Numerical Problems in General Relativity

The construction of gravitational wave observatories is one of the greatest scientific efforts of our time. As a result, there is presently a strong need to numerically simulate the emission of gravitation radiation from massive astronomical events such as black hole collisions. This entails the numerical solution of the Einstein field equations. We briefly describe the field equations in their natural setting, namely as statements about the geometry of space time. Next we describe the complicated system that arises when the field equations are recast as partial differential equations, and discuss procedures for deriving from them a more tractable system consisting of constraint equations to be satisfied by initial data and together with evolution equations. We present some applications of modern finite element technology to the solution of the constraint equations in order to find initial data relevant to black hole collisions. We conclude by enumerating some of the many computational challenges that remain.