Department of Applied Mathematics, California Institute of Technology
Numerical Treatment of Einstein's Field Equations
Abstract
In this talk, we consider the numerical treatment
of Einstein's field equations of general relativity. These
equations describe the geometry of space and time by determining
the evolution of a Riemannian metric on the spacetime manifold
in the presence of matter sources. In the first part of the
talk, we briefly review the covariant form of the field
equations, along with a canonical foliation of spacetime into
space-like hypersurfaces. Following the ADM formalism, the
equations are reformulated as a Hamiltonian system on an initial
hypersurface. Technical problems are avoided by employing the
York conformal decomposition, giving rise to a closed set of
twelve first-order hyperbolic equations for the evolution of
the spatial metric and its conjugate, along with four elliptic
constraint equations which must hold on each foliated slice.
This constrained system has many similarities to Maxwell's
equations.
In the second part of the talk, we focus on the numerical
solution of the coupled nonlinear elliptic constraint equations
on an arbitrary space-like hypersurface; this system must be
solved for example to produce consistent initial data for a
numerical integration of the full system. We employ the
computer program MC (Manifold Code) for this task, which is
designed to solve such nonlinear partial differential equations
on manifolds. MC is a 2D/3D simplex-based finite element code,
implementing a posteriori error estimation, adaptive bisection
of simplices, piecewise-linear/quadratic elements, global
inexact-Newton methods, Gummel methods, multilevel and Krylov
methods, and supporting domains with manifold structure. We
describe some of the algorithms in the code and some of its
other features, and present some initial numerical experiments
for two star-like objects in circular orbit in the presence of
matter sources.
Various parts of this work were done in collaboration with
David Bernstein, Stefan Vandewalle, and Peter Schroeder.
