Simulations of gravitational collapse in null coordinates: III. Hyperbolicity

TL;DR

This paper examines the hyperbolicity of Einstein equations in Bondi-like coordinates, showing they are not strongly hyperbolic in second-order form but establishing well-posedness through a first-order symmetric hyperbolic formulation in a linearized setting.

Contribution

It defines strong hyperbolicity for PDE systems, demonstrates non-hyperbolicity of second-order Einstein equations in Bondi coordinates, and provides a first-order symmetric hyperbolic formulation for linearized equations.

Findings

Second-order Einstein equations in Bondi coordinates are not strongly hyperbolic.

A first-order symmetric hyperbolic formulation of linearized Einstein equations is established.

The initial-boundary value problem is shown to be well-posed in a specific energy norm.

Abstract

We investigate the well-posedness of the characteristic initial-boundary value problem for the Einstein equations in Bondi-like coordinates (including Bondi, double-null and affine). We propose a definition of strong hyperbolicity of a system of partial differential equations of any order, and show that the Einstein equations in Bondi-like coordinates in their second-order form used in numerical relativity do not meet it, in agreement with results of Giannakopoulos et al for specific first-order reductions. In the principal part, frozen coefficient approximation that one uses to examine hyperbolicity, we explicitly construct the general solution to identify the solutions that obstruct strong hyperbolicity. Independently, we present a first-order symmetric hyperbolic formulation of the Einstein equations in Bondi gauge, linearised about Schwarzschild, thus completing work by Frittelli. This establishes an energy norm ($L^2$ in the metric perturbations and selected first and second derivatives), in which the initial-boundary value problem, with initial data on an outgoing null cone and boundary data on a timelike cylinder or an ingoing null cone, is well-posed, thus verifying a conjecture by Giannakopoulos et al. Unfortunately, our method does not extend to the pure initial-value problem on a null cone with regular vertex.