Global Studies of Einstein Spacetimes

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The ongoing effort to produce gravitational wave templates by the accurate numerical modeling of astrophysically interesting sources remains stymied by numerical instabilities revealed by the lack of satisfaction of the constraint equations. It appears as though these difficulties are most likely to be resolved by a close link between formal mathematics and hard-core numerical methods. The resolution to this quandary proposed is that the source for the instabilities is the constraint equations when not adequately enforced. The proposal is to develop a fully constrained evolution code by starting from an already existing finite element code for the Einstein constraint equations. The principle aim of the first of these projects is to solve the instability problems that plague many of the existing black hole collisions studies. If black holes are stable objects then the instabilities found in previous numerical computations must be artifacts of the approximations made and not features of genuine black hole evolutions. It is hoped that the approach of the PI will solve this crucial problem. A second topic of investigation will be Light cone estimates for Einstein's equations. This approach will be used to show under what conditions time evolution of Einstein's equations will lead to the curvature becoming singular.