Gravitation: Tensor Calculus

Transcription

1 An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013

2 Manifolds Gravity as geometry Manifolds Vectors and Tensors The Metric Causality Tensor Densities Differential Forms Integration

3 Gravity as Geometry According to Einstein: The metric tensor describing the curvature of spacetime is the dynamical field responsible for gravitation. Gravity is not a field propagating through spacetime. Gravitational interactions are universal (Principle of equivalence)

4 Weak Principle of Equivalence (WEP) The inertial mass and the gravitational mass of any object are equal F = m i a F g = m g Φ with m i and m g the inertial and gravitational masses, respectively. According to the WEP: m i = m g for any object. Thus, the dynamics of a free-falling, test-particle is universal, independent of its mass; that is, a = Φ Weak Principle of Equivalence (WEP) The motion of freely-falling particles are the same in a gravitational field and a uniformly accelerated frame, in small regions of spacetime

5 Einstein Equivalence Principle In small regions of spacetime, the laws of physics reduce to those of special relativity; it is impossible to detect the existence of a gravitational field by means of local experiments. Due to the presence of the gravitational field, it is not possible to built, as in SR, a global inertial frame that stretches through spacetime. Instead, only locally inertial frames are possible; that is, inertial frames that follow the motion of individual free-falling particles in a small enough region of spacetime. Spacetime is a mathematical structure that locally looks like Minkowski or flat spacetime, but may posses nontrivial curvature over extended regions.

6 Manifolds Generally speaking, a manifold is a space that with curvature and complicated topology that locally looks like R n. Examples: R n itself. R is a line and R 2 a plane. The n-sphere, S n ; that is, the locus of all points some fixed distance from the origin in R n+1. S 1 is a circle and S 2 sphere. The n-torus T n. T 2 is the surface of a doughnut. A Riemann surface of genus g. That is, a n-torus with g holes. A set of continuous transformations such as rotations in R n. The direct product of two manifolds is a manifold.

7 Manifolds identify opposite sides genus 0 genus 1 genus 2

8 Not manifolds

9 Map: Given two sets M and N, a map φ : M N is a relationship which assigns, to each element of M, exactly one element of N. Composition: Given two maps φ : A B and ψ : B C, we define the composition ψ φ : A C by the operation (ψ φ)(a) = ψ(φ(a)). So a A, φ(a) B, and thus (ψ φ)(a) C. One-to-one map: A map φ : M N such that each element of N has at most one element of M mapped into it. Onto map: A map φ : M N such that each element of N has at least one element of M mapped into it.

10 Examples: Consider a function φ : R R. Then φ(x) = e x is one-to-one, but not onto; φ(x) = x 3 x is onto, but not one-to-one; φ(x) = x 3 is both; and φ(x) = x 2 is neither. Given a map φ : M N, the set M is known as the domain of the map φ, and the set of points in N which M gets mapped into is called the image of φ. For some subset U N, the set of elements of M which get mapped to U is called the preimage of U under φ, or φ 1 (U). A map which is both one-to-one and onto is known as invertible and there exists a corresponding inverse map φ 1 : N M by (φ 1 φ)(a) = a.

11 Consider the maps φ : R m R n that takes an m-tuple (x 1, x 2,..., x m ) to an n-tuple (y 1, y 2,..., y n ) such that: y 1 = φ 1 (x 1, x 2,..., x m ) y 2 = φ 2 (x 1, x 2,..., x m ) y n = φ n (x 1, x 2,..., x m ) The functions φ i are C p if they are continuous and p-times differentiable. The entire map φ : R m R n is C p if each of its component functions are at least C p.

12 A C 0 map is continuous but not necessarily differentiable. A C or smooth map is continuous and can be differentiated as many times as one likes. Diffeomorphisms Two sets M and N are diffeomorphic if there exists a C map φ : M N with a C inverse φ 1 : N M; the map φ is then called a diffeomorphism.

13 Open ball: The set of all points x R n such that x y < r for some fixed y R n and r R. Open set: in R n is a set constructed from an arbitrary union of open balls. In other words, V R n is open if, for any y V, there is an open ball centered at y which is completely inside V. A chart or coordinate system consists of a subset U of a set M, along with a one-to-one map φ : U R n, such that the image φ(u) is open in R.

14 A C atlas is an indexed collection of charts {(U α, φ α )} such that 1 The union of the U α is equal to M 2 The charts are smoothly sewn together. That is, if two charts overlap, U α U β, then the map (φ α φ 1 β ) takes points in φ β (U α U β ) R n onto φ α (U α U β ) R n, and all of these maps must be C where they are defined. M U # U! "! n R " # -1 " "! # " ( U )!! n R " ( U ) # # -1 " " #! these maps are only defined on the shaded regions, and must be smooth there.

15 Manifold A C n-dimensional manifold is a set M along with a maximal atlas, one that contains every possible compatible chart. The requirement of a maximal atlas is needed so two equivalent spaces with different atlases do not count as different manifolds. Notice that the existence of a manifold does not depend on an embedding.

16 Most manifolds cannot be covered by a single chart. S 1 U 1 U 2 x3 x2 x 1 (x 1, x 2, x 3) (y 1, y 2) x 3 = -1

17 Chain Rule Consider two maps f : R m R n and g : R n R l, and their composition (g f ) : R m R l and each space in terms of coordinates: x a R m, y b R n, and z c R l. Chain rule: or x a (g f )c = b x a = b f b g c x a y b. y b x a y b.

18 Vectors Consider: The set of all parameterized curves γ(λ) through p, i.e. all maps γ : R M such that p is in the image of γ. The space F of all the C smooth maps f : M R. Tangent Space T p Each curve γ(λ) through p defines a directional derivative operator, which maps f df /dλ. T p is the space of directional derivative operators along curves through p

19 Vectors Notice: also they obey the Leibniz rule ( a d dλ + b d ) dη d dξ = a d dλ + b d dη (fg) = af dg df dg + ag + bf dλ dλ dη ( = a df dλ + b df ) g + dη + bg df dη ( a dg dλ + b dg dη ) f

20 T p basis Given a coordinate chart {x µ } in an n-dimensional manifold M, there is a set of n directional derivatives at p given by the partial derivatives µ at p. p! 2! 1 x 1 x 2

21 T p basis { µ } at p form a basis for the tangent space T p. Proof: Consider a coordinate chart φ : M R n, a curve γ : R M, and a function f : M R such that f " R " M f R! -1!! " R n f! -1 x µ

22 T p basis f " R " M f R! -1!! " R n f! -1 x µ Let λ be parameter along γ, then d dλ f = d d (f γ) = dλ dλ [(f φ 1 ) (φ γ)] d(φ γ)µ (f φ 1 ) = dλ x µ = dx µ dλ µf

23 T p basis Thus, d dλ f = dx µ dλ µf implies that µ is a good coordinate basis for the tangent space T p Coordinate Basis d dλ = dx µ dλ µ ê (µ) = µ

24 Vector transformation law Given then µ = x µ x µ µ. so V µ µ = V µ µ µ x µ = V µ, x µ V µ = x µ x µ V µ Lorentz transformations V µ = Λ µ µv µ are a special case.

25 Commutators [X, Y ](f ) = X(Y (f )) Y (X(f )) [X, Y ](a f + b g) = a[x, Y ](f ) + b[x, Y ](g) [X, Y ](f g) = f [X, Y ](g) + g[x, Y ](f ) [X, Y ] µ = X α α Y µ Y α α X µ

26 One-forms A Cotangent space T p is the set of linear maps ω : T p R. A one-form df is the gradient of a function f. Action of df on a vector d/dλ: ( ) d df dλ Recall ˆθ (µ) (ê (ν) ) = δ µ ν, thus dx µ ( ν ) = x µ = df dλ. x ν = δµ ν {dx µ } are a set of basis one-forms; that is, ω = ω µ dx µ. Transformation properties: and dx µ = x µ x µ dx µ, ω µ = x µ x µ ω µ.

27 Tensors A (k, l) tensor T can be expanded T = T µ 1 µ k ν1 ν l µ1 µk dx ν 1 dx ν l, Under a coordinate transformation the components of T change according to T µ 1 µ k ν 1 ν l = x µ 1 x µ x µ k x ν 1 1 x µ x ν l T µ 1 µ k ν1 ν k x ν 1 x ν l. l The partial derivative of a tensor is not a new tensor W ν x µ = x µ x µ = x µ x µ ( x ν x µ x ν x ν ) W ν x ( ν ) x µ W x µ ν + W ν x µ x ν x µ x ν

28 The Metric: g µν The metric g µν : (0, 2) tensor, g µν = g νµ (symmetric) g = g µν 0 (non-degenerate) g µν (inverse metric) g µν is symmetric and g µν g νσ = δ σ. µ g µν and g µν are used to raise and lower indices on tensors.

29 g µν properties The metric: provides a notion of past and future allows the computation of path length and proper time: ds 2 = g µν dx µ dx ν determines the shortest distance between two points replaces the Newtonian gravitational field provides a notion of locally inertial frames and therefore a sense of no rotation determines causality, by defining the speed of light faster than which no signal can travel replaces the traditional Euclidean three-dimensional dot product of Newtonian mechanics

30 g µν canonical form Canonical form: g µν = diag ( 1, 1,..., 1, +1, +1,..., +1, 0, 0,..., 0) If g µν is the metric in a n-dimension of the manifold M, and s is the number of +1 s in the canonical form, and t is the number of 1 s, then s t is the signature of g µν and s + t rank. If g µν is nondegenerate, the rank is equal to the dimension n. If g µν is continuous, the rank and signature of the metric tensor field are the same at every point. If all of the signs are positive (t = 0), g µν is called Euclidean or Riemannian or positive definite If there is a single minus (t = 1), g µν is called Lorentzian or pseudo-riemannian, If g µν with some +1 s and some 1 s is called indefinite The spacetimes of interest in general relativity have Lorentzian metrics.

31 Riemann normal coordinates: At any point p there exists a coordinate system such that: g µν takes its canonical form the first derivatives σ g µν all vanish the second derivatives ρ σ g µν cannot be made in general to all vanish the associated basis vectors constitute a local Lorentz frame. Notice: In Riemann normal coordinates, g µν at p looks, to first order, like the flat or Minkowski metric η µν. That is, in a small enough regions, the spacetime looks like flat or Minkowski space (local flatness theorem).

32 Proof: local flatness theorem g µ ν = xµ x µ Taylor expand both sides with x µ (p) = x µ (p) = 0 ν x gµν xν x µ = + ( ) x µ x µ x µ + 1 ( ) 2 x µ x µ 2 p x µ 1 x µ 1 x µ 2 2 p ( 1 3 x µ ) x µ 6 x µ 1 x µ 2 x µ 1 x µ 2 x µ 3 +, 3 p Thus to second order (g ) ( + g ) ( p p x + g ) p x x = + ( x x ( x x ) ( x x 2 x g x + p x x x g + x ) x x x g x p 3 x x x x g + 2 x x x 2 x x x g + x 2 x x x x g + x x x x g ) p x x.

33 Proof: local flatness theorem (g ) ( + g ) ( p p x + g ) p x x = + ( x x ( x x ) ( x x 2 x g x + p x x x g + x ) x x x g x p 3 x x x x g + 2 x x x 2 x x x g + x 2 x x x x g + x x x x g ) p x x. 16 numbers in ( x µ / x µ ) p to bring g µ ν (p) into a canonical form 40 numbers in ( 2 x µ / x µ 1 x µ 2 ) p to zero out the 40 components in σ g µ ν (p) 80 number in ( 3 x µ / x µ 1 x µ 2 x µ 3 ) p to zero out 80 of the 100 components in ρ σ g µ ν (p) Thus, the deviation from flatness is measured by 20 coordinate-independent degrees of freedom representing the second derivatives of the metric tensor field (Riemann curvature tensor).

34 Causality Initial-value problem or formulation: Given the appropriate initial data or state state of a system, the subsequent dynamical evolution of the system is uniquely determined. Causality: Future events are a consequence of past events. Fundamental principle: no signals can travel faster than the speed of light. Let (M, g µν ) be a spacetime, if a continuous choice of future and past is possible as one varies p in M, one says that M is time orientable. Lema: Let (M, g µν ) be time orientable, then there exists a smooth non-vanishing tim-elike vector field t µ on M.

35 Causality Causal curve: a curve γ that at every point p γ has a tangent t µ that is time-like or null. Causal future J + (S): Given any S M, the causal future of S is given by J + (S) M that can be reached from S by following a future-directed causal curves. Chronological future I + (S): Given any S M, the causal future of S is given by I + (S) M that can be reached from S by following a future-directed time-like curves. Notice: I + (S) J + (S) The causal past J (S) and chronological past I (S) are defined analogously.

36 Causality Achronal: A subset S M is called achronal if no two points in S are connected by a time-like curve. Future domain of dependence D + (S): Given a closed achronal set S, D + (S) is the set of points p M such that every past directed inextendible (goes on forever) causal curve through p intersects S. Future Cauchy horizon H + (S): The boundary of D + (S) (null surface). Notice: S D + (S) J + (S)

37 Causality Information at S is sufficient to predict the situation in p. Domain of dependence: D(S) = D + (S) D (S) Cauchy surface Σ: A closed achronal surface Σ is said to be a Cauchy surface if the domain of dependence is the entire manifold, i.e. D(Σ) = M A space-time (M, g µν ) which possesses a Cauchy surface is said to be globally hyperbolic.

38 Closed Timelike Curves Closed Timelike Curve: A forward directed curve that is everywhere time-like and intersects itself. Example: Consider the 2-dimensional spacetime (M, g µν ) with coordinates {t, x} with topology R S 1 ; that is, with coordinates (t, x) and (t, x + 1) identified, one can show that the metric ds 2 = cos (λ) dt 2 2 sin (λ) dt dx + cos (λ) dx 2 with λ = cot 1 t has closed time-like curves for t > 0 and a Cauchy horizon surface at t = 0.

39 Tensor Densities Levi-Civita symbol: +1 if µ 1 µ 2 µ n is an even permutation of 01 (n 1), ɛ µ1 µ 2 µ n = 1 if µ 1 µ 2 µ n is an odd permutation of 01 (n 1), 0 otherwise. It has the same components in any coordinate system It is not a tensor since it does not to change under coordinate transformations.

40 Tensor Densities Given some n n matrix M µ µ, the determinant M obeys ɛ µ 1 µ 2 µ n M = ɛ µ 1 µ 2 µ n M µ 1 µ 1 M µ 2 µ 2 M µn µ n. If M µ µ = x µ, x µ ɛ µ 1 µ = x µ 2 µ n x µ ɛ x µ1 x µ 2 µ 1 µ 2 µ n x µ 1 x µ 2 x µn x µ n Notice: It transforms almost as a tensor. Objects which transform in this way are known as tensor densities..

41 Tensor Densities g µ ν = x µ x ν g µν x µ x ν by taking the determinant in both sides one gets thus, g is not a tensor. x µ g(x µ ) = x µ 2 g(x µ ). Tensor density weight: The weight of a density is given by the power of the Jacobian. E.g. the Levi-Civita symbol is a tensor density of weight 1 and the determinant of the metric g is a scalar density of weight -2.

42 Levi-Civita tensor Given the Levi-Civita symbol, we can then define Levi-Civita tensor: ɛ µ1 µ 2 µ n = g ɛ µ1 µ 2 µ n. which will transform like a tensor.

43 Differential Forms Differential forms: A differential p-form is a (0, p) tensor which is completely antisymmetric. Examples: Scalars are 0-forms Dual vectors are 1-forms The electromagnetic tensor F µν is a 2-form ɛ µνρσ is a 4-form.

44 Differential Forms Λ p (M) is the space of all p-form fields over a manifold M. The number of linearly independent p-forms on an n-dimensional vector space is n!/(p!(n p)!). Thus, in a 4-dimensional spacetime there is one linearly independent 0-form, four 1-forms, six 2-forms, four 3-forms, and one 4-form. There are no p-forms for p > n.

45 Wedge Product: Given a p-form A and a q-form B, the wedge product A B as (A B) µ1 µ p+q = the result is a (p + q)-form. (p + q)! A [µ1 µ p! q! p B µp+1 µ p+q]. Example: (A B) µν = 2A [µ B ν] = A µ B ν A ν B µ. Notice: A B = ( 1) pq B A, so you can alter the order of a wedge product if you are careful with signs.

46 Exterior Derivative: (da) µ1 µ p+1 = (p + 1) [µ1 A µ2 µ p+1 ]. the result is a (p + 1)-form and thus a tensor. Example: (dφ) µ = µ φ. d(da) = 0, for any form A, a consequence that α β = β α. Property: d(ω η) = (dω) η + ( 1) p ω (dη) with ω a p-form and η a q-form.

47 Hodge duality: Hodge star operator on an n-dimensional manifold as a map from p-forms to (n p)-forms, ( A) µ1 µ n p = 1 p! ɛν 1 ν p µ1 µ n p A ν1 ν p, The Hodge dual does depend on the metric of the manifold If s is the number of minus signs in the eigenvalues of the metric, A = ( 1) s+p(n p) A, If A (n p) is an (n p)-form and B (p) is a p-form, (A (n p) B (p) ) R.

48 Cross Product If we restrict (A (n p) B (p) ) R. to the case of 3-dimensional Euclidean space, we get (U V ) i = ɛ i jk U j V k. which is the conventional cross product. It only exists in 3-dimensions because only in 3-dimensions do we have a map like this from two dual vectors to a third dual vector.

49 E&M revised µ F νµ = 4πJ ν [µ F νλ] = 0 thus df = 0. There must therefore be a one-form A µ such that F = da. The one-form A µ is the familiar vector potential of electromagnetism, Gauge invariance: The theory is invariant under A A + dλ for some scalar (zero-form) λ. The inhomogeneous Maxwell s equations are given by d( F ) = 4π( J),

50 Integration In ordinary calculus on R n the volume element d n x transforms as d n x x µ = x µ d n x. Identify thus d n x dx 0 dx n 1. dx 0 dx n 1 = 1 n! ɛ µ 1 µ n dx µ 1 dx µn, under a coordinate transformation, we have that ɛ µ1 µ n dx µ 1 x µ 1 µn x dx µn = ɛ µ1 µ n dx µ 1 dx µ n x µ 1 x µ n = x µ ɛ µ dx µ 1 µ 1 n dx µ n. x µ therefore volume element d n x transforms as a density, not a tensor.

51 Invariant Volume Element : g d n x = g dx 0 dx n 1 = g dx 0 dx (n 1) Integral: of a scalar function φ(x) in a n-dimensional manifold M I = φ(x) g d n x Stokes Theorem: Consider n-manifold M with boundary M, and an (n 1)-form ω on M, then dω = ω. M M