General Relativity: An Introduction for Physicists · Problem 11.19

Schwarzschild black holes — Hobson, Efstathiou & Lasenby

Physics · hawking-radiation-lifetime

Problem

A Schwarzschild black hole of mass

M

radiates as a blackbody of temperature

T = \hbar c^3/(8\pi k_B GM)

. Show from first principles that the black hole has a lifetime

\tau = M^3/(3\alpha\hbar)

, where

\alpha = c^4/(15360\pi G^2)

. In its last second, calculate the total energy radiated and estimate the typical energy of each radiated particle.