Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 357 / 2 / b

Table of contents
- Solution b

Solution

With

R = r + M

and

d R = d r

,

1 - \frac{2 M}{R} = \frac{r - M}{r + M}, \frac{R ^{2}}{( R - M ) ^{2}} = (\frac{r + M}{r})^{2} .

Defining the conformal factor by

ψ^{4} = (\frac{r + M}{r})^{2},

also gives

R^{2} = ψ^{4} r^{2}

. Hence

d s^{2} = - \frac{r - M}{r + M} d t^{2} + \frac{2 M}{r} d t d r + ψ^{4} (d r^{2} + r^{2} d Ω^{2}) .

The constant-time spatial metric is therefore conformally flat.

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