A hyperbolic tetrad approach to numerical relativity

Abstract

An in-depth numerical study using 3 + 1 formulations of the Einstein equations for 1D colliding plane waves reveals factors which increase accuracy and stability: using "mixed" variables, hyperbolicity, satisfying the energy and momentum constraint equations at the boundaries, and changing the speeds of "constraint" eigenmodes so that they are not in step with dominant sources of error. In order to generalize these results, a novel tetrad approach to vacuum numerical relativity has been developed, based on the formalism of Estabrook, Robinson, and Wahlquist [24]. Clear advantages of this approach are that the variables are more naturally related to physical quantities, and the Minkowski metric is used to raise and lower indices on the tetrad variables. A potential disadvantage is that tetrads as well as coordinates and hypersurfaces must be evolved. A lapse function and a shift vector are introduced, which allow for a general choice of coordinates. The evolution equations expressed as coordinate-based partial differential equations are symmetrizable hyperbolic for Nester, Lorentz, and fixed gauge conditions. Implementation of this tetrad formulation for 1D colliding plane waves results in significantly greater accuracy than results using comparable 3 + 1 approaches. Tests in spherically symmetric Schwarzschild spacetime indicate that the Nester gauge gives a poor evolution of the congruence for non-stationary initial conditions. With the Lorentz gauge, the solution settles down to a stationary state, but second order convergence holds only to about 40M. By changing the speeds of the "constraint" eigen-modes so that they propagate quickly through the region of instability around the event horizon, solid stability is achieved to at least 500M. The black hole and plane wave results suggest that having the "constraint" eigenmodes travel away from dominant sources of error is key to stable and accurate numerical evolutions. Future work will involve understanding how this insight may be implemented in 3D.