Some statistical mechanical properties of photon black holes

Abstract

We show that if the total internal energy of a black hole is constructed as the sum of $N$ photons all having a fixed wavelength chosen to scale with the Schwarzschild radius as $\lambda=\alpha R_{s}$, then $N$ will scale with $R_{s}^{2}$. A statistical mechanical calculation of the configuration proposed yields \( \alpha = 4 \pi^2 / \ln(2) \) and a total entropy of the system $S=k_{B}N \ln(2)$, in agreement with the Bekenstein entropy of a black hole. It is shown that the critical temperature for Bose-Einstein condensation for relativistic particles of $\lambda=\alpha R_{s}$ is always well below the Hawking temperature of a black hole, in support of the proposed internal configuration. We then examine our results from the point of view of recent loop quantum gravity ideas and find that a natural consistency of both approaches appears. We show that the Jeans criterion for gravitational instability can be generalised to the special and general relativistic regimes and holds for any type of mass-energy distribution.