Resumo: The duration of our Universe, all of its content, and whether it had a beginning or
has always existed has long been the subject of intense investigation. As free particles
trajectories are given by geodesics, the inquisition of godesic completeness and extensions
become crucial for the investigation of possible eternal Universes. Nevertheless, General
Relativity’s invariance under diffeomorphisms imposes an additional difficulty to realize
whether the incompleteness has physical significance or if it is merely an inappropriate
coordinate choice. In this context, the singularity theorems provide sufficient conditions
for geodesic incompleteness without recurring to coordinate charts. However, kinematic
alternatives for classifying incomplete space-times that are expanding have been proposed,
leading to the Borde-Guth-Vilenkin (BGV) theorem, where no restriction on the matter
fields are necessary, such as energy conditions. Notwithstanding, whether the space time
admits a metric extension that is compatible with General Relativity, i.e, a 𝒞2 extension,
needs to be addressed.
In this dissertation, using the pivotal example of the flat patch of the de Sitter space,
we manage to find a new global chart for this space - which without considerations of
extensibility would be diagnosed as geodesically incomplete. Furthermore, we developed
a general protocol for a 𝐶2 extension of a flat Friedmann-Lemaître-Robertson-Walker
metric, and the necessary conditions for its application by exhausting all the possible cases
in the asymptotic limit, finding necessary and sufficient conditions for extensibility. The
incomplete spaces that violate the assumptions have either a scalar curvature singularity or
a parallelly propagated singularity, in which cases no 𝒞2 extension is allowed. Moreover,
we discuss results for possible cyclic scenarios proposed in the literature. The results
obtained in this work were published in Phys. Rev. D 111, 123531 (2025). |
