The catastrophic fate of Schwarzschild black holes in a thermal bath

Abstract

Using the Schwazchild metric as a rudimentary toy-model, we pedagogically revisit the curious prediction that the mass of a classical black hole in a constant temperature thermal bath diverges in a finite amount of time. We study in details how this instability behaves if the temperature of the bath is allowed to vary with time and conclude that whatever the background behavior (but for a zero-measure subspace of the initial conditions), the black hole mass either diverges or vanishes in a finite time if the Hawking radiation is taken into account. The competition between both effects is subtle and not entirely governed by the hierarchy of the relevant temperatures. This instability is also shown to be reached before the background singularity in a contracting universe, which has implications for bouncing models. The results are generalized to spaces with extra-dimensions and the main conclusions are shown to remain true. The limitations of the model are reviewed, both from the point of view of the dynamical black hole horizon and from the point of view of the background space expansion. Comparisons with other approaches are suggested and possible developments are underlined.

Black hole in a heat bath

“…The thermal properties of black holes possess a subtle and an intriguing feature, which is often overlooked, but was recently brought to a central stage by Barrau, Martineau and Renevey in Ref. [1]. As it is well known, the energy density of thermal radiation is proportional to the fourth power of its absolute temperature T .…”

“…In Ref. [1] the effective radius of the Schwarzschild black hole was taken to be, in the natural units, 3 √ 3M , and the authors found that Eq. (1.1) takes the form:…”

“…Although this result, based on an approximation of a real, growing black hole by means of a Schwarzschild black hole, has little astrophysical relevance -even for a supermassive black hole with mass M i equal to around 10 The problem of the singularity in the black hole mass received a detailed investigation in Ref. [1] (see also the references there). However, the treatment in Ref.…”

“…However, the treatment in Ref. [1] was mostly classical. In this paper we shall attempt a quantummechanical approach.…”