# A Complementary Third Law for Black Hole Thermodynamics

## TL;DR

This paper introduces a 'complementary third law' for black hole thermodynamics, showing that black holes with finite temperature cannot fully evaporate under certain conditions, implying they become effective remnants with infinite evaporation time.

## Contribution

It proposes a new thermodynamic principle, the 'complementary third law,' explaining why black holes cannot completely evaporate despite no lower size bound.

## Key findings

- Black holes with finite temperature cannot fully evaporate under certain conditions.
- Such black holes become effective remnants with infinite evaporation time.
- The result challenges traditional views on black hole evaporation and thermodynamics.

## 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.