemann tensor are independent and one can show that the components must obey, R ρσµν + R σρµν = 0 (3.83 R ρσµν R µνρσ = 0 (3.84 R ρσµν + R ρµνσ + R ρνσµ = 0. (3.85 which reduces the number of independent components to only 20 out of 4 4 =256. (The number is 0, 1 and 6 in 1, 2 and 3 dimensions respectively. In addition algebraic identities there is a differential Bianchi identity, λ R ρσµν + ρ R σλµν + σ R λρµν = 0 (3.86 which is closely related to the Jacobi identity since it is nothing but [[ λ, ρ ], σ ] + [[ ρ, σ ], λ ] + [[ σ, λ ], ρ ]=0. (3.87 Some other useful tensor that can be formed from the Riemann tensor are: Ricci tensor: which is symmetric for Christoffel connections. Ricci scalar: R µν = R λ µλν (3.88 R = R µ µ (3.89 which can be shown (using Bianchi identity to obey µ R ρµ = 1 2 ρr. (3.90 Einstein tensor: G µν = R µν 1 2 Rg µν (3.91 which is also symmetric for Christoffel connections and can be shown to obey µ G µν =0. (3.92 Weyl tensor: C ρσµν = R ρσµν 1 2 ( gρ[µ R ν]σ g σ[µ R ν]ρ Rg ρ[µg ν]σ (3.93 which is essentially the Riemann tensor with all of the contractions removed (i.e. with zero traces while maintaining its symmetries.

14 CHAPTER 3. CURVATURE 42 Consider an example of two dimensional sphere with metric ds 2 = a ( 2 dθ 2 + sin 2 θdφ 2 (3.94 whose non-zero connection coefficients are Γ θ φφ = sin θ cos θ Γ φ θφ =Γφ φθ = cotθ. (3.95 Then R θ φθφ = θ Γ θ φφ φ Γ θ θφ +Γ θ θλγ λ φφ Γ θ φλγ λ θφ = = (sin 2 θ cos 2 θ ( sin θ cos θ cot θ = sin 2 θ (3.96 and R θφθφ = a 2 sin 2 θ. (3.97 It follows that R θθ = g φφ R φθφθ =1 R θφ = R φθ =0 R φφ = g θθ R θφθφ = sin 2 θ (3.98 and R = g θθ R θθ + g φφ R φφ =2a 2. (3.99 Note that the Ricci scalar is positive as it should be for a positively curved space such as the two-sphere, but for negatively curved spaces such as a saddle it would be negative. The considered two-sphere is an example of a maximally symmetric space generically defined by R ρσµν = a 2 (g ρµ g σν g ρν g σµ. (3.100 Consider a one parameter family of geodesics γ s (t on a manifold M such that for each s R, γ s is parametrized by the affine parameter t. For a non-crossing family of geodesics (which may or may not be the case we can choose the affine parameters in such a way that t and s form a system of coordinates on a two-dimensional surfaces embedded in M, i.e. There are two vector fields tangent vector, x µ (s, t M. (3.101 T µ = xµ t (3.102

15 CHAPTER 3. CURVATURE 43 and deviation vector whose commutator vanishes, S µ = xµ s (3.103 [S, T ]=0. (3.104 This means that they can be used as basis vectors for a coordinate system of our two-dimensional surface. Then the quantities defined as V µ =( T S µ = T ρ ρ S µ (3.105 and a µ =( T V µ = T ρ ρ V µ (3.106 we can call the relative velocity and relative acceleration of the geodesics. For a connection with vanishing torsion equation (3.104 implies and the relative acceleration is a µ = T ρ ρ (T σ σ S µ = T ρ ρ (S σ σ T µ S ρ ρ T µ = T ρ ρ S µ = (T ρ ρ S σ σ T µ + T ρ S σ ( ρ σ T µ = (S ρ ρ T σ σ T µ + T ρ S σ ( σ ρ T µ + R µ νρσ T ν = (S ρ ρ T σ σ T µ + S σ σ (T ρ ρ T µ (S σ σ T ρ ( ρ T µ +R µ νρσt ν T ρ S σ = S σ σ (T ρ ρ T µ +R µ νρσt ν T ρ S σ = R µ νρσ T ν T ρ S σ (3.107 where in the last line we used that T ρ is a tangent vector to geodesic and thus T ρ ρ T µ =0. (3.108 The resulting equation is known as the geodesic deviation equation, a µ = D2 dt 2 Sµ = R µ νρσ T ν T ρ S σ (3.109 which can be interpreted as the gravitational tidal force due to curvature R µ νρσ.

16 CHAPTER 3. CURVATURE Pullback and pushforward Consider a map φ : M N (3.110 where the manifolds M and N with respective coordinates x µ and y α might have different dimensions. Then one can think of φ as a function which maps function on N, f : N R (3.111 to functions on M, φ f : M R, (3.112 where φ f defined as a composite map φ f f φ. (3.113 Such map is called the pullback since it is defined by pulling back the action of function f from manifold N to manifold M (i.e. in the direction opposite to the way φ was defined. Clearly, it is not be possible to define a pushforward of a function, but it is possible to define a pushforward of a vector V which, as we know, is nothing but a map from functions on manifolds to real numbers. Then the pushforward of the vector field defined on manifold M, i.e. is a vector field on manifold N, i.e. V : F(M R. (3.114 φ V : F(N R (3.115 defined by the action of vector V on the pullback of functions, i.e. (φ V (f V (φ f. (3.116 In coordinate basis, the pushforward of a vector can be written as or (φ V α α f = V µ µ (φ f = V µ µ (f φ = V µ yα x µ αf (3.117 (φ V α = V µ yα x µ. (3.118

17 CHAPTER 3. CURVATURE 45 Then we can defined the pushforward matrix operator as so that (φ α µ = yα x µ (3.119 (φ V α =(φ α µ V µ. (3.120 Note that the pushforward matrix reduces to the general coordinate transformations matrix for M = N. Similarly to how it is not possible to define a pushforward of a function it is not possible to define a pullback of a vector, but it is possible to define a pullback of a one-form which is nothing but a map from vectors to real numbers. Then the the pullback of a one-form ω on manifold N, is a oneform φ ω on manifold M, defined by how it acts on pushforward vectors, i.e. (φ ω(v =ω (φ V. (3.121 Then we can also define the pullback matrix operator as so that (φ α µ = yα x µ (3.122 (φ ω µ =(φ α µ ω α. (3.123 The pushforward and pullback can respectively be generalized to arbitrary (k, 0 and (0,ltensors, (φ S ( ω (1,..., ω (k = S ( φ ω (1,..., φ ω (k, (3.124 or in indices notation (φ T ( V (1,..., V (l = T ( φ V (1,..., φ V (l (3.125 (φ S α1...α k = yα1... yαk µ1 x x µ k S µ1...µ k (3.126 (φ T µ1...µ l = yα1... yαl T µ1 x x µ α1...α l. (3.127 l The pushforward and pullback of other tensors is defined only if the map φ is invertible and then (φ S ( ω (1,..., ω (k,v (1,..., V (l = S ( φ ω (1,..., φ ω (k, ( φ 1 V (1,..., ( φ 1 V (l (3.128

18 CHAPTER 3. CURVATURE 46 or (φ T ( ω (1,..., ω (k,v (1,..., V (l = T (( φ 1 ω(1,..., ( φ 1 ω(k,φ V (1,..., φ V (l. (3.129 Clearly the pullback of φ is nothing but a pushforward of φ 1. Consider an example of a two-sphere S 2 with coordinates (θ, ϕembedded into a three dimensional Euclidean space R 3 with coordinates (x, y, z such that there is a natural map φ : S 2 R 3 (3.130 defined by φ(θ, ϕ = (sin θ cos ϕ, sin θ sin ϕ, cos θ. (3.131 Then the metric on R 3 as a (0, 2 tensor can be pullback to S 2. Then the pullback matrix operator ( cos θ cos ϕ cos θ sin φ sin θ φ = (3.132 sin θ sin ϕ sin θ cos φ 0 can be used to determine the pullback of the metric on R 3, i.e g = ( to the metric on S 2, i.e. φ g = = = ( cos θ cos ϕ cos θ sin ϕ sin θ sin θ sin ϕ sin θ cos ϕ 0 ( cos θ cos ϕ cos θ sin ϕ sin θ sin θ sin ϕ sin θ cos ϕ 0 ( sin 2 θ cos θ cos ϕ sin θ sin ϕ cos θ sin ϕ sin θ cos ϕ sin θ 0 cos θ cos ϕ sin θ sin ϕ cos θ sin ϕ sin θ cos ϕ sin θ 0 (3.134 As a result we will obtain a (0, 2 tensor on S 2 induced metric, (φ g µν. which is also called the 3.6 Killing vectors Given a one parameter family of diffeomorphism φ t : M M (3.135

19 CHAPTER 3. CURVATURE 47 such that φ s+t = φ s φ t (3.136 we can consider trajectory of a given point p under φ t. These trajectories x µ (t can used to define vector fields by evaluating tangent vector at t =0: [ ] dx V µ µ (x =. (3.137 dt t=0 Similarly, we can define a one parameter family of diffeomorphism (or integral curves from a given vector field (or generator of the diffeomorphism by solving the differential equation dx µ dt = V µ. (3.138 Then for any tensor field we can ask how it changes as we travel along the integral curves, i.e. t T µ1...µ k (p =φ t ( T µ 1...µ kν1...ν l (p T µ1...µ k (p (3.139 as well as define the so-called Lie derivative, i.e. ( ( L V T µ 1...µ t kν1...ν T µ1...µ k ν1...ν l = lim l t 0 t (3.140 which is a map from (k, l to(k, l. The Lie derivative is linear and obeys product rule L V (at + bs =al V T + bl V S (3.141 L V (T S =(L V T S + T (L V S. (3.142 Although the Lie derivative does not depend on the coordinate system it does not require specification of connections. It is easy to show that the Lie derivative reduces to an ordinary derivatives for scalars L V f = V (f =V µ µ f (3.143 and to a commutator (also called the Lie bracket for vectors L V U µ =[V,U] µ. (3.144

20 CHAPTER 3. CURVATURE 48 Then the Lie derivative of a one-form can be derived by considering the Lie derivative of a scalar L V (ω µ U µ = V ν ν (ω µ U µ (L V ω µ U µ + ω µ (L V U µ = V ν U µ ν ω µ + V ν ω µ ν U µ (L V ω µ U µ + ω µ (V ν ν U µ U ν ν V µ = V ν U µ ν ω µ + V ν ω µ ν U µ (L V ω µ U µ = V ν U µ ν ω µ + ω µ U ν ν V µ For an arbitrary tensor the Lie derivative is given by L V ω µ = V ν ν ω µ + ω ν µ V ν. (3.145 L V T µ1...µ k = V σ σ T µ1...µ k ( λ V µ1 T λµ2...µ k...+ ( ν1 V λ T µ1...µ k λν2...ν l... (3.146 Although the above expression is coordinate independent one can rewrite it in a more covariant form L V T µ1...µ k = V σ σ T µ1...µ k ( λ V µ1 T λµ2...µ k...+ ( ν1 V λ T µ1...µ k λν2...ν l... (3.147 A diffeomorphism φ is called a symmetry of some tensor T if it leaves the tensor invariant, i.e. φ T = φ T = T. (3.148 If the symmetry is generated by a vector field V µ (x thenthesymmetry implies that the Lie derivative vanishes L V T =0. (3.149 In that case one find a coordinate system in which the components of T are independent of one of the coordinates (pointing in the direction of the integral curves of V. One of the most important symmetries are of the metric φ g µν = g µν (3.150 in which case the diffeomorphism φ is called an isometry. The vector field V µ (x is called a Killing vector field if it generates the isometries, i.e. or L V g µν = 0 ( = V σ σ g µν +( µ V σ g σν +( ν V σ g µσ 0 = µ V ν + ν V µ 0 = (µ V ν (3.152

21 CHAPTER 3. CURVATURE 49 which is the Killing equations. The maximal number of Killing vector field in n-dimentional manifold is the sum of n translations and n(n 1/2 rotations (from any axis n to any other axis n 1 but without doublecounting. The sum of rotations and translations is n(n + 1/2ortenforthe four-dimensional space-time. The space-time of general relativity is called maximally symmetric if its metric tensor has ten independent Killing vectors which is the same as the number of independent parameters in a symmetric (0, 2 tensor. For example, ds 2 =dx 2 +dy 2 (3.153 has three Killing vectors (i.e. satisfy the Killing equations (3.152 corresponding to two translation and one rotation X = (1, 0 (3.154 Y = (0, 1 (3.155 R µ =( y, x (3.156 Consider a free particle moving along a geodesic x µ (λ whosetangent vector is U µ = dxµ. (3.157 Then for a Killing vector V µ we have U ν ν (V µ U µ = U ν U µ ν V µ + V µ (U ν ν U µ = 1 2 (U ν U µ ν V µ + U µ U ν µ V ν = 0. (3.158 which implies that the quantity V µ U µ is conserved along the trajectory of motion. 3.7 Tetrads So far we were using the coordinate basis for the tangent space ê (µ = µ (3.159 defined as partial derivatives with respect to coordinate functions and for dual space ˆθ (µ =dx µ (3.160

22 CHAPTER 3. CURVATURE 50 defined as gradients of the coordinate functions. Of course, we are free to use any basis we like and there is one particularly useful set of bias vectors, known as tetrads or vierbein, defined at each point by g ( ê (a, ê (b = ηab. (3.161 The coordinate basis vectors can be expressed in tetrad basis as ê (µ = e a µê(a (3.162 where the e a µ matrix an invertible matrix whose inverse eµ a basis vectors in coordinate basis expresses tetrad ê (a = e µ aê(µ. (3.163 As usual (with the abuse of notations the component e a µ and eµ a are often called the tetrads and inverse tetrads respectively. Clearly, they must satisfy or e µ ae a ν = δ µ ν (3.164 e a µ eµ b = δa b, (3.165 and (3.161 can be rewritten in the indices notation as or g µν e µ ae ν b = η ab (3.166 g µν = e a µ eb ν η ab (3.167 which is why tetrads are sometimes called the square root of the metric. After defining the orthonormal basis for vectors we can define an orthonormal basis for one-forms by setting ˆθ (a ( ê (b = δ a b. (3.168 Then one can show that using the very same inverse tetrads e µ a we can respectively express e a µ and tetrads ˆθ (µ = e µ a ˆθ (a (3.169 and ˆθ (a = e a µ ˆθ (µ. (3.170

23 CHAPTER 3. CURVATURE 51 Of course, no only the basis vectors but any vectors expressed in terms of the coordinate basis can be re-expressed in terms of tetrad basis. In other words V = V a ê (a = V µ ê (µ (3.171 implies Similarly for tensors V a = e a µ V µ. (3.172 V a b = ea µ V µ b = eν b V a ν = ea µ eν b V µ ν (3.173 and one example of such transformation we had already seen before for metric tensor ( For a non-coordinate tetrad basis we can illustrate how the change of tetrad basis would change components of tensors even without changing coordinates. If we go from one tetrad basis to another the orthonormality must be preserved and thus the metric η ab must remain flat. Of course the group of such transformation is known as Lorentz group and the transformation matrices are known as Lorentz matrices. In contrast to the global Lorentz transformations in special relativity we are now free to have different changes of basis at different points (local Lorentz transformations, i.e. ê (a ê (a =Λ a a (xê (a (3.174 where Λa a (x (actually an inverse Lorentz transformation is a function of position such that Λa a (xλ b b (xη ab = η a b (3.175 at every point. But since the local Lorentz transformations only transform basis, we can still transform the coordinates using the general coordinate transformations. So even if the tensor is expressed in mixed basis, we know exactly how to transform it ( ( T a µ x µ x b ν =Λ a a Λ b ν x µ b T aµ x ν bν. (3.176 Things become a little bit more complicated with covariant derivatives where the connection coefficients Γ λ µν are replaced with the so-called spin connections ω a µ b. For example, the covariant derivative of a (1,1 tensors components in tetrads basis is given by µ X a b = µx a b + ω a µ c Xc b ω c µ b Xa c. (3.177

24 CHAPTER 3. CURVATURE 52 Of course in mixed basis one can get terms with connection coefficients and spin coefficients in the same expression, e.g. µ X aν = µ X aν + ω a µ c Xcν +Γ ν µλ Xaλ. (3.178 But since the same object (e.g. written in coordinate basis covariant derivative of vector can be X = ( µ X λ dx µ λ = ( µ X λ +Γ λ µνx ν dx µ λ (3.179 as well as in tetrads basis X = ( µ X a dx µ ê (a = ( µ X a + ωµ a b Xb dx µ ê (a = ( µ (e a ν Xν +ωµ b( a e b ν X ν dx µ ( e λ a λ = ( e a ν µx ν + µ e a ν Xν + ωµ a b eb ν Xν dx µ ( e λ a λ = ( µ X λ + e λ a µe a ν Xν + ωµ a b eλ a eb ν Xν dx µ λ (3.180 we can express the connection coefficients in terms of spin coefficients and which is equivalent to Γ λ µν = eλ a µe a ν + ω a µ b eλ a eb ν (3.181 ω a µ b = ea ν eλ b Γν µλ eλ b µe a λ (3.182 µ e a ν =0. (3.183 Just like the connection coefficients the spin connections are not legitimate tensors, but the lower Greek index does transform as a one-form. The transformation law for other indices is given by ω a µ b =Λa a Λ b b ω a µ b Λ b c µλ a c. (3.184 which can be derived by demanding that covariant derivatives transform as tensors. It is sometime useful to view objects with mixed indices as tensor-valued (described by Latin indices differential from (described by Greek indices. For example, A a µν b which in antisymmetric in µ and ν can be thought of as (1,1 tensor valued differential two-form, or Xµ a can be thought of as a vector valued one form. Then one can show that the exterior derivative a (dx µν = µ Xν a ν X a µ (3.185

25 CHAPTER 3. CURVATURE 53 does not transform as a tensor, but a (dx µν +(ω X a µν = µ Xν a ν Xµ a + ω a µ b X ν b ω a ν b X µ b (3.186 is a legitimate tensor. Then one can view torsion as a vector valued twoform Tµν a and Riemann curvature tensor as a (1,1 valued two-form R a bµν. By suppressing indices of the differential forms the torsions and curvature tensors we can written as and T a =de a + ω a b eb (3.187 R a b =dω a b + ω a c ω c b (3.188 known as Maurer-Cartan structure equations. Similarly the Bianchi identities (3.85 and(3.86 can be written respectively as and dt a + ω a b T b = R a b eb (3.189 dr a b + ωa c Rc b Ra c ωc b =0. (3.190 The metric compatibility (i.e. µ g νλ =0 0= µ η ab = µ η ab ω c µa η cb ω c µb η ac = ω µba ω µab (3.191 implies the antisymmetric of the spin connection ω µab = ω µba. (3.192 For Christoffel connection the torsion (3.187 vanishes and thus de a = ω a b eb = ω ab e b (3.193 The two equations can be used to solve for the spin connections in terms of tetrads.