# Black hole radiation: mass as a function of time

Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 16; Problem P16.6.

We can get an expression for the mass of a black hole as a function of time, taking into account its evaporation through radiation. The total lifetime ${t_{0}}$ of the black hole of initial mass ${M_{0}}$ is given by

$\displaystyle M_{0}^{3}=\frac{3\sigma\hbar^{4}}{256\pi^{3}k_{B}^{4}G^{2}}t_{0} \ \ \ \ \ (1)$

From the equation we derived earlier (eqn. 11 in this post, with variables renamed to match what we’re doing here), the mass ${M}$ at time ${t}$ is

 $\displaystyle M\left(t\right)$ $\displaystyle =$ $\displaystyle \left[M_{0}^{3}-\frac{3\sigma\hbar^{4}}{256\pi^{3}k_{B}^{4}G^{2}}t\right]^{1/3}\ \ \ \ \ (2)$ $\displaystyle$ $\displaystyle =$ $\displaystyle M_{0}\left(1-\frac{t}{t_{0}}\right)^{1/3} \ \ \ \ \ (3)$

A plot of ${M/M_{0}}$ versus ${t/t_{0}}$ looks like this:

The rate of mass loss increases towards the end of the black hole’s life.