Physics Letters B 793 (2019) 234–239
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Thermal fluctuations of AdS black holes in three-dimensional rainbow
gravity
M. Dehghani
Department of Physics, Razi University, Kermanshah, Iran
a r t i c l e i n f o a b s t r a c t
Article history:
Received
12 February 2019
Received
in revised form 19 April 2019
Accepted
23 April 2019
Available
online 30 April 2019
Editor:
N. Lambert
Keywords:
Three-dimensional
black hole
Nonlinear
electrodynamics
Thermal
fluctuations
In this work, we explore the charged black holes with the power-law modified electromagnetic theory
in a three-dimensional energy dependent space-time. Through exact solution of the field equations,
we introduce a new class of nonlinearly charged black holes which are asymptotically anti-de Sitter
(AdS). The black hole entropy, temperature and electric potential are calculated from the geometrical
approaches. The counterterm method and Gauss’s electric law are utilized for calculating the black
hole mass and electric charge, respectively. By use of the Smarr formula, which states the black hole
mass as the function of thermodynamic extensive parameters, we prove the validity of the first law of
thermodynamics for the new AdS black holes. By use of the canonical ensemble method, the black hole
remnant or phase transitions are investigated regarding the signature of black hole heat capacity. We
show that the AdS black hole solutions, we just obtained, are thermodynamically stable if their horizon
radii are greater than a minimum value. Then, by considering the black hole thermal fluctuations, we
examine the quantum gravitational effects on the thermodynamic properties of the new AdS black holes.
We prove that, when the thermal fluctuations are taken into account, the thermodynamical first law is
no longer valid. Also, the thermal stability of the black holes gets some corrections.
© 2019 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
Gravity’s rainbow just like the Horava-Lifshitz gravity theory is
the ultraviolet completion of general relativity. Both of them, as the
attempts for constructing the quantum theory of gravity, are based
on promoting the usual dispersion relation to the so-called mod-
ified
dispersion relation. This modification is proposed by almost
all of the quantum gravity models [1–3]. The modified dispersion
relation is written as [4,5]
E
2
f
2
(ε) − p
2
g
2
(ε) = m
2
, (1.1)
where f (ε) and g(ε), in order, are known as the temporal and spa-
tial
rainbow functions and ε is a dimensionless quantity defined as
the ratio of the energy of the test particle E to the Planck energy
E
P
. The infrared version of dispersion relation is recovered by the
requirements
lim
ε→0
f (ε) = 1, and lim
ε→0
g(ε) = 1. (1.2)
E-mail address: m.dehghani@razi.ac.ir.
There are several proposed functional forms for the rainbow func-
tions
which are obtained from different motivations, among them
are [6]
f (ε) = 1, and g(ε) =
1 − ηε
n
, (1.3)
f
(ε) =
e
ζ ε
− 1
ζ ε
, and g(ε) = 1, (1.4)
f
(ε) = g(ε) =
1
1 − βε
. (1.5)
The coefficients η, ζ and β, known as the rainbow parameters, are
of the order of unity, ε ≤ 1 and the power n is a positive integer.
In general, the rainbow functions have a magnitude equal to or
slightly different from unity [7,8].
Obviously,
modified dispersion relation (1.1)is not Lorentz in-
variant.
Doubly/or deformed special relativity which, instead of
usual Lorentz transformations, is based on the nonlinear Lorentz
transformations that preserve Lorentz invariance of modified dis-
persion
relation. In the doubly special relativity, in addition to the
speed of light Planck energy is another constant quantity, which
are the upper bound of the speed and energy that a particle can
attain. In 2004 this theory was extended to the curved manifolds
https://doi.org/10.1016/j.physletb.2019.04.058
0370-2693/
© 2019 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
.