Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 4 / c / i

Table of contents
- Solution i

Solution

The horizon generator must be null at

r = r_{+}

. Substitution of

ξ = \partial_{v} + Ω_{H} \partial_{χ}

in the Kerr black hole metric gives

Ω_{H} = \frac{a}{r _{+}^{2} + a ^{2}} = \frac{a}{2 M r _{+}} .

The surface gravity and Hawking temperature are

κ = \frac{r _{+} - r _{-}}{2 ( r _{+}^{2} + a ^{2} )}, T_{H} = \frac{κ}{2 π} = \frac{r _{+} - r _{-}}{4 π ( r _{+}^{2} + a ^{2} )} = \frac{M ^{2} - a ^{2}}{4 π M r _{+}} .

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