Cosmological Spacetimes with Sign-Changing Spatial Curvature and Topological Transitions

Abstract

Observational evidence, together with practical computations and modeling, supports a Euclidean spatial sector in the current cosmological model based on the FLRW metric. This, however, would imply that the total amount of matter and energy immediately after the Big Bang must have been infinite, an implication that could only be avoided through a transition from a closed to an open universe, a process forbidden in standard FLRW models. In this article, we investigate the spacetimes resulting from promoting the spatial curvature $k$ in FLRW spacetimes to a time-dependent function, $k \to k(t)$, allowing it to change sign and thereby allowing changes in the topology of the constant-$t$ slices. Although previously dismissed due to a classical theorem by Geroch, such transitions are shown to be consistent with global hyperbolicity when the comoving time is distinct from a Cauchy time, as recent work by one of the authors demonstrates. We construct three distinct geometries exhibiting this behavior using different representations of constant-curvature spaces. We analyze their global properties and identify mild conditions under which they remain globally hyperbolic. Furthermore, we characterize their Killing vectors, proving a general result for spherically symmetric spacetimes and compare them with known geometries in the literature.

Figures (9)

• Figure 1: Foliations of the de Sitter spacetime ($n \geq 2$) represented in a Penrose diagram: the spherical foliation by constant-$T$ slices (left), which covers the entire spacetime, and the foliation by flat constant-$t$ slices (right), which covers only half of it. Note that, in the latter case, there exists an analogous foliation of the other half of the spacetime, but the two cannot be smoothly joined to form a global one, since the surface $X^0 + X^1 = 0$ is null (solid black line in the right panel). The arrows represent the corresponding comoving observers. Dashed lines represent constant-$\vartheta$ ($r$) in the left (right) panel.
• Figure 2: Spacetime diagram that shows the situation of the cosmological comoving observer of the $k(t)$-conformal metric hitting the curvature singularity at $r\to 1/\sqrt{-k(t)}$ whenever $|k(t)|$ increases in a certain interval. As indicated in the picture, such a singularity is always at an infinite spatial distance (along the $\partial_r$ direction) from any of these observers but is reached in a finite proper time for any of them at sufficiently large distance from the origin $r=0$.
• Figure 3: Spacetime diagram that shows the situation of the cosmological comoving observer of the $k(t)$-radial metric hitting the curvature singularity at $r\to 1/\sqrt{k(t)}$ whenever $k(t)$ increases in a certain interval. As indicated in the picture, the singularity is hit in a finite proper time by the observers $\partial_t$ at sufficiently large distance from the origin $r=0$.
• Figure 4: Causal structure of the $k(t)$-radial spacetime with $k(t)>0$.
• Figure 5: Cauchy slices for different $k(t)$-warped spacetimes with topology changes.

Theorems & Definitions (76)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Proposition 3.5
• proof