Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 38E / c / Solution

{c}

Boost a Schwarzschild mass with speed $v$ and hold $p=m\gamma$ fixed as $v\to1$. Away from the null plane $x=t$, the isotropic-coordinate expression tends to
$$
ds^2=ds_{\rm Minkowski}^2+\frac{4p}{|x-t|}(dt-dx)^2.
$$
The complete distributional limit on the null plane requires the additional coordinate treatment that produces the Aichelburg-Sexl shock wave.


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 Incoming links: Ultrarelativistic limit of the boosted Schwarzschild metric away from the null plane