Geodesic structure of spacetime near singularities

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.

💡 Research Summary

The paper investigates how two fundamental bi‑scalars – Synge’s world function Ω(x,y) and the van Vleck determinant Δ(x,y) – behave when the base point lies at a spacetime singularity. For regular points, Ω and Δ admit well‑known covariant expansions in terms of the curvature tensor and its derivatives, reflecting local flatness. At singular points, however, these expansions break down, and the authors set out to obtain meaningful expressions that capture the non‑trivial geometry of geodesic flows near singularities.

The authors first treat the Friedmann‑Lemaître‑Robertson‑Walker (FLRW) family with an arbitrary scale factor a(t). By exploiting the three spatial Killing vectors, they reduce the proper‑time integral for a timelike geodesic to a quadrature involving an integration constant α. Proper time τ(t,T;α) is expressed as an integral over a(t) and α, and the spatial geodesic distance ℓ(t,T;α) is related to α via a Lagrange‑Gauss inversion. Eliminating α yields Ω as an infinite series in ℓ, which can be summed into a compact closed‑form involving hyperbolic cosine and sine operators acting on a generating function G(α). This representation (Eq. 11) is non‑perturbative in ℓ and remains valid for any a(t).

From Ω the van Vleck determinant Δ is obtained by taking the mixed second derivative of Ω. The authors expand Δ up to O(ℓ²) and then examine two limits: (i) the coincidence limit ε = t−T → 0, and (ii) the singularity limit T → 0 (with t fixed). For a matter‑dominated FLRW universe (a∝t^{2/3}) they find Δ≈1+ε²/(9T²) in the coincidence regime, while near the singularity Δ exhibits terms proportional to ℓ² t^{9/2}/T^{5/3} and higher‑order corrections. In a radiation‑dominated case (a∝t^{1/2}) the singular behavior includes logarithmic terms, and intriguingly 2Δ^{1/2} vanishes both at coincidence and at the singularity, reflecting the vanishing Ricci scalar. These results demonstrate that while Ω stays regular, Δ encodes the caustic‑like focusing of geodesics near singularities.

The second part of the paper addresses the anisotropic Bianchi I (Kasner) spacetime, with three independent scale factors a_i(t). Three spatial Killing vectors provide three integration constants α_i. The world function is written as a proper‑time integral analogous to the FLRW case, and a multi‑variable Lagrange‑Gauss inversion yields a power‑series expansion of Ω in the coordinate differences (x−X, y−Y, z−Z). The leading terms reproduce the familiar (t−T)²/2 piece plus direction‑dependent quadratic corrections weighted by the Kasner exponents. Higher‑order terms involve integrals I_n(a_i)=∫T^t a_i(t′)^n dt′. The corresponding Δ and its square‑root are derived similarly, showing that anisotropy leads to direction‑dependent divergences in the singular limit, while the coincidence limit still matches the universal curvature expansion Δ≈1+R{ab}Ω^aΩ^b/6.

Overall, the paper establishes that at curvature singularities the scaling of Ω and Δ changes dramatically: Ω remains finite and provides a robust geometric probe, whereas Δ (and especially its square‑root) develops power‑law or logarithmic divergences that signal geodesic focusing. These findings refine the notion of “local flatness” by pinpointing precisely where it fails. Moreover, because Ω and Δ appear in point‑splitting regularization, heat‑kernel coefficients, and the construction of the effective metric q_{ab}(x;x₀) used in emergent gravity approaches, the results furnish concrete tools for extending quantum‑field‑theoretic analyses into singular regimes. The authors suggest that future work will exploit these bi‑scalars to explore quantum gravity effects near cosmological and black‑hole singularities.