Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the vacuum case

TL;DR

This paper develops a method using conformal Einstein field equations and wave coordinates to construct anti-de Sitter-like spacetimes with boundary conditions, facilitating numerical studies of such spacetimes.

Contribution

It introduces a new initial-boundary value formulation for anti-de Sitter-like spacetimes using the metric conformal Einstein equations, enabling systematic boundary condition analysis.

Findings

Formulation using conformal wave equations and generalised wave coordinates.

Boundary conditions derived from the conformal boundary metric and Weyl tensor components.

Discussion on the propagation of constraints ensuring solutions satisfy Einstein equations.

Abstract

We make use of the metric version of the conformal Einstein field equations to construct anti-de Sitter-like spacetimes by means of a suitably posed initial-boundary value problem. The evolution system associated to this initial-boundary value problem consists of a set of conformal wave equations for a number of conformal fields and the conformal metric. This formulation makes use of generalised wave coordinates and allows the free specification of the Ricci scalar of the conformal metric via a conformal gauge source function. We consider Dirichlet boundary conditions for the evolution equations at the conformal boundary and show that these boundary conditions can, in turn, be constructed from the 3-dimensional Lorentzian metric of the conformal boundary and a linear combination of the incoming and outgoing radiation as measured by certain components of the Weyl tensor. To show that a solution to the conformal evolution equations implies a solution to the Einstein field equations we also provide a discussion of the propagation of the constraints for this initial-boundary value problem. The existence of local solutions to the initial-boundary value problem in a neighbourhood of the corner where the initial hypersurface and the conformal boundary intersect is subject to compatibility conditions between the initial and boundary data. The construction described is amenable to numerical implementation and should allow the systematic exploration of boundary conditions.