Mathematical Problems in General Relativity

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ABSTRACTMATHEMATICAL PROBLEMS IN GENERAL RELATIVITYThis proposal deals with mathematical aspects of the problem ofgravitational collapse and the formation of black holes. Analyticallyspeaking, this is the study of the initial value problem forappropriate Einstein-matter systems for asymptotically flat initialdata. The central questions in the subject are the celebrated weak andstrong cosmic censorship conjectures of Penrose. In previous work,the author was able to resolve the issue of strong cosmic censorship inthe setting of the spherically symmetric Einstein-Maxwell scalar fieldsystem, confirming a heuristic picture that had been the subject of debatein the physics community. The present proposal concerns the developmentof 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 issimulated through charge. Mathematically, a rigorous formulation is givenby the initial value problem in the large for a charged self-gravitating scalarfield, i.e. for the Einstein-Maxwell-charged scalar field system. Thestudy can be divided roughly into three analytically distinct familiesof problems: the study of the formation of trapped surfaces,the study of the decay of fields on the event horizon, and the study ofthe instability of the Cauchy horizon.All problems require methods which go well beyond standard techniques ofthe theory of quasilinear hyperbolic p.d.e.'s in two dimensions.In particular, the interaction between geometric features characteristicof black holes with non-linear wave equations will certainly play a bigrole. Understanding these issues in the spherically symmetric case willhopefully point to the correct framework where these issues can eventuallybe studied in the absense of symmetry.The collapse of a star under the force of its own gravity andthe possible subsequent formation of a black hole is one of themost exciting predictions of the general theory of relativity. Thisprocess, however, is not well understood. Most of our currentintuition derives from so-called ``explicit solutions'' of theEinstein equations, the governing equations of the theory.These ``explicit'' solutions, however, in addition to exhibitingbehavior which is hoped to be physically relevant, exhibit alsobehavior widely considered to be pathological. For example, the Kerrsolution, on which our intuition for the final state of gravitationalcollapse 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 thissolution, or is this a general property of black holes? In viewof the strong non-linearity of the Einstein equations, and the difficultyof making accurate predictions on the basis of numerical or heuristicwork, this seems to be an area where the techniques of rigorousmathematical analysis can make a tremendous impact.In this project, it is proposed to undertake a rigorous mathematicalstudy of a realistic model of gravitational collapse, in particular onewhich takes account of angular momentum, the mechanism that leads to thefailure of ``causality'' in the Kerr solution. Partial progress hasalready been made in the author's previous work, and in particular, thequestion of the stability of the picture alluded to above has beenresolved in a very restricted setting. As this problem addresses aspectacular failure of Newtonian determinism in the setting of acompletely classical (i.e. non-quantum)physical theory, this work could impact current viewson some of the fundamental principles of physics.