On the Behavior of Solutions of Einstein's Equations and Other Geometric Nonlinear Partial Differential Equations

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

Einstein's gravitational field theory (general relativity), besides providing the most accurate current model for the study of gravitational physics on the astrophysical and cosmological scales, is also the source of a rich collection of mathematically interesting questions. This award will support a program of research which focusses on some of these, including the following: 1) Study of the behavior of the gravitational field very close to the big bang in cosmological solutions of Einstein's equations. Remarkably, it seems that even for solutions which appear to be very different some time after the big bang, the behavior close to the big bang is very similar: It oscillates in a characteristic way. We can show this, numerically or analytically, for simple families of solutions, and we hope to show it more generally. 2) Construction of new sets of initial conditions for the gravitational field which juxtapose two known sets. This is not easy because the initial conditions must satisfy certain nonlinear partial differential equations--the Einstein "constraints". We are developing techniques for smoothly joining ( or "gluing") two solutions of the constraints into a single solution. The first of these projects, although it ignores any quantum gravitational effects, could be very useful in our drive to understand the nature of the very early universe. The second is of interest both mathematically and physically. The mathematical interest stems from the fact that the Einstein constraint system is among the most complicated to which gluing studies have been applied. The physical interest comes from the possiblility of using the developed techniques to set up initial data for black hole collisions and other astrophysical problems which are being intensely studied numerically in anticipation of future data from LIGO.