Journal of High Energy Physics

, 2018:80 | Cite as

Extremal black holes, Stueckelberg scalars and phase transitions

  • Alessio Marrani
  • Olivera Miskovic
  • Paula Quezada Leon
Open Access
Regular Article - Theoretical Physics
  • 7 Downloads

Abstract

We calculate the entropy of a static extremal black hole in 4D gravity, non-linearly coupled to a massive Stueckelberg scalar. We find that the scalar field does not allow the black hole to be magnetically charged. We also show that the system can exhibit a phase transition due to electric charge variations. For spherical and hyperbolic horizons, the critical point exists only in presence of a cosmological constant, and if the scalar is massive and non-linearly coupled to electromagnetic field. On one side of the critical point, two extremal solutions coexist: Reissner-Nordström (A)dS black hole and the charged hairy (A)dS black hole, while on the other side of the critical point the black hole does not have hair. A near-critical analysis reveals that the hairy black hole has larger entropy, thus giving rise to a zero temperature phase transition. This is characterized by a discontinuous second derivative of the entropy with respect to the electric charge at the critical point. The results obtained here are analytical and based on the entropy function formalism and the second law of thermodynamics.

Keywords

Black Holes Black Holes in String Theory 

Notes

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.

References

  1. [1]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a holographic superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].
  2. [2]
    S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    S.S. Gubser, Breaking an abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].
  4. [4]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic superconductor/insulator transition at zero temperature, JHEP 03 (2010) 131 [arXiv:0911.0962] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  5. [5]
    G.T. Horowitz and B. Way, Complete phase diagrams for a holographic superconductor/insulator system, JHEP 11 (2010) 011 [arXiv:1007.3714] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  6. [6]
    M. Astorino, Magnetised Kerr/CFT correspondence, Phys. Lett. B 751 (2015) 96 [arXiv:1508.01583] [INSPIRE].
  7. [7]
    J. Bičák and F. Hejda, Near-horizon description of extremal magnetized stationary black holes and Meissner effect, Phys. Rev. D 92 (2015) 104006 [arXiv:1510.01911] [INSPIRE].
  8. [8]
    M. Astorino, CFT duals for accelerating black holes, Phys. Lett. B 760 (2016) 393 [arXiv:1605.06131] [INSPIRE].
  9. [9]
    A. Sen, Black hole entropy function and the attractor mechanism in higher derivative gravity, JHEP 09 (2005) 038 [hep-th/0506177] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    A. Sen, Entropy function for heterotic black holes, JHEP 03 (2006) 008 [hep-th/0508042] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    D. Astefanesei, O. Mišković and R. Olea, Attractor horizons in six-dimensional type IIB supergravity, Phys. Lett. B 714 (2012) 331 [arXiv:1205.2099] [INSPIRE].
  12. [12]
    B.R. Majhi, Entropy function from the gravitational surface action for an extremal near horizon black hole, Eur. Phys. J. C 75 (2015) 521 [arXiv:1503.08973] [INSPIRE].
  13. [13]
    S. Aretakis, Nonlinear instability of scalar fields on extremal black holes, Phys. Rev. D 87 (2013) 084052 [arXiv:1304.4616] [INSPIRE].
  14. [14]
    S. Aretakis, Horizon instability of extremal black holes, Adv. Theor. Math. Phys. 19 (2015) 507 [arXiv:1206.6598] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    J. Lucietti, K. Murata, H.S. Reall and N. Tanahashi, On the horizon instability of an extreme Reissner-Nordström black hole, JHEP 03 (2013) 035 [arXiv:1212.2557] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  16. [16]
    P. Zimmerman, Horizon instability of extremal Reissner-Nordström black holes to charged perturbations, Phys. Rev. D 95 (2017) 124032 [arXiv:1612.03172] [INSPIRE].
  17. [17]
    E.C.G. Stueckelberg, Interaction forces in electrodynamics and in the field theory of nuclear forces, Helv. Phys. Acta 11 (1938) 299 [INSPIRE].Google Scholar
  18. [18]
    S. Franco, A. Garcia-Garcia and D. Rodriguez-Gomez, A general class of holographic superconductors, JHEP 04 (2010) 092 [arXiv:0906.1214] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  19. [19]
    V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    O. Mišković and R. Olea, Topological regularization and self-duality in four-dimensional Anti-de Sitter gravity, Phys. Rev. D 79 (2009) 124020 [arXiv:0902.2082] [INSPIRE].
  21. [21]
    B. Bertotti, Uniform electromagnetic field in the theory of general relativity, Phys. Rev. 116 (1959) 1331 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    I. Robinson, A solution of the Maxwell-Einstein equations, Bull. Acad. Polon. 7 (1959) 351.MathSciNetMATHGoogle Scholar
  23. [23]
    K. Hristov, S. Katmadas and V. Pozzoli, Ungauging black holes and hidden supercharges, JHEP 01 (2013) 110 [arXiv:1211.0035] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    A.S. Chou et al., Critical points and phase transitions in 5 − D compactifications of M-theory, Nucl. Phys. B 508 (1997) 147 [hep-th/9704142] [INSPIRE].
  25. [25]
    R. Kallosh, A.D. Linde and M. Shmakova, Supersymmetric multiple basin attractors, JHEP 11 (1999) 010 [hep-th/9910021] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    M. Wijnholt and S. Zhukov, On the uniqueness of black hole attractors, hep-th/9912002 [INSPIRE].
  27. [27]
    G.W. Moore, Attractors and arithmetic, Les Houches Lectures on Strings and Arithmetic, hep-th/9807056 [INSPIRE].
  28. [28]
    A. Giryavets, New attractors and area codes, JHEP 03 (2006) 020 [hep-th/0511215] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    P. Dominic, T. Mandal and P.K. Tripathy, Multiple single-centered attractors, JHEP 12 (2014) 158 [arXiv:1406.7147] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    A. Marrani, P.K. Tripathy and T. Mandal, Supersymmetric black holes and freudenthal duality, Int. J. Mod. Phys. A 32 (2017) 1750114 [arXiv:1703.08669] [INSPIRE].
  31. [31]
    M. Iihoshi, Mutated hybrid inflation in f (R,R)-gravity, JCAP 02 (2011) 022 [arXiv:1011.3927] [INSPIRE].
  32. [32]
    R. Kallosh, A. Linde and A. Westphal, Chaotic inflation in supergravity after Planck and BICEP2, Phys. Rev. D 90 (2014) 023534 [arXiv:1405.0270] [INSPIRE].
  33. [33]
    S. Ferrara, R. Kallosh, A. Linde and M. Porrati, Minimal supergravity models of inflation, Phys. Rev. D 88 (2013) 085038 [arXiv:1307.7696] [INSPIRE].
  34. [34]
    S. Ferrara, R. Kallosh, A. Linde and M. Porrati, Higher order corrections in minimal supergravity models of inflation, JCAP 11 (2013) 046 [arXiv:1309.1085] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    R. Kallosh, A. Linde and D. Roest, Universal attractor for inflation at strong coupling, Phys. Rev. Lett. 112 (2014) 011303 [arXiv:1310.3950] [INSPIRE].
  36. [36]
    R. Kallosh, A. Linde and D. Roest, Superconformal inflationary α-attractors, JHEP 11 (2013) 198 [arXiv:1311.0472] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    S. Ferrara, R. Kallosh and A. Marrani, Degeneration of groups of type E 7 and minimal coupling in supergravity, JHEP 06 (2012) 074 [arXiv:1202.1290] [INSPIRE].
  38. [38]
    L. Andrianopoli, R. D’Auria and S. Ferrara, Supersymmetry reduction of N extended supergravities in four-dimensions, JHEP 03 (2002) 025 [hep-th/0110277] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    L. Andrianopoli, R. D’Auria and S. Ferrara, Consistent reduction of N = 2 → N = 1 four-dimensional supergravity coupled to matter, Nucl. Phys. B 628 (2002) 387 [hep-th/0112192] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”RomaItaly
  2. 2.Dipartimento di Fisica e Astronomia “Galileo Galilei”Università di PadovaPadovaItaly
  3. 3.INFN — Sezione di PadovaPadovaItaly
  4. 4.Instituto de FísicaPontificia Universidad Católica de ValparaísoValparaísoChile

Personalised recommendations