General Relativity: An Introduction for Physicists · Problem 17.19

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · gravitational-potentials-stationary-source-angular-momentum

Problem

Use your answer to Exercise 17.18 to show that, for a stationary non-relativistic source, the gravitational scalar and vector potentials respectively are given to leading order in 1/∣x⃗∣1/|\vec{x}| by Φg(x⃗)=−GM∣x⃗∣andA⃗g(x⃗)=−2Gc2∣x⃗∣3J⃗×x⃗,\Phi_\mathrm{g}(\vec{x}) = -\frac{GM}{|\vec{x}|} \qquad \text{and} \qquad \vec{A}_\mathrm{g}(\vec{x}) = -\frac{2G}{c^2|\vec{x}|^3}\vec{J}\times\vec{x}, where J⃗=∫(y⃗×p⃗) d3y⃗\vec{J} = \int(\vec{y}\times\vec{p})\, d^3\vec{y} is the total angular momentum vector of the source. Show further that these expressions are exact in the linear theory for a spherically symmetric source.

Related theory

Browse the problem library