General Relativity: An Introduction for Physicists · Problem 17.21

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · second-order-einstein-tensor-gauge-invariance

Problem

g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}

, show that the terms in the Einstein tensor that are second order in

h_{\mu\nu}

are given by

G^{(2)}_{\mu\nu} = R^{(2)}_{\mu\nu} - \tfrac{1}{2}\eta_{\mu\nu}R^{(2)} - \tfrac{1}{2}h_{\mu\nu}R^{(1)} + \tfrac{1}{2}\eta_{\mu\nu}h^{\rho\sigma}R^{(1)}_{\rho\sigma},

where

R^{(2)}_{\mu\nu}

denotes the terms in the Ricci tensor that are second order in

h_{\mu\nu}

, and

R^{(1)}

and

R^{(2)}

denote the terms in the Ricci scalar that are first and second order in

h_{\mu\nu}

respectively. Show further that this quantity is not invariant under a gauge transformation of the form (17.5).