Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 4 / d

Table of contents
- Solution d

Solution

On a constant-

t

asymptotically flat hypersurface, transform the large-

r

spatial metric to asymptotically Cartesian coordinates. Its leading perturbation is

h_{ij} = δ_{ij} + q_{ij} + O (r^{- 2}), q_{ij} = \frac{2 M}{r} n_{i} n_{j} = \frac{2 M x _{i} x _{j}}{r ^{3}} .

The rotation parameter

a

first affects the spatial metric at orders that vanish in the Arnowitt-Deser-Misner energy surface limit. Direct differentiation gives

\partial_{j} q_{ij} = \frac{2 M}{r ^{2}} n_{i}, \partial_{i} q_{jj} = - \frac{2 M}{r ^{2}} n_{i} .

Hence

n^{i} (\partial_{j} h_{ij} - \partial_{i} h_{jj}) = \frac{4 M}{r ^{2}} + O (r^{- 3}) .

Using

\int_{S_{r}^{2}} d A = 4 π r^{2}

in the ADM surface integral yields

E_{A D M} = \frac{1}{16 π} (4 π r^{2}) \frac{4 M}{r ^{2}} = M .

Thus the Kerr parameter $M$ is its total mass in the rest frame.

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