This chapter discusses the construction of anti-de Sitter-like spacetimes, that is, solutions to the vacuum Einstein field equations with an anti-de Sitter-like value of the cosmological constant λ. Following the general discussion in Chapter 10, an anti-de Sitter-like value of the cosmological constant implies a timelike conformal boundary. This feature of anti-de Sitter-like spacetimes marks the essential difference between the analysis contained in this chapter and the ones given in Chapters 15 and 16 for de Sitter-like and Minkowski-like spacetimes, respectively.

While the de Sitter and Minkowski spacetimes are both globally hyperbolic, and, accordingly, perturbations thereof can be constructed by means of suitable initial value problems, the anti-de Sitter spacetime is not-globally hyperbolic; see the discussion in Section 14.5. Consequently, anti-de Sitter-like spacetimes cannot be solely reconstructed from initial data. One needs to prescribe some boundary data on the conformal boundary. Thus, the proper setting for the construction of anti-de Sitter-like spacetimes is that of an initial boundary value problem. In this spirit, one of the key objectives of this chapter is to identify suitable boundary data for the conformal Einstein field equations.

For both the de Sitter and Minkowski spacetimes it is possible to obtain conformal representations which are compact in time so that global existence of perturbations can be analysed in terms of problems which are local in time. However, the conformal representations of the anti-de Sitter spacetime discussed in Chapter 6 involve an infinite range of time. As a consequence, the main result of this chapter is local in time and makes no assertions about the stability of the anti-de Sitter spacetime. The main result of this chapter can be formulated as follows:

Theorem (local existence of anti-de Sitter-like spacetimes).Given smooth anti-de Sitter-like initial data for the Einstein field equations on a three-dimensional manifold S with boundary and a smooth three-dimensional Lorentzian metric ℓ on a cylinder [0, τ•) ×∂S for some τ• > 0, and assuming that these data satisfy certain corner conditions, there exists a local-in-time solution to the Einstein field equations with an anti-de Sitter-like cosmological constant such that on {0}×S it implies the given anti-de Sitter-like initial data. Moreover, this solution to the Einstein field equations admits a conformal completion such that the intrinsic metric of the resulting (timelike) conformal boundary belongs to the conformal class[ℓ].