The horizon generator must be null at $r=r_+$. Substitution of $\xi=\partial_v+\Omega_H\partial_\chi$ in the <Kerr black hole> metric gives
$$
\boxed{\Omega_H=\frac{a}{r_+^2+a^2}
=\frac{a}{2Mr_+}}.
$$
The <surface gravity> and <Hawking temperature> are
$$
\kappa=\frac{r_+-r_-}{2(r_+^2+a^2)},
\qquad
\boxed{T_H=\frac{\kappa}{2\pi}
=\frac{r_+-r_-}{4\pi(r_+^2+a^2)}
=\frac{\sqrt{M^2-a^2}}{4\pi Mr_+}}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-311/4/c/i/solution