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

Solution

Stationarity makes

\partial_{t} α = 0

, while

\partial_{r} α = \frac{M}{( r + M ) ^{2}} .

The left side of the Bona--Masso slicing condition is therefore

- β^{r} \partial_{r} α = - \frac{M ^{2} r}{( r + M ) ^{4}} .

Its right side is

- α^{2} f (α) K = - \frac{M r ^{2}}{( r + M ) ^{4}} f (α) .

Equality requires

f = \frac{M}{r} .

Since

α = r / (r + M)

f (α) = \frac{1 - α}{α} .

Consequently

lim_{α \to 1} f (α) = 0

, as expected in the asymptotically flat region

r \to \infty

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