Geodesic structure of Lifshitz black holes in 2+1 dimensions

  • Norman Cruz
  • Marco Olivares
  • J. R. VillanuevaEmail author
Regular Article - Theoretical Physics


We present a study of the geodesic equations of a black hole space-time which is a solution of the three-dimensional NMG theory and is asymptotically Lifshitz with z=3 and d=1 as found in Ayon-Beato et al. (Phys. Rev. D 80:104029, 2009). By means of the corresponding effective potentials for massive particles and photons we find the allowed motions by the energy levels. Exact solutions for radial and non-radial geodesics are given in terms of the Weierstrass elliptic ℘, σ, and ζ functions.


Black Hole Event Horizon Effective Potential Black Hole Solution Massless Particle 
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M.O. thanks to PUCV. This work was supported by DICYT-USACH Grant No. 041331CM (NC).


  1. 1.
    E. Ayon-Beato, A. Garbarz, G. Giribet, M. Hassaine, Lifshitz black hole in three dimensions. Phys. Rev. D 80, 104029 (2009). arXiv:0909.1347 MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    K. Balasubramanian, J. McGreevy, An analytic Lifshitz black hole. Phys. Rev. D 80, 104039 (2009). arXiv:0909.0263 MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    M. Bañados, C. Teitelboim, J. Zanelli, The black hole in three dimensional space time. Phys. Rev. Lett. 69, 1849 (1992). arXiv:hep-th/9204099 MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    M. Bañados, M. Henneaux, C. Teitelboim, J. Zanelli, Geometry of the 2+1 black hole. Phys. Rev. D 48, 1506 (1993). arXiv:gr-qc/9302012 MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    E. Bergshoeff, O. Hohm, P. Townsend, Massive gravity in three dimensions. Phys. Rev. Lett. 102, 201301 (2009). arXiv:0901.1766 MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    E. Brynjolfsson, U. Danielsson, L. Thorlacius, T. Zingg, Holographic superconductors with Lifshitz scaling. J. Phys. A 43, 065401 (2010). arXiv:0908.2611 MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1983) zbMATHGoogle Scholar
  8. 8.
    N. Cruz, C. Martínez, L. Peña, Geodesic structure of the (2+1)-dimensional BTZ black hole. Class. Quantum Gravity 11, 2731 (1994). arXiv:gr-qc/9401025 ADSzbMATHCrossRefGoogle Scholar
  9. 9.
    N. Cruz, M. Olivares, J.R. Villanueva, The geodesic structure of the Schwarzschild anti-de Sitter black hole. Class. Quantum Gravity 22, 1167–1190 (2005). arXiv:hep-ph/0408016 MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    U. Danielsson, L. Thorlacius, Black holes in asymptotically Lifshitz spacetime. J. High Energy Phys. 0903, 070 (2009). arXiv:0812.5088 MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    C. Farina, J. Gamboa, A.J. Seguí-Santonja, Motion and trajectories of particles around three-dimensional black holes. Class. Quantum Gravity 10, 193 (1993). arXiv:hep-lat/9303005 ADSCrossRefGoogle Scholar
  12. 12.
    H.A. Gonzalez, D. Tempo, R. Troncoso, Field theories with anisotropic scaling in 2D, solitons and the microscopic entropy of asymptotically Lifshitz black holes. J. High Energy Phys. 1111, 066 (2011). arXiv:1107.3647 MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    S. Kachru, X. Liu, M. Mulligan, Gravity duals of Lifshitz-like fixed points. Phys. Rev. D 78, 106005 (2008). arXiv:0808.1725 MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    R.B. Mann, Lifshitz topological black holes. J. High Energy Phys. 06, 075 (2009). arXiv:0905.1136 ADSCrossRefGoogle Scholar
  15. 15.
    Y.S. Myung, Phase transitions for the Lifshitz black holes. Eur. Phys. J. C 72, 2116 (2012). arXiv:1203.1367 ADSCrossRefGoogle Scholar
  16. 16.
    M. Nakasone, I. Oda, On unitarity of massive gravity in three dimensions. Prog. Theor. Phys. 121, 1389 (2009). arXiv:0902.3531 ADSzbMATHCrossRefGoogle Scholar
  17. 17.
    S. Deser, Ghost-free, finite, fourth order D=3 (alas) gravity. Phys. Rev. Lett. 103, 101302 (2009). arXiv:0904.4473 MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    M. Olivares, G. Rojas, Y. Vásquez, J.R. Villanueva, Particles motion on topological Lifshitz black holes in 3+1 dimensions. arXiv:1304.4297 (2013)

Copyright information

© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Authors and Affiliations

  • Norman Cruz
    • 1
  • Marco Olivares
    • 2
  • J. R. Villanueva
    • 3
    • 4
    Email author
  1. 1.Departamento de Física, Facultad de CienciaUniversidad de Santiago de ChileSantiagoChile
  2. 2.Instituto de FísicaPontificia Universidad de Católica de ValparaísoValparaísoChile
  3. 3.Departamento de Física y Astronomía, Facultad de CienciasUniversidad de ValparaísoValparaísoChile
  4. 4.Centro de Astrofísica de ValparaísoValparaísoChile

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