Abstract
We investigate three-dimensional (3D) Anti-de Sitter (AdS) gravity coupled to a complex scalar field ϕ with self-interaction potential V (|ϕ|). We show that the mass of static, rotationally symmetric, AdS black hole with scalar hairs is determined algebraically by the scalar charges. We recast the field equations as a linear system of first order differential equations. Exact solutions, describing 3D AdS black holes with real spherical scalar hairs and vortex-black hole solutions are derived in closed form for the case of a scalar field saturating the Breitenlohner-Freedman (BF) bound and for a scalar field with asymptotic zero mass. The physical properties of these solutions are discussed. In particular, we show that the vortex solution interpolates between two different AdS3 vacua, corresponding respectively to a U(1)-symmetry-preserving maximum and to a symmetrybreaking minimum of the potential V.
Similar content being viewed by others
References
J.D. Bekenstein, Transcendence of the law of baryon-number conservation in black hole physics, Phys. Rev. Lett. 28 (1972) 452 [SPIRES].
C. Teitelboim, Nonmeasurability of the quantum numbers of a black hole, Phys. Rev. D 5 (1972) 2941 [SPIRES].
J.D. Bekenstein, Novel ’no scalar hair’ theorem for black holes, Phys. Rev. D 51 (1995) 6608 [SPIRES].
T. Hertog, Towards a Novel no-hair Theorem for Black Holes, Phys. Rev. D 74 (2006) 084008 [gr-qc/0608075] [SPIRES].
K. Hotta, Y. Hyakutake, T. Kubota, T. Nishinaka and H. Tanida, The CFT-interpolating Black Hole in Three Dimensions, JHEP 01 (2009) 010 [arXiv:0811.0910] [SPIRES].
M. Cadoni and M.R. Setare, Near-horizon limit of the charged BTZ black hole and AdS 2 quantum gravity, arXiv:0806.2754 [SPIRES].
M. Cadoni, M. Melis and P. Pani, Microscopic entropy of black holes and AdS 2 quantum gravity, arXiv:0812.3362 [SPIRES].
C. Martinez, C. Teitelboim and J. Zanelli, Charged rotating black hole in three spacetime dimensions, Phys. Rev. D 61 (2000) 104013 [hep-th/9912259] [SPIRES].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [SPIRES].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic Superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [SPIRES].
N. Kim, Y. Kim and K. Kimm, Global vortex and black cosmic string, Phys. Rev. D 56 (1997) 8029 [gr-qc/9707056] [SPIRES].
D. Maity, S. Sarkar, N. Sircar, B. Sathiapalan and R. Shankar, Properties of CFTs dual to Charged BTZ black-hole, arXiv:0909.4051 [SPIRES].
P. Breitenlohner and D.Z. Freedman, Positive Energy in anti-de Sitter Backgrounds and Gauged Extended Supergravity, Phys. Lett. B 115 (1982) 197 [SPIRES].
M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Asymptotically anti-de Sitter spacetimes and scalar fields with a logarithmic branch, Phys. Rev. D 70 (2004) 044034 [hep-th/0404236] [SPIRES].
M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [SPIRES].
D. Ruelle, Statistical Mechanics: Rigorous Results, Addison-Wesley, Reading, MA U.S.A. (1989).
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 0911.3573
Rights and permissions
About this article
Cite this article
Cadoni, M., Pani, P. & Serra, M. Scalar hairs and exact vortex solutions in 3D AdS gravity. J. High Energ. Phys. 2010, 91 (2010). https://doi.org/10.1007/JHEP01(2010)091
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2010)091