# Black Holes and Solution Generating Symmetries in Gravity

[Examensarbete på avancerad nivå]

A gravity theory in *D* dimensions with a spacetime metric that admits *n* commuting Killing vectors can be dimensionally reduced to an effectively (*D − n*)-
dimensional theory. The reduction is performed by using Kaluza-Klein techniques and, as it turns out, the lower dimensional theory reveals a hidden global symmetry,
described by a group *G*, on the space of solutions. This symmetry is not seen in the original *D*-dimensional theory. When the solutions admit a suffcient number of Killing vectors we can dimensionally reduce down to *D* = 3 where we get a non-linear sigma-model. This sigma-model contains scalar fields which originate from the *D*-dimensional metric and whatever other possible *D*-dimensional fields the theory may contain. These scalar fields can also be described as parameters in a coset space *G/H* which depends on the particular *D*-dimensional theory we started from. If one rewrites the sigma-model in terms of coset representatives the hidden symmetry emerges and becomes manifest; the sigma-model is invariant under global *G* transformations. That is, given a solution to the equations of motion we can transform it to get a new solution. Thus, the technique utilizes the symmetries which become manifest upon dimensional reduction to generate new solutions.

For the case when four-dimensional pure gravity is reduced over the time dimension to three-dimensions the symmetry is described by *SL*(2,R) and the coset space by *SL*(2,R)/*SO*(1, 1). We demonstrate how the Reissner-Nordströom solution and the Schwarzschild solution are related by a *SO*(1, 1) transformation and identity the subgroup *SO*(1, 1) as the generator of electric charge. For the stationary
axisymmetric solutions in *D* = 4 we can reduce down to two dimensions. The remarkable property of two-dimensional gravity is that the symmetry group *G* enlarges to an infinite-dimensional symmetry group. In terms of group theory this corresponds to the affine Kac-Moody group associated to the group *G*. In this thesis we explicitly show how *SL*(2,R) enlarges to its affine extension *SL*(2,R)^{+}. The coset space *G/H* has to be extended to a coset space *G*^{+}/*H*^{+} which requires an introduction of a spectral parameter and the so called monodromy matrix. This matrix encodes all the information about the spacetime metric and the key
problem is to factorize this matrix. This amounts to a certain infinte-dimensional Riemann-Hilbert problem.

The main goal of this thesis has been to solve this for the case of minimal supergravity in five dimensions where the symmetry is given by *G*_{2(2)} and the coset space by *G*_{2(2)}/*SO*(2, 2). As a result of this thesis, we have constructed the seed monodromy matrix for Schwarzschild and generated the five-dimensional Reissner-Nordströom metric via a *SO*(2,2) transformation

Publikationen registrerades 2014-12-30. Den ändrades senast 2015-01-07

CPL ID: 209215

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