Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 2 / c

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- Solution c

Solution

Substituting the two differential coordinate transformations cancels every coefficient singular as

Δ^{- 1}

, so the ingoing Kerr-like coordinates are regular at

r = r_{+}

. The inverse metric applied to the normal covector

d r

has

g^{ab} (d r)_{b} = \frac{r ^{2} + a ^{2}}{Σ} \partial_{v}^{a} + \frac{a}{Σ} \partial_{χ}^{a} + \frac{Δ}{Σ} \partial_{r}^{a} .

Its norm is

g^{rr} = Δ/Σ

, which vanishes at

r = r_{+}

. Thus that level set is a null hypersurface. On it the raised normal is proportional to

\partial_{v} + \frac{a}{r _{+}^{2} + a ^{2}} \partial_{χ} .

Since the transformed stationary and axial Killing fields are

K = \partial_{v}

and

m = \partial_{χ}

, the horizon is a Killing horizon generated by

ξ = K + Ω_{H} m, Ω_{H} = \frac{a}{r _{+}^{2} + a ^{2}} .

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