# Greybody factor for black holes in dRGT massive gravity

## Abstract

In general relativity, greybody factor is a quantity related to the quantum nature of a black hole. A high value of greybody factor indicates a high probability that Hawking radiation can reach infinity. Although general relativity is correct and has been successful in describing many phenomena, there are some questions that general relativity cannot answer. Therefore, general relativity is often modified to attain answers. One of the modifications is the ‘massive gravity’. The viable model of the massive gravity theory belongs to de Rham, Gabadadze and Tolley (dRGT). In this paper, we calculate the gravitational potential for the de Sitter black hole and for the dRGT black hole. We also derive the rigorous bound on the greybody factor for the de Sitter black hole and the dRGT black hole. It is found that the structure of potentials determines how much the rigorous bound on the greybody factor should be. That is, the higher the potential, the lesser the bound on the greybody factor will be. Moreover, we compare the greybody factor derived from the rigorous bound with the greybody factor derived from the matching technique. The result shows that the rigorous bound is a true lower bound because it is less than the greybody factor obtained from the matching technique.

## 1 Introduction

General relativity was formulated in 1915. It offers profound insights into the concept of gravity. General relativity has succeeded in describing many gravitational phenomena such as the gravitational deflection of light, gravitational radiation, the anomalous perihelion of Mercury, and the behavior of black holes.

According to cosmological observations, todays the universe is expanding with acceleration [1, 2]. There are a number of cosmological models accounting for this current acceleration. Among these models, the simplest one is the Lambda cold dark matter (\(\Lambda \)CDM) model in which the cosmological constant drives the acceleration of the universe. The model assumes that general relativity is the correct theory of gravity on cosmological scales. Although the \(\Lambda \)CDM model is in excellent agreement with observations, the cosmological constant itself encounters a theoretical problem that the observational value of the cosmological constant is much smaller than the theoretical one. It may be possible that general relativity may not be the most suitable theory to describe the universe on a large scale, at least on a scale larger than the solar system. The dark side of the universe is evidence that some modifications of general relativity at a large scale is required [3].

Such modifications should be correct on the largest scale and then reduce to general relativity on a smaller scale. One of the candidates for such modifications is massive gravity. Massive gravity is a theory of gravity in which a graviton has a mass. The most successful theory of massive gravity is popularly known by the de Rham–Gabadadze–Tolley (dRGT) model [4, 5]. Because of the Vainshtein mechanism, adding mass to graviton keeps physics on a small scale equivalent to general relativity, with some small corrections [6]. However, this leads to the modification of gravity on a larger scale.

Regarding the cosmological solutions in the dRGT massive gravity theory, even though all the solutions cannot provide a viable cosmological model, for example, the solutions do not admit flat-FLRW metric [7, 8] or the model encounters instabilities [9, 10, 11], a class of solutions can provide a viable cosmological model [12, 13, 14]. The solutions for the dRGT massive gravity are not only investigated in a cosmological background, but also in a spherically symmetric background [15]. For spherically symmetric solutions, the black hole solutions have been investigated in both analytical [16, 17, 18, 19, 20, 21, 22, 23, 24, 25] and non-analytical [26, 27] forms, depending on the fiducial metric form. However, they still share the same property, which is represented as an asymptotic AdS/dS behavior.

A black hole can emit thermal radiation if the quantum effects are considered. This thermal radiation is known as Hawking radiation [28]. Hawking radiation propagates on spacetime, which is curved by the black hole. The curvature of spacetime acts as a gravitational potential. Therefore, Hawking radiation is scattered from this potential. One part of the Hawking radiation is reflected back into the black hole, while the other part is transmitted to spatial infinity. The transmission probability in this context is also known as the greybody factor.

There are many methods to calculate the greybody factor. For example, one can obtain an approximate greybody factor using the matching technique [29, 30, 31]. If the gravitational potential is high enough, one can use the WKB approximation to derive the greybody factor [32, 33, 34]. Other than approximation, the greybody factor can also be obtained using the rigorous bound [35, 36, 37]. The bound can give a qualitative description of a black hole.

In this work, we investigate the greybody factor using the analytical black hole solution in dRGT massive gravity. In Sect. 2, the structure of the horizons of the solution is analyzed in order to generate a suitable form for the analysis of the properties of the greybody factor. In Sect. 3, the properties of the gravitational potential are investigated for both the de Sitter black hole and dRGT black hole. The height of their potentials are determined by the parameters of the model. In Sect. 4, we derive the rigorous bound on the greybody factor, and the reflection probability for the de Sitter black hole and the dRGT black hole. The value of the rigorous bound on the greybody factor corresponds to the structure of potentials. In addition, the effects of the graviton mass, cosmological constant and angular momentum quantum number on the greybody factors will also be explored. Finally, concluding remarks are provided in Sect. 5.

## 2 dRGT black hole background

*R*is a Ricci scalar corresponding to a physical metric \(g_{\mu \nu }\), \(m^2_g\) to the square of the graviton mass, with \(\mathcal {L}_i\)s representing the interactions of the

*i*th order of the massive graviton. In particular, those interactions of the massive graviton are constructed from two kinds of metrics and can be expressed as follows,

*c*is a constant. It can be shown in [38] that the theory of massive gravity with the choice of the singular fiducial metric such as one in equation (6) is absent of Boulware–Deser (BD) ghost. The static and spherically symmetric black hole solution satisfying this theory can be written as [18]

*M*is an integration constant related to the mass of the black hole. The parameters above can be written in terms of the original parameters as

### 2.1 Horizon structure for AdS/dS-like solutions

*f*must be increased and then decreased where

*r*is increased. This means that \(f(r) \rightarrow -\infty \) again when \(r\rightarrow \infty \). Therefore, in order to obtain two horizons, \(\Lambda \) must be negative. This corresponds to the de Sitter (dS) spacetime, while in the case of anti-de Sitter (AdS) spacetime, \(\Lambda > 0\), there exists only one horizon. Now let us find the conditions for which there are two horizons for the de Sitter spacetime, where \(\Lambda < 0\). If two horizons exist, the maximum value of

*f*must be positive. The maximum point of

*f*can be found by solving \(f' = 0\). As a result, the maximum point is

*f*(

*r*) in Eq. (13), the maximum value of

*f*can be written as

*f*can be rewritten as

### 2.2 Horizon structure for the dRGT massive gravity solutions

For the complete massive gravity solution, it is significantly difficult and complicated to find the horizon analytically. One of the conditions for having three horizons is that \(\Lambda > 0\). Therefore, we can separate our consideration into two classes; the asymptotic AdS solutions for \(\Lambda > 0\) and the asymptotic de Sitter solution for \(\Lambda < 0\) [43].

*c*is not set to be zero. This means that we have to introduce a scale to the theory. It is useful to work out the solution using dimensionless variable, \(\tilde{r} = r/c\), and then find out what scale

*c*would assume. As a result, function

*f*can be written in terms of a dimensionless variable as

*c*takes place at \(\tilde{M} \sim \alpha _g\). Therefore, one can choose the parameter

*c*as

*r*through the equation

*f*using the equation \(f' = 0\) or

*f*at the extremum point can be written as

*f*at the extremum points can be written as

## 3 Equations of motion of massless scalar field

*V*(

*r*) is the potential given by

To see the effect of the parameter \(c_2\) on the potential, let us fix \(\beta _m = \beta _{mc}\). The peak of the potential is plotted as shown in the left panel of Fig. 5. The potential is also plotted with various values of \(c_2\) as shown in the right panel of Fig. 5. The parameter \(c_2\) characterizes the strength of the graviton mass. Therefore, the graviton mass will enhance the potential in contrast to the effect of the cosmological constant in the de Sitter black hole.

In this section, we explore the behavior of the gravitational potential for both the de Sitter black hole and the dRGT black hole of a massless scalar field. By making a comparison with the potential in the Schwarzschild black hole, we found that the local maximum of the de Sitter potential is always less than one of the Schwarzschild potential. For the dRGT black hole, the local maximum of the potential depend on the model parameters; \(\beta _m\) characterizing the existence of two horizon (\(0< \beta _m < 1 \)) and \(c_2\) characterizing the strength of the graviton mass. In contrast to the de Sitter potential, we found that the local maximum of the dRGT potential will be larger than ones for the Schwarzschild and the de Sitter potential by setting parameter \(\beta _m \ll 1\) or \(c_2 \ll -\,1\). In the same fashion as quantum theory, the shape of the potential has an effect on the transmission amplitude or the the greybody factor in this context. We will use the information of the potential to analyze the behavior of the greybody factor in the next section.

## 4 The rigorous bounds on the greybody factors

From a classical point of view, a black hole is believed to be black because nothing, when having entered the black hole, can escape, not even light. From a quantum point of view, however, a black hole is no longer considered ‘black’ since it has been proven to emit a type of thermal radiation known as Hawking radiation. At the event horizon of a black hole, Hawking radiation is exactly a blackbody spectrum. While Hawking radiation propagates out from the event horizon, it is, however, modified by the spacetime curvature generated by its black hole source. Thus, an observer at an infinite distance observes the modified form of Hawking radiation, which is different from the original Hawking radiation at the event horizon. This difference can be measured by the so-called greybody factor.

### 4.1 de Sitter black holes

### 4.2 dRGT black holes

By fixing \(c_2\), one can see that the bound crucially depends on \(|\tilde{R}_H - \tilde{r}_H|\), which is proportional to \(1/\beta _m\). Therefore, one finds that the larger the value of \(\beta _m\), the higher is the value of the bound. This can be seen explicitly by numerically plotting \(T_b\) with various values of \(\beta _m\) as illustrated in the right panel of Fig. 7. Moreover, this behavior can also be seen by analyzing the potential. From Fig. 4, we found that the larger the value of \(\beta _m\), the lower is the peak of the potential. Therefore, one finds that the larger the value of \(\beta _m\), the higher is the value of the bound.

## 5 Conclusion

In this paper, we obtain the gravitational potential from Schwarzschild black holes, de Sitter black holes, and dRGT black holes. We also derive the rigorous bound on the greybody factor for the de Sitter black hole and the dRGT black hole. It is found that the structure of potentials determines how much the rigorous bound on the greybody factor should be. Since \(V_{\max }\) contributed from a de Sitter black hole is always less than one in a Schwarzschild black hole, the bound for a de Sitter black hole is greater than one for a Schwarzschild black hole. In case of potentials with the same height, the result shows that the bound from a dRGT black hole is always larger than the bound from a de Sitter black hole. Otherwise, the bound from a dRGT black hole can be larger or smaller than ones from both de Sitter and Schwarzschild black holes due to different effects of the parameter \(c_{2}\) on de Sitter and dRGT spacetimes. Furthermore, we compare the greybody factor derived from the rigorous bound with the greybody factor derived from the matching technique. The results show that the greybody factor obtained from the rigorous bound is less than the one from the matching technique, which means that the rigorous bound is a true lower bound.

## Notes

### Acknowledgements

This project was funded by the Ratchadapisek Sompoch Endowment Fund, Chulalongkorn University (Sci-Super 2014-032), by a Grant for the professional development of new academic staff from the Ratchadapisek Somphot Fund at Chulalongkorn University, by the Thailand Research Fund (TRF), and by the Office of the Higher Education Commission (OHEC), Faculty of Science, Chulalongkorn University (RSA5980038). PB was additionally supported by a scholarship from the Royal Government of Thailand. TN was also additionally supported by a scholarship from the Development and Promotion of Science and Technology Talents Project (DPST). PW was also additional supported by the Naresuan University Research Fund through Grant no. R2559C235 and the ICTP through Grant no. OEA-NET-01

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