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 r0, 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, Volhτ(Bhτ(pτ,r)) ≥ κL(τ) r3, where κL(τ) = κ0 e−3Kτ exp (∫0τΘ(s) ds), K = Θ0/3 + Σ0 . 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, κL(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.
Versions
Related
- Landau Levels of Landau-Raychaudhuri Equation and Wheeler-DeWitt Crystals.
- Scale-invariant bounds on thin sets for measures satisfying a first-order PDE and the anisotropic Michael-Simon inequality
- Unbalanced Minimal Matchings for Measurable Scale Invariant Costs on ℝd
- Regularity of critical immersions for scale-invariant curvature energies in even dimensions
- Unbiased estimation of normalized scale-invariant indices under the Gamma distribution
BibTeX
@misc{dhormare2026scaleinvariant,
title = {{Scale-Invariant Non-Collapse from Raychaudhuri Comparison}},
author = {Rohit Dhormare},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07879},
eprint = {2610.07879},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07879},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Scale-Invariant Non-Collapse from Raychaudhuri Comparison
AU - Rohit Dhormare
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07879
UR - https://arxiv.org/abs/2610.07879
AB - 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, Vol_h_τ(B_h_τ(p_τ,r)) ≥ κ_L(τ) r^3, where κ_L(τ) = κ_0 e^−3Kτ exp (∫_0^τΘ(s) ds), K = Θ_0/3 + Σ_0 . 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_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.
ER -