Black holes in higher dimensions

Einstein's theory of General Relativity is a mathematical theory which currently provides the most accurate description of the force of gravity as observed in our universe. It predicts the existence of objects so massive that they distort space (and time) in such a way as to create a region from which nothing can escape -- such a region is called a black hole. Black holes provide such extreme settings for the study of gravitation that effects studied in quantum mechanics, the theory of the fundamental particles, become important. Therefore the study of black holes is an ideal arena within which to further our understanding of how gravity and quantum mechanics might be unified leading to a long sought after theory of quantum gravity.Superficially it appears that our universe has three spatial dimensions and one time dimension: together these are referred to as four dimensional space-time. A fundamental question is whether there are in fact more dimensions which we cannot see, either because they are too small or due to some other mechanism. In fact, String Theory, currently one of our best attempts at unifying the theory of gravity with the other forces, predicts the existence of extra dimensions. There are also more abstract reasons why it is important to investigate gravitation and objects such as black holes in more than four space-time dimensions. An important recent theoretical development is the realization that certain five dimensional theories of quantum gravity are in fact equivalent to particle physics theories in ordinary four dimensional space-time. This leads to the exciting possibility that we can learn about particle physics theories in regimes which are inaccessible using standard techniques, by studying five dimensional theories of gravity.The open problems within higher dimensional general relativity I propose to investigate are: (1) classification of all possible equilibrium (time independent) black holes and (2) dynamical stability of these black holes. In four space-time dimensions, both of these mathematical problems have been solved. It turns out that there is a unique black hole once one specifies its mass, spin and electric charge and it is stable. Furthermore, its event horizon (i.e. the boundary of the black hole) cannot have any holes - just like the surface of a (squashed) ball. These results are of clear astrophysical importance.In higher dimensions these problems are much more complicated. One reason for this is the existence of the black ring : a five dimensional black hole solution with a doughnut-like event horizon. It shows that an event horizon can have holes and thus need not be spherical. Furthermore, unlike in four dimensions, the mass, spin and electric charge are insufficient to specify a black hole as one can have both spherical and ring-like horizons. I intend to make progress on problems (1) and (2) by focusing on certain subsets of black holes which, while still physically interesting, are mathematically more amenable to analysis. I have already been developing new methods aimed at problem (1) by focusing on certain subsets of black holes and have used these successfully to answer certain open problems. I have also already worked on problem (2) by providing the first stability analysis for a certain subset of highly symmetric black holes. This has given me extensive experience with both (1) and (2) and I intend to use this to address these problems for more generic subsets of black holes.The results of these problems of higher dimensional general relativity have direct applications to string theory and quantum gravity, which I hope to study. They will contribute to the solution of important open problems such as the quantum description of black rings within string theory and the quantum description of certain black holes in terms of ordinary four dimensional particle physics theories. Solving such problems will deepen our understanding of quantum gravity.