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A black hole is a spacetime that solves Einstein equations of general relativity characterized by the fact that it posseses an event horizon hypersurface (or several of them) which has a number of special characteristics; for example the light can not escape from the space confined by the horizon hypersurface due to gravitational effects. Much of the theoretical considerations are about the entropy of black holes (cf. Bekenstein-Hawking entropy) and the information paradox.
Black holes are considered theoretically for gravitational theories in various number $d$ of dimension. For $d \geq 5$ a black hole spacetime may have nontrivial topology, e.g. black rings are possible.
In usual asymptotically 3+1-dimensional Minkowski spacetime, the first black hole solution that was found is the Schwarzschild black hole solution; such a black hole posses a single horizon hypersurface and seems to be stable under various perturbations.
Another solution with finite angular momentum is called the Kerr spacetime, and there is a simple generalization having also the electric charge, the Newman solution or the Kerr-Newman black hole. This solution differs pretty much from the Schwarzschild solution and its structure is unstable under various physical mechanisms and perturbations; it possesses two horizons, inner and outer.
Hawking’s Theorem of Black Hole topology asserts that the in case of $d = 4$ asymptotically flat stationary black holes satisfying the suitable dominant energy condition, the cross sections of the event horizon are spherical.
Galloway and Schoen extended this theorem to higher dimensions; they showed that the cross sections of event horizon (stationary case) and the outer (apparent) horizon (general case) are of Yamabe type.
See
Some candidate astrophysical? objects which seem to point to black hole have been observed.
wikipedia: black hole
Barrett O’Neill, The geometry of Kerr black holes
S. Chandrasekhar, The mathematical theory of black holes
G. T. Horowitz, A. Strominger, Counting states of near-extremal black holes, Phys. Rev. Lett. 77 (1996) 2368–2371, hep-th/9602051.
R. D’Auria, P. Fre’, BPS black holes in supergravity, hep-th/9812160
Gregory Galloway, Richard Schoen, A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions Commun. Math. Phys. (2006) (pdf)
R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, University of Chicago Press 1994; The back reaction effect in particle creation in curved spacetime, Commun. Math. Phys. 54, 1 (1977)
Ahmed Almheiri, Donald Marolf, Joseph Polchinski, James Sully, Black holes: complementarity or firewalls?, http://arxiv.org/abs/arXiv:1207.3123
Discussion of black holes in the context of the holographic principle and the AdS-CFT correspondence is in
Monica Guica, Thomas Hartman, Wei Song, Andrew Strominger, The Kerr/CFT Correspondence (arXiv:0809.4266)
Alejandra Castro, Alexander Maloney, Andrew Strominger, Hidden Conformal Symmetry of the Kerr Black Hole (arXiv:1004.0996)
The nature of the event horizon, specifically, is discussed in