General Relativity: An Introduction for Physicists · Problem 7.8

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

Physics · weak-field-equation-of-motion

Problem

Consider the limit of a weak gravitational field in a coordinate system in which

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

, with

|h_{\mu\nu}| \ll 1

, and

\partial_0 g_{\mu\nu} = 0

. Keeping only terms that are first order in

v/c

, show that the equation of motion for a slowly moving test particle takes the form

\frac{d^2 x^i}{dt^2} \approx -\tfrac{1}{2}c^2 \delta^{ij}\partial_j h_{00} + c\delta^{ik}(\partial_j h_{0k} - \partial_k h_{0j})v^j.

Give a physical interpretation of the second term on the right-hand side.