General Relativity: An Introduction for Physicists · Problem 7.17

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

Physics · killing-vectors-curvature

Problem

Show that any Killing vector

v^a

, as defined in Exercise 4.11, satisfies the relations

\nabla_c\nabla_b v^a = R^a{}_{bcd}v^d,

v^a\nabla_a R = 0.