General Relativity: An Introduction for Physicists · Problem 17.9

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · plane-gravity-waves-null-wavevector-eigenvector

Problem

From your answer to Exercise 17.7, show that for

k^2 = 0

one requires

k^{\rho}\bar{h}_{\mu\rho} = 0

. Hence show that the wavevector

k^{\rho}

is an eigenvector of the Riemann tensor in the sense that

R_{\sigma\mu\nu\rho}k^{\rho} = 0