Enveloping space of globally hyperbolic conformally flat spacetimes

We prove that any simply-connected globally hyperbolic conformally flat spacetime <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> can be conformally embedded in a bigger conformally flat spacetime, called enveloping space of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> , containing all the conformally flat Cauchy extensions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> , in particular its <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝒞</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -maximal extension. As a result, we establish a new proof of the existence and the uniqueness of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝒞</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -maximal extension of a globally hyperbolic conformally flat spacetime. Furthermore, this approach allows us to prove that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝒞</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -maximal extensions respect inclusion.