In general relativity, the universe is modeled by a four-dimensional Lorentzian manifold satisfying Einstein’s equations. A fundamental result by Choquet-Bruhat and Geroch (1969) establishes the existence and uniqueness of a maximal development associated with given initial data. These solutions fall within the framework of globally hyperbolic spacetimes, which are naturally endowed with a partial order relation, leading to the notion of maximal extension.
In this talk, I will focus on these questions in the context of conformally flat globally hyperbolic spacetimes. In 2013, C. Rossi proved the existence and uniqueness of a maximal extension in this setting. However, her proof does not provide an explicit description of this extension.
I will present an alternative and constructive approach, based on the notion of an enveloping space, within which the maximal extension can be realized explicitly. I will illustrate this construction with several examples.
