Summary in General Relativity

Transcription

1 Summary in General Relativity Jacob Shapiro October 17, 2014 Abstract A collection of things I deemed necessary to memorize for the final oral exam in general relativity held in ETHZ in January The main source for this summary has been Robert Wald s General Relativity textbook, as well as my own lecture notes and the official lecture notes by Professor Jetzer. I have tried to avoid, whenever possible, a reference to any particular chart on a manifold, and also to avoid using indices. I admit this may have been a bad idea, but it was an experiment. Please do refer to the definitions of T R, C, A, S, P and E, which are operators replacing the corresponding index notation. This summary has now been extended to include also the topics covered in the Applications of General Relativity class given in the following semester by Prof. Jetzer as well. The topics covered include the post- Newtonian approximation, gravitational waves, the Kerr-Newman solution, among other things. Note that these notes include only definitions and claims, but no proofs, which are left as an exercise to the reader. Contents I Differential Geometry 4 1 Manifolds Topological Manifolds Differentiable Manifolds Notes Diffeomorphisms Notes Tangent Vectors Preliminary Notions The Tangent Vector Space A Chart Basis for the Tangent Space at a Point Curves Tangent vector to a curve The Tangent Bundle Tangent Fields

2 1.4.8 One Parameter Groups of Diffeomorphisms Integral Curves The Commutator Cotangent Dual Vectors Tensors Multilinear Maps Tensors Tensor Bundles Metrics The Covariant Derivative The Derivative Operator / Covariant Derivative Useful Formulas The Ordinary Derivative Comparing Different Derivative Operators The Metric Covariant Derivative Parallel Transport The Riemann Curvature Tensor and its By Products The Riemann Curvature Tensor as a Measure of Change in Vector along a Parallely Transported Tour The Ricci Tensor The Scalar Curvature The Weyl Tensor The Einstein tensor Geodesics Affine Parametrization for Geodesics The Geodesic equation Principle of Extremal Action The Exponential map Riemann Normal Coordinates Gaussian Normal Coordinates / Synchronous Coordinates Geodesic Deviation Equation Differential Forms Things to note Exterior Product or Wedge Product The Grassman Algebra Exterior Derivative Dual Forms / Hodge Star Integration on Manifolds Oreintable Manifolds Measurable Forms Handedness of Charts Integration on an open set on the Manifold Locally Finite A Refintement of an Open Cover Paracompactness Partition of Unity Integration over the whole Manifold Stokes Theorem Volume Element

3 Integral of a Scalar Function Gauss Theorem Pull-Backs and Push-Forwards Pull Back of Scalars Push Forward of Tangent Vectors Pull Back of Dual Cotangent Vectors Pull Back and Push Forward for Diffeomorphisms Symmetry Transformations and Isometries The Lie Derivative Properties Killing s Vectors Conserved Quantities Determination of a Killing Vector field from Initial Conditions II Physics 30 2 Prerelativity Physics The Principle of General Covariance The Principle of Special Covariance General Covariance implies Special Covairance Some Principles of Newtonian Physics Problems with Newtonian Gravity Special Relativity Some Principles of Relativity The Paths which Material Particles Trace Proper Time Velocity The Geodesic Hypothesis Rest Mass Momentum Energy The Stress-Energy-Momentum Tensor Perfect Fluid Electromagentism Gravitation General Relativity Acceleration Electromagnetism Revisited Einstein s Equation Notes The Hilbert-Einstein Action Geodesic Precession Out of Plane Precession of S2 Orbit Linearized Metric Newtonian Limit Gravitomagnetic Potential

4 4.4.3 Gravitational Radiation Quadrupole Radiation The Post-Newtonian Approximation The FRW Solution Spatially Homogeneous Spacetime Spatially Isotropic Spacetime Spaces of Constant Curvature Robertson-Walker Metric Solutions for the Robertson-Walker Metric The Cosmological Constant The Big Bang ΛCDM Universe Dynamics of a (τ) Redshift in the Robertson-Walker Metric f (R) Gravity The Schwarzschild Solution Stationary Spacetimes Static Spacetimes Spherically Symmetric Spacetimes A good chart for the metric Robertson Parameters (Robertson Expansion) Solution in Vacuum Solution in the Interior Cosmological Constant Birkhoff s Theorem Gravitational Redshift Geodesics Light Bending Perihelion Precession Gravitational Time Delay / Shapiro Time Delay The Reissner-Nordstrom Metric The Sun s Quadrupole Moment The Kruskal Extension The Kerr-Newmann Solution Part I Differential Geometry 1 Manifolds 1.1 Topological Manifolds Let m N. A topological m-manifold is a Hausdorff second-countable topological space, M, such that p M, O p Open (M) : p O p U Open (R m ) : O p U. 4

5 1.2 Differentiable Manifolds Let {m, k} N. An m-dimensional-c k -real-manifold M is a Hausdorff paracompact topological space together with a collection of open sets {O α } α A Open (M) such that: 1. p M α A : p O α 2. α A U α Open (R m ) : O α U α. We shall label these homeomorphisms as Ψ α and call them charts or coordinate systems. 3. {α, β} A : O α O β, Ψ β Ψ 1 α Ψ β (O α O β ) Ψα(Oα O β) is C k Notes 1. Instead of k N we can speak of a smooth manifold, in which case, we understand the charts are C. In fact we shall always assume onward that the manifold is smooth rather than C k for simplicity. 2. By convention we define the manifold to have the maximal collection of compatible {O α } so that we don t get new manifolds on the same set M simply by adding maps. 1.3 Diffeomorphisms If M and N are two differentiable manifolds with respective collections of charts {Ψ α } α A and {Φ β } β B, and if f N M, f is said to be C iff (α, β) A B, Φ β f Ψα 1 is C. If f N M is C, bijective, and f 1 is also C, then f is called a diffeomorphism between M and N Notes 1. Since R is a smooth manifold, we can determine if a map f R M is C using the above criteria. We denote the set of all such functions by F (M), called the set of scalars on M. 1.4 Tangent Vectors Preliminary Notions 1. A map v R F(M) is linear iff v (af + bg) = av (f) + bv (g) {a, b} R {f, g} F (M). 2. A map v R F(M) is Leibnitz at p M iff v (fg) = g (p) v (f) + f (p) v (g) {f, g} F (M). 3. A map v R F(M) is a tangent vector at p M iff it is linear and Leibnitz at p M. 5

6 Notes 1. The value of any tangent vector at some point on any constant scalar will always be zero The Tangent Vector Space The set of all tangent vectors at p M is called the tangent space at p M and is denoted by V p (M). With addition and scalar multiplication defined in a natural way, this set forms a vector space. Thus at each point on the manifold we get a copy of a vector space constructed with the tangent vectors A Chart Basis for the Tangent Space at a Point Given a point p M we know that there exists (at least one) chart which covers this point, label it with α p A, where A was the indexing set for the charts on M. f F (M), i {1,..., m}, define the following tangent vector at p M (it is indeed a tangent vector) by how it acts on an arbitrary f F (M): X αp i (f) := [ i ( f ( Ψ αp ) 1 )] Ψαp (p) where i is the ordinary partial derivative with respect to the ith component of the R m R map, f ( ) 1. Ψ αp Then we may establish a basis of V p (M) using the following set: { } X αp m i i=1. Thus it is clear that dim (V p (M)) = dim (M) Useful Formulas 1. For an aribtrary v V p (M), the components of v with respect to the basis { } X αp m i i=1 are given by v ( ) π i Ψ αp where π i is the projection map from R m R onto the ith component. Then we have v = m i=1 v ( π i Ψ αp ) X α p i where we understand that v ( π i Ψ αp ) are simpy numbers multiplying vectors, on the right hand side of the equation. 2. If we pick another chart (which also covers p M), labelled by β A, then by exactly { the } same manner we could get a new basis for V p (M), m denote it by X β i. The change of basis formula for the basis vectors i=1 is given by: X β i = m ( )] [ i π j Ψ αp Ψ 1 β j=1 Ψβ (p) X αp j 6

7 whereas the change of basis formula for the components of a vector v is given by: v ( π i ) m ( )] Ψ β = [ j π i Ψ β Ψα 1 p v ( π j ) Ψ αp j=1 Ψαp (p) Observe the reciprocal nature of these laws: the actual basis vectors transform with the initial chart in the middle, whereas the components transform with the new chart in the middle Curves A smooth curve is a C map from R M Tangent vector to a curve Let γ be a smooth curve. τ 0 dom (γ), define a tangent vector at γ (τ 0 ) M, T γ(τ0) by the following: f F (M), Notes 1. f γ is a map in R R. T γ(τ0) (f) := [ d dτ (f γ)] 2. The tangent vectors to curves defined by varying only one component in a given chart are exactly the basis vectors associated with the chart, where the varying component defines the particular basis vector Useful Formulas The components of T γ(τ0) in a chart (labeled by α γ(τ0) A) basis are given by: ( ) [ γ] d T γ(τ0) π i Ψ αγ(τ0 = ) dτ πi Ψ αγ(τ0 ) The Tangent Bundle The tangent bundle of M, denoted by V (M) is defined as the following disjoint union: V (M) := {(p, v) M V p (M) v V p (M)} p M There is a natural topology on V (M) which makes it into a smooth manifold in its own right. τ 0 τ 0 7

8 1.4.7 Tangent Fields A tangent field is a map v V (M) M. Even though v (p) = (p, u) is a pair, where u V p (M), we would usually denote u simply as v p and thus ) v (p) = (p, v p Notes 1. A tangent field is defined to be smooth iff f F (M), the map v p (f) : M R is smooth in the sense defined above (the argument to this map is now p M, and f is kept fixed). 2. Within the domain of a given chart, the basis vectors defined by the chart establish a set of basis vector fields, all of which are smooth One Parameter Groups of Diffeomorphisms A one parameter group of diffeomorphisms, denoted by φ t, is a smooth map R M M, t p φ t (p) which is a group action on M, and each φ t is a diffeomorphism. This is also called a flow sometimes Notes 1. The orbits of each point define curves, these curves in turn have tangent fields. Thus we can think of a tangent field associated with φ t. 2. By the same token, it s possible to construct a tangent field which generates φ t (its Lie algebra) Integral Curves If v is a smooth tangent field, the integral curves of v is the family of curves in M such that p M,! curve from the family which passes through p and the tangent to this curve at p is exactly v p. Note that the integral curves to a given tangent field always exist and are unique. This stems from the the fact that the system of ordinary differential equations for functions π i Ψ γ : R R: d ( π i Ψ γ ) ( = v dt γ(t) π i Ψ ) ( always has a unique solution (Observe that v γ(t) π i Ψ ) is just some function π 1 Ψ γ of t: Ψ γ (t) = π 2 Ψ γ... (t)). π m Ψ γ 8

9 The initial conditions determine where the curve passes through, together, the set of all initial conditions must cover the whole domain of a chart. Additionally note that the integral curves {γ i (t)} i I of a given tangent field v define naturally a one-parameter group of diffeomorphisms: φ t (p) := γ ip (t p + t) where i p I is the label of the integral curve which passes through p, and t p domain ( γ ip ) is the value of the parameter of the curve such that γ ip (t p ) = p The Commutator Given two tangent vectors at a point p M, {u, v} V p (M), the commutator of u and v, denoted by [u, v], is yet another tangent vector, whose action on f F (M) is defined as follows: Notes [u, v] (f) := u (v (f)) v (u (f)) 1. The commutator of two chart basis vectors is zero because partial derivatives in R m commute. 2. If {X i } m i=1 is a set of non-vanishing vector fields such that [X i, X } j ] = 0 {i, j} {1,..., m} and such that p M, {X 1 p,..., X m p are linearly independent, a chart Ψ such that the chart basis vectors of this chart is exactly {X i } m i= Cotangent Dual Vectors Because V p (M) is a vector space, we may define its dual vector space, V p (M) which is by definitions of linear algebra the set { f R Vp(M) f is linear }. There is also a naturally defined basis for Vp (M), given a basis of V p (M). If {X i } m i=1 is a basis of V p (M), then define i {1,..., m} the dual vector f i Vp (M) such that v V p (M), f i (v) := v i where v i is the component of the tangent vector v expanded in the basis {X i } m i=1. This means that f i (X j ) = δj i. 1.6 Tensors Multilinear Maps Let n N\ {0}. Let {A j } n j=1 be an arbitrary collection of sets. Let f : n j=1 A j R. Let j 0 {1,..., n} be given. 9

10 Let (a 1, a 2,..., a j0 1, a j0+1,..., a n ) j 0 1 j=1 A j n j=j 0+1 A j be given. Define g j0 : A j0 R by g j0 (b) := f (a 1, a 2,..., a j0 1, b, a j0+1,..., a n ) b A j0. The map f is multilinear iff the map g j0 is linear, for any possible combination of choices of j 0 {1,..., n} and (a 1, a 2,..., a j0 1, a j0+1,..., a n ) j0 1 j=1 A j n j=j A 0+1 j Tensors Let (k, l) N N. A tensor of type (k, l) at p M is a multilinear map Vp (M) Vp (M) Vp (M) V p (M) V p (M) V p (M) }{{}}{{} k times l times R. Denote the set (naturally a vector space) of all tensors of type (k, l) at p M as T p (k, l) (M) Notes 1. A tensor of type (0, 1) is a dual vector. 2. Via the cannonical isomorphism of the dual dual with the original vector space, a tensor of type (1, 0) is a tangent vector. 3. dim (T p (k, l) (M)) = dim (M) k+l 4. Define the contraction of a tensor of type (k, l) on the ith cotangent slot and jth tangent slot to be a new tensor, of type (k 1, l 1), given by the following: If {X i } m i=1 is a basis of V p (M) and { f i} m is its dual i=1 basis, define the new tensor as, by its action on { µ j} k 1 j=1 V p (M) and {v j } l 1 j=1 V p (M): ( C i j (T ) ) ( ) µ 1,... µ k 1 ; v 1,... v l 1 := m q=1 T ( ) µ 1,..., µ i 1, f q, µ i+1,... µ k 1 ; v 1,..., v j 1, X q, v j+1,..., v l 1 Though defined via an actual basis, the Cj i (T ) is in fact indepedent of the basis with which it is defined. If we omit the indices and write C (T ) instead of Cj i (T ) it is because it is hopefully obvious which slots are to be contracted, for instance, when applied on a tensor in T p (1, 1) (M), or when working on a T p (1, 2) (M) which is symmetric in its vector slots, so the choice is not important anyway. 5. Define the outer product or tensor product of two tensors, T T (k, l) (M) and S T (n, o) (M) to be a new tensor in T (k + n, l + o) (M), denoted by (T S) ( µ 1,..., µ k+n ; v 1,..., v l+o ) := T ( µ 1,..., µ k ; v 1,..., v l ) S ( µ k+1,..., µ k+n ; v l+1,..., v l+o ). 10

11 6. A tensor which can be expanded as the outer product of vectors and dual vectors is called a simple tensor. 7. { X i1 X ik f j1 f j l i a {1,..., m} a {1,..., k}, j b {1,..., m} b {1,..., l} } is a basis of simple tensors of T p (k, l) (M). 8. If we choose a chart basis for V p (M) and then the natural dual basis for Vp (M) then we get a chart basis for T p (k, l) (M). Thus the components of a tensor transform according to the following formula: T (dy α1,..., Y βl ) = n µ i,ν i=1 T (dx µ1,..., X νl ) [ µ1 ( π α 1 φ Ψ 1)] Ψ(p)... [ βl ( π ν l Ψ φ 1)] φ(p) Where {X i } is the basis associated with the chart Ψ and {Y i } is the basis associated with the chart φ. 9. T T p (0, 2) (M) is nondegenerate iff [T (u, v) = 0 v V p (M)] u = Let σ S l. Define a map, called the permutator, P σ : T p (k, l) (M) T p (k, l) (M) by the following formula: (P σ (T )) (ω 1,..., ω k ; v 1,... v l ) := T ( ω 1,..., ω k ; v σ(1),..., v σ(l) ) This map applies the permutation σ on vector slots of a tensor. Similarly we define P σ for all σ S k which applies the permutation σ on dual vector slots of a tensor. We will often make use of cycle notation to notate particular permutations. 11. Define a map, called the antisymmetrizor, A {i, j,... } : T p (k, l) (M) T p (k, l) (M) by the following formula: A {i, j,... } (T ) := 1 {i, j,... }! σ S sign (σ) P σ (T ) where S is the subgroup of S l which is isomorphic to S {i, j,... }. This map antisymmetrizes the i, j,... vector slots of a tensor. Similarly we define A {i, j,... } which anti-symmetrizes the i, j,... dual vector slots of a tensor, and also agree that when {i, j,... } are omitted, all slots (dual and vector) are to be acted on. 12. Define a map, called the symmetrizor, S {i, j,... } : T p (k, l) (M) T p (k, l) (M) by the following formula: S {i, j,... } (T ) := 1 {i, j,... }! σ S P σ (T ) where S is the subgroup of S l which is isomorphic to S {i, j,... }. This map symmetrizes the i, j,... vector slots of a tensor. Similarly 11

12 we define S {i, j,... } which symmetrizes the i, j,... dual vector slots of a tensor, and also agree that when {i, j,... } are omitted, all slots (dual and vector) are to be acted on. 13. i {1,..., l}, define a map E i : T p (k, l) (M) T p (k, l) (M) by the following formula: E i (T ) := P (i, l, l 1,...,i+1) (T ) where (i, l, l 1,..., i + 1) S l is (in cycle notation) the permutation that shifts the ith item to the end. Similarly define E i. 14. i {1,..., l}, define a map B i : T p (k, l) (M) T p (k, l) (M) by the following formula: B i (T ) := P (1, 2,..., i) (T ) where (1, 2,..., i) S l is the permutation that shifts the ith item to the beginning. Similarly define B i Tensor Bundles Define the tensor bundle of type (k, l) to be the disjoint union: T (k, l) (M) := {(p, T ) T T p (k, l) (M)} p M Define tensor bundle to be the vector field direct sum: T (M) := k=0 l=0 T (k, l) (M) A tensor field is a map from M T (k, l) (M). This map is defined to be smooth in a natural way Metrics A metric is a smooth tensor field in T (0, 2) (M) which is nondegenrate and symmetric in its two slots (g = S (g)). If {X µ } is a basis field of V (M) with its dual basis field {dx µ } then in components the metric can be written as: g = m µ=1 ν=1 m g (X µ, X ν ) dx µ dx ν 12

13 Notes 1. An orthonormal basis of V p (M) is a basis of V p (M), {X µ } m µ=1, such that g (X µ, X ν ) = ±δ µν {µ, ν} {1,..., m}. Given a metric, there is an orthonormal basis. 2. The number of plus and minus signs appearing in the orhtonormal basis is called the signature of the metric. Though defined via an actual basis, the signature will not depend on the basis. 3. A metric with a siganture that is only pluses is called Riemannian whereas a metric with a signature with only one minus and the rest pluses is called Lorentzian. 4. g can be thought of as a map from V p (M) into Vp (M): v g (v,.) R Vp(M) = Vp (M). Since it is nondegenerate, it is bijective, and thus g 1. Thus g exhibits an isomorphism between V p (M) and Vp (M), and thus given a metric, we don t need to distinguish between the two anymore. We thus identify the cotangent vector corresponding to v V p (M) as g (v,.). We notate this by: v (u) := g (v, u) u V p (M) and ω (µ) := g 1 (ω, µ). 5. If γ is a curve with tangent vector field v then the length of γ is defined as: ˆ g (v, v) dt t domain(γ) This length is indepedent of parametrization of the curve. 6. If γ is a curve with tangent vector field v then we say that: γ is timelike if g (v, v) < 0 t domain (γ) γ is null if g (v, v) = 0 t domain (γ) γ is spacelike if g (v, v) > 0 t domain (γ) 7. A tensor density is a tensor T such that there exists another tensor T where T = det (g (X µ, X ν )) T, where T is the same for any chart. Then T will transform like a tensor under diffeomorphisms or change of charts. 8. Define a new map, called the trace, T R i,j : T p (k, l) (M) T p (k, l 2) (M) by the following: T R i,j (T ) := Cj 1 ( ( C 1 i g 1 T )) We may sometimes supress the i, j arguments when they are clear (for instance, when T T p (0, 2) ). Similarly define T R i,j : T p (k, l) (M) T p (k 2, l) (M) by the following: T R i,j (T ) := C j ( 1 C i 1 (g T ) ) 13

14 1.7 The Covariant Derivative The Derivative Operator / Covariant Derivative T (k, l) A derivative operator or covariant derivative is a map T (k, l + 1) which obeys the following four conditions: 1. It is linear. 2. It is Leibnitz. 3. It commutes with the contraction map C. 4. It obeys the following rule: f F (M) v V (M), v (f) = C 1 1 (v (f)) Notation always takes the first vector argument in an expression. It is customary to denote C1 1 (v (T )) as v (T ). Thus our fourth rule is v (f) = v (f) Torsion A derivative operator is called torsion free iff f F (M), ( (f)) = S ( ( (f))). Most of the times we shall indeed assume our derivative operator is torsion free. If this condition is not held, we may define a tensor in T (1, 2) (M) called the torsion tensor, T, such that: Then it can be shown that: 2A ( ( (f))) = C 1 1 ( (f) T ) T ( ; u, v) = u (v) v (u) [u, v] Examples We will work with merely three examples of a covariant derivative in general relativity: the ordinary derivative, the metric covariant derivative (called the Levi-Civita connection), and the Lie derivative Useful Formulas [u, v] = v u u v assuming no torsion The Ordinary Derivative Given a chart Ψ with associated chart basis {X µ } of V (M) and dual basis {dx µ }, define the ordinary derivative : T (k, l + 1) T (k, l) by the following rule: (T ) := σ,µ 1,...,ν l σ ( T (dx µ1,..., X νl ) Ψ 1 ) X µ1 X µk dx σ dx ν1 dx νl 14

15 Notes 1. is in fact a concrete example of a particular derivative operator, which we can construct given a chart. It is torsion free by the commutativity of partial derivatives. 2. Each chart gives a different ordinary derivative. Thus this derivative operator crucially depends on the chart and cannot be given an intrinsic manifold meaning Comparing Different Derivative Operators 1. v V (M), f F (M) we have v (f) = v (f). However, because v (f) is defined without reference to any particular derivative operator, all derivative operators must agree on this value: v (f) = v (f) for any two derivative operators and. From which we get (plugging in as v a basis vector): (f) = (f) 2. For any f F (M) and any ω V (M) we have ( ) (f ω) (f ω) = f (ω) (ω) 3. For any ω V (M) we see that the tensor (ω) (ω) depends only on ω p and not on its neighborhood at all (if two dual vector fields are the same on one particular point but otherwise different, the tensor will still be the same). Thus defines a map from Vp (M) T p (0, 2) (M) and so we can think of as lying inside T p (1, 2) (M) in the following way: Define a tensor, called the generalized Christoffel tensor, Chris T p (1, 2) (M) by the following: ( ) Chris (ω; u, v) := (ω) (ω) (u, v) From which we get the formula: ( ) ( (ω)) (u, v) = (ω) (u, v) Chris (ω; u, v) If we assume that is torsion free, we can additionally see that: Chris (ω; u, v) = Chris (ω; v, u) 15

16 And thus we can calculate, via Chris, the difference working on vectors: ( ) ( (v)) (ω; u) = (v) (ω; u) + Chris (ω; u, v) And via induction we can generalize this formula to arbitrary tensors T T p (k, l) (M): (T ) = (T ) + k i=1 B i C i+1 2 (Chris T ) Thus Chris completely determines from. l ( P(2, 3,..., j+1) Cj+2 1 (Chris T ) ) 4. If is torsion free, Chris T p (1, 2) (M) will be symmetric in its two vector slots. Thus Chris will have 1 2 n2 (n + 1) free components. 5. If torsion, then the torsion tensor mention above is equal to: T = 2A (Chris) 6. For the special case where, we call Chris the Christoffel symbols, and denote it by Γ = Chris. Observe that in that case, since stems from a particular chart, Γ is chart dependent and thus is not an intrinsic object of the manifold (such as the metric), but rather, a construct built using a chart. As such it cannot appear in any physical law by itself (since we assume physics doesn t care about which charts we use). 7. In conclusion, the specification of Γ is the specification of in a particular chart The Metric Covariant Derivative Given a metric g, there is a unique torsion free covariant derivative operator,, such that (g) = 0. We call this particular covariant derivative the metric covariant derivative or the Levi-Civita connection. There is an explicit formula for in terms of Chris: Chris = j=1 ( S C ( 2 1) C2 g 1 ) (g) We may also construct a covariant derivative with torsion, given a particular torsion tensor. Then the above formula becomes more complicated. From now onwards we assume implicitly that all covariant derivatives we deal with are indeed metric. 16

17 1.7.6 Parallel Transport Let γ be a smooth curve and let γ be its tangent vector field. A tensor field T T (k, l) (M) is said to be parallely transported along γ iff: γ (T ) γ(τ) = 0 τ domain (γ) This equation (or the equation for the components in a particular chart) defines a way for us to obtain the value of a vector field along γ given an initial value somewhere on γ (τ 0 ). Thus this defines a natural way to move tensors around different points on the manifold (As we know that T p (k, l) (M) T q (k, l) (M). This mapping from T γ(τ) (k, l) (M) T γ(τ+τ ) (k, l) (M) is called a connection, which is depdent on our choice of: both and γ Notes 1. Given a connection (that is, a map T γ(τ) (k, l) (M) T γ(τ+τ ) (k, l) (M)), there exists a covariant derivative which gives rise to this connection. 2. If two vectors {u, v} V (M) are parallely transported along a curve γ, then we have the following: γ (g (u, v)) = 0 Or in words: the inner product of parallely transported vectors is unchanged along the curve. 1.8 The Riemann Curvature Tensor and its By Products Define a tensor, Riemann T p (1, 3) (M), called the Riemann curvature tensor defined by the following formula: Riemann (ω; v 1, v 2, v 3 ) := ( A {1,2} ( ( (ω))) ) (; v 1, v 2, v 3 ) This is a well defined tensor field because Riemann (f ω; v 1, v 2, v 3 ) = f Riemann (ω; v 1, v 2, v 3 ) for any f F (M). Note that if has torsion then we have another definition for the Riemann curvature tensor: Riemann (ω; v 1, v 2, v 3 ) := ( A {1,2} ( ( (ω))) ) (; v 1, v 2, v 3 ) ( [v1, v 2] (ω) ) (v 3 ) Useful Formulas 1. ( A {1,2} ( ( (v 3 ))) ) (ω; v 1, v 2 ) = Riemann (ω; v 1, v 2, v 3 ) for all ω T p (0, 1) (M) and {v 1, v 2, v 3 } T p (1, 0) (M). 17

18 2. And thus by induction we have for a general tensor T T p (k, l) (M): 2A {1, 2} ( ( (T ))) = 3. S {1,2} (Riemann) = 0 4. A {1, 2, 3} (Riemann) = 0 k i=1 ( B i C 1+i 3 (Riemann T ) ) + 5. As we ve mentioned before, using g we have an isomorphism between V (M) and V (M): v g (v,.). Thus we could just as well feed into Riemann four tangent vectors, we it would be understood that: l ( P(3, 4,..., j+2) Cj+3 1 (Riemann T ) ) j=1 Riemann (; v 1, v 2, v 3, v 4 ) Riemann (g (v 4, ) ; v 1, v 2, v 3 ) Thus when there is reference to the 4th vector slot, the above is what is meant. Using this notation we have the following: 6. Riemann = P (13)(24) Riemann 7. The Bianchi identity: S {3,4} (Riemann) = 0 A {1,2,3} ( (Riemann)) = 0 8. Using all the symmetries above, we may deduce that the Riemann curvature tensor has: 1 12 m2 ( m 2 1 ) indepedent components, where m = dim (M). Thus in one dimension the curvature is identically zero, in two dimensions there is only one free component (manifold of constant curvature), in three dimensions there are six, and in four dimensions there 20 free components The Riemann Curvature Tensor as a Measure of Change in Vector along a Parallely Transported Tour If we take a tangent vector v at a point p M, and parallely transport v along a small rectangle defined by two components of a chart Ψ (with chart basis {X µ } and {dx µ }) and if we denote the parameter of of the integral curve of X 1 by t and the parameter of the integral curve of X 2 by s, then we have that the change in the component of v is (the symbol denotes change): [v (π µ Ψ)] = t s Riemann (dx µ ; X 1, X 2, v) + O ( t 3) + O ( s 3) 18

19 1.8.2 The Ricci Tensor Define the Ricci tensor, Ricci T (0, 2) (M) by the following formula: Ricci := C 1 2 (Riemann) Note that the Ricci tensor is symmetric: The Scalar Curvature Ricci = S (Ricci) Define the scalar curvature, R F (M), by the following formula: The Weyl Tensor R := T R (Ricci) If dim (M) 3, define the Weyl tensor, W eyl T (0, 4) (M): W eyl := Riemann 1 ( ) P(243) P (24) P (1432) + P (142) (g Ricci) n 2 1 ( ) + P(243) P (24) (R g g) (n 1) (n 2) Notes 1. S {1,2} (W eyl) = 0 2. S {3,4} (W eyl) = 0 3. A {1,2,3} (W eyl) = 0 4. The trace of any two slots on the W eyl tensor is zero: T R ij (W eyl) = 0 i {1, 2, 3, 4}, j {1, 2, 3, 4} \ {i} 5. The W eyl tensor is invariant under conformal transformations The Einstein tensor Define the Einstein tensor as Einstein T (0, 2) (M) by the following: Einstein := Ricci 1 2 R g Observe that: 1. T R ( (Einstein)) = 0 19

20 1.9 Geodesics A smooth curve γ M R with tangent vector field v V (M) is called a geodesic iff there exists a function α F (im (γ)) such that: Notes v (v) = α v 1. A geodesic doesn t change its type (timelike, null, spacelike) along the curve, and the length is indepdendent of parametrization Affine Parametrization for Geodesics A geodesic is said to be affinely parametrized iff the function α above in the definition is zero, and then we have: v (v) = 0 This is called geodesic equation (in coordinate free expression). Given a geodesic with an arbitrary parametrization, we can always reparametrize it affinely. Define a new geodesic, γ f where f is given by: ˆ [ ˆ ] f (t) = C 1 + exp C 2 α (t) dt dt Then γ f is an affinely parametrized geodesic The Geodesic equation In components of a chart Ψ and its associated chart basis {X µ }, the geodesic equation is: d 2 dt 2 (πµ Ψ γ) + m σ,λ=1 Γ (dx µ ; X σ, X λ ) d dt (πσ Ψ γ) d dt ( π λ Ψ γ ) = 0 This is the geodesic equation in the components of a certain chart. This is a coupled system of m second order differential equations for the functions π µ Ψ γ : R R and it always has a unique solution given a set of initial values of π µ Ψ γ (basically a point in M) and d dt (πµ Ψ γ) (a tangent vector at that point). Thus given p M and a v V p (M), we can always find a geodesic which passes through p and has v as its tangent vector at that point Principle of Extremal Action The geodesic equation can also be derived from the principle of extremal action, where the action is defined as the length of the geodesic. 20

21 1.9.4 The Exponential map Let p M and v p V p (M). Let γ be the unique geodesic passing through p with tangent vector v p then. Without loss of generality we may assume that: (1) γ is affinely parametrized (otherwise take the reparamaterized geodesic). (2) γ (0) = p. Define a map, exp M V (M) by the following formula: )) exp ((p, v p := γ (1) This map is not necessarily injective nor well defined for arbitrary v p Riemann Normal Coordinates There will be a neighborhoud of 0 V p (M) in which exp is well defined and injective. Since V p (M) is isomorphic to R m (being just a finite dimensional vector space over R), in the neighborhoud in which exp is well defined and injective, we have thus established a homeomorphism between a neighborhoud of R m (where exp is well defined) and M (the image of exp). Thus, exp 1 (which exists via injectivity) defines a special chart on M. This chart is called the Riemann normal coordinates at p M. In Riemann normal coordinates, all geodesics through p get mapped onto straight lines through the origin in R m. In Riemann normal coordinates, the Christoffel symbols vanish and the geodesic equation becomes a simple differential equation of a straight line Gaussian Normal Coordinates / Synchronous Coordinates Let S be a given hypersurface, that is an m 1-dimensional embedded submanifold of the m-dimensional manifold M. We may view V (S) as a subset of V (M), in which case, there is a vector field n V (M) which is orthogonal to all of S. Pick some chart on S. Define a chart on M by the following procedure: At each point p S, there is a unique geodesic passing through p with tangent vector n. At each point q M in the vicinity of S, there is thus passing a unique geodesic which emanated from some point on S. Then we give q the same coordinates as the point p which is the source of the geodesic, plus an additional coordinate, which is the parameter along the geodesic on which q lies. These coordinates thus define new hypersurfaces (parametrized by the geodesics) where S 0 is our original surface and S t are surfaces of constant geodesic parameter. The coordinates satisfy the property that the geodesics remain orthogonal to all hypersurfaces S t of constant geodesic parameter. 21

22 1.9.7 Geodesic Deviation Equation Let γ s (t) be a one parameter family of geodesic, indexed by s, affinely paramterized by t. The parameter for selecting different members of the family is thus s. Let Σ be the two-dimensional submanifold of M spanned by γ s (t) and let s and t be coordinates for Σ, that is, define a chart Ψ on Σ defined by: [ ] t such that γs (t) = p Ψ (p) := s such that γ s (t) = p which is a good chart because γ s (t) is injective in both its parameters. Then we have the two basis vector fields X t and X s, which we can pull up to M as basis vector fields of V (M). We will have Xt (X t ) = 0 because γ s (t) is a geodesic. Because these are two basis vectors, they commute: Xt (X s ) = Xs (X t ). Then, it turns out that Xt ( Xt (X s )) = Riemann ( ; X t, X s, X t ). If we interpret X s as a measure of the separation between nearby geodesics in our family, then this equation states that the acceleration along the geodesic γ s (t) is exactly the Riemann curvature tensor Differential Forms Let p M and let n N {0}. A differential n-form is a tensor ω T p (0, n) which obeys the following equation: ω = A (ω) The vector space of differential n-forms at p is denoted by Λ n p. differential n-form fields is denoted by Λ n. The set of Things to note 1. Λ n p = {0} n > dim (M) 2. dim ( Λ n p ) = m! n!(m n)! where m = dim (M) Exterior Product or Wedge Product Define a map, called the wedge product, : Λ k p Λ l p Λ k+l p formula: by the following ω µ := (k + l)! A (ω µ) k!l! 22

23 Notes 1. ω µ = ( 1) nω+nµ µ ω. 2. The wedge product is associative The Grassman Algebra If we define a vector space as Λ p := n=0 Λn p, then it turns out that Λ Λp The tuple (Λ p, ) is called the Grassman algebra over Λ 1 p Exterior Derivative If is a given covariant derivative operator, define a map d : Λ n Λ n+1 by the following formula: d (ω) := (n + 1) A ( (ω)) p. If there is no torsion, then this definition is indepdent of choice of (that is, and will give rise to the same dω). Poincare s Lemma: d d = 0. A closed form ω is one such that d (ω) = 0 An exact n-form ω is one such that there exists an n 1-form, µ such that ω = d (µ). Thus an exact form is automatically closed. The converse of Poincare s lemma: If α is closed then there exists a neighborhoud in which it is also exact Dual Forms / Hodge Star Let α Λ n (M) where dim (M) = m. Let ε be the natural volume element on M, which is derived from the metric g. Define the dual of α, α, to be the following element of Λ m n (M): α := 1 n! (T R 1,2 T R 1,n T R 1,n+1 ) (α ε) so we are left with m n free slots from ε. α = ( 1) s+n(m n) α where s is the number of zeros in the signature of the metric. 23

24 1.11 Integration on Manifolds Oreintable Manifolds M is orientable iff a continous nowhere vanishing ε Λ m (M). ε is called an orientation on M. Two orientations ε and ε are equivalent iff a strictly positive function f such that ε = fε Any orientable manifold has two orientations up to equivalence. These are called (arbitrarily) right-handed and left-handed. Every simply-connected manifold is orientable. The product of two orientable manifolds is orientable Measurable Forms An m-form α (where dim (M) = m) is measurable iff chart Ψ, the component of α (remember it has only one component) in Ψ are Lebesgue measurable in R m, that is, if the function: α (X 1,..., X m ) Ψ 1 : R m R is measurable Handedness of Charts Let Ψ be a chart on M with domain U Open (M) and let ε be an orientation on M. Since ε is nonvanishing and continuous, it will have only one sign over the whole manifold. If ε (X 1,..., X m ) > 0 then Ψ is called right-handed with respect to ε Otherwise Ψ is called left-handed with respect to ε Integration on an open set on the Manifold Let Ψ be a chart and let U Open (M) be such that U domain (Ψ). Let α be a measurable m-form on M. Define the integral of α with respect to the orientation ε over U to be: ˆ ˆ [ α := ± α (X1,..., X m ) Ψ 1] dm U Ψ(U) where dm is the usual Lebesgue measure on R m and the sign is chosen according to the handedness of Ψ: right handedness is plus. Note that α is indepedent of the choice of Ψ, as long as we always choose U charts of the same handedness Locally Finite Let X be a topological space. A collection of subsets, {V β } β B 2 X is called locally finite iff x X ( W Open (X) : x W ( {β B : V β W } < )). 24

25 A Refintement of an Open Cover Let X be a topological space, and let {O α } α A be an open cover for X, that is, {O α } Open (X) and α A O α = X. A refinement of {O α } α A is another collection, {V β } β B such that β B α A such that V β O α Paracompactness A topological space X is paracompact iff every open cover admits an open cover refinement that is locally finite Partition of Unity If {O α } α A is a locally finite open cover of M, a partition of unity subordinate to {O α } α A is a collection {f α } α A F (M) such that: 1. supp (f α ) O α 2. f α (O α ) [0, 1] 3. α A f α = 1 Note that since {O α } α A is locally finite, this sum is finite and thus defined Integration over the whole Manifold Let ω be an m-form, and let {f α } be some partition of unity subordinate to {O α }, which is a collection of domains corresponding to a collection of charts which are assumed to be locally finite with compact closures. If α A O α f α ω <, define: ˆ M ω := α A ˆ O α f α ω In which case we say ω is integrable. Note that this definition is indepdent of a particular choice of {f α }! Stokes Theoremˆ ˆ d (α) = int(m) M α 25

26 Volume Element A volume element is an orientation on M. However, we don t identify together different volume elements which have the same sign. If we have a metric, a natural choice for a volume element given a chart Ψ and its associated chart basis {X µ } is: ε = det (g (X µ, X ν )) dx 1 dx m Integral of a Scalar Function Let ε be a volume element on M. Define the integral of f F (M) as: ˆ f := M ˆ M f ε which makes sense because f ε is again an m-form Gauss Theorem ˆ ˆ T R ( (v)) = int(m) M C 1 1 (ε v) where ε is a volume element on M and thus C1 1 (ε v) is a volume element on M. 1 [ ( It is possible to prove that T R ( (v)) = i det (g) v π i Ψ ) Ψ 1] det (g) for some chart Ψ, in which case we recover the simple form of the theorem Pull-Backs and Push-Forwards Let φ N M be a smooth map between two manifolds (of not necessarily the same dimension) Pull Back of Scalars f F (N ), define the pull back of f, which is then an element of F (M), to be: φ (f) := f φ Push Forward of Tangent Vectors p M, v V p (M), define the push forward of v, φ (v) V φ(p) (N ) by its action on an arbitrary function: (φ (v)) (f) := v (f φ) f F (N ) 26

27 Thus φ ( V φ(p) (N ) ) V p(m). It is a linear map. The components of φ are equal to the Jacobian matrix of φ: If Ψ is a chart on M, If χ is a chart on N, then we have: (φ (v)) (π ν χ) = dim(m) µ=1 (φ ) ν µ v (πµ Ψ) where (φ ) ν µ = ( µ π ν χ φ Ψ 1) Pull Back of Dual Cotangent Vectors The Push Forward above can be inverted into a map φ : T φ(p) (0, l) (N ) T p (0, l) (M) for arbitrary l N {0}: If T T φ(p) (0, l) (N ), then (φ (T )) (; v 1,..., v k ) := T (φ (v 1 ),..., φ (v l )) for all {v j } l j=1 V (M) In particular, if ω V φ(p) (N ), then (φ (ω)) (v) = ω (φ (v)) for all v V p (M). Now we may invert the map yet again: For arbitrary k N {0}, define φ : T p (k, 0) (M) T φ(p) (k, 0) (N ) by the following: φ (T ) (ω 1,..., ω k ; ) := T (φ (ω 1 ),..., φ (ω k )) for all {ω i } V φ(p) (N ), T T p (k, 0) (M). However, we have no way to go from V φ(p) (N ) V p (M) or from Vp (M) (N ). V φ(p) Pull Back and Push Forward for Diffeomorphisms When φ N M is a diffeomorphism, then φ 1 M N exists, and then we can construct the maps from V φ(p) (N ) V p (M) or from Vp (M) Vφ(p) (N ). Thus define φ : T p (k, l) (M) T φ(p) (k, l) (N ) by the following: ( (φ (T )) (ω 1,... ; v 1,... ) := T φ (ω 1 ),... ; ( φ 1) ) (v1 ),... for all ω j V φ(p) (N ) and for all v j V φ(p) (N ). Observe that φ = ( φ 1) as so we can always deal only with push forwards. 27

28 Symmetry Transformations and Isometries If φ M M is a diffeomorphism and T is some tensor field, then if T p = ) φ (T φ 1 (p) then φ is a symmetry transformation for T. If φ is a symmetry transformation for the metric then it is an isometry The Lie Derivative Let v be a vector field and let φ t be the one-parameter group of diffeomorphisms which v generates. Let T be a tensor field in T (k, l) (M). Thus T p T p (k, l) (M). ) Therefore φ t (T p T φ t(p) (k, l) (M). ( ) Thus φ t T φt(p) T p (k, l) (M). Define the Lie derivative of T at p with respect to v to be the following tensor at T p (k, l) (M): ) [ d (L v (T ) p (ω 1,... ; v 1,... ) := dt ( ( )) (φ t ) T φt(p) (ω 1,... ; v 1,... )] t=0 for all ω j Vp (M) and for all v j V p (M), where d dt is the ordinary derivative of a R R function Properties ) (L v (T ) p T p (k, l) (M) due to the linearity of the ordinary derivative d dt. The map T L v (T ) is: 1. Linear. 2. Obeys the Leibnitz rule for tensor products. 3. L v (f) = v (f) 4. Commutes with contractions and is thus another example of a type of covariant derivative. L v (T ) p = 0 p M φ t is a symmetry transformation of T for all t domain (φ t ). 28

29 If we use a chart where v is one of the basis vectors, then the components of L v (T ) p are simply the partial derivative with respect to the parameter of v. Thus φ t will be a symmetry transformation of T iff in that special chart the components of T are indepdent of the coordinate corresponding to v. L v (u) = [v, u] L v1+v 2 (u) = L v1 (u) + L v2 (u) L [v1, v 2] (T ) = [L v1 (T ), L v2 (u)] L v1 (L v2 (T )) L v2 (L v1 (T )) This property is nice because it implies that if v 1 and v 2 are Killing vectors then [v 1, v 2 ] is also a Killing vector (see below). L v1 d = d L v1 L v (ω) = v (ω ( )) ω ([v, ]) for all v V (M) and ω V (M) If T T (k, l) (M) then (L v (T )) (ω 1,... ; v 1,... ) = v (T (ω 1,... ; v 1,... )) T (v (ω 1 ( )) ω 1 ([v, ]), ω 2,... ; v 1,... )... T (ω 1,... ; v 1,..., [v, v l ]) and so we have completely determined L v (T ) for an arbitrary tensor using more primitive constructions. Note: the in the formula denotes a map that takes as its paramter (a tangent vector). We may also express L v (T ) using a given covariant derivative. The formula is: L v (T ) = v (T ) k E i Cl+1 i (T (v)) + i=1 l i=1 In particular, if is metric, L v (g) = 2S ( (v)) E i C k+1 i (T (v)) 1.14 Killing s Vectors A Killing vector ξ is a vector field which generates a one parameter group of isometries φ t. Thus ξ is a Killing vector field iff L ξ (g) = 0. Thus ξ is a Killing vector field iff S ( (ξ)) = 0, which is called Killing s equation. 29

30 Conserved Quantities If γ is a geodesic with tangent vector field γ, then g (ξ, γ) is a constant along the geodesic. In particular, if T is a symmetric tensor type T (0, 2) (M) such that T R ( (T )) = 0 then C (T ξ) is a conserved charge Determination of a Killing Vector field from Initial Conditions Observe that ( (ξ)) = P (13) ( T R 1,2 (Riemann ξ) ) And so given the value of ξ and ξ at a particular point, we can determine ξ at all points on the manifold. Thus there is a maximum of 1 2m (m + 1) indepedent Killing vectors on an m dimensional manifold. Part II Physics boxed statements In this section, are physical postulates which are not mathematically derivable truths nor arbitrary definitions, but rather axiomatic postulates corresponding to the physical nature which we observe in the world to some degree of accuracy of measurements. 2 Prerelativity Physics Space is the manifold R 3 with a flat Riemannian metric defined on it. 2.1 The Principle of General Covariance The metric of space is the only quantity pertaining to space that can appear in any law of physics. 2.2 The Principle of Special Covariance Any physically possible set of measurements obtained by some family of observers is also a physically possible set of measurements for another family of observers, related to the first family by an isometry. 30

31 Observer By observer we mean a timelike curve, whereas an inertial observer is a timelike geodesic General Covariance implies Special Covairance IF the physical quantities are descriable by tensor fields, and satisfy tensor equations, special covariance follows from general covariance. We could say that general covariance implies the invariance of the equations under general diffeomorphisms whereas special covariance implies the invariance of the equations only for diffeomorphisms which are isometries. 2.3 Some Principles of Newtonian Physics 1. Newton s first law: Every body continues in its state of rest or uniform motion in a straight line unless it is compelled to change that state by forces acting on it. Galilean transformations (10 parameter group of isometries 4 translation parameters, 3 rotation parameters, 3 boost parameters): { x = x + vt t = t 2.4 Problems with Newtonian Gravity The problem with Newtonian gravity is that it allows instantaneous action at a distance. 3 Special Relativity Spacetime is the manifold R 4 with a flat Lorentzian metric. The geodesics of this metric define global interial coordinates. We denote the metric in special relativity by η instead of g. In the coordinates defined above, η (X µ, X ν ) = diag ( 1, 1, 1, 1). Both general and special covariance hold for special relativity, with the exception of time reversal and parity transformations, which are isometries, but we don t assume the equations are invariant under these isometries. Practically we define an inertial reference system as a system which moves with constant speed with respect to the distant (and thus fixed) stars. The isometry group of special relativity is the Poincare group: Λ T ηλ = η. A 10 parameter group like the Galilean group. For a boost along 31

32 cosh (Ψ) the x-axis, we have Λ = sinh (Ψ) arctanh ( v c ) is called the rapidity. sinh (Ψ) cosh (Ψ) 1 where Ψ = Some Principles of Relativity 1. Restricted principle of special relativity: All inertial observers are equivalent as far as dynamical experiments are concerned. 2. Postulate I. Principle of Special Relativity: All inertial observers are equivalent. 3. Postulate II. Constancy of Velocity of Light: The velocity of light is the same in all inertial systems. 4. Special Lorentz transformations: { x = x vt 1 v 2 t = t vx 1 v 2 t depends on x. 5. Lengths in motion contract (a rod of length 1m at rest will be measured as 0.86m when it moves at 0.5c), times in motion dilate (The lifetime of a particle is 1sec when it is at rest and will be measured as 1.15sec when moving at 0.5c). 6. Equivalence principle: There is a local neighborhoud around each point where SR rules hold. 7. Mach s principles: (1) The matter distribution determines the geometry of the universe. (2) Without matter (an empty universe) has no geometry. (3) A singular body in an otherwise empty universe will never feel any inertial forces. 8. Inertial mass is equal to gravitational mass. 9. Strong principle of equivalence: The motion of a gravitational test particle in a gravitational field is indepedent of its mass and composition. 10. Weak principle of equivalence: The gravitational field is coupled to everything (any form of energy and matter). 11. Alternative statement of the principle of equivalence: There are no local experiments which can distinguish non-rotating free fall in a gravitational field from uniform motion in space in the absence of a gravitational field. 32