# Greybody factor for black string in dRGT massive gravity

## Abstract

The greybody factor from the black string in the de Rham–Gabadadze–Tolley (dRGT) massive gravity theory is investigated in this study. The dRGT massive gravity theory is one of the modified gravity theories used in explaining the current acceleration in the expansion of the universe. Through the use of cylindrical symmetry, black strings in dRGT massive gravity are shown to exist. When quantum effects are taken into account, black strings can emit thermal radiation, called Hawking radiation. The Hawking radiation at spatial infinity differs from that at the source by the so-called greybody factor. In this paper, we examine the rigorous bounds on the greybody factors from the dRGT black strings. The results show that the greybody factor crucially depends on the shape of the potential, which is characterised by the model parameters. The results agree with ones in quantum mechanics; the higher the potential, the harder it is for the waves to penetrate, and also the lower the bound for the rigorous bounds.

## 1 Introduction

Based on cosmological observations, our universe is expanding with an acceleration [1, 2]. However, the explanation for this phenomenon remains unclear. Many authors propose the existence of exotic matter called dark energy to explain this observed cosmic acceleration. On the other hand, some authors modify gravity without dark energy. One of the modifications of gravity is to give mass to the graviton. The de Rham–Gabadadze–Tolley (dRGT) models [3, 4] are successful and viable models of massive gravity. Reviews of the theory of massive gravity can be found in [5, 6]. For spherical symmetry, the black hole solutions have also been found, and their thermodynamics properties extensively investigated [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34].

When quantum effects are taken into account, black holes can emit thermal radiation called Hawking radiation [35]. The original Hawking radiation emitted from a black hole is blackbody radiation. Due to the curvature of spacetime, the Hawking radiation is modified, while propagating to spatial infinity. The radiation at spatial infinity differs from that at the emitter by the so-called greybody factor. There are various methods to find the greybody factors, such as the matching technique and the WKB approximation [36, 37, 38, 39, 40, 41, 42, 43, 44]. Another interesting method is to bound the greybody factor from below [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55].

Besides the solution with spherical symmetry, the solution to the Einstein field equation in the case of cylindrical symmetry has also been investigated and is known as the black string solution [56, 57]. This solution can be achieved by introducing the cosmological constant into the Einstein field equation. The charge and the rotating black string solutions can also be found [58]. The quasinormal modes [59] and the greybody factor of the black string have been investigated [60].

As is well known, the dRGT massive gravity theory can provide a more general solution than the Schwarzschild-dS/AdS. Therefore, it is possible to obtain the cylindrical solution in the dRGT massive gravity theory [61]. The rotating solutions and their thermodynamic properties are also investigated [62]. The quasinormal mode for the dRGT black string solution have been investigated as well [63], while the greybody factor has not been investigated yet. In the present work, the rigorous bounds on the greybody factor from the dRGT black strings are examined.

This paper is organised as follows. In Sect. 2, the background of the dRGT black string is presented. The horizon structures are analysed in Sect. 3. The equation of motion of the massless scalar field emitted from a dRGT black hole and the gravitational potential which modifies the scalar field are derived in Sect. 4. The rigorous bounds on the greybody factors are calculated in Sect. 5, and the conclusions are given in Sect. 6.

## 2 dRGT black string background

*R*is the Ricci scalar, \(\mathcal{U}\) is a potential term used in characterising the behaviour of the mass term of graviton, and \(m_g\) is the parameter interpreted as the graviton mass. The suitable form of the potential \(\mathcal{U}\) in four-dimensional spacetime is given by

*f*(

*r*) in the physical metric can be written as [61]

The solution in Eq. (11), including the function *f* in Eq. (13), is an exact black string solution in dRGT massive gravity which, in the limit \( c_2 = \alpha _g^2\) and \(c_0=c_1=0\), naturally goes over to Lemos’ black string in GR with cosmological constant [56, 57]. In particular, it incorporates the cosmological constant term (\(c_2\) term) naturally in terms of the graviton mass. Moreover, this solution also provides a global monopole term (\(c_0\) term) and another non-linear scale term (\(c_1\) term).

It is important to note that the strong coupling scale of the dRGT massive gravity theory is \({\varLambda }_3^{-1} = (m_g^2 M_{Pl})^{1/3} \sim 10^{3} ~\text {km} \ll r_V \sim 10^{16} ~\text {km}\), so that we do not have to worry about the strong coupling issue in dRGT massive gravity for a system of scale below \({\varLambda }_3\) (or of a length scale beyond \(\sim 10^3\) km), where \(r_V\) is the Vainshtein radius characterised by the non-linear scale of the massive gravity theory [61].

One can see that the horizon structure depends on the sign of \(c_2\). If \(c_2>0\), corresponding to the anti-de Sitter-like solution, the maximum number of horizons is three. If \(c_2<0\), corresponding to the de Sitter-like solution, the maximum number of horizons is two. This behaviour is explicitly shown in the next section.

## 3 Horizon structure

In order to investigate the structure of the horizons for the solution in Eq. (11), where *f* is in Eq. (13), one has to find the number of possible extremum points. As a result, this depends on the asymptotic behaviour of the solution. For the asymptotic dS solution, \(c_2 < 0\), the solution becomes the dS black string for the large-*r* limit, while the solution becomes the AdS black string for the large-*r* limit of the asymptotic AdS solution, \(c_2 > 0\). As a result, one can find the conditions to obtain one positive real maximum of *f* for the asymptotic dS solution. For the asymptotic AdS solution, one can find the conditions to have one positive real maximum and one negative real minimum *f*. We will investigate this behaviour separately in the following subsection.

It is important to note that by choosing the fiducial metric as \(h_0 = 0\), the solution becomes AdS/dS black string solution. This is not surprising since the potential term becomes a constant.

### 3.1 Asymptotic dS solution

*r*, by

*f*at the extremum can be written as

*f*, and then finding the solution of \(f =0\) for

*r*, one obtains two horizons as follows:

One can see that we now have two parameters, \(c_2\) and \(\beta _m\), controlling the behaviour of the horizons. The parameter \(c_2\) controls the strength of the graviton mass or the cosmological constant, while \(\beta _m\) controls the existence of the horizons. For \(0< \beta _m <1\) there are no horizons, while for \(\beta _m > 1\) there are two horizons. Two such horizons become closer and closer when \(\beta _m\) approaches 1, and thus the two horizons merge at \(\beta _m = 1\), as shown in Fig. 1.

It is useful to emphasise here that our choice, \(c_1 = 6 (M c_2^2)^{1/3}\), provides only a class of conditions characterising the existence of the horizons. It is not valid in general. For example, for \(c_0 =0\), corresponding to \(\beta _m=0\), it is still possible to find the parameter space for \(c_2\) and \(c_1\) so as to have two horizons. Even though this choice and the set of parameters (\(c_2, \beta _m\)) provide a loss of generality of parameter space, it provides us with a qualitative way to analyse the effects of the horizon structure on the potential and the greybody factor. This will be explicitly shown in Sects. 4 and 5.

It is important to note that the existence of parameters, \(c_1\) and \(c_0\), is characterised by the structure of the dRGT massive gravity theory, which provides an additional part to the usual dS black string solution [56, 57]. From Eq. (13), one can see that without these parameters (\(c_2 <0, c_1=0, c_0=0\)), it is not possible to have a horizon since *f* is always negative; therefore, it is not possible to investigate the thermodynamics of the black string or find the greybody factor for the dS black string solution. This is a crucial issue for the dRGT massive gravity black string solution, and we will investigate this issue in the next section.

### 3.2 Asymptotic AdS solution

*f*at the extrema can be written as

*f*, and then finding the solution of \(f =0\) for

*r*, one obtains three horizons as follows:

## 4 Equations of motion of the massless scalar field

*Y*satisfies the equation

*V*(

*r*) is the potential given by

*f*from Eq. (13), and then reparametrise the parameters in terms of \(\beta _m\) and \(c_2\). As a result, by fixing \(c_2\), and then varying \(\beta _m\), the behaviour of the potential in both the asymptotic dS and the asymptotic AdS solutions can be illustrated by Fig. 3. From the left panel of this figure (the asymptotic dS case), one can see that the potential becomes lower when the parameter \(\beta _m\) approaches 1. In other words, when the horizons become closer, the potential becomes lower and lower. This gives a hint to us that the greybody factor bound will be higher when the horizons become closer. This analysis is also valid for the asymptotic AdS case. We will consider this analysis in detail in the next section.

## 5 The rigorous bounds on the greybody factors

*f*in Eq. (13), the potential is

*f*(

*r*) is given by Eq. (13). From Eq. (35), the rigorous bound on the greybody factor given by Eq. (39) becomes

Now, let us consider the asymptotic AdS solution. As we have discussed, it is possible to obtain the three horizons for this kind of solutions. In this case, one may have to assume the place of the observer. As a result, we can divide our consideration into two cases; the observer being between the first and the second horizons, and the observer being between the second and the third horizons. From Fig. 2, one finds that three horizons exist if \(\sqrt{3}/2< \beta _m < 1\). For \(\beta _m = \sqrt{3}/2\), the first and the second horizons are sunk together, and for \(\beta _m = 1\), the second and the third horizons are sunk together.

## 6 Conclusion

In this paper, we investigated the greybody factor of the black string in dRGT massive gravity theory by using the rigorous bound. In order to properly study the dRGT black string, we first investigated the horizon structures of the dRGT black string. We defined the new model parameter \(\beta _m\) to characterise the existence of the horizons. The results show that, for the asymptotic dS solution, there are two horizons when \(\beta _{m} > 1\), and for the asymptotic AdS solution, there are three horizons when \(\sqrt{3}/2< \beta _{m} < 1\). By considering a massless uncharged scalar field emitted from the dRGT black string as Hawking radiation, a Schrödinger-like equation is obtained for the radial part of the solution. This allows us to consider the behaviour of the potential for investigating the greybody factor. It is found that the height of the potential becomes lower when the parameter \(\beta _m\) approaches 1 for the asymptotic dS solution, while \(\beta _m\) approaches \(1, \sqrt{3}/2\) for the asymptotic AdS solution where two horizons are merged. Moreover, rigorous bounds on the greybody factors have also been calculated. It is found that the greybody factor bound can be qualitatively analysed by using a certain form of the potential; the higher the value of the potential, the more difficult it is for the waves to be transmitted and then the lower the bound of the greybody factor. This result is valid for both the asymptotic AdS solution and the asymptotic dS solution, and also it was checked by numerical methods. Since our analysis/results are similar to ones in quantum mechanics, it provides us with an easier way to deal with the quantum nature of black holes or black strings, even though a complicated form of spacetime is considered.

## 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 Ratchada pisek 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 supported by the Thailand Research Fund (TRF) through grant no. MRG6180003 and partially supported by the ICTP through grant no. OEA-NT-01.

## References

- 1.Supernova Search Team Collaboration, A. G. Riess et al., Astron. J.
**116**, 1009–1038 (1998). arXiv:astro-ph/9805201 - 2.Supernova Cosmology Project Collaboration, S. Perlmutter et al., Astrophys. J.
**517**, 565–586 (1999). arXiv:astro-ph/9812133 - 3.C. de Rham, G. Gabadadze, Generalization of the Fierz-Pauli action. Phys. Rev. D
**82**, 044020 (2010). arXiv: 1007.0443 [hep-th]ADSCrossRefGoogle Scholar - 4.C. de Rham, G. Gabadadze, A.J. Tolley, Resummation of massive gravity. Phys. Rev. Lett.
**106**, 231101 (2011). arXiv: 1011.1232 [hep-th]ADSCrossRefGoogle Scholar - 5.K. Hinterbichler, Theoretical aspects of massive gravity. Rev. Mod. Phys.
**84**, 671–710 (2012). arXiv: 1105.3735 [hep-th]ADSCrossRefGoogle Scholar - 6.C. de Rham, Massive gravity. Living Rev. Relat.
**17**, 7 (2014). https://doi.org/10.12942/lrr-2014-7. arXiv:1401.4173 [hep-th]ADSCrossRefzbMATHGoogle Scholar - 7.D. Vegh, Holography without translational symmetry. (2013) arXiv:1301.0537 [hep-th]
- 8.R.G. Cai, Y.P. Hu, Q.Y. Pan, Y.L. Zhang, Thermodynamics of black holes in massive gravity. Phys. Rev. D
**91**, 024032 (2015). arXiv:1409.2369 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 9.S.G. Ghosh, L. Tannukij, P. Wongjun, A class of black holes in dRGT massive gravity and their thermodynamical properties. Eur. Phys. J. C
**76**(3), 119 (2016). arXiv:1506.07119 [gr-qc]ADSCrossRefGoogle Scholar - 10.A. Adams, D.A. Roberts, O. Saremi, Hawking-Page transition in holographic massive gravity. Phys. Rev. D
**91**(4), 046003 (2015). arXiv:1408.6560 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 11.J. Xu, L.M. Cao, Y.P. Hu, P-V criticality in the extended phase space of black holes in massive gravity. Phys. Rev. D
**91**, 124033 (2015). arXiv:1506.03578 [gr-qc]ADSCrossRefGoogle Scholar - 12.T.M. Nieuwenhuizen, Exact Schwarzschild–de Sitter black holes in a family of massive gravity models. Phys. Rev. D
**84**, 024038 (2011). arXiv:1103.5912 [gr-qc]ADSCrossRefGoogle Scholar - 13.R. Brito, V. Cardoso, P. Pani, Black holes with massive graviton hair. Phys. Rev. D
**88**, 064006 (2013). arXiv:1309.0818 [gr-qc]ADSCrossRefGoogle Scholar - 14.L. Berezhiani, G. Chkareuli, C. de Rham, G. Gabadadze, A.J. Tolley, On black holes in massive gravity. Phys. Rev. D
**85**, 044024 (2012). arXiv:1111.3613 [hep-th]ADSCrossRefGoogle Scholar - 15.Y.F. Cai, D.A. Easson, C. Gao, E.N. Saridakis, Charged black holes in nonlinear massive gravity. Phys. Rev. D
**87**, 064001 (2013). arXiv:1211.0563 [hep-th]ADSCrossRefGoogle Scholar - 16.E. Babichev, A. Fabbri, A class of charged black hole solutions in massive (bi)gravity. JHEP
**1407**, 016 (2014). arXiv:1405.0581 [gr-qc]ADSCrossRefGoogle Scholar - 17.M.S. Volkov, Self-accelerating cosmologies and hairy black holes in ghost-free bigravity and massive gravity. Class. Quantum Gravity
**30**, 184009 (2013). arXiv:1304.0238 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 18.E. Babichev, R. Brito, Black holes in massive gravity. Class. Quantum Gravity
**32**, 154001 (2015). arXiv:1503.07529 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 19.F. Capela, P.G. Tinyakov, Black hole thermodynamics and massive gravity. JHEP
**1104**, 042 (2011). arXiv:1102.0479 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 20.M.S. Volkov, Hairy black holes in the ghost-free bigravity theory. Phys. Rev. D
**85**, 124043 (2012). arXiv:1202.6682 [hep-th]ADSCrossRefGoogle Scholar - 21.Y.P. Hu, X.M. Wu, H. Zhang, Generalized vaidya solutions and misner-sharp mass for \(n\)-dimensional massive gravity. Phys. Rev. D
**95**(8), 084002 (2017). arXiv:1611.09042 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 22.Y.P. Hu, X.X. Zeng, H.Q. Zhang, Holographic thermalization and generalized vaidya-AdS solutions in massive gravity. Phys. Lett. B
**765**, 120 (2017). arXiv:1611.00677 [hep-th]ADSCrossRefGoogle Scholar - 23.D.C. Zou, R. Yue, M. Zhang, Reentrant phase transitions of higher-dimensional AdS black holes in dRGT massive gravity. Eur. Phys. J. C
**77**(4), 256 (2017). arXiv:1612.08056 [gr-qc]ADSCrossRefGoogle Scholar - 24.S.H. Hendi, R.B. Mann, S. Panahiyan, B.E. Panah, Van der Waals like behavior of topological AdS black holes in massive gravity. Phys. Rev. D
**95**(2), 021501 (2017)ADSMathSciNetCrossRefGoogle Scholar - 25.S.H. Hendi, G.H. Bordbar, B.E. Panah, S. Panahiyan, Neutron stars structure in the context of massive gravity. JCAP
**1707**, 004 (2017). arXiv:1701.01039 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 26.S.H. Hendi, B.E. Panah, S. Panahiyan, M.S. Talezadeh, Geometrical thermodynamics and P-V criticality of black holes with power-law Maxwell field. Eur. Phys. J. C
**77**(2), 133 (2017). arXiv:1612.00721 [hep-th]ADSCrossRefGoogle Scholar - 27.B.E. Panah, S. Panahiyan, S.H. Hendi, Entropy spectrum of charged BTZ black holes in massive gravity’s rainbow. PTEP
**2019**, 013 (2019). arXiv:1611.10151 [hep-th]MathSciNetGoogle Scholar - 28.S.H. Hendi, S. Panahiyan, S. Upadhyay, B.E. Panah, Charged BTZ black holes in the context of massive gravitys rainbow. Phys. Rev. D
**95**(8), 084036 (2017). arXiv:1611.02937 [hep-th]ADSCrossRefGoogle Scholar - 29.S.H. Hendi, N. Riazi, S. Panahiyan, Holographical aspects of dyonic black holes: massive gravity generalization. Ann. Phys.
**530**(2), 1700211 (2018). arXiv:1610.01505 [hep-th]CrossRefGoogle Scholar - 30.S.H. Hendi, G.Q. Li, J.X. Mo, S. Panahiyan, B.E. Panah, New perspective for black hole thermodynamics in Gauss–Bonnet–Born–Infeld massive gravity. Eur. Phys. J. C
**76**(10), 571 (2016). arXiv:1608.03148 [gr-qc]ADSCrossRefGoogle Scholar - 31.I. Arraut, The black hole radiation in massive gravity. Universe
**4**(2), 27 (2018). arXiv:1407.7796 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 32.I. Arraut, Komar mass function in the de Rham–Gabadadze–Tolley nonlinear theory of massive gravity. Phys. Rev. D
**90**, 124082 (2014). arXiv:1406.2571 [gr-qc]ADSCrossRefGoogle Scholar - 33.I. Arraut, “On the apparent loss of predictability inside the de-Rham–Gabadadze–Tolley non-linear formulation of massive gravity: the Hawking radiation effect. EPL
**109**(1), 10002 (2015). arXiv:1405.1181 [gr-qc]ADSCrossRefGoogle Scholar - 34.H. Kodama, I. Arraut, Stability of the Schwarzschildde Sitter black hole in the dRGT massive gravity theory. PTEP
**2014**, 023E02 (2014). arXiv:1312.0370 [hep-th]zbMATHGoogle Scholar - 35.S.W. Hawking, Particle creation by black holes. Commun. Math. Phys.
**43**, 199 (1975)ADSMathSciNetCrossRefGoogle Scholar - 36.M.K. Parikh, F. Wilczek, Hawking radiation as tunneling. Phys. Rev. Lett.
**85**, 5042–5045 (2000). arXiv:hep-th/9907001 ADSMathSciNetCrossRefGoogle Scholar - 37.C.H. Fleming, Hawking radiation as tunneling. (2005). http://www.physics.umd.edu/grt/taj/776b/fleming.pdf
- 38.S. Fernando, Greybody factors of charged dilaton black holes in 2 + 1 dimensions. Gen. Relativ. Gravit.
**37**, 461–481 (2005)ADSMathSciNetCrossRefGoogle Scholar - 39.P. Lange, Calculation of Hawking radiation as quantum mechanical tunneling. Thesis, Uppsala Universitet (2007)Google Scholar
- 40.W. Kim, J.J. Oh, Greybody factor and Hawking radiation of charged dilatonic black holes. JKPS
**52**, 986–991 (2008)ADSCrossRefGoogle Scholar - 41.J. Escobedo, Greybody factors Hawking radiation in disguise. Masters Thesis, University of Amsterdam (2008)Google Scholar
- 42.T. Harmark, J. Natario, R. Schiappa, Greybody factors for d-dimensional black holes. Adv. Theor. Math. Phys.
**14**, 727 (2010). arXiv:0708.0017 [hep-th]MathSciNetCrossRefGoogle Scholar - 43.P. Kanti, T. Pappas, N. Pappas, Greybody factors for scalar fields emitted by a higher-dimensional Schwarzschild–de-Sitter black-hole. Phys Rev D
**90**, 124077 (2014). arXiv:1409.8664 [hep-th]ADSCrossRefGoogle Scholar - 44.R. Dong, D. Stojkovic, Greybody factors for a black hole in massive gravity. Phys. Rev. D
**92**, 084045 (2015). arXiv:1505.03145 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 45.M. Visser, Some general bounds for 1-D scattering. Phys. Rev. A
**59**, 427438 (1999). arXiv:quant-ph/9901030 CrossRefGoogle Scholar - 46.P. Boonserm, M. Visser, Bounding the Bogoliubov coefficients. Ann. Phys.
**323**, 2779–2798 (2008). arXiv:0801.0610 [quant-ph]ADSMathSciNetCrossRefGoogle Scholar - 47.P. Boonserm, M. Visser, Bounding the greybody factors for Schwarzchild black holes. Phys. Rev. D
**78**, 101502 (2008). arXiv:0806.2209 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 48.P. Boonserm, Rigorous bounds on transmission, reflection, and Bogoliubov coefficients. Ph.D. Thesis, Victoria University of Wellington (2009). arXiv:0907.0045 [mathph]
- 49.T. Ngampitipan, P. Boonserm, Bounding the greybody factors for non-rotating black holes. Int. J. Mod. Phys. D
**22**, 1350058 (2013). arXiv:1211.4070 [math-ph]ADSCrossRefGoogle Scholar - 50.P. Boonserm, T. Ngampitipan, M. Visser, Regge–Wheeler equation, linear stability and greybody factors for dirty black holes. Phys. Rev. D
**88**, 041502 (2013). arXiv:1305.1416 [gr-qc]ADSCrossRefGoogle Scholar - 51.P. Boonserm, T. Ngampitipan, M. Visser, Bounding the greybody factors for scalar field excitations of the Kerr–Newman spacetime. J. High Energy Phys.
**2014**, 113 (2014). arXiv:1401.0568 [gr-qc]CrossRefGoogle Scholar - 52.P. Boonserm, A. Chatrabhuti, T. Ngampitipan, M. Visser, Greybody factors for Myers–Perry black holes. J. Math. Phys.
**55**, 112502 (2014). https://doi.org/10.1063/1.4901127. arXiv:1405.5678 [gr-qc]ADSMathSciNetCrossRefzbMATHGoogle Scholar - 53.T. Ngampitipan, Rigorous bounds on greybody factors for various types of black holes. Ph.D. Thesis, Chulalongkorn University (2014)Google Scholar
- 54.T. Ngampitipan, P. Boonserm, P. Wongjun, Bounding the greybody factor, temperature and entropy of black holes in dRGT massive gravity. Am. J. Phys. Appl.
**4**, 64 (2016)Google Scholar - 55.P. Boonserm, T. Ngampitipan, P. Wongjun, Greybody factor for black holes in dRGT massive gravity. Eur. Phys. J. C
**78**, 492 (2018). arXiv:1705.03278 [gr-qc]ADSCrossRefGoogle Scholar - 56.J.P.S. Lemos, Cylindrical black hole in general relativity. Phys. Lett. B
**353**, 46 (1995). arXiv:gr-qc/9404041 ADSMathSciNetCrossRefGoogle Scholar - 57.J.P.S. Lemos, Two-dimensional black holes and planar general relativity. Class. Quantum Gravity
**12**, 1081 (1995). arXiv:gr-qc/9407024 ADSMathSciNetCrossRefGoogle Scholar - 58.J.P.S. Lemos, V.T. Zanchin, Rotating charged black string and three-dimensional black holes. Phys. Rev. D
**54**, 3840 (1996). arXiv:hep-th/9511188 ADSMathSciNetCrossRefGoogle Scholar - 59.V. Cardoso, J.P.S. Lemos, Quasinormal modes of toroidal, cylindrical and planar black holes in anti-de Sitter space-times. Class. Quantum Gravity
**18**, 5257 (2001). arXiv:gr-qc/0107098 ADSCrossRefGoogle Scholar - 60.J. Ahmed, K. Saifullah, Greybody factor of scalar fields from black strings. Eur. Phys. J. C
**77**, 885 (2017). arXiv:1712.07574 [gr-qc]ADSCrossRefGoogle Scholar - 61.S.G. Ghosh, L. Tannukij, P. Wongjun, Black string in dRGT massive gravity. Eur. Phys. J. C
**77**(12), 846 (2017). arXiv:1701.05332 [gr-qc]ADSCrossRefGoogle Scholar - 62.S.G. Ghosh, R. Kumar, L. Tannukij, P. Wongjun, Rotating black string in dRGT massive gravity (2019). arXiv:1903.08809 [gr-qc]
- 63.S. Ponglertsakul, P. Burikham, L. Tannukij, Quasinormal modes of black strings in de RhamGabadadzeTolley massive gravity. Eur. Phys. J. C
**78**(7), 584 (2018). arXiv:1803.09078 [gr-qc]ADSCrossRefGoogle Scholar - 64.T. Chullaphan, L. Tannukij, P. Wongjun, Extended DBI massive gravity with generalized fiducial metric. JHEP
**06**, 038 (2015). arXiv:1502.08018 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 65.L. Tannukij, P. Wongjun, Mass-varying massive gravity with k-essence. Eur. Phys. J. C
**76**(1), 17 (2016)ADSCrossRefGoogle Scholar - 66.R. Nakarachinda, P. Wongjun, Cosmological model due to dimensional reduction of higher-dimensional massive gravity theory. Eur. Phys. J. C
**78**(10), 827 (2018). arXiv:1712.09349 [gr-qc]ADSCrossRefGoogle Scholar

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