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

Solution

The spatial metric is stationary and only

β^{r} = b (r) = M r / (r + M)^{2}

is nonzero. The evolution equation therefore says

2 α K_{ij} = (L_{β} γ)_{ij},

the Lie derivative of the spatial metric along the shift vector. For the radial component,

2 α K_{rr} = b \partial_{r} γ_{rr} + 2 γ_{rr} \partial_{r} b = - \frac{2 M}{r ( r + M )},

K_{rr} = - \frac{M}{r ^{2}} .

For the angular components,

2 α K_{θθ} = b \partial_{r} (r + M)^{2} = \frac{2 M r}{r + M},

K_{θθ} = M, K_{ϕϕ} = M sin^{2} θ .

All off-diagonal components vanish.

Contracting with the inverse spatial metric gives the mean curvature

K = γ^{rr} K_{rr} + γ^{θθ} K_{θθ} + γ^{ϕϕ} K_{ϕϕ} = - \frac{M}{( r + M ) ^{2}} + \frac{M}{( r + M ) ^{2}} + \frac{M}{( r + M ) ^{2}} .

Thus

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

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