Skip to main content

Dynamically and thermodynamically stable black holes in Einstein-Maxwell-dilaton gravity

A preprint version of the article is available at arXiv.

Abstract

We consider Einstein-Maxwell-dilaton gravity with the non-minimal exponential coupling between the dilaton and the Maxwell field emerging from low energy heterotic string theory. The dilaton is endowed with a potential that originates from an electromagnetic Fayet-Iliopoulos (FI) term in \( \mathcal{N} \) = 2 extended supergravity in four spacetime dimensions. For the case we are interested in, this potential introduces a single parameter α. When α → 0, the static black holes (BHs) of the model are the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) solutions. When α → ∞, the BHs become the standard Reissner-Nordström (RN) solutions of electrovacuum General Relativity. The BH solutions for finite non-zero α interpolate between these two families. In this case, the dilaton potential regularizes the extremal limit of the GMGHS solution yielding a set of zero temperature BHs with a near horizon AdS2 × S2 geometry. We show that, in the neighborhood of these extremal solutions, there is a subset of BHs that are dynamically and thermodynamically stable, all of which have charge to mass ratio larger than unity. By dynamical stability we mean that no growing quasi-normal modes are found; thus they are stable against linear perturbations (spherical and non-spherical). Moreover, non-linear numerical evolutions lend support to their non-linear stability. By thermodynamical stability we mean the BHs are stable both in the canonical and grand-canonical ensemble. In particular, both the specific heat at constant charge and the isothermal permittivity are positive. This is not possible for RN and GMGHS BHs. We discuss the different thermodynamical phases for the BHs in this model and comment on what may allow the existence of both dynamically and thermodynamically stable BHs.

References

  1. [1]

    W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164 (1967) 1776 [INSPIRE].

    ADS  Google Scholar 

  2. [2]

    T. Regge and J.A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108 (1957) 1063 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  3. [3]

    F.J. Zerilli, Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett. 24 (1970) 737 [INSPIRE].

    ADS  Google Scholar 

  4. [4]

    S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].

  5. [5]

    S.W. Hawking, Black holes and thermodynamics, Phys. Rev. D 13 (1976) 191 [INSPIRE].

    ADS  Google Scholar 

  6. [6]

    W. Israel, Event horizons in static electrovac space-times, Commun. Math. Phys. 8 (1968) 245 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  7. [7]

    V. Moncrief, Odd-parity stability of a Reissner-Nordström black hole, Phys. Rev. D 9 (1974) 2707 [INSPIRE].

    ADS  Google Scholar 

  8. [8]

    V. Moncrief, Stability of Reissner-Nordström black holes, Phys. Rev. D 10 (1974) 1057 [INSPIRE].

    ADS  Google Scholar 

  9. [9]

    P.C.W. Davies, The thermodynamic theory of black holes, Proc. Roy. Soc. Lond. A 353 (1977) 499.

    ADS  Google Scholar 

  10. [10]

    G.W. Gibbons, Antigravitating black hole solitons with scalar hair in n = 4 supergravity, Nucl. Phys. B 207 (1982) 337.

    ADS  MathSciNet  Google Scholar 

  11. [11]

    G.W. Gibbons and K.-I. Maeda, Black holes and membranes in higher dimensional theories with dilaton fields, Nucl. Phys. B 298 (1988) 741 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  12. [12]

    D. Garfinkle, G.T. Horowitz and A. Strominger, Charged black holes in string theory, Phys. Rev. D 43 (1991) 3140 [Erratum ibid. 45 (1992) 3888] [INSPIRE].

  13. [13]

    A. Anabalón, D. Astefanesei, A. Gallerati and M. Trigiante, Hairy black holes and duality in an extended supergravity model, JHEP 04 (2018) 058 [arXiv:1712.06971] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  14. [14]

    A. Anabalon, D. Astefanesei and R. Mann, Exact asymptotically flat charged hairy black holes with a dilaton potential, JHEP 10 (2013) 184 [arXiv:1308.1693] [INSPIRE].

    ADS  Google Scholar 

  15. [15]

    D. Astefanesei, D. Choque, F. Gómez and R. Rojas, Thermodynamically stable asymptotically flat hairy black holes with a dilaton potential, JHEP 03 (2019) 205 [arXiv:1901.01269] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  16. [16]

    J.L. Blázquez-Salcedo, S. Kahlen and J. Kunz, Quasinormal modes of dilatonic Reissner-Nordstr¨om black holes, Eur. Phys. J. C 79 (2019) 1021 [arXiv:1911.01943] [INSPIRE].

    ADS  Google Scholar 

  17. [17]

    A. Jansen, A. Rostworowski and M. Rutkowski, Master equations and stability of Einstein-Maxwell-scalar black holes, JHEP 12 (2019) 036 [arXiv:1909.04049] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  18. [18]

    C.A.R. Herdeiro, E. Radu, N. Sanchis-Gual and J.A. Font, Spontaneous scalarization of charged black holes, Phys. Rev. Lett. 121 (2018) 101102 [arXiv:1806.05190] [INSPIRE].

    ADS  Google Scholar 

  19. [19]

    P.G.S. Fernandes, C.A.R. Herdeiro, A.M. Pombo, E. Radu and N. Sanchis-Gual, Spontaneous scalarisation of charged black holes: coupling dependence and dynamical features, Class. Quant. Grav. 36 (2019) 134002 [Erratum ibid. 37 (2020) 049501] [arXiv:1902.05079] [INSPIRE].

  20. [20]

    P.G.S. Fernandes, C.A.R. Herdeiro, A.M. Pombo, E. Radu and N. Sanchis-Gual, Charged black holes with axionic-type couplings: classes of solutions and dynamical scalarization, Phys. Rev. D 100 (2019) 084045 [arXiv:1908.00037] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  21. [21]

    C.W. Misner and D.H. Sharp, Relativistic equations for adiabatic, spherically symmetric gravitational collapse, Phys. Rev. 136 (1964) B571 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  22. [22]

    D. Astefanesei, R. Ballesteros, D. Choque and R. Rojas, Scalar charges and the first law of black hole thermodynamics, Phys. Lett. B 782 (2018) 47 [arXiv:1803.11317] [INSPIRE].

    ADS  MATH  Google Scholar 

  23. [23]

    C.A.R. Herdeiro and E. Radu, Asymptotically flat black holes with scalar hair: a review, Int. J. Mod. Phys. D 24 (2015) 1542014 [arXiv:1504.08209] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  24. [24]

    K. Goldstein, N. Iizuka, R.P. Jena and S.P. Trivedi, Non-supersymmetric attractors, Phys. Rev. D 72 (2005) 124021 [hep-th/0507096] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  25. [25]

    A. Sen, Black hole entropy function and the attractor mechanism in higher derivative gravity, JHEP 09 (2005) 038 [hep-th/0506177] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  26. [26]

    D. Astefanesei, K. Goldstein, R.P. Jena, A. Sen and S.P. Trivedi, Rotating attractors, JHEP 10 (2006) 058 [hep-th/0606244] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  27. [27]

    A. Sen, Black hole entropy function, attractors and precision counting of microstates, Gen. Rel. Grav. 40 (2008) 2249 [arXiv:0708.1270] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  28. [28]

    A. Anabalón and D. Astefanesei, On attractor mechanism of AdS4 black holes, Phys. Lett. B 727 (2013) 568 [arXiv:1309.5863] [INSPIRE].

    ADS  MATH  Google Scholar 

  29. [29]

    I. Robinson, A solution of the Maxwell-Einstein equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 7 (1959) 351.

    MathSciNet  MATH  Google Scholar 

  30. [30]

    B. Bertotti, Uniform electromagnetic field in the theory of general relativity, Phys. Rev. 116 (1959) 1331 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  31. [31]

    J.L. Blázquez-Salcedo, D.D. Doneva, J. Kunz and S.S. Yazadjiev, Radial perturbations of the scalarized Einstein-Gauss-Bonnet black holes, Phys. Rev. D 98 (2018) 084011 [arXiv:1805.05755] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  32. [32]

    K.D. Kokkotas and B.G. Schmidt, Quasinormal modes of stars and black holes, Living Rev. Rel. 2 (1999) 2 [gr-qc/9909058] [INSPIRE].

  33. [33]

    H.-P. Nollert, Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars, Class. Quant. Grav. 16 (1999) R159 [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  34. [34]

    E. Berti, V. Cardoso and A.O. Starinets, Quasinormal modes of black holes and black branes, Class. Quant. Grav. 26 (2009) 163001 [arXiv:0905.2975] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  35. [35]

    R.A. Konoplya and A. Zhidenko, Quasinormal modes of black holes: from astrophysics to string theory, Rev. Mod. Phys. 83 (2011) 793 [arXiv:1102.4014] [INSPIRE].

    ADS  Google Scholar 

  36. [36]

    V. Ferrari, M. Pauri and F. Piazza, Quasinormal modes of charged, dilaton black holes, Phys. Rev. D 63 (2001) 064009 [gr-qc/0005125] [INSPIRE].

  37. [37]

    J.L. Blázquez-Salcedo et al., Quasinormal modes of compact objects in alternative theories of gravity, Eur. Phys. J. Plus 134 (2019) 46 [arXiv:1810.09432] [INSPIRE].

    Google Scholar 

  38. [38]

    E.W. Leaver, Quasinormal modes of Reissner-Nordström black holes, Phys. Rev. D 41 (1990) 2986 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  39. [39]

    J.L. Blázquez-Salcedo et al., Perturbed black holes in Einstein-dilaton-Gauss-Bonnet gravity: stability, ringdown and gravitational-wave emission, Phys. Rev. D 94 (2016) 104024 [arXiv:1609.01286] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  40. [40]

    J.L. Blázquez-Salcedo, F.S. Khoo and J. Kunz, Quasinormal modes of Einstein-Gauss-Bonnet-dilaton black holes, Phys. Rev. D 96 (2017) 064008 [arXiv:1706.03262] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  41. [41]

    N. Sanchis-Gual, J.C. Degollado, P.J. Montero, J.A. Font and C. Herdeiro, Explosion and final state of an unstable Reissner-Nordström black hole, Phys. Rev. Lett. 116 (2016) 141101 [arXiv:1512.05358] [INSPIRE].

    ADS  Google Scholar 

  42. [42]

    N. Sanchis-Gual, J.C. Degollado, C. Herdeiro, J.A. Font and P.J. Montero, Dynamical formation of a Reissner-Nordström black hole with scalar hair in a cavity, Phys. Rev. D 94 (2016) 044061 [arXiv:1607.06304] [INSPIRE].

    ADS  Google Scholar 

  43. [43]

    E.W. Hirschmann, L. Lehner, S.L. Liebling and C. Palenzuela, Black hole dynamics in Einstein-Maxwell-dilaton theory, Phys. Rev. D 97 (2018) 064032 [arXiv:1706.09875] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  44. [44]

    I. Cordero-Carrion and P. Cerda-Duran, Partially implicit Runge-Kutta methods for wave-like equations in spherical-type coordinates, arXiv:1211.5930 [INSPIRE].

  45. [45]

    I. Cordero-Carrión and P. Cerdá-Durán, Partially implicit Runge-Kutta methods for wave-like equations, in Advances in differential equations and applications, Springer, Cham, Switzerland (2014), pg. 267.

  46. [46]

    Einstein toolkit: open software for relativistic astrophysics webpage, http://einsteintoolkit.org/.

  47. [47]

    F. Löffler et al., The Einstein toolkit: a community computational infrastructure for relativistic astrophysics, Class. Quant. Grav. 29 (2012) 115001 [arXiv:1111.3344] [INSPIRE].

    ADS  MATH  Google Scholar 

  48. [48]

    G.W. Gibbons, R. Kallosh and B. Kol, Moduli, scalar charges and the first law of black hole thermodynamics, Phys. Rev. Lett. 77 (1996) 4992 [hep-th/9607108] [INSPIRE].

    ADS  Google Scholar 

  49. [49]

    H. Lü, Y. Pang and C.N. Pope, AdS dyonic black hole and its thermodynamics, JHEP 11 (2013) 033 [arXiv:1307.6243] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  50. [50]

    H. Lü, C.N. Pope and Q. Wen, Thermodynamics of AdS black holes in Einstein-scalar gravity, JHEP 03 (2015) 165 [arXiv:1408.1514] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  51. [51]

    A. Anabalon, D. Astefanesei and C. Martinez, Mass of asymptotically anti-de Sitter hairy spacetimes, Phys. Rev. D 91 (2015) 041501 [arXiv:1407.3296] [INSPIRE].

    ADS  Google Scholar 

  52. [52]

    A. Anabalon, D. Astefanesei, D. Choque and C. Martinez, Trace anomaly and counterterms in designer gravity, JHEP 03 (2016) 117 [arXiv:1511.08759] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  53. [53]

    J. Brown and J.W. York, Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [INSPIRE].

  54. [54]

    T. Hertog and G.T. Horowitz, Designer gravity and field theory effective potentials, Phys. Rev. Lett. 94 (2005) 221301 [hep-th/0412169] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

Author information

Affiliations

Authors

Corresponding author

Correspondence to Dumitru Astefanesei.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 1912.02192

Rights and permissions

Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Astefanesei, D., Blázquez-Salcedo, J.L., Herdeiro, C. et al. Dynamically and thermodynamically stable black holes in Einstein-Maxwell-dilaton gravity. J. High Energ. Phys. 2020, 63 (2020). https://doi.org/10.1007/JHEP07(2020)063

Download citation

Keywords

  • Black Holes
  • Black Holes in String Theory