Abstract
There are some examples in the literature, in which despite the fact that the underlying theory or model does not impose a lower bound on the size of black holes, the final temperature under Hawking evaporation is nevertheless finite and nonzero. We show that under some loose conditions, the black hole is necessarily an effective remnant, in the sense that its evaporation time is infinite. That is, the final state that there is nonzero finite temperature despite having no black hole remaining cannot be realized. We discuss the limitations, subtleties, and the implications of this result, which is reminiscent of the third law of black hole thermodynamics, but with the roles of temperature and size interchanged. We therefore refer to our result as the “complementary third law” for black hole thermodynamics.
Highlights
A popular way of obtaining a black hole remnant is via the generalized uncertainty principle (GUP), which incorporates the effect of gravity into the Heisenberg’s uncertainty principle
Since GUP arises from various general considerations involving gravity and quantum mechanics, as well as string theory [3,4,5,6,7,8,9,10,11], it is usually treated as a phenomenological approach to study various properties of quantum gravity
Such a choice of sign of α may seem unusual, but it is consistent with some quantum gravity models in which physics at the Planck scale “classicalized” and becomes deterministic [13,14,15,16]
Summary
In the usual picture of Hawking evaporation, an asymptotically flat Schwarzschild black hole evaporates completely in finite time, the time scale is extremely long for a stellar mass black hole. Since the Hawking temperature is inversely proportional to the mass, the black hole becomes hotter as it shrinks. It has been argued that this leads to a correction in the Hawking temperature, resulting in a black hole remnant [12] In this scenario as the evaporation stops at some finite mass, the temperature stops at a finite, nonzero value, see Fig. 1 below. This is the complementary third law of black hole thermodynamics We found that this behavior is rather general, and can be stated as finite time under Hawking evaporation if the temperature is nonzero. Note that Hawking temperature of the usual kind T ∝ 1/M is not differentiable at rh = 0 This result reminded us of the (“Nernst version” of) third law of black hole thermodynamics: zero temperature (extremal) black hole, which is of nonzero size, is unattainable in finite number of steps. We have the opposite scenario, zero mass/size black hole is unattainable in
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