Black Holes and Solution Generating Symmetries in Gravity

Examensarbete för masterexamen
Master Thesis
Physics and astronomy (MPPAS), MSc
Radnäs, Axel
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 G2(2) and the coset space by G2(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
Grundläggande vetenskaper , Fysik , Matematisk fysik , Basic Sciences , Physical Sciences , Mathematical physics
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