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

## TL;DR

This paper develops a method to construct anti-de Sitter-like spacetimes with tracefree matter using conformal Einstein equations, ensuring local existence and uniqueness of solutions through a wave equation approach.

## Contribution

It introduces a systematic procedure for initial-boundary value problems in conformal Einstein equations with tracefree matter, including explicit models like scalar, Maxwell, and Yang-Mills fields.

## Key findings

- Established local existence and uniqueness of solutions.
- Analyzed conformal constraints for initial and boundary data.
- Identified data requirements for coupling matter models.

## Abstract

Using a metric conformal formulation of the Einstein equations, we develop a construction of 4-dimensional anti-de Sitter-like spacetimes coupled to tracefree matter models. Our strategy relies on the formulation of an initial-boundary problem for a system of quasilinear wave equations for various conformal fields by exploiting the conformal and coordinate gauges. By analysing the conformal constraints we show a systematic procedure to prescribe initial and boundary data. This analysis is complemented by the propagation of the constraints, showing that a solution to the wave equations implies a solution to the Einstein field equations. In addition, we study three explicit tracefree matter models: the conformally invariant scalar field, the Maxwell field and the Yang-Mills field. For each one of these we identify the basic data required to couple them to the system of wave equations. As our main result, we establish the local existence and uniqueness of solutions for the evolution system in a neighbourhood around the corner, provided compatibility conditions for the initial and boundary data are imposed up to a certain order.