 
Summary: NUMERICAL PROBLEMS IN GENERAL RELATIVITY
DOUGLAS N. ARNOLD
Department of Mathematics, Penn State University, University Park, PA 16802,
USA
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 sim
ulate 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 compli
cated system that arises when the field equations are recast as partial differential
equations, and discuss procedures for deriving from them a more tractable sys
tem 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.
1 General relativity and gravity waves: the scientific problem
In the theory of general relativity spacetime is a smooth fourdimensional
manifold endowed with a pseudoRiemannian metric.a
