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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 37D / c / Solution

The station's proper rest mass is

M_{st} = 4 π r_{st}^{2} ρ .

A static local energy

d E_{loc}

at radius

r_{st}

contributes the redshifted energy

d E_{\infty} = 1 - \frac{2 M}{r _{st}} d E_{loc}

to the mass measured at infinity. To first order in

M_{st} / M

, the exterior mass parameter is therefore

M_{out} = M + 4 π r_{st}^{2} ρ 1 - \frac{2 M}{r _{st}} .

Terms quadratic in

ρ

describe the shell's self-gravity and are neglected.

The exterior of the shell is vacuum and spherically symmetric. The Birkhoff theorem therefore makes it a Schwarzschild spacetime with mass

M_{out}

:

d s^{2} = - (1 - \frac{2 M _{out}}{r}) d t^{2} + (1 - \frac{2 M _{out}}{r})^{- 1} d r^{2} + r^{2} d Ω^{2}, r > r_{st} .

This is the redshifted gravitational mass of a static thin shell.

Solved by gpt-5.6-sol high.

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