黑洞的量子温度下界：Bekenstein-Hawking界限新证据

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"黑洞的贝肯斯坦-霍金温度下界：量子界限与 gedanken 实验" 这篇论文《黑洞的贝肯斯坦-霍金温度下界》探讨了黑洞的热力学性质，尤其是其Bekenstein-Hawking温度的量子下限。Bekenstein-Hawking温度是黑洞物理学中的一个核心概念，由雅各布·贝肯斯坦和斯蒂芬·霍金提出，它描述了黑洞作为热辐射源的温度。这个温度与黑洞的事件视界半径成反比，即T_BH = ħ/(4πM k_B)，其中T_BH是Bekenstein-Hawking温度，ħ是约化普朗克常数，M是黑洞的质量，k_B是玻尔兹曼常数。作者Shahar Hoda通过一个 gedanken（思想实验）提供了证据，表明存在一个量子力学的下限，限制了黑洞的Bekenstein-Hawking温度。在这个实验中，设想一个带电粒子被投入一个Kerr黑洞（旋转的黑洞）。Kerr黑洞在吸收了带电粒子后会转变为Kerr-Newman黑洞，这是一种带有电荷和角动量的黑洞。作者证明，这种最终配置的黑洞温度TBH与视界半径rH的关系满足TBH × rH > (ħ / rH)^2。这意味着即使黑洞的温度降低，它也不会无限制地减小，而是受到一个量子力学效应设定的下限。这个发现对理解黑洞的信息悖论和量子引力理论有重要意义。黑洞的热辐射，即霍金辐射，暗示了信息可能以某种方式从黑洞中泄漏出来，而这个下限可能与这个过程有关。此外，它也对黑洞熵的理解提出了新的挑战，因为熵通常与表面积而不是体积成正比，这与热力学第二定律相符。然而，这个新的量子下限可能需要我们在黑洞热力学和信息守恒的框架内重新考虑这些基本原理。文章在2016年由Elsevier B.V.出版，并在Physics Letters B期刊上发表，属于开放访问文章，受Creative Commons Attribution (CC BY) 许可证保护，并由SCOAP3资助。该研究不仅加深了我们对黑洞物理的理解，还可能为未来的量子引力理论研究提供新的方向。

Physics Letters B 759 (2016) 541–545

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

A lower bound on the Bekenstein–Hawking temperature of black holes

Shahar Hod

a,b,∗

The Ruppin Academic Center, Emeq Hefer 40250, Israel

The Hadassah Institute, Jerusalem 91010, Israel

a r t i c l e i n f o a b s t r a c t

Article history:

Received

14 April 2016

Received

in revised form 9 June 2016

Accepted

9 June 2016

Available

online 15 June 2016

Editor:

M. Cveti

We present evidence for the existence of a quantum lower bound on the Bekenstein–Hawking

temperature of black holes. The suggested bound is supported by a gedanken experiment in which a

charged particle is dropped into a Kerr black hole. It is proved that the temperature of the ﬁnal Kerr–

Newman

black-hole conﬁguration is bounded from below by the relation T

×r

h/r

)

, where r

the horizon radius of the black hole.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

It is well known [1,2] that, for mundane physical systems of

spatial size R, the thermodynamic (continuum) description breaks

down in the low-temperature regime T ∼

h/R (we shall use grav-

itational

units in which G = c = k

= 1). In particular, these low

temperature systems are characterized by thermal ﬂuctuations

whose wavelengths λ

thermal

∼

h/T are of order R, the spatial size

of the system, in which case the underlying quantum (discrete) na-

ture

of the system can no longer be ignored. Hence, for mundane

physical systems of spatial size R, the physical notion of temper-

ature

is restricted to the high-temperature thermodynamic regime

[1,2]

T × R 

h . (1)

Interestingly, black holes are known to have a well-deﬁned no-

tion

of temperature in the complementary regime of low tem-

peratures.

In particular, the Bekenstein–Hawking temperature of

generic Kerr–Newman black holes is given by [3,4]

h(r

−r

−

)

4π(r

)

(2)

where

= M +(M

−a

− Q

)

1/2

(3)

are the radii of the black-hole (outer and inner) horizons (here M,

J ≡ Ma, and Q are respectively the mass, angular momentum,

Correspondence to: The Ruppin Academic Center, Emeq Hefer 40250, Israel.

E-mail

address: shaharhod@gmail.com.

and electric charge of the Kerr–Newman black hole). The rela-

tion

(2) implies that near-extremal black holes in the regime

−r

−

)/r

 1are characterized by the strong inequality [5]

×r



h . (4)

It is quite remarkable that black holes have a well deﬁned no-

tion

of temperature in the regime (4) of low temperatures, where

mundane physical systems are governed by ﬁnite-size (quantum)

effects and no longer have a self-consistent thermodynamic de-

scription.

One

naturally wonders whether black holes can have a physi-

cally

well-deﬁned notion of temperature all the way down to the

extremal (zero-temperature) limit T

×r

h → 0? In order to ad-

dress

this intriguing question, we shall analyze in this paper a

gedanken experiment which is designed to bring a Kerr–Newman

black hole as close as possible to its extremal limit. We shall show

below that the results of this gedanken experiment provide com-

pelling

evidence that the Bekenstein–Hawking temperature of the

black holes is bounded from below by the quantum inequality

×r

 (

h/r

)

2. The gedanken experiment

We consider a spherical body of proper radius R, rest mass μ,

and electric charge q which is slowly lowered towards a Kerr black

hole of mass M and angular momentum J = Ma along the sym-

metry

axis of the black hole (we shall assume q > 0 and a > 0

without

loss of generality). The black-hole spacetime is described

by the line element [6,7]

=−



(dt −a sin

θdφ)



+ρ

dθ

http://dx.doi.org/10.1016/j.physletb.2016.06.021

0370-2693/

SCOAP

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