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.

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