D-dimensional charged Anti-de-Sitter black holes in f (T) gravity

  • A.M. Awad
  • S. Capozziello
  • G.G.L. NashedEmail author
Open Access
Regular Article - Theoretical Physics


We present a D-dimensional charged Anti-de-Sitter black hole solutions in f (T) gravity, where f (T) = T + βT 2 and D ≥ 4. These solutions are characterized by flat or cylindrical horizons. The interesting feature of these solutions is the existence of inseparable electric monopole and quadrupole terms in the potential which share related momenta, in contrast with most of the known charged black hole solutions in General Relativity and its extensions. Furthermore, these solutions have curvature singularities which are milder than those of the known charged black hole solutions in General Relativity and Teleparallel Gravity. This feature can be shown by calculating some invariants of curvature and torsion tensors. Furthermore, we calculate the total energy of these black holes using the energy-momentum tensor. Finally, we show that these charged black hole solutions violate the first law of thermodynamics in agreement with previous results.


Black Holes Spacetime Singularities 


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.


  1. [1]
    S. Carlip, Black hole thermodynamics, Int. J. Mod. Phys. D 23 (2014) 1430023 [arXiv:1410.1486] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  2. [2]
    G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].
  3. [3]
    L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    C. Rovelli, Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K., (2007).Google Scholar
  6. [6]
    T. Jacobson, Thermodynamics of space-time: the Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [INSPIRE].
  7. [7]
    E.P. Verlinde, On the origin of gravity and the laws of Newton, JHEP 04 (2011) 029 [arXiv:1001.0785] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    A. Einstein, Einheitliche Feldtheorie von Gravitation und Elektrizität Unified Field Theory of Gravity and Electricity (in German), Pruss. Acad. Sci. (1925) 414.Google Scholar
  9. [9]
    A. Einstein, Riemanngeometrie mit Aufrechterhaltung des Begriffes des Fern-Parallelismus (in German), Pruss. Acad. Sci. (1928) 217.Google Scholar
  10. [10]
    A. Einstein, Auf die Riemann-Metrik und den Fern-Parallelismus gegründete einheitliche Feldtheorie (in German), Math. Ann. 102 (1930) 685.Google Scholar
  11. [11]
    A. Unzicker and T. Case, Translation of Einstein’s attempt of a unified field theory with teleparallelism, physics/0503046 [INSPIRE].
  12. [12]
    K. Hayashi and T. Shirafuji, New general relativity, Phys. Rev. D 19 (1979) 3524 [Addendum ibid. D 24 (1982) 3312] [INSPIRE].
  13. [13]
    R. Ferraro and F. Fiorini, Modified teleparallel gravity: inflation without inflaton, Phys. Rev. D 75 (2007) 084031 [gr-qc/0610067] [INSPIRE].
  14. [14]
    R. Ferraro and F. Fiorini, On Born-Infeld gravity in Weitzenböck spacetime, Phys. Rev. D 78 (2008) 124019 [arXiv:0812.1981] [INSPIRE].ADSGoogle Scholar
  15. [15]
    G.R. Bengochea and R. Ferraro, Dark torsion as the cosmic speed-up, Phys. Rev. D 79 (2009) 124019 [arXiv:0812.1205] [INSPIRE].ADSGoogle Scholar
  16. [16]
    Y.-F. Cai, S. Capozziello, M. De Laurentis and E.N. Saridakis, f (T) teleparallel gravity and cosmology, Rept. Prog. Phys. 79 (2016) 106901 [arXiv:1511.07586] [INSPIRE].
  17. [17]
    G.G.L. Nashed, Stationary axisymmetric solutions and their energy contents in teleparallel equivalent of Einstein theory, Astrophys. Space Sci. 330 (2010) 173 [arXiv:1503.01379] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  18. [18]
    R. Aldrovandi, J.G. Pereira and K.H. Vu, Selected topics in teleparallel gravity, Braz. J. Phys. 34 (2004) 1374 [gr-qc/0312008] [INSPIRE].
  19. [19]
    R. Aldrovandi and J.G. Pereira, Teleparallel gravity: an introduction, Springer, Dordrecht The Netherlands, (2012).Google Scholar
  20. [20]
    R. Aldrovandi and J.G. Pereira, Teleparallel gravity,
  21. [21]
    J.W. Maluf, Localization of energy in general relativity, J. Math. Phys. 36 (1995) 4242 [gr-qc/9504010] [INSPIRE].
  22. [22]
    J.W. Maluf, The teleparallel equivalent of general relativity, Annalen Phys. 525 (2013) 339 [arXiv:1303.3897] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    K. Bamba, S. Capozziello, S. Nojiri and S.D. Odintsov, Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests, Astrophys. Space Sci. 342 (2012) 155 [arXiv:1205.3421] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  24. [24]
    S. Basilakos, S. Capozziello, M. De Laurentis, A. Paliathanasis and M. Tsamparlis, Noether symmetries and analytical solutions in f (T )-cosmology: a complete study, Phys. Rev. D 88 (2013) 103526 [arXiv:1311.2173] [INSPIRE].ADSGoogle Scholar
  25. [25]
    Y.-F. Cai, S.-H. Chen, J.B. Dent, S. Dutta and E.N. Saridakis, Matter bounce cosmology with the f (T) gravity, Class. Quant. Grav. 28 (2011) 215011 [arXiv:1104.4349] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    R. Ferraro and F. Fiorini, Spherically symmetric static spacetimes in vacuum f (T) gravity, Phys. Rev. D 84 (2011) 083518 [arXiv:1109.4209] [INSPIRE].ADSGoogle Scholar
  27. [27]
    G.G.L. Nashed, Vacuum nonsingular black hole solutions in tetrad theory of gravitation, Gen. Rel. Grav. 34 (2002) 1047 [gr-qc/0109033] [INSPIRE].
  28. [28]
    P.A. Gonzalez, E.N. Saridakis and Y. Vasquez, Circularly symmetric solutions in three-dimensional teleparallel, f (T) and Maxwell-f (T) gravity, JHEP 07 (2012) 053 [arXiv:1110.4024] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    G.G.L. Nashed and W. El Hanafy, A built-in inflation in the f (T)-cosmology, Eur. Phys. J. C 74 (2014) 3099 [arXiv:1403.0913] [INSPIRE].CrossRefGoogle Scholar
  30. [30]
    M.E. Rodrigues, M.J.S. Houndjo, J. Tossa, D. Momeni and R. Myrzakulov, Charged black holes in generalized teleparallel gravity, JCAP 11 (2013) 024 [arXiv:1306.2280] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    G.G.L. Nashed, Kerr-Newman solution in modified teleparallel theory of gravity, Int. J. Mod. Phys. D 24 (2014) 1550007 [INSPIRE].ADSzbMATHGoogle Scholar
  32. [32]
    E.L.B. Junior, M.E. Rodrigues and M.J.S. Houndjo, Regular black holes in f (T) gravity through a nonlinear electrodynamics source, JCAP 10 (2015) 060 [arXiv:1503.07857] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    K. Bamba, G.G.L. Nashed, W. El Hanafy and S.K. Ibraheem, Bounce inflation in f (T) cosmology: a unified inflaton-quintessence field, Phys. Rev. D 94 (2016) 083513 [arXiv:1604.07604] [INSPIRE].
  34. [34]
    R. Ferraro and F. Fiorini, Cosmological frames for theories with absolute parallelism, Int. J. Mod. Phys. Conf. Ser. 3 (2011) 227 [arXiv:1106.6349] [INSPIRE].CrossRefGoogle Scholar
  35. [35]
    R. Ferraro and F. Fiorini, On Born-Infeld gravity in Weitzenböck spacetime, Phys. Rev. D 78 (2008) 124019 [arXiv:0812.1981] [INSPIRE].ADSGoogle Scholar
  36. [36]
    P. Wu and H.W. Yu, Observational constraints on f (T) theory, Phys. Lett. B 693 (2010) 415 [arXiv:1006.0674] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    A. Awad and G. Nashed, Generalized teleparallel cosmology and initial singularity crossing, JCAP 02 (2017) 046 [arXiv:1701.06899] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    G. Farrugia, J.L. Said and M.L. Ruggiero, Solar system tests in f(T) gravity, Phys. Rev. D 93 (2016) 104034 [arXiv:1605.07614] [INSPIRE].ADSMathSciNetGoogle Scholar
  39. [39]
    C. Bejarano, R. Ferraro and M.J. Guzmán, Kerr geometry in f (T) gravity, Eur. Phys. J. C 75 (2015) 77 [arXiv:1412.0641] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    G.G.L. Nashed, Vacuum nonsingular black hole in tetrad theory of gravitation, Nuovo Cim. B 117 (2002) 521 [gr-qc/0109017] [INSPIRE].
  41. [41]
    M. Krššák, Holographic renormalization in teleparallel gravity, Eur. Phys. J. C 77 (2017) 44 [arXiv:1510.06676] [INSPIRE].ADSGoogle Scholar
  42. [42]
    M. Krššák and E.N. Saridakis, The covariant formulation of f (T) gravity, Class. Quant. Grav. 33 (2016) 115009 [arXiv:1510.08432] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    C.G. Boehmer, A. Mussa and N. Tamanini, Existence of relativistic stars in f (T) gravity, Class. Quant. Grav. 28 (2011) 245020 [arXiv:1107.4455] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    H. Dong, Y.-B. Wang and X.-H. Meng, Extended Birkhoff ’s theorem in the f (T ) gravity, Eur. Phys. J. C 72 (2012) 2002 [arXiv:1203.5890] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    S. Capozziello, O. Luongo and E.N. Saridakis, Transition redshift in f (T ) cosmology and observational constraints, Phys. Rev. D 91 (2015) 124037 [arXiv:1503.02832] [INSPIRE].ADSMathSciNetGoogle Scholar
  46. [46]
    L. Iorio and E.N. Saridakis, Solar system constraints on f (T) gravity, Mon. Not. Roy. Astron. Soc. 427 (2012) 1555 [arXiv:1203.5781] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    G.G.L. Nashed, Spherically symmetric solutions on a non-trivial frame in f (T ) theories of gravity, Chin. Phys. Lett. 29 (2012) 050402.Google Scholar
  48. [48]
    K. Bamba, S. Nojiri and S.D. Odintsov, Trace-anomaly driven inflation in f (T ) gravity and in minimal massive bigravity, Phys. Lett. B 731 (2014) 257 [arXiv:1401.7378] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    Y. Xie and X.-M. Deng, f (T ) gravity: effects on astronomical observation and solar system experiments and upper-bounds, Mon. Not. Roy. Astron. Soc. 433 (2013) 3584 [arXiv:1312.4103] [INSPIRE].
  50. [50]
    G.G.L. Nashed and W. El Hanafy, Analytic rotating black hole solutions in N -dimensional f (T) gravity, Eur. Phys. J. C 77 (2017) 90 [arXiv:1612.05106] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    G.G.L. Nashed, Spherically symmetric charged-dS solution in f (T) gravity theories, Phys. Rev. D 88 (2013) 104034 [arXiv:1311.3131] [INSPIRE].ADSGoogle Scholar
  52. [52]
    S. Capozziello, P.A. González, E.N. Saridakis and Y. Vásquez, Exact charged black-hole solutions in D-dimensional f (T) gravity: torsion vs curvature analysis, JHEP 02 (2013) 039 [arXiv:1210.1098] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    G.G.L. Nashed, A special exact spherically symmetric solution in f (T) gravity theories, Gen. Rel. Grav. 45 (2013) 1887 [arXiv:1502.05219] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    R.B. Mann, Topological black holes: outside looking in, Annals Israel Phys. Soc. 13 (1997) 311 [gr-qc/9709039] [INSPIRE].
  55. [55]
    J.P.S. Lemos, Cylindrical black hole in general relativity, Phys. Lett. B 353 (1995) 46 [gr-qc/9404041] [INSPIRE].
  56. [56]
    A.M. Awad, Higher dimensional charged rotating solutions in (A)dS space-times, Class. Quant. Grav. 20 (2003) 2827 [hep-th/0209238] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  57. [57]
    A.M. Awad, Higher dimensional Taub-NUTS and Taub-Bolts in Einstein-Maxwell gravity, Class. Quant. Grav. 23 (2006) 2849 [hep-th/0508235] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    A.M. Awad and C.V. Johnson, Holographic stress tensors for Kerr-AdS black holes, Phys. Rev. D 61 (2000) 084025 [hep-th/9910040] [INSPIRE].ADSMathSciNetGoogle Scholar
  59. [59]
    A.M. Awad and C.V. Johnson, Scale versus conformal invariance in the AdS/CFT correspondence, Phys. Rev. D 62 (2000) 125010 [hep-th/0006037] [INSPIRE].ADSMathSciNetGoogle Scholar
  60. [60]
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  61. [61]
    R. Weitzenböck, Invariance theorie, Nordhoff, Groningen The Netherlands, (1923).Google Scholar
  62. [62]
    S.C. Ulhoa and E.P. Spaniol, On the gravitational energy-momentum vector in f (T ) theories, Int. J. Mod. Phys. D 22 (2013) 1350069 [arXiv:1303.3144] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    J.W. Maluf, J.F. da Rocha-Neto, T.M.L. Toribio and K.H. Castello-Branco, Energy and angular momentum of the gravitational field in the teleparallel geometry, Phys. Rev. D 65 (2002) 124001 [gr-qc/0204035] [INSPIRE].
  64. [64]
    F.J. Tipler, Singularities in conformally flat spacetimes, Phys. Lett. A 64 (1977) 8 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  65. [65]
    C. Clarke and A. Krolak, Conditions for the occurence of strong curvature singularities, J. Geom. Phys. 2 (1985) 127.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    R.-X. Miao, M. Li and Y.-G. Miao, Violation of the first law of black hole thermodynamics in f (T) gravity, JCAP 11 (2011) 033 [arXiv:1107.0515] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    M. Akbar and R.-G. Cai, Thermodynamic behavior of Friedmann equations at apparent horizon of FRW universe, Phys. Rev. D 75 (2007) 084003 [hep-th/0609128] [INSPIRE].ADSGoogle Scholar
  68. [68]
    A. Awad and A.F. Ali, Minimal length, Friedmann equations and maximum density, JHEP 06 (2014) 093 [arXiv:1404.7825] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    K. Bamba, S. Capozziello, M. De Laurentis, S. Nojiri and D. Sáez-Gómez, No further gravitational wave modes in F (T) gravity, Phys. Lett. B 727 (2013) 194 [arXiv:1309.2698] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Center for Theoretical PhysicsBritish University of EgyptSherouk CityEgypt
  2. 2.Department of Physics, School of Sciences and EngineeringAmerican University in CairoCairoEgypt
  3. 3.Department of Mathematics, Faculty of ScienceAin Shams UniversityCairoEgypt
  4. 4.Dipartimento di Fisica “E. Pancini”Università di Napoli “Federico II”, Complesso Universitario di Monte Sant’AngeloNapoliItaly
  5. 5.Istituto Nazionale di Fisica Nucleare (INFN), Sezione di NapoliComplesso Universitario di Monte Sant’AngeloNapoliItaly
  6. 6.Gran Sasso Science InstituteL’AquilaItaly

Personalised recommendations