Physics Letters B 803 (2020) 135335
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Nonlinearly charged AdS black hole solutions in three-dimensional
massive 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 14 January 2020
Received in revised form 16 February 2020
Accepted 21 February 2020
Available online 26 February 2020
Editor: N. Lambert
Keywords:
Three-dimensional black hole
Nonlinear electrodynamics
Massive gravity theory
Gravity’s rainbow
Two new families of nonlinearly charged and asymptotically anti-de Sitter (AdS) rainbow black holes
have been introduced, as the exact solutions to the coupled electromagnetic and gravitational field
equations, in massive gravity theory and in the presence of power-law nonlinear electrodynamics. The
conserved and thermodynamic quantities such as black hole mass, charge, temperature, entropy and
electric potential have been calculated from geometric and thermodynamic approaches. Despite the fact
that some of them receive corrections from rainbow functions and nonlinear electrodynamics, it has
been proved that they fulfill the first law of black hole thermodynamics. Thermal stability of the black
holes has been studied by use of the canonical ensemble and geometrical thermodynamics methods,
separately. Regarding the black hole heat capacity and thermodynamic Ricci scalars, the points of type-
one
and type-two phase transitions and the conditions under which the black holes remain locally stable
have been determined. Also, global stability of the novel rainbow black holes has been investigated from
the viewpoint of the grand canonical ensemble. Through calculation of Gibbs free energy of the black
holes, the points of Hawking-Page phase transition and the size of the black holes which are globally
stable or in the radiative phase have been determined.
© 2020 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
Einstein’s general relativity, as the relativistic theory of gravity, predicts the existence of massless spin-2 particles which are named as
gravitons. Alternative models of the so-called modified gravity theory are proposed with the aim of solving the failures of this theory. It is
well-known that the accelerated expansion of the universe in the large scale structure cannot be explained by Einstein’s theory of gravity.
Nowadays, it is believed that this phenomena can be explained if an unknown source of energy, named as dark energy, exist [1–4].
Recent studies on the gravitational waves show that propagation speed of gravitational waves is slightly different from the speed of
light. Observation of the cosmic rays, originated in our Galaxy, on the earth constraints the gravitational wave speed to c − v
g
< 10
−15
c.
Also, for the cosmic rays with extragalactic origin the speed of gravitational wave is bounded to c − v
g
< 10
−19
c [5–7]. This implies that
gravitons carry a nonzero amount of mass. According to the reports of LIGO scientific Collaboration and Virgo Collaboration, by assumption
that gravitons are dispersed in vacuum like massive particles, the graviton mass is bounded to m
g
≤ 7.7 × 10
−23
ev/c
2
[8,9]. Massive
gravity theory is one of the alternatives that modifies Einstein gravity by giving the graviton a mass and provides a possible explanation
for the accelerated expansion of the universe without requirement of dark energy component [10–12]. It is evident that massive gravity
reduces to the usual gravity theory when the mass of graviton is taken equal to zero. Massive gravity theory is an interesting research
topic and its gravitational and cosmological aspects have been studied by many authors [3,13,14] (See also [15] and references therein).
In addition, gravity’s rainbow, as the generalization of doubly special relativity to the case of curved spacetimes, is an attempt to
constructing the quantum theory of gravity. In this theory, the spacetime geometry depends on the energy of the test particle which is
used to probe the gravity. Thus, the particles with different amount of energy experience different geometry. In this regard, there is a
rainbow of energy-dependent metrics, and this theory of doubly general relativity is named as gravity’s rainbow [16,17]. It must be noted
that all the energy dependency of the spacetime is given by the rainbow functions. This theory is considerable in the high energy regime
when the amount of energies are in the order of Planck energy. In the infrared limit, when the energies are negligible in comparison to
E-mail address: m.dehghani@razi.ac.ir.
https://doi.org/10.1016/j.physletb.2020.135335
0370-2693/© 2020 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
.