Mathematical methods in higher dimensional gravity

The development of string theory as a possible framework for theunification of the fundamental interactions and for a quantum theory of gravity raises the question of the dimensionality of space-time. Indeed, a consistent string theory requires a space-time with ten dimensions, which means six dimensions more than the usual four dimensions in which standard physics is formulated. Recently, the special role of gravity in such higher dimensional space-times has been investigated through brane world models, in which our four dimensional universe is seen as a subspace, i.e. a brane, on which ordinary matter is confined and only gravity is higher dimensional. The project that I propose concerns the development of mathematical methods to improve our understanding of gravity in more than four dimensions. Unlike in four dimensions where it has been proved that Einstein's theory is the unique theory compatible with the postulates of general relativity, in higher dimensions, it has been shown that Einstein's equations must be completed by a specific set of terms of increasing complexity. As is already the case in Einstein gravity, the resulting equations, i.e. Lovelock's equations, can hardly be solved by straightforward calculations, except in some very simple cases. It is therefore necessary to develop alternative approaches. My project consists of four parts concerning related questions. The first question concerns the stability of the solutions. Given an arbitrary solution, how will it behave if we deform it slightly? Will it evolve towards another solution, i.e. unstable solution, or will it return to the initial one, i.e. stable solution? Unlike in Einstein gravity, the answer to this question seems to be related with algebraic properties of the given solution. These algebraic properties could therefore be used as a basis to classify the different types of solutions of Lovelock's equations. In four dimensional general relativity, confronted by the complexity of Einstein's equations, physicists and mathematicians have developed a set of construction methods. These are techniques which permit one to get new solutions out of known ones. This means that, though we can only get a few solutions by direct calculation, we can extend the catalogue of known solutions by performing such techniques without needing to solve the equations any further. It would therefore be interesting to extend such methods to higher dimensions.The importance of black hole physics in four dimensions has also motivated a lot of research in this area. In particular, there exist theorems constraining the shape of the black hole's horizons in Einstein gravity. In more than four dimensions, one is interested in black holes but also in black brane solutions, which represent extended objects, and subsequently one is interested in the allowed shapes of these objects. What I propose to study in this context is the role of the Lovelock higher order terms on the shape of the black hole solutions. Do they exclude shapes that are authorized in pure Einstein gravity? This could, for example, exclude singular or unstable solutions from the catalogue of solutions of the Lovelock equations.The last project that I propose concerns the relation between gravity in higher dimensions and string theory. In particular, it is related to the problem of exact, non-perturbative solutions in string theory. Indeed, string theory can only be solved exactly, on some specific backgrounds, otherwise, it is generally treated in perturbations. The point is that one of the non-perturbative backgrounds of string theory, i.e. the pp-wave solution, also has the property of solving Lovelock's equations to all orders. What I propose is to study the relation between both properties by finding the other solutions of Lovelock's equations to all orders. This could give some new non-perturbative solutions in string theory or some criteria to find them.