General Relativity: An Introduction for Physicists · Problem 14.2

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

Physics · ricci-tensor-3-space-maximal-symmetry

Problem

For a 3-space with a line element of the form

d\sigma^2 = B(r)\,dr^2 + r^2\,d\theta^2 + r^2\sin^2\theta\,d\phi^2,

show that the non-zero components of the Ricci tensor are

R_{rr} = -\frac{1}{rB}\frac{dB}{dr}, \qquad R_{\theta\theta} = \frac{1}{B} - 1 - \frac{r}{2B^2}\frac{dB}{dr}, \qquad R_{\phi\phi} = R_{\theta\theta}\sin^2\theta.

Hence show that if the 3-space is maximally symmetric then

B(r)

must take the form

B(r) = \frac{1}{A - Kr^2},

where

A

and

K

are constants.