Physics Letters B 750 (2015) 306–311
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Entropy of an extremal electrically charged thin shell and the extremal
black hole
José P.S. Lemos
a,∗
, Gonçalo M. Quinta
a
, Oleg B. Zaslavskii
b,c
a
Centro Multidisciplinar de Astrofísica – CENTRA, Departamento de Física, Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Avenida Rovisco Pais 1,
1049-001 Lisboa, Portugal
b
Astronomical Institute of Kharkov, V.N. Karazin National University, 35 Sumskaya St., Kharkov, 61022, Ukraine
c
Institute of Mathematics and Mechanics, Kazan Federal University, 18 Kremlyovskaya Street, Kazan 420008, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
11 August 2015
Received
in revised form 28 August 2015
Accepted
31 August 2015
Available
online 2 September 2015
Editor:
M. Cveti
ˇ
c
Keywords:
Black
holes
Quasiblack
holes
Extremal
horizon
Entropy
Thermodynamics
There is a debate as to what is the value of the entropy S of extremal black holes. There are approaches
that yield zero entropy S = 0, while there are others that yield the Bekenstein–Hawking entropy S =
A
+
/4, in Planck units. There are still other approaches that give that S is proportional to r
+
or even that
S is a generic well-behaved function of r
+
. Here r
+
is the black hole horizon radius and A
+
= 4πr
2
+
is its horizon area. Using a spherically symmetric thin matter shell with extremal electric charge, we
find the entropy expression for the extremal thin shell spacetime. When the shell’s radius approaches its
own gravitational radius, and thus turns into an extremal black hole, we encounter that the entropy is
S = S(r
+
), i.e., the entropy of an extremal black hole is a function of r
+
alone. We speculate that the
range of values for an extremal black hole is 0 ≤ S(r
+
) ≤ A
+
/4.
© 2015 The Authors. 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
The entropy S and thermodynamics of black holes have been
worked out first by Bekenstein [1] and Hawking and collabora-
tors
[2,3]. The Bekenstein–Hawking entropy is given by S = A
+
/4,
where A
+
= 4πr
2
+
, A
+
and r
+
are the horizon area and the hori-
zon
radius, respectively, and we are putting all the natural con-
stants
equal to one, i.e., we use Planck units. York and collabora-
tors
[4–6] (see also [7,8]) have further worked out the black hole
thermodynamic properties by using canonical and grand canonical
thermodynamic ensembles. There are several other methods that
can be used to study black hole thermodynamics, one that suits
us here uses matter shells [9–11]. In this method, one studies the
generic thermodynamics of the shells at any shell radius, and as
one sends the shell to its own gravitational radius one recovers
the S = A
+
/4 Bekenstein–Hawking entropy. This is the quasiblack
hole method, the evident power of it was displayed in [12].
A
particular class of black holes is the extremal black hole class.
Electrically charged black holes in general relativity, the ones we
are interested here, have m ≥ Q , and the extremal black holes
*
Corresponding author.
E-mail
addresses: joselemos@ist.utl.pt (J.P.S. Lemos), goncalo.quinta@ist.utl.pt
(G.M. Quinta),
ozaslav@kharkov.ua (O.B. Zaslavskii).
are characterized by having their mass m equal to their electric
charge Q , m = Q . The extremal black holes seem to have distinct
properties. For instance, according to the Hawking temperature
formula, extremal black holes have zero temperature. In addition
the entropy of an extremal black hole is a subject of a wide de-
bate
as there are different reasonings that can be applied which
lead to different values for the entropy. Hawking and collabora-
tors
[13] and Teitelboim [14] have given topological arguments
which point to the conclusion that extremal black holes have zero
entropy. Further evidence from other arguments for S = 0for ex-
tremal
black holes was provided in [15–17], see also [18,19]. One
could also argue, naively, that since the Hawking temperature is
zero, then according to one of the formulations of the third law
of thermodynamics as many textbooks present it should have zero
entropy.
However,
there remain doubts why the Bekenstein–Hawking
formula does not hold. After all, working out the entropy of non-
extremal
black holes and taking the extremal limit m = Q yields
S = A
+
/4, see, e.g., [2,3,5,10]. In this case, the thermodynamic ar-
gument
would not hold, the extremal black hole could be a system
of minimum energy and degenerate ground state and such systems
can have entropy even at zero temperature. Moreover, in string
theory, there are arguments, other than geometrical, that make
use of a direct counting of string and D-brane states in composite
http://dx.doi.org/10.1016/j.physletb.2015.08.065
0370-2693/
© 2015 The Authors. 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
.