Eur. Phys. J. C (2019) 79: 513
https://doi.org/10.1140/epjc/s10052-019-7003-1

Regular Article - Theoretical Physics

A complementary third law for black hole thermodynamics

¹ Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, 180 Siwangting Road, Yangzhou, Jiangsu, 225002, China
² School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, 200240, China
³ Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 106 91, Stockholm, Sweden

^* e-mail: ycong@yzu.edu.cn

Received: 15 December 2018
Accepted: 2 June 2019
Published online: 15 June 2019

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.