General Relativity: An Introduction for Physicists · Problem 7.13

The equivalence principle and spacetime curvature — Hobson, Efstathiou & Lasenby

Physics · frw-spacetime-connection-ricci-einstein

Problem

A spacetime has the metric

ds^2 = c^2\,dt^2 - a^2(t)(dx^2 + dy^2 + dz^2).

Show that the only non-zero connection coefficients are

\Gamma^0{}_{11} = \Gamma^0{}_{22} = \Gamma^0{}_{33} = a\dot{a} \qquad \text{and} \qquad \Gamma^1{}_{10} = \Gamma^2{}_{20} = \Gamma^3{}_{30} = \dot{a}/a.

Deduce that particles may be at rest in such a spacetime and that for such particles the coordinate

t

is their proper time. Show further that the non-zero components of the Ricci tensor are

R_{00} = 3\ddot{a}/a \qquad \text{and} \qquad R_{11} = R_{22} = R_{33} = -a\ddot{a} - 2\dot{a}^2.

Hence show that the 00-component of the Einstein tensor is

G_{00} = -3\dot{a}^2/a^2