Extreme Black Holes and Near-Horizon Geometries
Li, Ka Ki
In this thesis we study near-horizon geometries of extreme black holes. We first consider stationary extreme black hole solutions to the Einstein-Yang-Mills theory with a compact semi-simple gauge group in four dimensions, allowing for a negative cosmological constant. We prove that any axisymmetric black hole of this kind possesses a near-horizon AdS2 symmetry and deduce its near-horizon geometry must be that of the abelian embedded extreme Kerr-Newman (AdS) black hole. We show that the near-horizon geometry of any static black hole is a direct product of AdS2 and a constant curvature space. We then consider near-horizon geometry in Einstein gravity coupled to a Maxwell field and a massive complex scalar field, with a cosmological constant. We prove that assuming non-zero coupling between the Maxwell and the scalar fields, there exists no solution with a compact horizon in any dimensions where the massive scalar is non-trivial. This result generalises to any scalar potential which is a monotonically increasing function of the modulus of the complex scalar. Next we determine the most general three-dimensional vacuum spacetime with a negative cosmological constant containing a non-singular Killing horizon. We show that the general solution with a spatially compact horizon possesses a second commuting Killing field and deduce that it must be related to the BTZ black hole (or its near-horizon geometry) by a diffeomorphism. We show there is a general class of asymptotically AdS3 extreme black holes with arbitrary charges with respect to one of the asymptotic-symmetry Virasoro algebras and vanishing charges with respect to the other. We interpret these as descendants of the extreme BTZ black hole. However descendants of the non-extreme BTZ black hole are absent from our general solution with a non-degenerate horizon. We then show that the first order deformation along transverse null geodesics about any near-horizon geometry with compact cross-sections always admits a finite-parameter family of solutions as the most general solution. As an application, we consider the first order expansion from the near-horizon geometry of the extreme Kerr black hole. We uncover a local uniqueness theorem by demonstrating that the only possible black hole solutions which admit a U(1) symmetry are gauge equivalent to the first order expansion of the extreme Kerr solution itself. We then investigate the first order expansion from the near-horizon geometry of the extreme self-dual Myers-Perry black hole in 5D. The only solutions which inherit the enhanced SU(2) X U(1) symmetry and are compatible with black holes correspond to the first order expansion of the extreme self-dual Myers-Perry black hole itself and the extreme J = 0 Kaluza-Klein black hole. These are the only known black holes to possess this near-horizon geometry. If only U(1) X U(1) symmetry is assumed in first order, we find that the most general solution is a three-parameter family which is more general than the two known black hole solutions. This hints the possibility of the existence of new black holes.