General Relativity: An Introduction for Physicists · Problem 17.1

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · linearised-general-relativity-metric-perturbation

Problem

In a region of spacetime with a weak gravitational field, there exist coordinates in which the metric takes the form

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

. Show that

h_{\mu\nu}

is not a tensor under a general coordinate transformation. Show further that, to first order in

h_{\mu\nu}

g^{\mu\nu} = \eta^{\mu\nu} - h^{\mu\nu}

where

h^{\mu\nu} = \eta^{\mu\rho}\eta^{\nu\sigma}h_{\rho\sigma}