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,∗
a
The Ruppin Academic Center, Emeq Hefer 40250, Israel
b
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
ˇ
c
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
BH
×r
H
>(
¯
h/r
H
)
2
, where r
H
is
the horizon radius of the black hole.
© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
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
B
= 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]
T
BH
=
¯
h(r
+
−r
−
)
4π(r
2
+
+a
2
)
,
(2)
where
r
±
= M +(M
2
−a
2
− Q
2
)
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
−
)/r
+
1are characterized by the strong inequality [5]
T
BH
×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
BH
×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
T
BH
×r
+
(
¯
h/r
+
)
2
.
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]
ds
2
=−
ρ
2
(dt −a sin
2
θdφ)
2
+
ρ
2
dr
2
+ρ
2
dθ
2
http://dx.doi.org/10.1016/j.physletb.2016.06.021
0370-2693/
© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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