Scale-Invariant Non-Collapse from Raychaudhuri Comparison

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 $r_0$, 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}{hτ}\bigl(B_{h_τ}(p_τ,r)\bigr) \geq κ{\mathrm{L}}(τ)\, r^3, \end{equation*} where \begin{equation*} κ{\mathrm{L}}(τ) = κ0 e^{-3Kτ} \exp!\left(\int_0^τΘ(s)\,ds\right), \qquad K = \frac{Θ_0}{3} + Σ_0 . \end{equation*} Here $Θ$ denotes the solution of the scalar Raychaudhuri comparison equation and $K$ controls the spatial metric distortion. For every finite time strictly before the comparison barrier, $κ</em>{\mathrm{L}}(τ)$ 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 $κ$-noncollapsing theorem for Ricci flow.