General Relativity: An Introduction for Physicists · Problem 7.23

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

Physics · newtonian-tidal-forces

Problem

In Newtonian gravity, consider two nearby particles with trajectories

x^i(t)

and

\bar{x}^i(t)

(i = 1, 2, 3)

respectively in Cartesian coordinates. Show that the components of the separation vector

\zeta^i = x^i - \bar{x}^i

evolve as

\frac{d^2\zeta^i}{dt^2} = -\left(\frac{\partial^2\Phi}{\partial x^i \partial x^j}\right)\zeta^j,

where

\Phi

is the Newtonian gravitational potential.