General Relativity: An Introduction for Physicists · Problem 17.18

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · metric-perturbation-stationary-source-multipole

Problem

For a non-relativistic stationary source, show that, in centre-of-momentum coordinates, hˉ00(x⃗)=−4GMc2∣x⃗∣+O ⁣(1∣x⃗∣2),\bar{h}^{00}(\vec{x}) = -\frac{4GM}{c^2|\vec{x}|} + \mathcal{O}\!\left(\frac{1}{|\vec{x}|^2}\right), hˉ0i(x⃗)=2Gc3∣x⃗∣3 xjJij+O ⁣(1∣x⃗∣3),\bar{h}^{0i}(\vec{x}) = \frac{2G}{c^3|\vec{x}|^3}\, x_j J^{ij} + \mathcal{O}\!\left(\frac{1}{|\vec{x}|^3}\right), hˉij(x⃗)=0,\bar{h}^{ij}(\vec{x}) = 0, where the quantities MM and JijJ^{ij} are given by M=∫Vρ(y⃗) d3y⃗andJij=∫V[yipj(y⃗)−yjpi(y⃗)]d3y⃗,M = \int_V \rho(\vec{y})\, d^3\vec{y} \qquad \text{and} \qquad J^{ij} = \int_V \left[y^i p^j(\vec{y}) - y^j p^i(\vec{y})\right] d^3\vec{y}, in which ρ(y⃗)\rho(\vec{y}) is the proper density distribution of the source and pi(y⃗)=ρ(y⃗)ui(y⃗)p^i(\vec{y}) = \rho(\vec{y})u^i(\vec{y}) is the momentum density distribution of the source. Give a physical interpretation of JijJ^{ij}. Hint: You will find your answer to Exercise 17.17 useful.
Intuitive explanation

The key is solving the linearized Einstein equations in the weak-field limit using the retarded Green's function, then extracting the leading multipole moments of the source. For a stationary source in the center-of-momentum frame, the metric perturbations hˉμν\bar{h}^{\mu\nu} depend only on the mass MM (monopole) and angular momentum tensor JijJ^{ij} (dipole); the 0000 component falls off as 1/r1/r because gravity is long-range, the 0i0i components vanish faster due to momentum conservation, and spatial components vanish to leading order because there's no time-varying quadrupole. The quantity JijJ^{ij} is the source's total angular momentum tensor—it encodes how the source's internal momentum distribution creates frame-dragging effects in spacetime.

Intuitive explanation

The key is solving the linearized Einstein equations in the weak-field limit using the retarded Green's function, then extracting the leading multipole moments of the source. For a stationary source in the center-of-momentum frame, the metric perturbations hˉμν\bar{h}^{\mu\nu} depend only on the mass MM (monopole) and angular momentum tensor JijJ^{ij} (dipole); the 0000 component falls off as 1/r1/r because gravity is long-range, the 0i0i components vanish faster due to momentum conservation, and spatial components vanish to leading order because there's no time-varying quadrupole. The quantity JijJ^{ij} is the source's total angular momentum tensor—it encodes how the source's internal momentum distribution creates frame-dragging effects in spacetime.

Related theory

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