Paper · Jul 10, 2026

Entropy and Non-Collapse in Lorentzian Geometry

Rohit Dhormare

gr-qcarXiv:2607.09940
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As on arXiv

Abstract

In this paper, we establish a geometric correspondence between the Lorentzian Raychaudhuri equation and Perelman's non-collapsing theorem for the Ricci flow. By interpreting the Raychaudhuri equation as a Lorentzian analogue of Ricci flow, we connect geodesic focusing in general relativity to the monotonicity and entropy functionals of geometric analysis. Using this correspondence, we derive a Lorentzian non-collapsing theorem and introduce a covariant entropy functional governing causal volume evolution. Finally, we propose the concept of geodesic entropy capacity, a curvature-bounded limit on the information that can be stored in spacetime regions, providing a unified geometric framework linking gravitation, thermodynamics, and information.

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Submitted
Jul 10, 2026
Subjects
gr-qc, math-ph
Journal
Physics Letters B Date: April 2026 Article: 140355 Volume: Volume 875
DOI
10.1016/j.physletb.2026.140355

Comments: We establish, using comparison geometry, a finite entropy bound below a critical threshold and prove that the corresponding geometric evolution remains smooth for all proper times for which the entropy stays below the critical value. The paper contains one theorem, proof, and references

Ricci flow(Non)collapsingPhysics

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper On the improved no-local-collapsing theorem of Ricci flow (Wangjian Jian, 2021)2Jian 2021 Local Conformal Instability and Local Non-Collapsing in the Ricci flow of Quantum Spacetime (M. J. Luo, 2022)3Luo 2022 General Relativity and the Ricci Flow (Mohammed Alzain, 2022)7Alzain 2022 Stochastic Ricci Flow dynamics of the gravitationally induced wave-function collapse (Matteo Lulli, Antonino Marciano and Kristian Piscicchia, 2023)5Lulli 2023 Well-posedness of Ricci Flow in Lorentzian Spacetime and its Entropy Formula (M. J. Luo, 2025)6Luo 2025 Functional Renormalization Group as a Ricci Flow: An 𝓕-Entropy Perspective on Information Metric Dynamics (Jin Mo Bok and Ki-Seok Kim, 2026)4Bok 2026 Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow (Tianyin Ren and Chong Song, 2026)8Ren 2026 Scale-Invariant Non-Collapse from Raychaudhuri Comparison (Rohit Dhormare, 2026)1Dhormare 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Scale-Invariant Non-Collapse from Raychaudhuri Comparison

    Rohit Dhormare

    Topics (Non)collapsing · Same author Rohit Dhormare · Shared terms “Raychaudhuri”, “collapse”, “Perelman”

  2. On the improved no-local-collapsing theorem of Ricci flow

    Wangjian Jian

    Topics (Non)collapsing · Shared terms “collapsing theorem”, “Perelman”

  3. Local Conformal Instability and Local Non-Collapsing in the Ricci flow of Quantum Spacetime

    M. J. Luo

    Topics (Non)collapsing, Physics · Shared terms “spacetime region”, “collapsing theorem”, “collapse”

  4. Functional Renormalization Group as a Ricci Flow: An -Entropy Perspective on Information Metric Dynamics

    Jin Mo Bok and Ki-Seok Kim

    Topics Physics · Shared terms “information”, “geometric correspondence”, “entropy functional”

  5. Stochastic Ricci Flow dynamics of the gravitationally induced wave-function collapse

    Matteo Lulli, Antonino Marciano and Kristian Piscicchia

    Topics Physics · Shared terms “collapse”, “relativity”

  6. Well-posedness of Ricci Flow in Lorentzian Spacetime and its Entropy Formula

    M. J. Luo

    Topics Physics · Shared terms “Lorentzian”, “entropy functional”, “Perelman”

  7. General Relativity and the Ricci Flow

    Mohammed Alzain

    Topics Physics · Shared terms “relativity”, “analogue”

  8. Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow

    Tianyin Ren and Chong Song

    Topics Physics · Shared terms “correspondence”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.