Skip to main content
SpringerLink
Log in

Gauss-bonnet modification to Hawking evaporation of AdS black holes in massive gravity

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

The Stefan-Boltzmann law can estimate particle emission power and lifetime of a black hole. However, in modified gravity theories, new parameters in the action can cause qualitative changes in thermodynamic quantities, thus obtaining specific thermodynamic properties often requires complicated calculation with higher degree equations. In this work, we aim to provide a general model-independent description of the evolution of AdS black holes, using Gauss-Bonnet massive gravity as an example. We prove that the impact factor of an infinitely large AdS black hole is equal to the effective AdS radius, and the black hole is able to evaporate an infinite amount of mass in finite time, so that the lifetime of the black hole depends on the final state temperature in the evaporation process. The black hole will evaporate out when the final state temperature diverges and will transform into a remnant when the temperature is zero. Since we have analyzed massive gravity in detail, we can introduce the Gauss-Bonnet term to study how it affects the thermodynamic quantities. We obtain the final states for different parameter intervals, study the qualitative properties of black hole evaporation, and classify the associated cases. This method can also be applied to other models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Data Availability

No Data associated in the manuscript.

Notes

  1. We can not discuss infinitely large black hole in de Sitter spacetime. Given the relevant parameters, there is an upper limit to the mass of black holes in de Sitter spacetime, so increasing the radius of a black hole causes its radius to converge to the cosmological horizon, eventually reaching the Nariai limit [45, 46]

  2. Note that our action here is not exactly the same as in the [55]. In particular, our cosmological constant are introduced by hand, whereas in [55] the effective cosmological constant term emerged from the graviton mass. But this does not matter, because the final form of the metric and the black hole thermodynamics are still the same.

  3. Here we assume that \(r_p\) must exists. If \(r_p\) does not exist, the impact factor is directly \(\ell _{\text {eff}}\).

  4. In the Schwarzschild AdS case, we have \(a=0\) and \(n=1\), so the \(r_p\sim M \sim r_+^3\).

  5. We could also get the Gauss-Bonnet result first and then introduce massive gravity as a correction. However, the massive gravity itself is more complicated and has several parameters, while there is only one parameter \(\alpha \) in the Gauss-Bonnet gravity, so it is more convenient to use the Gauss-Bonnet term as a correction.

  6. For \(\alpha =0\), in \(\gamma ^2\ell ^2<3(\varepsilon +1)\), the roots are not real, and in \(\gamma ^2\ell ^2>3(\varepsilon +1)\), \(\gamma >0\), \(\varepsilon +1>0\), the roots are negative [55].

  7. The Gauss-Bonnet coefficient \(\alpha \) is supposed to be postive in string theory [16], but for generality we consider both cases of \(\alpha >0\) and \(\alpha <0\) in the present work. See e.g. [59] for more discussion in the issue.

  8. Under certain parameter choices, points b and c may no longer exist, and M will become a monotonically increasing function of \(r_+\). This is similar to the case of \(\sqrt{-2\alpha }>r_c\), where the final states of the black hole are \(T = \infty \) and the black hole will have completely evaporated.

References

  1. M.E. Fierz, W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field. Proc. Roy. Soc. Lond. A 173, 211 (1939). https://doi.org/10.1098/rspa.1939.0140

    Article  ADS  MathSciNet  Google Scholar 

  2. P.V. Nieuwenhuizen, On ghost-free tensor lagrangians and linearized gravitation. Nucl. Phys. B 60, 478 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  3. H. van Dam, M.J.G. Veltman, Massive and massless yang-mills and gravitational fields. Nucl. Phys. B 22, 397 (1970)

    Article  ADS  Google Scholar 

  4. V.I. Zakharov, Linearized gravitation theory and the graviton mass. JETP Lett. 12, 312 (1970)

    ADS  Google Scholar 

  5. A.I. Vainshtein, To the problem of nonvanishing gravitation mass. Phys. Lett. B 39, 393 (1972)

    Article  ADS  Google Scholar 

  6. D.G. Boulware, S. Deser, Can gravitation have a finite range? Phys. Rev. D 6, 3368 (1972). https://doi.org/10.1103/PhysRevD.6.3368

    Article  ADS  Google Scholar 

  7. C. de Rham, G. Gabadadze, Generalization of the fierz-pauli action. Phys. Rev. D 82, 044020 (2010). https://doi.org/10.1103/PhysRevD.82.044020

    Article  ADS  Google Scholar 

  8. C. de Rham, G. Gabadadze, A.J. Tolley, Resummation of massive gravity. Phys. Rev. Lett. 106, 231101 (2011)

    Article  ADS  Google Scholar 

  9. S.F. Hassan, R.A. Rosen, On non-linear actions for massive gravity. JHEP 07, 009 (2011). https://doi.org/10.1007/JHEP07(2011)009

    Article  ADS  MathSciNet  Google Scholar 

  10. S.F. Hassan, R.A. Rosen, Resolving the ghost problem in nonlinear massive gravity. Phys. Rev. Lett. 108, 041101 (2012). https://doi.org/10.1103/PhysRevLett.108.041101

    Article  ADS  Google Scholar 

  11. S.F. Hassan, R.A. Rosen, A. Schmidt-May, Ghost-free massive gravity with a general reference metric. JHEP 02, 026 (2012). https://doi.org/10.1007/JHEP02(2012)026

    Article  ADS  MathSciNet  Google Scholar 

  12. S. Deser, K. Izumi, Y.C. Ong, A. Waldron, Massive gravity acausality redux. Phys. Lett. B 726, 544 (2013)

    Article  ADS  Google Scholar 

  13. S. Deser, K. Izumi, Y.C. Ong, A. Waldron, Problems of massive gravities. Mod. Phys. Lett. A 30, 1540006 (2015). https://doi.org/10.1142/S0217732315400064

    Article  ADS  MathSciNet  Google Scholar 

  14. G. D’Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava, A.J. Tolley, Massive cosmologies. Phys. Rev. D 84, 124046 (2011). https://doi.org/10.1103/PhysRevD.84.124046

    Article  ADS  Google Scholar 

  15. A. De Felice, A. Emir Gumrukcuoglu, C. Lin, S. Mukohyama, On the cosmology of massive gravity. Class. Quant. Grav. 30, 184004 (2013). https://doi.org/10.1088/0264-9381/30/18/184004/meta

    Article  ADS  MathSciNet  Google Scholar 

  16. D.G. Boulware, S. Deser, String generated gravity models. Phys. Rev. Lett. 55, 2656 (1985). https://doi.org/10.1103/PhysRevLett.55.2656

    Article  ADS  Google Scholar 

  17. C.G. Callan Jr., R.C. Myers, M.J. Perry, Black holes in string theory. Nucl. Phys. B 311, 673–698 (1989). https://doi.org/10.1016/0550-3213(89)90172-7

    Article  ADS  MathSciNet  Google Scholar 

  18. R.G. Cai, Gauss-Bonnet black holes in AdS spaces. Phys. Rev. D 65, 084014 (2002). https://doi.org/10.1103/PhysRevD.65.084014

    Article  ADS  MathSciNet  Google Scholar 

  19. M. Cvetic, S. Nojiri, S.D. Odintsov, Black hole thermodynamics and negative entropy in de Sitter and anti-de Sitter Einstein-Gauss-Bonnet gravity. Nucl. Phys. B 628, 295–330 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  20. T. Torii, H. Maeda, Spacetime structure of static solutions in Gauss-Bonnet gravity: neutral case. Phys. Rev. D 71, 124002 (2005). https://doi.org/10.1103/PhysRevD.71.124002

    Article  ADS  MathSciNet  Google Scholar 

  21. R.A. Konoplya, A. Zhidenko, Long life of Gauss-Bonnet corrected black holes. Phys. Rev. D 82, 084003 (2010). https://doi.org/10.1103/PhysRevD.82.084003

    Article  ADS  Google Scholar 

  22. W. Xu, H. Xu, L. Zhao, Gauss-Bonnet coupling constant as a free thermodynamical variable and the associated criticality. Eur. Phys. J. C 74, 2970 (2014). https://doi.org/10.1140/epjc/s10052-014-2970-8

    Article  ADS  Google Scholar 

  23. H. Xu, W. Xu, L. Zhao, Extended phase space thermodynamics for third order Lovelock black holes in diverse dimensions. Eur. Phys. J. C 74(9), 3074 (2014). https://doi.org/10.1140/epjc/s10052-014-3074-1

    Article  ADS  Google Scholar 

  24. H. Xu, Z.M. Xu, Maxwell’s equal area law for Lovelock thermodynamics. Int. J. Mod. Phys. D 26(04), 1750037 (2016)

    Article  ADS  Google Scholar 

  25. Y. Sun, H. Xu, L. Zhao, Thermodynamics and holographic entanglement entropy for spherical black holes in 5D Gauss-Bonnet gravity. JHEP 1609, 060 (2016). https://doi.org/10.1007/JHEP09(2016)060

    Article  ADS  MathSciNet  Google Scholar 

  26. R.A. Konoplya, A. Zhidenko, Eikonal instability of Gauss-Bonnet-(anti-)-de Sitter black holes. Phys. Rev. D 95(10), 104005 (2017). https://doi.org/10.1103/PhysRevD.95.104005

    Article  ADS  MathSciNet  Google Scholar 

  27. R.A. Konoplya, A. Zhidenko, Quasinormal modes of Gauss-Bonnet-AdS black holes: towards holographic description of finite coupling. JHEP 09, 139 (2017). https://doi.org/10.1007/JHEP09(2017)139

    Article  ADS  MathSciNet  Google Scholar 

  28. D. Glavan, C. Lin, Einstein-gauss-bonnet gravity in four-dimensional spacetime. Phys. Rev. Lett. 124(8), 081301 (2020). https://doi.org/10.1103/PhysRevLett.124.081301

    Article  ADS  MathSciNet  Google Scholar 

  29. H. Lu, Y. Pang, Horndeski gravity as \(D \rightarrow 4\) limit of Gauss-Bonnet. Phys. Lett. B 809, 135717 (2020)

    Article  MathSciNet  Google Scholar 

  30. M. Guo, P.C. Li, Innermost stable circular orbit and shadow of the \(4D\) Einstein-Gauss-Bonnet black hole. Eur. Phys. J. C 80(6), 588 (2020). https://doi.org/10.1140/epjc/s10052-020-8164-7

    Article  ADS  Google Scholar 

  31. C.Y. Zhang, P.C. Li, M. Guo, Greybody factor and power spectra of the Hawking radiation in the \(4D\) Einstein-Gauss-Bonnet de-Sitter gravity. Eur. Phys. J. C 80(9), 874 (2020). https://doi.org/10.1140/epjc/s10052-020-08448-z

    Article  ADS  Google Scholar 

  32. R.A. Konoplya, A.F. Zinhailo, Grey-body factors and Hawking radiation of black holes in 4D Einstein-Gauss-Bonnet gravity. Phys. Lett. B 810, 135793 (2020). https://doi.org/10.1016/j.physletb.2020.135793

    Article  MathSciNet  Google Scholar 

  33. S. Mahapatra, A note on the total action of 4D Gauss-Bonnet theory. Eur. Phys. J. C 80(10), 992 (2020). https://doi.org/10.1140/epjc/s10052-020-08568-6

    Article  ADS  Google Scholar 

  34. J. Arrechea, A. Delhom, A. Jiménez-Cano, Inconsistencies in four-dimensional Einstein-Gauss-Bonnet gravity. Chin. Phys. C 45(1), 013107 (2021). https://doi.org/10.1088/1674-1137/abc1d4

    Article  ADS  Google Scholar 

  35. R. Kumar, S.G. Ghosh, Rotating black holes in \(4D\) Einstein-Gauss-Bonnet gravity and its shadow. JCAP 07(07), 053 (2020). https://doi.org/10.1088/1475-7516/2020/07/053

    Article  ADS  MathSciNet  Google Scholar 

  36. C.Y. Zhang, S.J. Zhang, P.C. Li, M. Guo, Superradiance and stability of the regularized 4D charged Einstein-Gauss-Bonnet black hole. JHEP 08, 105 (2020). https://doi.org/10.1007/JHEP08(2020)105

    Article  ADS  MathSciNet  Google Scholar 

  37. K. Yang, B.M. Gu, S.W. Wei, Y.X. Liu, Born-Infeld black holes in 4D Einstein-Gauss-Bonnet gravity. Eur. Phys. J. C 80(7), 662 (2020). https://doi.org/10.1140/epjc/s10052-020-8246-6

    Article  ADS  Google Scholar 

  38. M. Gürses, T. Şişman, B. Tekin, Is there a novel Einstein-Gauss-Bonnet theory in four dimensions? Eur. Phys. J. C 80(7), 647 (2020). https://doi.org/10.1140/epjc/s10052-020-8200-7

    Article  ADS  Google Scholar 

  39. J. Arrechea, A. Delhom, A. Jiménez-Cano, Comment on einstein-gauss-bonnet gravity in four-dimensional spacetime. Phys. Rev. Lett. 125(14), 149002 (2020). https://doi.org/10.1103/PhysRevLett.125.149002

    Article  ADS  MathSciNet  Google Scholar 

  40. M. Gurses, T. Şişman, B. Tekin, Comment on Einstein-Gauss-Bonnet Gravity in 4-Dimensional Space-Time. Phys. Rev. Lett. 125(14), 149001 (2020). https://doi.org/10.1103/PhysRevLett.125.149001

    Article  ADS  MathSciNet  Google Scholar 

  41. F.W. Shu, Vacua in novel 4D Einstein-Gauss-Bonnet Gravity: Pathology and instability? Phys. Lett. B 811, 135907 (2020). https://doi.org/10.1016/j.physletb.2020.135907

    Article  MathSciNet  Google Scholar 

  42. D.N. Page, Particle emission rates from a black hole: massless particles from an uncharged, nonrotating hole. Phys. Rev. D 13, 198 (1976). https://doi.org/10.1103/PhysRevD.13.198

    Article  ADS  Google Scholar 

  43. J.M. Maldacena, The Large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113–1133 (1999). https://doi.org/10.1023/A:1026654312961

    Article  MathSciNet  Google Scholar 

  44. E. Witten, Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253–291 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  45. H. Nariai, On a new cosmological solution of einstein’s field equations of gravitation. Gen. Relativ. Gravit. 31, 963–971 (1999). https://doi.org/10.1023/A:1026602724948

    Article  ADS  Google Scholar 

  46. P. Ginsparg, M.J. Perry, Semiclassical perdurance of de Sitter space. Nucl. Phys. B 222, 245–268 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  47. D.N. Page, Finite upper bound for the hawking decay time of an arbitrarily large black hole in anti-de sitter spacetime. Phys. Rev. D 97, 024004 (2018). https://doi.org/10.1103/PhysRevD.97.024004

    Article  ADS  MathSciNet  Google Scholar 

  48. W.A. Hiscock, L.D. Weems, Evolution of charged evaporating black holes. Phys. Rev. D 41, 1142 (1990). https://doi.org/10.1103/PhysRevD.41.1142

    Article  ADS  Google Scholar 

  49. H. Xu, Y.C. Ong, M.H. Yung, Cosmic censorship and the evolution of d-dimensional charged evaporating black holes. Phys. Rev. D 101, 064015 (2020). https://doi.org/10.1103/PhysRevD.101.064015

    Article  ADS  MathSciNet  Google Scholar 

  50. R. Ling, H. Xu, Y.C. Ong, How anti-de Sitter black holes reach thermal equilibrium. Phys. Lett. B 826, 136896 (2022)

    Article  MathSciNet  Google Scholar 

  51. H. Xu, M.H. Yung, Black hole evaporation in lovelock gravity with diverse dimensions. Phys. Lett. B 794, 77 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  52. H. Xu, M.H. Yung, Black hole evaporation in conformal (Weyl) gravity. Phys. Lett. B 793, 97 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  53. H. Xu, Y.C. Ong, Black hole evaporation in Ho\(\check{r}\)ava-Lifshitz gravity. Eur. Phys. J. C 80(7), 679 (2020). https://doi.org/10.1140/epjc/s10052-020-8249-3

    Article  ADS  Google Scholar 

  54. C.H. Wu, Y.P. Hu, H. Xu, Hawking evaporation of einstein-gauss-bonnet AdS black holes in \(D\geqslant 4\) dimensions. Eur. Phys. J. C 81(4), 351 (2021). https://doi.org/10.1140/epjc/s10052-021-09140-6

    Article  ADS  Google Scholar 

  55. M.S. Hou, H. Xu, Y.C. Ong, Hawking evaporation of black holes in massive gravity. Eur. Phys. J. C 80(11), 1090 (2020). https://doi.org/10.1140/epjc/s10052-020-08678-1

    Article  ADS  Google Scholar 

  56. S. Upadhyay, D.V. Singh, Black hole solution and thermal properties in 4\(D\)\(AdS\) Gauss-Bonnet massive gravity. Eur. Phys. J. Plus 137(3), 383 (2022). https://doi.org/10.1140/epjp/s13360-022-02569-y

    Article  Google Scholar 

  57. 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). https://doi.org/10.1103/PhysRevD.91.024032

    Article  ADS  MathSciNet  Google Scholar 

  58. P. Landsberg, A. De Vos, The stefan-boltzmann constant in n-dimensional space. J. Phys. Math. Gen. 22, 1073 (1989). https://doi.org/10.1088/0305-4470/22/8/021

    Article  ADS  MathSciNet  Google Scholar 

  59. Y.C. Ong, Holographic consistency and the sign of the Gauss-Bonnet parameter. Nucl. Phys. B 984, 115939 (2022)

    Article  MathSciNet  Google Scholar 

  60. J. Maldacena, A simple quantum system that describes a black hole. arXiv:2303.11534 [hep-th]

  61. H. Breuer, F. Petruccione, The theory of open quantum systems (Oxford University Press, Oxford, 2002)

    Google Scholar 

  62. L. Hodgkinson, J. Louko, A.C. Ottewill, Static detectors and circular-geodesic detectors on the Schwarzschild black hole. Phys. Rev. D. 89(10), 104002 (2014). https://doi.org/10.1103/PhysRevD.89.104002

    Article  ADS  Google Scholar 

  63. K.K. Ng, R.B. Mann, E. Martín-Martínez, Unruh-DeWitt detectors and entanglement: the anti-de Sitter space. Phys. Rev. D. 98(12), 125005 (2018). https://doi.org/10.1103/PhysRevD.98.125005

    Article  ADS  MathSciNet  Google Scholar 

  64. H. Xu, Decoherence and thermalization of Unruh-DeWitt detector in arbitrary dimensions. JHEP 03, 179 (2023). https://doi.org/10.1007/jhep03(2023)179

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

Hao Xu thanks National Natural Science Foundation of China (No.12205250) for funding support.

Author information

Authors and Affiliations

  1. Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou, 225009, China

    Hao Xu & Yun Du

Authors
  1. Hao Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yun Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hao Xu.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, H., Du, Y. Gauss-bonnet modification to Hawking evaporation of AdS black holes in massive gravity. Eur. Phys. J. Plus 139, 77 (2024). https://doi.org/10.1140/epjp/s13360-023-04853-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-023-04853-x

Search

Navigation