Volume-distance-ratio asymptote and spacetime inextendibility for spatially flat and hyperbolic FLRW spacetimes

TL;DR

This paper investigates the volume-distance-ratio asymptote in spatially flat and hyperbolic FLRW spacetimes, establishing criteria for their inextendibility based on asymptotic behavior, and identifying critical exponents related to spacetime inextendibility.

Contribution

It introduces a novel application of volume-distance-ratio asymptote analysis to determine inextendibility of FLRW spacetimes with specific scale factors.

Findings

Existence of a critical exponent  for  > 2 where VDR asymptote matches Minkowski value

The study establishes inextendibility criteria for FLRW spacetimes based on asymptotic analysis

Identification of critical exponents    in relation to spacetime inextendibility

Abstract

We study the volume-distance-ratio (VDR) asymptote at the past timelike boundary point for spatially flat FLRW spacetime with scale factor $a(t) = t^{\alpha}$, and spatially hyperbolic FLRW spacetime with scale factor $a(t) = a_0 t^{\alpha}$. We employ the spacetime inextendibility criteria via the volume-distance-ratio asymptote to deduce their inextendibility. We show that for spatial dimensions greater than $2$, there exists at least one critical exponent $\alpha \in (0,1)$ for which the VDR asymptote along the $\Sigma_t$-orthogonal geodesic equals the Minkowski value.