Constraint Damping in the Cauchy Problem in General Relativity

Abstract

The Cauchy Problem in General Relativity is an area of active research in partial differential equations (PDEs). Much progress has been made on the Cauchy Problem in General Relativity via the local existence theory of symmetric hyperbolic systems and the choice of generalized harmonic gauge. The choice of generalized harmonic gauge also allows for the introduction of constraint damping terms, which allow for control over any violation of the general harmonic gauge constraint. In this thesis, we first go through the symmetric hyperbolic theory of partial differential equations since it will establish the basis for our local existence result for Einstein’s vacuum equations. Afterwards, we look at the 3+1 formalism as well as generalized harmonic coordinates, before finally proving stability for the damped constraint evolution equations in Minkowski space.