This proposal deals with mathematical aspects of the problem of gravitational collapse and the formation of black holes. Analytically speaking, this is the study of the initial value problem for appropriate Einstein-matter systems for asymptotically flat initial data. The central questions in the subject are the celebrated weak and strong cosmic censorship conjectures of Penrose. In previous work, the author was able to resolve the issue of strong cosmic censorship in the setting of the spherically symmetric Einstein-Maxwell scalar field system, confirming a heuristic picture that had been the subject of debate in the physics community. The present proposal concerns the development of a complete theory for gravitational collapse in spherical symmetry, which at the same time takes into account the physics of angular momentum. In view of the constraint of spherical symmetry, angular momentum is simulated through charge. Mathematically, a rigorous formulation is given by the initial value problem in the large for a charged self-gravitating scalar field, i.e. for the Einstein-Maxwell-charged scalar field system. The study can be divided roughly into three analytically distinct families of problems: the study of the formation of trapped surfaces, the study of the decay of fields on the event horizon, and the study of the instability of the Cauchy horizon. All problems require methods which go well beyond standard techniques of the theory of quasilinear hyperbolic p.d.e.'s in two dimensions. In particular, the interaction between geometric features characteristic of black holes with non-linear wave equations will certainly play a big role. Understanding these issues in the spherically symmetric case will hopefully point to the correct framework where these issues can eventually be studied in the absense of symmetry.
The collapse of a star under the force of its own gravity and the possible subsequent formation of a black hole is one of the most exciting predictions of the general theory of relativity. This process, however, is not well understood. Most of our current intuition derives from so-called ``explicit solutions'' of the Einstein equations, the governing equations of the theory. These ``explicit'' solutions, however, in addition to exhibiting behavior which is hoped to be physically relevant, exhibit also behavior widely considered to be pathological. For example, the Kerr solution, on which our intuition for the final state of gravitational collapse is largely based, contains so-called ``closed timelike curves''. These allow observers who enter the black hole to subsequently travel in time. Is this an unhappy accident of the very special nature of this solution, or is this a general property of black holes? In view of the strong non-linearity of the Einstein equations, and the difficulty of making accurate predictions on the basis of numerical or heuristic work, this seems to be an area where the techniques of rigorous mathematical analysis can make a tremendous impact. In this project, it is proposed to undertake a rigorous mathematical study of a realistic model of gravitational collapse, in particular one which takes account of angular momentum, the mechanism that leads to the failure of ``causality'' in the Kerr solution. Partial progress has already been made in the author's previous work, and in particular, the question of the stability of the picture alluded to above has been resolved in a very restricted setting. As this problem addresses a spectacular failure of Newtonian determinism in the setting of a completely classical (i.e. non-quantum) physical theory, this work could impact current views on some of the fundamental principles of physics.
