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

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Einstein's gravitational field theory (general relativity), besidesproviding the most accurate current model for the study of gravitationalphysics on the astrophysical and cosmological scales, is also the source ofa rich collection of mathematically interesting questions. This award willsupport a program of research which focusses on some of these, includingthe following:1) Study of the behavior of the gravitational field very close to the bigbang in cosmological solutions of Einstein's equations. Remarkably, itseems that even for solutions which appear to be very different some timeafter the big bang, the behavior close to the big bang is very similar: Itoscillates in a characteristic way. We can show this, numerically oranalytically, for simple families of solutions, and we hope to show it moregenerally.2) Construction of new sets of initial conditions for the gravitationalfield which juxtapose two known sets. This is not easy because the initialconditions must satisfy certain nonlinear partial differentialequations--the Einstein "constraints". We are developing techniques forsmoothly joining ( or "gluing") two solutions of the constraints into asingle solution.The first of these projects, although it ignores any quantum gravitationaleffects, could be very useful in our drive to understand the nature of thevery early universe. The second is of interest both mathematically andphysically. The mathematical interest stems from the fact that the Einsteinconstraint system is among the most complicated to which gluing studieshave been applied. The physical interest comes from the possiblility ofusing the developed techniques to set up initial data for black holecollisions and other astrophysical problems which are being intenselystudied numerically in anticipation of future data from LIGO.