General Relativity: An Introduction for Physicists · Problem 14.13

The Friedmann–Robertson–Walker geometry — Hobson, Efstathiou & Lasenby

Physics · frw-connection-ricci-tensor

Problem

In the comoving coordinates

[x^\mu] = (t, r, \theta, \phi)

, the FRW metric takes the form

ds^2 = c^2\,dt^2 - R^2(t)\left[\frac{dr^2}{1-kr^2} + r^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right].

Using the 'Lagrangian' method, or otherwise, calculate the corresponding connection coefficients

\Gamma^\sigma{}_{\mu\nu}

. Hence calculate the non-zero elements of the Ricci tensor

R_{\mu\nu}