Enveloping space of a globally hyperbolic conformally flat spacetime

In 2013, C. Rossi established the existence and the uniqueness of the maximal extension of a globally hyperbolic conformally flat spacetime. Her proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice, specifically through Zorn's lemma, and does not provide a description of the maximal extension. In this talk, I will propose a constructive proof that avoids the need of Zorn's lemma. The key idea is that any simply-connected globally hyperbolic conformally flat spacetime M can be embedded into a bigger conformally flat spacetime E(M), called enveloping space of M, containing all the Cauchy-extensions of M, and in particular the maximal extension. It turns out that M and its Cauchy-extensions can be well described in M, a description that I will specify. In particular, the description of the maximal extension in E(M) involves the concept of eikonal functions, coming from PDE theory. I will illustrate this construction with examples and conclude with discussing some interesting consequences that arise from it.