Physics Letters B 773 (2017) 325–331
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Quantum fluctuations of a BTZ black hole in massive gravity
Behnam Pourhassan
a,∗
, Mir Faizal
b,c
, Zaid Zaz
d
, Anha Bhat
e
a
School of Physics, Damghan University, Damghan, 3671641167, Iran
b
Irving K. Barber School of Arts and Sciences, University of British Columbia, Kelowna, British Columbia, V1V 1V7, Canada
c
Department of Physics and Astronomy, University of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada
d
Department of Electronics and Communication Engineering, Institute of Technology, University of Kashmir, Srinagar, Kashmir, 190006, India
e
Department of Metallurgical and Materials Engineering, National Institute of Technology, Srinagar, Kashmir, 190006, India
a r t i c l e i n f o a b s t r a c t
Article history:
Received
20 May 2017
Received
in revised form 9 August 2017
Accepted
21 August 2017
Available
online 25 August 2017
Editor:
N. Lambert
Keywords:
Black
hole
Massive
gravity
Thermodynamics
In this work, we shall analyze the effects of quantum fluctuations on the properties of a BTZ black hole, in
a massive theory of gravity. We will analyze this for a charged BTZ black hole in asymptotically AdS and
dS space–times. The quantum fluctuations would produce thermal fluctuations in the thermodynamics of
this BTZ black hole. As these fluctuations would become relevant at a sufficiently small scale, we shall
discuss the effects of such thermal fluctuations on the entropy of a small charged BTZ black. We shall
also analyze the effects of these fluctuations on the stability of such a black hole.
© 2017 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
.
The second law of thermodynamics seems to be in conflict with
the physics of black holes, if a maximum entropy is not associated
with black holes [1,2]. This is because a spontaneous reduction in
the entropy of the universe would occur by an object with finite
entropy crossing the event horizon of a black hole. So, a condi-
tion
for the second law of thermodynamics and black holes to
co-exist is the association of maximum entropy with black holes.
This entropy associated with black holes is maximal, in the sense,
that black holes have more entropy than other objects with the
same volume [3,4]. This maximum entropy of a black hole S
0
,
is related to its area A, as S
0
= A/4. So, the maximum entropy
of a region scales with the area of the boundary enclosing that
region [5]. This scaling behavior of the maximum entropy has mo-
tivated
the development of the holographic principle [6,7]. It may
be noted that at very small scales (close to the Planck scale), the
holographic principle is expected to get violated due to quantum
gravitational effects [8,9]. So, the quantum gravitational effects are
also expected to modify this entropy-area relation, at very small
scales. The effect of quantum fluctuations, can be calculated from
a short distance correction to the area-entropy law. This is because,
the geometry of space–time can be obtained from thermodynam-
*
Corresponding author.
E-mail
addresses: b.pourhassan@du.ac.ir (B. Pourhassan),
mirfaizalmir@googlemail.com (M. Faizal), mohammadzaz@gmail.com (Z. Zaz),
anhajan1@gmail.com (A. Bhat).
ics in the Jacobson formalism [10,11]. So, thermal fluctuations in
the thermodynamics would correspond to quantum fluctuations in
the geometry [12]. Furthermore, these quantum fluctuations in the
geometry can be neglected for very large black holes. However,
such large black holes have a very small temperature, and the ef-
fects
of thermal fluctuations can also be neglected for such black
holes. Now these black holes evaporate with Hawking radiation,
and reduce in size. At sufficiently small size, the effects of quan-
tum
fluctuations cannot be neglected, and these fluctuations can
be analyzed perturbation. At such a small size, these black holes
will also have a large temperature, and the effects of thermal fluc-
tuations
can also not be neglected at this stage. So, at this stage
the thermal fluctuations can be analyzed as perturbations around
the equilibrium thermodynamics [13]. It may be noted that as the
black hole keeps reducing in size and it’s the temperature keeps
increasing, a stage is reached at which the manifold description
of space–time breaks down, and at this stage the equilibrium de-
scription
of thermodynamics also breaks down. So, the correction
to the equilibrium thermodynamics is only relevant at an interme-
diate
scale, such that the scale is not so small that the equilibrium
description breaks down, and it is not so large that effects by fluc-
tuations
can be neglected. In this paper, we shall analyze our sys-
tem
at such an intermediate scale, where the thermal fluctuations
can be expressed as perturbations around equilibrium thermody-
namics.
We
will analyze the effects of fluctuations on a black hole in
massive gravity. The massive gravity is constructed using the Vain-
http://dx.doi.org/10.1016/j.physletb.2017.08.046
0370-2693/
© 2017 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
.