GLOBAL STUDIES OF EINSTEIN SPACETIMES

Yale University, New Haven CT

Investigators

Abstract

One of the main open theoretical problems in Einstein's theory of general relativity is the occurrence, in many otherwise reasonable solutions to the field equations, of spacetime singularities at which the curvature can become infinite, or "blow up", or at which some more subtle breakdown in the spacetime geometry can arise instead. The fundamental issue is whether these singularities represent a genuine failure of classical determinism for Einstein's theory of gravitation, and thus a potentially fatal flaw of general relativity itself, or whether instead they are in fact consistent with determinism in some subtle way. A conjecture due to Roger Penrose, called the "cosmic censorship conjecture" would, if it could be proven true, resolve this fundamental question and assure the logical consistency of Einstein's theory at the classical (i.e., non-quantum) level. To shed light on this difficult problem requires a study of the evolution of disturbances in the spacetime geometry and of how these disturbances (which propagate at the speed of light) can focus and (sometimes) blow up to yield physical singularities. One of the main aims of this project is to develop theoretical techniques for the precise analysis of such propagation, focusing and blowup of curvature with a view towards providing mathematical tools for an ultimate proof of Penrose's conjecture. Another aspect of this project deals with a fascinating connection, that has come to light only recently, between the "Einstein spaces" studied by purely Riemannian geometers and the "Einstein spacetimes" studied by mathematical general relativists. From the dynamical point of view, in which spacetime is seen as the evolution of space through time, certain Einstein spaces provide natural attractors for this dynamical evolution. This occurs not only for spacetimes of the classical dimension, but also for those of higher or lower than 4 dimensions as well. While the physical significance of higher dimensional spacetimes is still largely speculative there is little doubt that the aforementioned property of Einstein's theory (in various possible) dimensions) is of genuine mathematical interest. A final aspect of this project involves the development of improved techniques for the numerical study of gravitational waves emitted in exotic processes like the collision of two black holes. Such waves will almost surely be detected eventually by current or proposed experimental searches but it is conceivable that the ultimate success of these efforts may hinge on the prior development of accurate theoretical templates for the hypothetical waveforms in question. An ongoing effort of this project is to study the feasibility of computing the wave signals and analyzing their properties all the way out to null infinity, the natural boundary for the theoretical evolution.