General Relativity: An Introduction for Physicists · Problem 3.6

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · connection-coefficients-diagonal-metric

Problem

For a diagonal metric

g_{ab}

, show that the connection coefficients are given by (with

a \neq b \neq c

and no summation over repeated indices)

\Gamma^a{}_{bc} = 0, \qquad \Gamma^b{}_{aa} = -\frac{1}{2g_{bb}} \partial_b g_{aa},

\Gamma^a{}_{ba} = \Gamma^a{}_{ab} = \partial_b\!\left(\ln\sqrt{|g_{aa}|}\right), \qquad \Gamma^a{}_{aa} = \partial_a\!\left(\ln\sqrt{|g_{aa}|}\right).