Physics Letters B 777 (2018) 351–360
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Thermodynamics of charged dilatonic BTZ black holes in rainbow
gravity
M. Dehghani
Department of Physics, Ilam University, Ilam, Iran
a r t i c l e i n f o a b s t r a c t
Article history:
Received
7 October 2017
Received
in revised form 17 December 2017
Accepted
21 December 2017
Available
online 27 December 2017
Editor:
N. Lambert
Keywords:
Three-dimensional
black hole
Charged
black hole with scalar hair
Maxwell’s
theory of electrodynamics
Rainbow
gravity
In this paper, the charged three-dimensional Einstein’s theory coupled to a dilatonic field has been
considered in the rainbow gravity. The dilatonic potential has been written as the linear combination
of two Liouville-type potentials. Four new classes of charged dilatonic rainbow black hole solutions,
as the exact solution to the coupled field equations of the energy dependent space time, have been
obtained. Two of them are correspond to the Coulomb’s electric field and the others are consequences
of a modified Coulomb’s law. Total charge and mass as well as the entropy, temperature and electric
potential of the new charged black holes have been calculated in the presence of rainbow functions.
Although the thermodynamic quantities are affected by the rainbow functions, it has been found that
the first law of black hole thermodynamics is still valid for all of the new black hole solutions. At the
final stage, making use of the canonical ensemble method and regarding the black hole heat capacity, the
thermal stability or phase transition of the new rainbow black hole solutions have been analyzed.
© 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
.
1. Introduction
One of the most outstanding achievements in the various the-
ories
of quantum gravity theory, such as string theory [1], loop
quantum gravity [2], noncommutative geometry [3] and Gedanken
experiments [4], is the prediction of a minimal measurable length
in the order of the Planck length [5]. The existence of such a
minimal length, which restricts the maximum energy that a par-
ticle
can attain to the Planck energy, is related to the modifica-
tion
of linear momentum and also quantum commutation rela-
tions.
Therefore, it can be captured by modification of the usual
uncertainty principle known as the generalized uncertainty princi-
ple
or by promoting the standard energy–momentum relation (i.e.
E
2
− p
2
= m
2
) to the modified dispersion relation. In addition, it
is well known that Einstein’s general relativity, as an effective the-
ory
of gravity, is valid in the infrared limit while it fails to produce
accurate results in ultraviolet regime. The gravity’s rainbow just
like the Horava–Lifshitz gravity theory is motivated by modification
of standard dispersion relation in the ultraviolet limit [6]. Such a
modification of the geometry at high energy scale can be regarded
as the ultraviolet completion of the general relativity. Therefore,
gravity’s rainbow can be regarded as an attempt to construct the
theory of quantum gravity [7].
E-mail address: m.dehghani@ilam.ac.ir.
On the other hand, the modified dispersion relation violates
Lorentz invariants. Doubly/Deformed special relativity, as a the-
ory
which predicts naturally the modified dispersion relation, is
an extension of the special theory of relativity. It has been estab-
lished
based on the nonlinear Lorentz transformations in momen-
tum
space. In this theory, the Planck-scale energy beside the speed
of light remains invariant. Also the Planck-scale corrected disper-
sion
relation preserves a deformed Lorentz symmetry [8,9]. It is
believed that the violation of Lorentz invariancy plays an essential
rule in constructing the quantum theory of gravity. Such a Lorentz
symmetry violation can occur in the string theory because of the
existence of an unstable perturbative string vacuum [10].
Now,
the doubly special relativity has been generalized to
curved space times and a doubly general relativity or gravity’s rain-
bow
has been arrived [11]. In this theory, the geometry of space
time depends on the energy of the test particle. Thus, it seems
different for the particles having different amounts of energy and
the energy dependent metrics form a rainbow of metrics. This is
why the double general relativity is named as gravity’s rainbow.
The modified dispersion relation can be written in the following
general form
E
2
f
2
(ε) − p
2
g
2
(ε) =m
2
, (1.1)
where, ε = E/E
P
, E
P
is the Planck-scale energy, E is the energy
of the test particle and the functions f (ε) and g(ε) are known as
https://doi.org/10.1016/j.physletb.2017.12.048
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
.