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Regarding the significant interests in massive gravity and combining it with gravity’s rainbow and also BTZ black holes, we apply the formalism introduced by Jiang and Han in order to investigate the quantization of the entropy of black holes. We show that the entropy of BTZ black holes in massive gravity’s rainbow is quantized with equally spaced spectra and it depends on the black holes’ properties including massive parameters, electrical charge, the cosmological constant, and also rainbow functions. In addition, we show that quantization of the entropy results in the appearance of novel properties for this quantity, such as the existence of divergences, non-zero entropy in a vanishing horizon radius, and the possibility of tracing out the effects of the black holes’ properties. Such properties are absent in the non-quantized version of the black hole entropy. Furthermore, we investigate the effects of quantization on the thermodynamical behavior of the solutions. We confirm that due to quantization, novel phase transition points are introduced and stable solutions are limited to only de Sitter black holes (anti-de Sitter and asymptotically flat solutions are unstable).

General relativity (GR) is a successful theory of gravity with certain shortcomings; for example, the accelerated expansion of the universe, massive gravitons, and the ultraviolet (UV) behavior cannot be explained with GR. To address these and other issues, GR needs to be modified. There are some modified theories, such as Horava–Lifshitz gravity [

In order to understand the UV behavior of GR, various attempts have been made to obtain different models of UV completion of GR such that they should reduce to GR in the infrared (IR) limit. The first attempt in this field is related to Horava–Lifshitz gravity [

In other words, the geometry of spacetime is modified to be energy dependent and this energy dependency of the metric is incorporated through the introduction of rainbow functions. The standard energy–momentum relation in gravity’s rainbow is given as

(

(

(

In the gravity’s rainbow context, and by combining various gravities, the black hole and cosmological solutions have been studied in some works. For example,

On the other hand, and in order to have massive gravitons, GR must be modified, because the gravitons are massless particles in GR. Therefore, Fierz and Pauli were the first to study the theory describing massive gravitons (FP massive theory) [

The first three-dimensional black hole solution in the presence of the cosmological constant was obtained by Bañados, Teitelboim, and Zanelli, and is known as the BTZ black hole [

As the first cornerstone, the concept of Hawking radiation of black holes improved our knowledge of the quantum theory of gravity. Then, Bekenstein showed that there is a lower bound for the event horizon area of black holes [

It is notable that we can obtain

It is notable that, when a photon travels across the horizon of a black hole, the radial null path (or radial null geodesic) is given by

Using the above equation and Eq. (

In order to solve the adiabatic invariant quantity of Eq. (

The area spectrum and also the entropy spectrum spacing change with respect to the change in coordinate transformation. In other words, the adiabatic invariant quantity (

The metric of three-dimensional spacetime in the presence of the gravity’s rainbow is given by

The only non-zero term of massive gravity is

The metric function is obtained in this gravity as [

The electric potential (

Using the standard definition of the Hawking temperature (

Therefore, the Hawking temperature of these black holes is [

The total mass of these solutions is given by [

Here, we want to quantize the entropy of this black hole using the adiabatic invariant quantity and Bohr–Sommerfeld quantization rule. Considering Eqs. (

In order to solve the above equation we use the near-horizon approximation, so

The first term is zero (

The Smarr formula for a BTZ black hole in massive gravity’s rainbow is

Therefore, Eq. (

On the other hand, the Bohr–Sommerfeld quantization rule is given by

Comparing Eq. (

Quantization of the entropy has the following specific physical results:

(1) The quantization results in the formation of a spectrum of the entropy characterized by

(2) The entropy spectrum is a function of the black hole’s properties (the electric field, the massive parameters, the cosmological constant, and the gravity’s rainbow generalizations and horizon radius).

(3) While the usual entropy of black holes, Eq. (

(4) Quantization of the entropy could also provide the possibility of the formation of a root for this quantity. The root of the entropy is given by

(5) If the relation

(6) One of the most important results of the quantization is non-zero entropy for

The limit

The asymptotic behavior of quantized entropy is given by

To further clarify the effects of quantization on the thermodynamics of black holes, we investigate the heat capacity. The heat capacity gives a detailed picture regarding thermal/thermodynamical behavior of the solutions. In general, for black holes with quantized entropy, this quantity is given by

Evidently, by quantizing the entropy, the heat capacity is consequently quantized. But here, we should be a little bit cautious. The reason is that quantization is only done for the entropy while the temperature is not quantized. In addition, we have considered an ensemble where the electric charge and the potential are both fixed (canonical ensemble). The obtained heat capacity highlights several important contributions of quantized entropy:

(1) Here as well, due to quantization, a spectrum is formed by the heat capacity characterized by

(2) The positivity/negativity of the heat capacity determines the thermal stability/instability of the solutions. Therefore, the stability conditions are given by the following set of conditions:

(3) The root of the heat capacity is given by

(4) The divergences in heat capacity are where thermal phase transitions take place. The divergences of heat capacity are given by

The quantization has also resulted in a modification in the place and number of the divergences in heat capacity compared to the non-quantized case. Evidently, it is possible for the heat capacity to have up to three divergences (see the right panel of Fig.

In this paper, we considered BTZ black holes in the presence of massive gravity’s rainbow. We studied the quantization of entropy of these black holes using an adiabatic invariant integral method put forwarded by Majhi and Vagenas with modifications proposed by Jiang and Han, and the Bohr–Sommerfeld quantization rule.

It was shown that quantization of the entropy results in the formation of a spectrum of entropy. In addition, the quantized entropy leads to the existence of divergences and roots for the entropy that were absent in the usual entropy. It was also shown that in the high-energy limit and/or for a vanishing horizon radius the entropy has a non-zero value, which again was in contrast to the usual entropy. In general, the behavior of quantized entropy depends on the parameter of horizon radius, massive gravity, the rainbow functions, the cosmological and Newton constants, and also the electric charge. But it was shown that for specific choices of different parameters, the effects of the black holes’ properties (both gravitational and matter field contributions) could be cancelled, resulting in a fixed entropy. In other words, for this specific case, the quantized entropy of black holes was independent of the size, electric field, and other characteristics of the black holes. In addition, the heat capacity of solutions was investigated. It was shown that quantization resulted in the appearance of novel phase transition points in the structure of the black holes. Also, it was shown that due to quantization, only dS black holes could be thermally stable while asymptotically flat and AdS solutions were unstable.

BE and SHH thank the Shiraz University Research Council. This work has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha, Iran.

Open Access funding: SCOAP