Constructing and classifying five-dimensional black holes using integrability

• Main Author: Tomlinson, Fred
• Format: Dissertation
• Language: English
• Published: University of Edinburgh 2022
• DOI: 10.7488/era/1993

In this thesis we look at the problem of nding and classifying stationary and biaxisymmetric solutions in ve-dimensional theories of gravity, using particular hidden symmetries. We consider three theories: the electrostatic sector of Einstein-Maxwell, vacuum gravity and minimal supergravity (Einstein-Maxwell gravity with a Chern-Simons term). For electrostatic solutions to Einstein-Maxwell theory, the equations on the metric and Maxwell eld possess a SL(2;R) symmetry. This allows one to derive transformations which either charge a solution or immerse it in an electric Melvin background. By considering a neutral static black lens seed and performing these two transformations with appropriately tuned transformation parameters, we construct the rst example of a regular black lens in Einstein Maxwell theory with topologically trivial asymptotics. For vacuum gravity we consider asymptotically at solutions. The vacuum Einstein equations are integrable in the sense that they can be reformulated as the integrability condition for an auxiliary linear system of PDEs. Taking these PDEs, one can integrate them over the event horizons, the axes of symmetry and in nity. By carefully considering continuity conditions between these solutions, one may actually solve for metric data on the horizons and the axes in terms of some geometrically de ned moduli, subject to a set of polynomial constraints. This represents a very useful tool for answering the existence problem, reducing it to the much more tractable question of whether a particular system of polynomials (subject to some inequalities) has any solutions. Using this polynomial system we provide a constructive uniqueness proof for the Kerr (using analogous four-dimensional results), Myers-Perry and black ring solutions. We also prove, through a combination of analytic and numerical methods, that the \simplest" L(n; 1) black lens cannot exist by showing that it must possess a conical singularity on one of the axes. Finally we consider the case of asymptotically at solutions in minimal supergravity. As with the vacuum, this is an integrable theory and so a similar analysis can be performed with exactly analogous results, although with rather more complicated polynomial systems determining the existence of solutions. A notable feature of minimal supergravity, not present in the vacuum theory, is the existence of regular solitons - in this context these are non-trivial solutions without black hole regions. We begin the exploration of the moduli space of these solitons by rst studying the case of at space.