Is there a black hole minimum mass?

Abstract

Applying the first and generalized second laws of thermodynamics for a realistic process of near critical black hole formation, we derive an entropy bound, which is identical to Bekenstein's one for radiation. Relying upon this bound, we derive an absolute minimum mass $\ensuremath{\sim}0.04\sqrt{{g}_{*}}{m}_{\mathrm{Pl}}$, where ${g}_{*}$ and ${m}_{\mathrm{Pl}}$ is the effective degrees of freedom for the initial temperature and the Planck mass, respectively. Since this minimum mass coincides with the lower bound on masses of which black holes can be regarded as classical against the Hawking evaporation, the thermodynamical argument will not prohibit the formation of the smallest classical black hole. For more general situations, we derive a minimum mass, which may depend on the initial value for entropy per particle. For primordial black holes, however, we show that this minimum mass can not be much greater than the Planck mass at any formation epoch of the Universe, as long as ${g}_{*}$ is within a reasonable range. We also derive a size-independent upper bound on the entropy density of a stiff fluid in terms of the energy density.