Simulation of general relativistic shock wave interactions by a locally inertial Godunov method featuring dynamical time dilation

Abstract

We introduce the locally inertial Godunov method with dynamical time dilation , and use it to give a definitive numerical simulation of a point of shock wave interaction in general relativity starting from a new initial dataset. Prior work of Groah and Temple justifies meeting the Einstein constraint equations for the initial data only at the weak level of Lipshitz continuity in the metric. The forward time simulations, presented here, resolve the secondary wave in the Smoller–Temple shock wave model for an explosion into a static, singular, isothermal sphere. The backward time solutions indicate black hole formation from a smooth solution via collapse associated with an incoming rarefaction wave. A new feature is that space–time is approximated as locally flat in each grid cell so that Riemann problems and the Godunov method can be implemented. Clocks are then dynamically dilated to simulate effects of space–time curvature. Such points of shock wave interaction are more singular than points on single shock surfaces because the coordinate systems that make space–time locally flat on single shock surfaces (Gaussian normal coordinates), break down at points of shock wave interaction.