Mathematical Problems in General Relativity

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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 cosmic censorship conjectures of Penrose. In previous work, the author was able to resolve the issue of strong cosmic censorship in the setting of a preliminary model, confirming a heuristic picture that had been the subject of debate in the physics community.

A central goal of the present proposal is 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 (Price's law), and the study of the instability of the Cauchy horizon. In parallel with this model, other self-gravitating systems will be studied, motivated in part by current research interests in the numerical relativity and high energy physics communities. 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 absence of symmetry.

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The Old Schools, Trinity Lane
CAMBRIDGE
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