Bekenstein's generalized second law of thermodynamics: The role of the hoop conjecture

Shahar Hod

doi:10.1016/j.physletb.2015.10.052

Shahar Hod

Open Access

https://doi.org/10.1016/j.physletb.2015.10.052

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Abstract

Bekenstein's generalized second law (GSL) of thermodynamics asserts that the sum of black-hole entropy, SBH=Ac3/4ħG (here A is the black-hole surface area), and the ordinary entropy of matter and radiation fields in the black-hole exterior region never decreases. We here re-analyze an intriguing gedanken experiment which was designed by Bekenstein to challenge the GSL. In this historical gedanken experiment an entropy-bearing box is lowered into a charged Reissner–Nordström black hole. For the GSL to work, the resulting increase in the black-hole surface area (entropy) must compensate for the loss of the box's entropy. We show that if the box can be lowered adiabatically all the way down to the black-hole horizon, as previously assumed in the literature, then for near-extremal black holes the resulting increase in black-hole surface-area (due to the assimilation of the box by the black hole) may become too small to compensate for the loss of the box's entropy. In order to resolve this apparent violation of the GSL, we here suggest to use a generalized version of the hoop conjecture. In particular, assuming that a physical system of mass M and electric charge Q forms a black hole if its circumference radius rc is equal to (or smaller than) the corresponding Reissner–Nordström black-hole radius rRN=M+M2−Q2, we prove that a new (and larger) horizon is already formed before the entropy-bearing box reaches the horizon of the original near-extremal black hole. This result, which seems to have been overlooked in previous analyzes of the composed black-hole-box system, ensures the validity of Bekenstein's GSL in this famous gedanken experiment.

Highlights

The legend says [1, 2] that it all began with a cup of tea and two genius physicists, Professor John Archibald Wheeler and his young student Jacob David Bekenstein, who tried to figure out what happens to the second law of thermodynamics when the cup goes down a black hole.In this gedanken experiment, the thermal entropy of the tea disappears behind the black-hole horizon
As emphasized by Bekenstein [3, 10], an entropy bound of the form (4) ensures that the generalized second law of thermodynamics (2) is respected in a physical process in which a spherical body with negligible self-gravity is captured by a black hole [17]
In order to resolve this apparent violation of the generalized second law (GSL), we shall concentrate on the dangerous regime (19) and examine the physical consequences of Thorne’s hoop conjecture [22] in the context of our gedanken experiment

Summary

INTRODUCTION

The legend says [1, 2] that it all began with a cup of tea and two genius physicists, Professor John Archibald Wheeler and his young student Jacob David Bekenstein, who tried to figure out what happens to the second law of thermodynamics when the cup goes down a black hole In this gedanken experiment, the thermal entropy of the tea disappears behind the black-hole horizon. Using the conjectured proportionality (1) between black-hole entropy and horizon area, Bekenstein proposed a generalized version of the second law of thermodynamics [3]: The sum of black-hole entropy, SBH, and the ordinary entropy of matter and radiation fields in the black-hole exterior region, S, cannot decrease This conjecture asserts that physical processes involving black holes are characterized by the relation. It is of physical interest to consider gedanken experiments in order to test the validity of the GSL in various physical situations

BEKENSTEIN’S UNIVERSAL ENTROPY BOUND

CHALLENGING THE GENERALIZED SECOND LAW OF THERMODYNAMICS

BEKENSTEIN’S GSL AND THORNE’S HOOP CONJECTURE

SUMMARY