Abstract
We consider four-dimensional conformal gravity coupled to the U(1) Maxwell and SU(2) Yang-Mills fields. We study the structure of general black hole solutions carrying five independent parameters: the mass, the electric U(1) and magnetic SU(2) charges, the massive spin-2 charge and the thermodynamical pressure associated with the cosmological constant, which is an integration constant in conformal gravity. We derive the thermodynamical first law of the black holes. We obtain some exact solutions including an extremal black hole with vanishing mass and entropy, but with non-trivial SU(2) Yang-Mills charges. We derive the remainder of the first law for this special solution. We also reexamine the colored black holes and derive their first law in Einstein-Yang-Mills gravity with or without a cosmological constant.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Bartnik and J. Mckinnon, Particle-like solutions of the Einstein-Yang-Mills equations, Phys. Rev. Lett. 61 (1988) 141 [INSPIRE].
P. Bizon, Colored black holes, Phys. Rev. Lett. 64 (1990) 2844 [INSPIRE].
B. Kleihaus and J. Kunz, Rotating hairy black holes, Phys. Rev. Lett. 86 (2001) 3704 [gr-qc/0012081] [INSPIRE].
B. Kleihaus, J. Kunz and F. Navarro-Lerida, Rotating Einstein-Yang-Mills black holes, Phys. Rev. D 66 (2002) 104001 [gr-qc/0207042] [INSPIRE].
T. Torii, K.-i. Maeda and T. Tachizawa, Cosmic colored black holes, Phys. Rev. D 52 (1995) 4272 [gr-qc/9506018] [INSPIRE].
E. Winstanley, Existence of stable hairy black holes in SU(2) Einstein-Yang-Mills theory with a negative cosmological constant, Class. Quant. Grav. 16 (1999) 1963 [gr-qc/9812064] [INSPIRE].
A.H. Chamseddine and M.S. Volkov, Non-Abelian Bogomol’nyi-Prasad-Sommerfield monopoles in N = 4 gauged supergravity, Phys. Rev. Lett. 79 (1997) 3343 [hep-th/9707176] [INSPIRE].
P. Meessen, Supersymmetric coloured/hairy black holes, Phys. Lett. B 665 (2008) 388 [arXiv:0803.0684] [INSPIRE].
M. Hübscher, P. Meessen, T. Ortín and S. Vaulà, N=2 Einstein-Yang-Mills’s BPS solutions, JHEP 09 (2008) 099 [arXiv:0806.1477] [INSPIRE].
F. Canfora, F. Correa, A. Giacomini and J. Oliva, Exact meron black holes in four dimensional SU(2) Einstein-Yang-Mills theory, Phys. Lett. B 722 (2013) 364 [arXiv:1208.6042] [INSPIRE].
P. Bueno, P. Meessen, T. Ortin and P.F. Ramirez, \( \mathcal{N}=2 \) Einstein-Yang-Mills’ static two-center solutions, JHEP 12 (2014) 093 [arXiv:1410.4160] [INSPIRE].
H. Lü and C.N. Pope, Critical gravity in four dimensions, Phys. Rev. Lett. 106 (2011) 181302 [arXiv:1101.1971] [INSPIRE].
J. Maldacena, Einstein gravity from conformal gravity, arXiv:1105.5632 [INSPIRE].
R.J. Riegert, Birkhoff ’s theorem in conformal gravity, Phys. Rev. Lett. 53 (1984) 315 [INSPIRE].
P.D. Mannheim and D. Kazanas, Exact vacuum solution to conformal Weyl gravity and galactic rotation curves, Astrophys. J. 342 (1989) 635 [INSPIRE].
P.D. Mannheim and D. Kazanas, Solutions to the Kerr and Kerr-Newman problems in fourth order conformal Weyl gravity, Phys. Rev. D 44 (1991) 417 [INSPIRE].
H.-S. Liu and H. Lü, Charged rotating AdS black hole and its thermodynamics in conformal gravity, JHEP 02 (2013) 139 [arXiv:1212.6264] [INSPIRE].
Y. Brihaye and Y. Verbin, Spherical non-Abelian solutions in conformal gravity, Phys. Rev. D 81 (2010) 044041 [arXiv:0910.0973] [INSPIRE].
Y. Brihaye and Y. Verbin, Spherical structures in conformal gravity and its scalar-tensor extension, Phys. Rev. D 80 (2009) 124048 [arXiv:0907.1951] [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
H. Lü, Y. Pang and C.N. Pope, Black holes in six-dimensional conformal gravity, Phys. Rev. D 87 (2013) 104013 [arXiv:1301.7083] [INSPIRE].
A. Corichi, U. Nucamendi and D. Sudarsky, Einstein-Yang-Mills isolated horizons: phase space, mechanics, hair and conjectures, Phys. Rev. D 62 (2000) 044046 [gr-qc/0002078] [INSPIRE].
H. Lü, Y. Pang, C.N. Pope and J.F. Vazquez-Poritz, AdS and Lifshitz black holes in conformal and Einstein-Weyl gravities, Phys. Rev. D 86 (2012) 044011 [arXiv:1204.1062] [INSPIRE].
H. Lü and Z.-L. Wang, Exact Green’s function and Fermi surfaces from conformal gravity, Phys. Lett. B 718 (2013) 1536 [arXiv:1210.4560] [INSPIRE].
J. Li, H.-S. Liu, H. Lü and Z.-L. Wang, Fermi surfaces and analytic Green’s functions from conformal gravity, JHEP 02 (2013) 109 [arXiv:1210.5000] [INSPIRE].
S. Deser and B. Tekin, Energy in generic higher curvature gravity theories, Phys. Rev. D 67 (2003) 084009 [hep-th/0212292] [INSPIRE].
N. Okuyama and J.-i. Koga, Asymptotically anti de Sitter spacetimes and conserved quantities in higher curvature gravitational theories, Phys. Rev. D 71 (2005) 084009 [hep-th/0501044] [INSPIRE].
Y. Pang, Brief note on AMD conserved quantities in quadratic curvature theories, Phys. Rev. D 83 (2011) 087501 [arXiv:1101.4267] [INSPIRE].
A. Ashtekar and A. Magnon, Asymptotically anti-de Sitter space-times, Class. Quant. Grav. 1 (1984) L39 [INSPIRE].
A. Ashtekar and S. Das, Asymptotically anti-de Sitter space-times: conserved quantities, Class. Quant. Grav. 17 (2000) L17 [hep-th/9911230] [INSPIRE].
S. Gao, The first law of black hole mechanics in Einstein-Maxwell and Einstein-Yang-Mills theories, Phys. Rev. D 68 (2003) 044016 [gr-qc/0304094] [INSPIRE].
H.-S. Liu, H. Lü and C.N. Pope, Thermodynamics of Einstein-Proca AdS black holes, JHEP 06 (2014) 109 [arXiv:1402.5153] [INSPIRE].
H.-S. Liu and H. Lü, Scalar charges in asymptotic AdS geometries, Phys. Lett. B 730 (2014) 267 [arXiv:1401.0010] [INSPIRE].
H. Lü, C.N. Pope and Q. Wen, Thermodynamics of AdS black holes in Einstein-scalar gravity, arXiv:1408.1514 [INSPIRE].
H.-S. Liu and H. Lü, Thermodynamics of Lifshitz black holes, JHEP 12 (2014) 071 [arXiv:1410.6181] [INSPIRE].
D. Kastor, S. Ray and J. Traschen, Enthalpy and the mechanics of AdS black holes, Class. Quant. Grav. 26 (2009) 195011 [arXiv:0904.2765] [INSPIRE].
M. Cvetič, G.W. Gibbons, D. Kubiznak and C.N. Pope, Black hole enthalpy and an entropy inequality for the thermodynamic volume, Phys. Rev. D 84 (2011) 024037 [arXiv:1012.2888] [INSPIRE].
C.N. Pope, The embedding of the Einstein-Yang-Mills equations in d = 11 supergravity, Class. Quant. Grav. 2 (1985) L77 [INSPIRE].
H. Lü and J.F. Vazquez-Poritz, Four-dimensional Einstein-Yang-Mills-de Sitter gravity from eleven-dimensions, Phys. Lett. B 597 (2004) 394 [hep-th/0308104] [INSPIRE].
R.B. Mann, E. Radu and D.H. Tchrakian, Non-Abelian solutions in AdS 4 and d = 11 supergravity, Phys. Rev. D 74 (2006) 064015 [hep-th/0606004] [INSPIRE].
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
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1411.5372
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), 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.
About this article
Cite this article
Fan, ZY., Lü, H. SU(2)-colored (A)dS black holes in conformal gravity. J. High Energ. Phys. 2015, 13 (2015). https://doi.org/10.1007/JHEP02(2015)013
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2015)013