Geodesic structure of spacetime near singularities

TL;DR

This work investigates the geodesic structure of spacetime near curvature singularities by deriving explicit representations and limits for Synge's world function $\Omega(x,y)$ and the van Vleck determinant $\Delta(x,y)$ in FLRW and Bianchi Type I (Kasner-like) spacetimes. By obtaining both nonperturbative and small-parameter expansions, the authors reveal pronounced differences between singular and regular neighborhoods, including singularity-specific scaling in $\Delta$ and $\Box\Delta^{1/2}$, as well as the behavior of equi-geodesic surfaces and lightcones. The results, including Kasner-type limits of the Schwarzschild singularity, provide new insights into classical spacetime structure and have potential implications for quantum gravity through effective metric constructions like the $q$-metric. The analysis highlights the qualitative impact of anisotropy and shear near singularities and offers a framework for studying Planck-scale regularization of spacetime geometry. Overall, the paper advances our understanding of how geodesic density and interval measures encode singular structures and their quantum implications.

Abstract

Geodesic flows emanating from an arbitrary point $\mathscr{P}$ in a manifold $\mathscr{M}$ carry important information about the geometric properties of $\mathscr{M}$. These flows are characterized by Synge's world function and van Vleck determinant - important bi-scalars that also characterize quantum description of physical systems in $\mathscr{M}$. If $\mathscr{P}$ is a regular point, these bi-scalars have well known expansions around their flat space expressions, quantifying \textit{local flatness} and equivalence principle. We show that, if $\mathscr{P}$ is a singular point, the scaling behavior of these bi-scalars changes drastically, capturing the non-trivial structure of geodesic flows near singularities. This yields remarkable insights into classical structure of spacetime singularities and provides useful tool to study their quantum structure.

Figures (4)

• Figure 1: The most basic non-local observables in an arbitrary curved manifold are Synge's world function and the van Vleck determinant. The former characterizes single geodesics (left), while the latter measures the density of geodesics emanating from a given point (right). $\lambda$ is the affine parameter along the geodesic. See text for more details.
• Figure 2: A local region of matter-dominated FLRW spacetime is shown with $t=$constant and $\Omega=$ constant anchored at a point $P(T, X, Y, Z)$. The extrinsic curvature $K_{ab}$ of $\Omega$=constant hypersurfaces characterizes the local expansion of these surfaces about point $P$. The $t=$constant hypersurfaces follow Hubble's law, whereas $\Omega=$constant hypersurfaces obey a generalized version of Hubble's law Kothawala:2018BGV.
• Figure 3: The equi-geodesic surfaces and lightcones are constructed using biscalar $\Omega(x^{a}, X^{a})$ about different points $P$ in the spacetime. Left: Matter-dominated FLRW spacetime with $p=2/3$, Right: Kasner form of Schwarzschild singularity. The zig-zag line shows the curvature singularity in both cases.
• Figure 4: The lightcones near a Kasner (Schwarzschild) singularity stretches due to non-zero shear and anisotropic nature of spacetime, whereas lightcones in FLRW spacetime expands isotropically. Left: Lightcones in Kasner (Schwarzschild) elongate (left) along the expanding direction and squeeze (right) along the contracting direction (here $z$). Right: General comparison of lightcone structure between Kasner (Schwarzschild) and FLRW (matter-dominated) spacetime.