Constraint preserving boundary conditions in Bondi-Sachs gauge: a numerical study of stability of pure AdS spacetime

TL;DR

This paper investigates the stability of pure Anti-de Sitter (AdS) spacetime under scalar field perturbations using a hyperbolic formulation of Einstein's equations in Bondi-Sachs gauge, demonstrating that small boundary perturbations can lead to collapse.

Contribution

It develops a constraint-preserving boundary scheme in Bondi-Sachs gauge and provides numerical evidence that small boundary perturbations can cause AdS spacetime collapse.

Findings

Small boundary perturbations can cause AdS collapse.

Numerical evidence of apparent horizon formation.

Constraint-preserving boundary conditions are effective.

Abstract

In the Bondi-Sachs gauge, the Einstein equations with a cosmological constant coupled to a scalar field in spherical symmetry are cast into a first order strongly hyperbolic formulation in which the lapse and shift are the fundamental variables. For this system of equations, the lapse and shift are ingoing characteristic fields, and the scalar field has three modes: ingoing, outgoing and static, respectively. A constraint-preserving initial boundary value problem is constructed by using Bianchi identity. Using this scheme, we find that any small perturbation of the scalar field at the boundary far away enough can cause the collapse of the pure AdS spacetime, and we provide the numerical evidence for the formation of apparent horizons. The numerical evolution is performed with a standard method of lines, second order in space and time. The evolution is performed using the standard second order Runge-Kutta method while the space discrete derivative is second order central difference with fourth order artificial dissipation.