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 Incoming links: Slowly rotating weak-field metric

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 3 / b / iii / Solution Created 2026-09-28 Updated 2026-10-03

For

r ≫ R

, the multipole expansion is

\frac{1}{∣ r - r ^{'} ∣} = \frac{1}{r} + \frac{r \cdot r ^{'}}{r ^{3}} + O (R^{2} / r^{3}) .

The center of mass condition removes the mass dipole. Axisymmetry makes the mixed second moments vanish and gives

\int ρ x^{'2} = \int ρ y^{'2}

. Undoing trace reversal therefore yields

d s^{2} = - (1 - \frac{2 m}{r}) d t^{2} + (1 + \frac{2 m}{r}) d r^{2} + \frac{4 ma}{r ^{3}} (y d x - x d y) d t,

where

m = \int ρ d^{3} r^{'}, ma = J_{z} = Ω \int ρ (x^{'2} + y^{'2}) d^{3} r^{'} .

Thus $a = J_{z} / m$ is the specific angular momentum, and the cross term is the dipole part of the slowly rotating weak-field metric.

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