# Kerr black hole evaporation time question

Is there any closed (even if complicated) formula por the Hawking evaporation time for a Kerr black hole (and more general black holes) just like the one $$\begin{equation} t_e=\dfrac{5120\pi G^2M_0^3}{\hbar c^4} \end{equation}$$ for Schwarzschild black hole? Any reference about those? I am writing a black hole vademecum document I will post in a weeks worldwide and I need those formulae... Remark: I presume that, in the Kerr black hole formula $$T_{BH}(Kerr)=\dfrac{\hbar c^3}{4\pi GMk_B}\left(\dfrac{\sqrt{\dfrac{G^2M^2}{c^4}-\dfrac{J^2}{c^2M^2}}}{\dfrac{GM}{c^2}+\sqrt{\dfrac{G^2M^2}{c^4}-\dfrac{J^2}{c^2M^2}}}\right)=\dfrac{\hbar c^3}{4\pi GMk_B}\left(\dfrac{\sqrt{1-a^2}}{1+\sqrt{1-a^2}}\right)$$ both mass $$M$$ and $$a$$ are varying (or $$M,J$$). So, I believe it is no so easy to calculate the evaporation time for Kerr black holes. Anyway, if someone knows references about that, I would also be pleased.

• That formula for the Schwarzschild hole is an approximation that neglects so-called gray-body factors. – G. Smith Jul 12 '20 at 16:35

## 1 Answer

I looked at Page’s 1976 paper on emission of massless particles from a Kerr hole. He numerically solves two differential equations for the hole’s changing mass and angular momentum. I didn’t find a simple formula for the resulting lifetime.

• One should note, that in 1976 people were certain that neutrinos were massless (and tau neutrino had not been observed yet), so the “Everything” curves in figures that include emission of neutrinos would be wrong. – A.V.S. Jul 13 '20 at 6:53