Exercises on General Relativity and Cosmology

Physikalisches Institut Exercise 07

Universit¨at Bonn 25 May 2011

Theoretische Physik SS 2011

Exercises on General Relativity and Cosmology

Priv. Doz. Dr. S. F¨orste

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Due 1 June 2011

Exercise 7.1: Riemann Tensor (14 credit s)

The Christoffel symbols arenottensor, and thus are not suitable to describe the physics of curved geometry in a coordinate-invariant way. The only tensor can be constructed from the metric and its first and second derivatives is the Riemann tensor:

R^λ_µνκ ≡∂_νΓ^λ_µκ−∂_κΓ^λ_µν + Γ^η_µκΓ^λ_ην−Γ^η_µνΓ^λ_ηκ

Through self-contractions we get theRicci tensorRµκ≡R^λµλκand thecurvature scalar R≡g^µκR_µκ.

(a) Prove the following identity:

[D_ν, D_κ]a_η =R^λ_ηνκa_λ

(3 credit s) (b) Using the metric, the Riemann tensor can be written in the form:

R_λµνκ =g_ληR^λ_µνκ Check its symmetry properties:

Rλµνκ =−Rλµκν

R_λµνκ = +R_νκλµ

. (3 credit s)

(c) Calculate the components of R^l_mnk , R_mk and the curvature scalar R for a space (θ, φ) and metricg_mn= diag(a², a² sin²θ) ie. a 2-sphere. (3 credit s) (d) Show that for a general two dimensional space, the Riemann tensor takes the form:

R_abcd =R g_a[cg_d]b

. (3 credit s)

Hint: You may use the fact that the number of independent components of a Riemann tensor in D-dimensions is D²(D²−1)/12

(e) What is the curvature of a circle with fixed radiusR? (2 credit s) 1

Exercise 7.2: Bianchi identities (6 credit s)

(a) Verify the Bianchi identities:

R_λµνκ;η+R_λµην;κ+R_λµκη;ν = 0

where, the covariant derivative of a generic tensor is given by (extending the results of ex-5.1(b)):

X^µ..._ν...;ρ=∂_ρX^µ..._ν...+ Γ^µ_ρσX^σ..._ν...+...−Γ^σ_νρX^µ..._σ...−...

(3 credit s) Hint: You may use locally inertial (Riemann normal) coordinates

(b) Contract indices in the above identities to arrive at:

R^µν− 1 2g^µνR

;µ

= 0

(3 credit s)

2