Initial value problems in general relativity

Abstract

The initial value formulation viewpoint is one of the main foci of research in general relativity. This thesis establishes results for two problems pertinent to it. In the first of the problems which form the contents of Chapters 3 and 4, the focus is on a class of five dimensional stationary asymptotically flat spacetimes. These naturally arise in high energy physics as possible microstates of black hole spacetimes. In Chapter 3, several spacetimes with non-trivial topology in the domain of outer communication are considered, of which the soliton spacetime is one such example. The mass variation formula previously established for such spacetimes is used to compute energy, angular momenta and charge for these spacetimes. It is shown that regularity is essential for the formula relating them to hold. In Chapter 4, the decay of the wave equation in a family of soliton spacetimes is studied and a slow decay rate is established, hinting at nonlinear instability. The second problem is establishing a horizon-based initial boundary value formulation with the goal of studying the near-horizon spacetime. The problem is addressed in the setting of four-dimensional general relativity. In Chapter 5, we establish that data specified on the horizon and a future null boundary determine the near horizon geometry and illustrate this for spherically symmetric spacetimes with a massless scalar field. In Chapter 6 we conclude with directions for future research.