Tutorials Geometry of General Relativity -School of Mathematics PDF

Preview

Summary

Geometry of General Relativity -School of Mathematics Lectured by James Lucietti. Introduction to the mathematical theory of general relativity. Healthy intro of differential geometry given....

Description

Geometry of General Relativity 2019

Workshop Sheet 1

Geometry of General Relativity 2019 Workshop sheet 1 Tangent vectors, vector fields and commutator The purpose of this workshop is to work with tangent vectors, vector fields and commutators both abstractly and in explicit examples. 1. Let Xp 2 Tp (M) and f, g 2 C 1 (M). Using the definition of a tangent vector given in lectures prove the Leibniz rule: Xp (f g) = Xp (f )g(p) + f (p)Xp (g) 2. Show that the components of a tangent vector Xp 2 Tp (M) relative to the coordinate basis defined by a chart φ = (xi ) are given by Xpi = Xp (xi ). 3. Let φ = (xi ) and φ0 = (x0i ) be two charts containing p 2 M. Relate the coordinate basis vectors ei = (∂/∂xi )p and ei0 = (∂/∂x0i )p . Now let X 2 Tp (M) and prove X ✓ ∂x0j ◆ 0j X = Xi , ∂xi φ(p) j

where by standard abuse of notation x0i (x) denotes the transition function x0i φ1 . 4. Let f 2 C 1 (M) and X, Y, Z be vector fields on smooth manifold M. Prove: (a) [X, f Y ] = f [X, Y ] + X(f )Y (Leibniz property) (b) [X, [Y, Z]] + [Y, [Z, X ]] + [Z, [X, Y ]] = 0 (Jacobi identity) 5. Let n be a nonnegative integer and consider the vector fields Xn = xn ∂x∂ on R. (a) Show that [Xm , Xn ] = (n  m)Xm+n1 . (b) Find the integral curves of Xn . For what values of n are they complete? 6. Consider the following vector fields on R2 X=x

∂ ∂ +y , ∂x ∂y

Y =x

∂ ∂ y ∂y ∂x

(a) Calculate the integral curves of X and Y through any point p = (a, b) 6= (0, 0). (b) Compute [X, Y ] by acting on an arbitrary function f . (c) Write X, Y in the coordinate basis defined by the polar coordinates (r, θ) given by (x, y) = (r cos θ, r sin θ). Hence give a quick proof for parts (a), (b). Question 6 is to be handed in on Friday 8 February at the lecture. Feedback will be returned on Friday 15 February at the workshop.

1

Geometry of General Relativity 2019

Workshop Sheet 2

Geometry of General Relativity 2019 Workshop sheet 2 Tensors and metric tensors The purpose of this workshop is to work with the definition of tensors and metric tensors. 1. Let T be a (1, 1) tensor over a vector space V . Let {ea} be a basis for V and {f a} be its dual basis. Derive the transformation law for its components under a change of basis f 0a = Aab f b , e0a = (A1 )baeb . Use this to show that: (a) The components of the Kronecker delta tensor δ are the same in any basis. (b) The contraction of T is basis independent. 2. Show that a (1, 1) tensor T over V is equivalent to a linear map V ⇤ → V ⇤ . Similarly, show that T is also equivalent to a linear map V → V . ∼ (V ⇤ )⇤ .] [Hint: recall the natural isomorphism V = 3. Let S be a (2, 3) tensor. Define a (1, 2) tensor T by the contraction, T (λ, X, Y ) = S(λ, f a, X, Y, ea), where λ is a covector and X, Y are vectors, {f a}, {ea} are dual bases. Show this definition is basis independent. Write out the components of T in terms of those of S. Write down all possible (1, 2) tensors that are contraction of S . 4. Let g be a Lorentzian metric on V and X, Y ∈ V orthogonal nonzero vectors. Prove: (i) If X is timelike then Y is spacelike; (ii) if X is null then Y is null and collinear to X, or spacelike; (iii) if X is spacelike then Y may be of any type. [Hint: choose an orthonormal basis of g and use Cauchy-Schwarz inequality.] 5. Let (M, gˆ) be a Riemannian manifold and suppose Z is a nowhere vanishing vector field on M. Show that g(X, Y ) = gˆ(X, Y ) −

2ˆ g (X, Z )ˆ g (Y, Z) gˆ(Z, Z)

where X, Y are vector fields, defines a Lorentzian metric on M. [Hint: consider an orthonormal basis for gˆ such that Z is as simple as possible.] 6. Consider a (2, 2) tensor T over V and λ, µ ∈ V ⇤ and X, Y ∈ V . Let {ea} be a basis of V and {f a} its dual basis. (a) Write down T (λ, µ, X, Y ) in terms of the components of T, λ, µ, X, Y . Hence prove that {ea ⊗ eb ⊗ f c ⊗ f d } is a basis for type (2, 2) tensors. (b) Derive the transformation law for the components of T under a change of basis f 0a = Aab f b , e0a = (A1 )b aeb . (c) Define a (2, 2) tensor by T (λ, µ, X, Y ) = λ(X )µ(Y ) − λ(Y )µ(X). Find all the contractions of T and express them in terms of the Kronecker delta tensor δ. Question 6 is to be handed in on Friday 1 March at the lecture. Feedback will be returned on Friday 8 March at the workshop. 1

Geometry of General Relativity 2019

Workshop Sheet 3

Geometry of General Relativity 2019 Workshop sheet 3 Affine connections and curvature The purpose of this workshop is to work with the definitions of connections and curvature. 1. Consider R2 with Cartesian coordinates (x, y) and def ine a connection in the coor∂ ∂ by ri ej = 0, i.e., Γij k = 0. , ey = ∂y dinate basis ex = ∂x ∂ ∂ where (r, θ) are (a) Find ri ej and Γji k for the coordinate basis er = ∂r , eθ = ∂θ polar coordinates. [Hint: relate the coordinate bases and use axioms for r.]

(b) Show that the Riemann tensor vanishes. Explain why this result was expected. 2. Prove that under a general basis change ea0 = (A1 )b aeb and f 0a = Aab f b , Γ0abc = Aad (A1 )g b (A1 )hc Γdgh + Aad (A1 )gc eg ((A1 )db ) . Specialise this to coordinate bases. Hence check your answer to 1 (a). 3. Using the Leibniz rule define rT for a (1, 1) tensor field T and show its components rc T ab = ec (T ab ) + ΓadcT db  ΓdbcT ad . Deduce the Kronecker delta tensor is covariantly constant rδ = 0. 4. Let r be a torsionless connection. By working in normal coordinates prove Rabcd = 32 (Ra(bc)d  Ra(bd)c ) , r[c Ra|b|de] = 0 . 5. Let r be a torsionless connection. Prove the Ricci identity for covectors: rarb λc  r b raλc = Rdcab λd [Hint: consider rarb (X c λc ) and use the Ricci identity for a vector f ield X a.] Hence prove Ra [bcd] = 0 by considering the covector field λc = rc f . ¯ be connections on a manifold M and define H(X, Y ) = r ¯ X Y  rX Y 6. Let r and r where X, Y are vector fields. a . How (a) Show that H defines a (1, 2) tensor field and find its components Hbc many connections are there on a given manifold?

(b) Show that if r is torsionless T¯ abc = 2H a[bc] ¯ abcd = Rabcd + 2r[c H ad]b + 2H e[d|b| H a R c]e ¯ where T¯ and R¯ are the torsion and Riemann curvature of r. Question 6 is to be handed in on Friday 15th March at the lecture. Feedback will be returned on Friday 22th March at the workshop. 1

Geometry of General Relativity 2019

Workshop Sheet 4

Geometry of General Relativity 2019 Workshop sheet 4 Riemannian geometry The purpose of this workshop is to work with metric tensors and the Levi-Civita connection both explicitly and more abstractly. 1. Consider R2 in polar coordinates (r, θ) with the metric g = dr 2 + r 2 dθ 2 . Show that r the Levi-Civita connection has non-vanishing components Γθθ = r, Γθrθ = r −1 . Deduce the Riemann tensor vanishes (recall Q1 of Workshop 3). Explain why this result was expected. 2. Prove the fundamental theorem of (pseudo-)Riemannian geometry by considering ri gj k + rj gki  rk gij . Verify directly that ri gj k = 0 using the Christoffel symbols. p 3. Show that for the Levi-Civita connection Γji i = ∂j log | det g| where det g is the dep terminant of the matrix of components gij . Deduce, ri X i = p 1 ∂i ( | det g|X i ). | det g|

[Hint: use the differential identity TrA−1 ∂i A = ∂i log | det A|, valid for any real invertible symmetric matrix A that is a function of coordinates (xi ).] 4. Consider the Levi-Civita connection of a metric g. Derive the Ricci identity for covectors from the Ricci identity for vectors by lowering the index Xa = gab X b . 5. Consider a pseudo-Riemannian manifold (M, g) of dimension n. (a) Suppose Rabcd = K(gac gbd  gadgbc ). Show that if n  3, then K is a constant. What happens if n = 2? [Hint: consider the contracted Bianchi identity.] (b) Suppose n = 2. Prove that Rabcd = 21R(gacgbd  gad gbc ). [Hint: find the dimension of the space of (0, 4) tensors with the same symmetries as Rabcd .] 6. Consider the sphere S 2 in polar coordinates (θ, φ) equipped with the unit round metric g = dθ 2 + sin2 θdφ2 . (a) Show that the non-vanishing components of the Levi-Civita connection are φ Γθφφ =  cos θ sin θ and Γθφ = cot θ.

(b) Show that k = sin2 θφ˙ is constant along a geodesic curve t 7! (θ (t), φ(t)). 2 Deduce θ˙ 2 + sink2 θ is also a constant. (c) Compute Rθφθφ and hence show that Rab = gab .

Question 6 is to be handed in on Friday 29th March at the lecture. Feedback will be returned on Friday 5th April at the workshop.

1

Geometry of General Relativity 2019

Workshop Sheet 5

Geometry of General Relativity 2019 Workshop sheet 5 Gravitational waves The purpose of this workshop is to show that the Einstein equations predict the existence of gravitational waves. Consider a spacetime (R4 , g) with cartesian coordinates xµ , µ = 0, 1, 2, 3, and a metric gµν = ⌘µν + hµν where ⌘µν = diag(−1, 1, 1, 1) is the Minkowski metric and the components hµν and @ρ hµν are small compared to 1. Thus our spacetime is a small perturbation of Minkowski spacetime. In all subsequent calculations neglect any terms which are of quadratic order in hµν and @ρ hµν . All indices are raised and lowered with respect to ⌘µν (except for g µν ). 1. (a) Show the inverse metric is g µν = ⌘ µν − hµν and the Christoffel symbols are 1 Γµνρ = ⌘ µσ (@ν hσρ + @ρ hσν − @σ hνρ ) 2 Hence show that the linearised Ricci tensor is 1 1 Rµν = − @ 2 hµν + @ ρ @(µ hν)ρ − @µ @ν h 2 2 where @ 2 = @ ρ @ρ and h = hµµ . (b) Define h¯µν = hµν − 12 h⌘µν and show that the linearised Einstein equation is 1 ¯ 1 ρ σ¯ ρ ¯ − @ 2h µν + @ @(µhν)ρ − ⌘µν @ @ hρσ = 8⇡GTµν 2 2 2. The diffeomorphism invariance of GR implies that the linearised Einstein equations are invariant under the transformations h → h + Lξ ⌘ where ⇠ is a vector field. (a) Verify the linearised Einstein equation is invariant under h → h + Lξ ⌘. (b) Argue this can be used to fix @ ρh¯ρµ = 0; what transformations preserve this condition? Deduce that the linearised Einstein equations reduces to @ 2 ¯hµν = −16⇡GTµν 3. The linearised vacuum Einstein equation reduces to the wave equation @ 2h¯µν = 0 where @ ρ ¯hρµ = 0. Consider a plane wave solution of the form ¯hµν = Re(✏µν eikµ xµ ) where the wave vector k µ and polarisation ✏µν are constant. (a) Show that wave vector k µ is null and the polarisation is transverse ✏µν k ν = 0. (b) Argue that gravitational waves only two have independent ‘physical’ polarisations. You will need to consider the residual transformations h → h + Lξ ⌘.

1...