Scale-Invariant Non-Collapse from Raychaudhuri Comparison
arXiv:2610.07879 · doi:10.1016/j.physletb.2026.140987
Abstract
We establish a scale-invariant spatial non-collapse estimate for hypersurface-orthogonal timelike geodesic congruences. The result extends a previous Raychaudhuri-based construction in which a comparison estimate for the expansion was used to control the volume of domains transported along the congruence. The additional ingredient here is a quantitative estimate of the spatial distance distortion induced by the expansion and shear. Assuming that the initial spatial geometry is non-collapsed up to a scale , and that the expansion, shear, and timelike Ricci curvature satisfy suitable uniform bounds, we obtain, before the first blow-up time of the Raychaudhuri comparison solution, \begin{equation*} \operatorname{Vol}_{h_τ}\bigl(B_{h_τ}(p_τ,r)\bigr) \geq κ_{\mathrm{L}}(τ)\, r^3, \end{equation*} where \begin{equation*} κ_{\mathrm{L}}(τ) = κ_0 e^{-3Kτ} \exp\!\left(\int_0^τΘ(s)\,ds\right), \qquad K = \frac{Θ_0}{3} + Σ_0 . \end{equation*} Here denotes the solution of the scalar Raychaudhuri comparison equation and controls the spatial metric distortion. For every finite time strictly before the comparison barrier, is positive. When the comparison solution has a finite blow-up time, the explicit comparison coefficient approaches zero as that time is approached from below. The comparison barrier is a boundary of the estimate and is not, by itself, a statement that the physical congruence develops a focal point or spacetime singularity. The result is structurally analogous to scale-invariant non-collapse estimates in geometric flow, but is mathematically distinct from Perelman's -noncollapsing theorem for Ricci flow.
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