The third law of black hole thermodynamics and the mass-to-charge ratio of scalar fields

Abstract

Kehle and Unger have recently shown that self-gravitating charged classical scalar fields can collapse to form extremal Reissner-Nordström (eRN) black holes in finite time, thus violating the third law of thermodynamics (TLT). Reall has subsequently proved that the formation of an eRN black hole in gravitational collapse is impossible if the mass-to-charge ratio of the field exceeds unity, $m/e\geq1$. In the present compact paper we conjecture, using quantum considerations, that the Reall bound is {\it not} sharp. In particular, we point out that the electric field of a charged system with charge $Q$ will spontaneously polarize the vacuum in the regime $(eQ/\hbar)^2>(mQ/\hbar)^2+1/4$, thereby creating pairs of charged particles that discharge the black hole and ensure the quantum validity of the TLT in this regime. Motivated by the TLT, we conjecture that the classical Reall bound can be strengthened such that classical considerations would prevent eRN black holes from forming dynamically in the regime $m/e\geq\sqrt{1-\big({{\hbar}/{2eQ}}\big)^2}$, where quantum polarization effects are unable to protect the TLT.