General Relativity: An Introduction for Physicists · Problem 17.7

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · plane-gravity-waves-riemann-ricci-tensor

Problem

For a plane gravity wave of the form

h_{\mu\nu} = A_{\mu\nu}\exp(ik_{\lambda}x^{\lambda})

, show that the linearised Riemann tensor is given by

R_{\sigma\mu\nu\rho} = \tfrac{1}{2}(k_{\nu}k_{\sigma}h_{\mu\rho} + k_{\rho}k_{\mu}h_{\sigma\nu} - k_{\nu}k_{\mu}h_{\sigma\rho} - k_{\rho}k_{\sigma}h_{\mu\nu}).

Hence show that the linearised Ricci tensor is given by

R_{\mu\nu} = \tfrac{1}{2}(k_{\nu}w_{\mu} + k_{\mu}w_{\nu} - k^2 h_{\mu\nu}),

where

k^2 = k_{\rho}k^{\rho}

and

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

. Hence show that the linearised Einstein field equations require that

k^2 h_{\mu\nu} = k_{\nu}w_{\mu} + k_{\mu}w_{\nu}.