General Relativity: An Introduction for Physicists · Problem 7.14

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

Physics · covariant-curvature-tensor-symmetries

Problem

Show that the covariant components of the curvature tensor are given by

R_{abcd} = \tfrac{1}{2}(\partial_d\partial_a g_{bc} - \partial_d\partial_b g_{ac} + \partial_c\partial_b g_{ad} - \partial_c\partial_a g_{bd}) - g_{ef}(\Gamma^e{}_{ac}\Gamma^f{}_{bd} - \Gamma^e{}_{ad}\Gamma^f{}_{bc}),

and hence verify its symmetries. Show further that, for an

N

-dimensional manifold, the number of independent components is

N^2(N^2-1)/12