Geodesic structure of spacetime near singularities

TL;DR

This paper investigates how geodesic flows and associated bi-scalars behave near spacetime singularities, revealing new insights into the classical and quantum structure of such singularities.

Contribution

It demonstrates that the scaling behavior of Synge's world function and van Vleck determinant changes near singularities, offering a novel approach to study their structure.

Findings

Bi-scalars exhibit altered scaling near singularities.

New insights into classical spacetime singularities.

Potential tools for quantum gravity research.

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.