Black hole radiation: energy of a particle from a solar mass black hole

Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 16; Problem P16.1.

The crude calculation presented earlier gives the energy at infinity of a particle created near the event horizon of a black hole as

\displaystyle  E_{\infty}=\frac{\hbar}{4GM} \ \ \ \ \ (1)

In GR units

\displaystyle   \hbar \displaystyle  = \displaystyle  3.5153\times10^{-43}\mbox{ kg m}\ \ \ \ \ (2)
\displaystyle  E_{\infty} \displaystyle  = \displaystyle  \frac{8.788\times10^{-44}}{GM}\mbox{ kg} \ \ \ \ \ (3)

For a solar mass black hole {GM=1477\mbox{ m}}, so

\displaystyle  E_{\infty}=5.95\times10^{-47}\mbox{ kg} \ \ \ \ \ (4)

In other units, this is (using {1\mbox{ eV}=1.602\times10^{-19}\mbox{ J}})

\displaystyle   E_{\infty} \displaystyle  = \displaystyle  mc^{2}\ \ \ \ \ (5)
\displaystyle  \displaystyle  = \displaystyle  5.355\times10^{-30}\mbox{ J}\ \ \ \ \ (6)
\displaystyle  \displaystyle  = \displaystyle  3.34\times10^{-11}\mbox{ eV} \ \ \ \ \ (7)

This is an almost unimaginably small energy. For comparison, the energy of a single photon of visible light (with a wavelength of 500 nm) is around 2.5 eV:

\displaystyle  E=h\nu=\frac{hc}{\lambda}=6.626\times10^{-34}\times\frac{3\times10^{8}}{5\times10^{-7}\times1.602\times10^{-19}}\mbox{ eV} \ \ \ \ \ (8)

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