Title:

Extreme black holes and nearhorizon geometries

In this thesis we study nearhorizon geometries of extreme black holes. We first consider stationary extreme black hole solutions to the EinsteinYangMills theory with a compact semisimple gauge group in four dimensions, allowing for a negative cosmological constant. We prove that any axisymmetric black hole of this kind possesses a nearhorizon AdS2 symmetry and deduce its nearhorizon geometry must be that of the abelian embedded extreme KerrNewman (AdS) black hole. We show that the nearhorizon geometry of any static black hole is a direct product of AdS2 and a constant curvature space. We then consider nearhorizon geometry in Einstein gravity coupled to a Maxwell field and a massive complex scalar field, with a cosmological constant. We prove that assuming nonzero coupling between the Maxwell and the scalar fields, there exists no solution with a compact horizon in any dimensions where the massive scalar is nontrivial. 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 threedimensional vacuum spacetime with a negative cosmological constant containing a nonsingular 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 nearhorizon 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 asymptoticsymmetry 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 nonextreme BTZ black hole are absent from our general solution with a nondegenerate horizon. We then show that the first order deformation along transverse null geodesics about any nearhorizon geometry with compact crosssections always admits a finiteparameter family of solutions as the most general solution. As an application, we consider the first order expansion from the nearhorizon 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 nearhorizon geometry of the extreme selfdual MyersPerry 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 selfdual MyersPerry black hole itself and the extreme J = 0 KaluzaKlein black hole. These are the only known black holes to possess this nearhorizon geometry. If only U(1) X U(1) symmetry is assumed in first order, we find that the most general solution is a threeparameter family which is more general than the two known black hole solutions. This hints the possibility of the existence of new black holes.
