Physics Letters B 785 (2018) 274–283
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Thermodynamics of novel charged dilaton black holes in gravity’s
rainbow
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
7 June 2018
Received
in revised form 6 August 2018
Accepted
8 August 2018
Available
online 30 August 2018
Editor:
N. Lambert
Keywords:
Four-dimensional
black hole
Charged
dilaton black hole
Maxwell’s
theory of electrodynamics
Gravity’s
rainbow
In the present work, we have studied the thermodynamical properties of black holes arising as the
solutions of the four-dimensional dilaton gravity coupled to Maxwell’s electrodynamics in gravity’s
rainbow. This theory allows three classes of asymptotically non-flat and non-AdS black hole solutions.
We showed that the self-interacting scalar function, as the solution to the scalar field equation, can
be written as the linear combination of three Liouville-type potentials. The thermodynamical quantities
are identified and in particular, a generalized Smarr formula is derived. It is shown that, although
the thermodynamic quantities are affected by the rainbow functions, the validity of the black hole
thermodynamical first law is supported. The thermodynamic stability of the solutions have been analyzed
through the black hole heat capacity. We have shown that, even in the presence of the rainbow functions,
the black holes can be locally stable in the sense that there exists a range of the horizon radiuses for
which the heat capacity is positive.
© 2018 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 more outstanding problems in the context of the-
oretical
physics is to combine the gravity with the quantum me-
chanics
and to construct the theory of quantum gravity. Despite
the attempts made through the string theory [1,2], loop quan-
tum
gravity [3], space time foam [4], Horava–Lifshitz gravity [5],
non-commutative geometry [6] and other approaches there is cur-
rently
no complete theory of quantum gravity and this problem
is still open. A common feature of almost all of these alternative
approaches is a strong interest on the modification of the usual
energy-momentum dispersion relation at the Planck-scale regime
[7,8]. It seems that this modification plays an important role in
establishing a full theory of quantum gravity, but it leads to the vi-
olation
of Lorentz invariance, as the most fundamental symmetry
in the universe. One of the approaches to overcome this problem
is the doubly or deformed special relativity [9,10]. The doubly spe-
cial
relativity, as a modified formalism of special relativity, has
been proposed to make the modified dispersion relation Lorentz
invariant [11–13]. In the high energy formalism of the special rel-
ativity,
known as the doubly special relativity, the speed of light
and the Planck energy are two Lorentz invariant quantities. Also,
the Planck energy and the speed of light are the upper bounds of
E-mail address: m.dehghani@razi.ac.ir.
the energy and velocity that a particle can attain [14]. This theory
is accomplished based on a nonlinear Lorentz transformation in
the momentum space in such a way that the energy-momentum
relation appears with the corrections in the order of the Planck
length and the Planck-scale corrected dispersion relation preserves
a deformed Lorentz symmetry [11,15].
The
doubly special relativity has now been extended to the
curved space times and a doubly general relativity or gravity’s rain-
bow
has been proposed by Magueijo and Smolin [16,17]. In the
rainbow gravity theory, the geometry of space time depends on the
energy of the test particle. Thus, it seems different for the parti-
cles
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 [18,19]
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 the rainbow functions. The rainbow functions are satisfied the
following conditions
lim
ε→0
f (ε) = 1, and lim
ε→0
g(ε) = 1. (1.2)
https://doi.org/10.1016/j.physletb.2018.08.045
0370-2693/
© 2018 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
.