General Relativity: An Introduction for Physicists · Problem 17.15

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · newtonian-gravitational-potential-multipole-expansion

Problem

In Newtonian gravity, the gravitational potential Φ(x⃗)\Phi(\vec{x}) produced by some density distribution ρ(x⃗)\rho(\vec{x}) is given by Φ(x⃗)=−G∫Vρ(y⃗)∣x⃗−y⃗∣ d3y⃗,\Phi(\vec{x}) = -G\int_V \frac{\rho(\vec{y})}{|\vec{x} - \vec{y}|}\, d^3\vec{y}, where the integral extends over the volume of the distribution. Show that 1∣x⃗−y⃗∣=1∣x⃗∣+x⃗⋅y⃗∣x⃗∣3+O ⁣(1∣x⃗∣3).\frac{1}{|\vec{x} - \vec{y}|} = \frac{1}{|\vec{x}|} + \frac{\vec{x}\cdot\vec{y}}{|\vec{x}|^3} + \mathcal{O}\!\left(\frac{1}{|\vec{x}|^3}\right). Hence show that the gravitational potential can be written as Φ(x⃗)=−GM∣x⃗∣−Gd⃗⋅x⃗∣x⃗∣3+O ⁣(1∣x⃗∣3),\Phi(\vec{x}) = -\frac{GM}{|\vec{x}|} - \frac{G\vec{d}\cdot\vec{x}}{|\vec{x}|^3} + \mathcal{O}\!\left(\frac{1}{|\vec{x}|^3}\right), where M=∫Vρ(y⃗) d3y⃗andd⃗=∫Vρ(y⃗) y⃗ d3y⃗.M = \int_V \rho(\vec{y})\, d^3\vec{y} \qquad \text{and} \qquad \vec{d} = \int_V \rho(\vec{y})\,\vec{y}\, d^3\vec{y}.

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