Simulations of gravitational collapse in null coordinates. III. Hyperbolicity

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.CONTENTS I. Introduction 1 II.Obstructions to strong hyperbolicity of the second-order form of the Einstein equations in Bondi-like gauges 3 A. Metric and field equations in twist-free axisymmetry 3 B. Gauges and solution algorithm 3 C. Definition of strong hyperbolicity 4 D. The principal symbol 4 E. Double-null and related gauges 5 F. Bondi and related gauges 6 G. Affine and related gauges 6 H. Linearisation about Minkowski spacetime and frozen coefficient approximation 6 I.A toy model for the obstruction to strong hyperbolicity 7 III.Symmetric hyperbolic first-order reduction of the linearised Einstein equations in Bondi gauge 9 A. Balance laws from symmetric hyperbolic first-order systems 9 B. Estimate on an arbitrary control volume 9 C.The scalar wave equation on Schwarzschild 10 D. The vacuum Einstein equations, linearised in Bondi gauge about Schwarzschild 11 E. Estimates for characteristic initial value and initial-boundary value problems 13 IV.Conclusions 15 Acknowledgments 16 A.A textbook example of a weakly hyperbolic system 16 B. Notes on [15] 16 C.Symmetric hyperbolic form of the toy model 17 D. Attempts to extend the estimates to r 0 = 0 17 E. Numerical experiments 18