## Constructing and classifying five-dimensional black holes using integrability

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17/03/2022##### Author

Tomlinson, Fred

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##### Abstract

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.