Physics Letters B 799 (2019) 135037
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Thermodynamic properties of novel dilatonic BTZ black holes under
the influence of 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
18 August 2019
Received
in revised form 6 October 2019
Accepted
18 October 2019
Available
online 24 October 2019
Editor:
M. Cveti
ˇ
c
Keywords:
Three-dimensional
black holes
Einstein-dilaton
gravity theory
Rainbow
gravity
Through exact solution of the coupled scalar and gravitational field equations, in an energy dependent
spacetime, two classes of novel dilatonic BTZ black holes have been found. The black hole solutions have
only one horizon and their asymptotic behaviors are non-flat and non-AdS. The black hole mass, entropy
and temperature have been calculated, as the conserved and thermodynamic quantities, and it has been
shown that, although these quantities get modified in the presence of rainbow functions, they satisfy the
first law of black hole thermodynamics in its standard form. The black hole heat capacity and Gibbs free
energy have been calculated and the local and global stabilities of the black holes have been analyzed
making use of the canonical and grand canonical ensembles, respectively. Then, by considering the black
hole thermal fluctuations, the quantum gravitational effects on the local and global stabilities have been
studied.
© 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
One of the outstanding achievements in perturbative string the-
ory
and loop quantum gravity is the prediction of a fundamental
measurable length which is of the order of Planck length. Based
on the existence of such observer independent fundamental length
there are some interest, in almost all of the various quantum
gravity approaches, to promote the usual energy momentum re-
lation
to the well-known modified dispersion relation [1–3]. The
modified dispersion relation violates the Lorentz invariance. De-
formed
(modified) special relativity, as the Planck-scale version of
the usual special relativity, has been pout forwarded initially based
on the nonlinear Lorentz transformations to make the modified
dispersion relation Lorentz invariant. In the deformed special rela-
tivity
theory in addition to the speed of light Planck energy is an
invariant quantity too. The light speed and Planck energy are the
upper limit of the amount of speed and energy that a particle can
attain. It is evident that the modified dispersion relation reduces
to its usual form when the infrared limit is taken [4–8].
Gravity’s
rainbow is considered as a simple extension of the de-
formed
special relativity to include gravity. Indeed, gravity’s rain-
bow
is a deformed general theory of relativity in which the im-
pacts
of string theory and loop quantum gravity are taken into
account by considering the minimal measurable length. Thus, ac-
E-mail address: m.dehghani@razi.ac.ir.
cording to the correspondence principle, this theory is expected
to recover the standard general relativity at low energy regime.
In this regard, it is believed that this theory can be successful
in explaining the well-known problems of the standard theory of
gravity [9–11]. Now the gravity’s rainbow has been the subject of
many interesting works and a lot of papers have appeared in which
the thermodynamic properties of the black holes have been stud-
ied
at the framework of high energy physics by considering the
impacts of rainbow functions [12–15].
On
the other hand, Hawking et al. showed that black holes
are thermodynamic systems having well-defined thermodynamic
quantities such as temperature and entropy. Nowadays, study of
the black hole thermodynamic properties, and especially thermo-
dynamic
stability of the black holes, have attracted an increasing
interest and it is an important subject area in the context of black
hole physics. In the context of canonical ensemble, one is able to
analyze the local stability of the black holes by use of the black
hole heat capacity, with the black hole charge as a constant, or
noting the signature of the Hessian determinant. Geometric ther-
modynamics
is the other approach for studying the black hole
local stability [16–18]. Global stability of the black holes can be
investigated regarding the signature of the Gibbs free energy. Ther-
modynamic
local stability or phase transition of black holes in the
three- and four-dimensional gravity’s rainbow have been studied in
refs. [4,7,19,20]. Local and global stabilities as well as the Hawking-
Page
phase transition of dilatonic black holes with power-law elec-
trodynamics
have been studied in our previous work [21]. Here,
https://doi.org/10.1016/j.physletb.2019.135037
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© 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
.