Paper · Sep 2, 2022

General Relativity and the Ricci Flow

Mohammed Alzain

gr-qcarXiv:2209.00952
View PDF View HTML arXiv page

As on arXiv

Abstract

In Riemannian geometry, the Ricci flow is the analogue of heat diffusion; a deformation of the metric tensor driven by its Ricci curvature. As a step towards resolving the problem of time in quantum gravity, we attempt to merge the Ricci flow equation with the Hamilton-Jacobi equation for general relativity.

View PDF View HTML arXiv page
Submitted
Sep 2, 2022
Subjects
gr-qc
Ricci flowPhysics

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 Stochastic Quantization of General Relativity à la Ricci-Flow (Matteo Lulli, Antonino Marciano and Xiaowen Shan, 2021)1Lulli 2021 Quantum Modified Gravity at Low Energy in the Ricci Flow of Quantum Spacetime (M. J. Luo, 2022)6Luo 2022 A path integral formula of quantum gravity emergent from entangled local structures (Jinglong Liu, Stephon Alexander, Antonino Marciano et al., 2023)2Liu 2023 Aspects of the Classical Double Copy (Rashid Alawadhi, 2023)8Alawadhi 2023 Stochastic Ricci Flow dynamics of the gravitationally induced wave-function collapse (Matteo Lulli, Antonino Marciano and Kristian Piscicchia, 2023)3Lulli 2023 Dark energy and dark matter configurations for wormholes and solitionic hierarchies of nonmetric Ricci flows and F(R,T,Q,Tₘ) gravity (Laurenţiu Bubuianu, Sergiu I. Vacaru, Elşen Veli Veliev et al., 2024)5Bubuianu 2024 Solutions to the Ricci Flow via Einstein Field Equations (Tommaso Morone and Roberto Tateo, 2024)7Morone 2024 The perturbative Ricci flow in gravity (Robert V. Harlander, Yannick Kluth, Jonas T. Kohnen et al., 2026)4Harlander 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stochastic Quantization of General Relativity à la Ricci-Flow

    Matteo Lulli, Antonino Marciano and Xiaowen Shan

    Topics Physics · Shared terms “relativity”, “metric tensor”

  2. A path integral formula of quantum gravity emergent from entangled local structures

    Jinglong Liu, Stephon Alexander, Antonino Marciano et al.

    Topics Physics · Shared terms “quantum gravity”, “relativity”

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

    Matteo Lulli, Antonino Marciano and Kristian Piscicchia

    Topics Physics · Shared terms “relativity”

  4. The perturbative Ricci flow in gravity

    Robert V. Harlander, Yannick Kluth, Jonas T. Kohnen et al.

    Topics Physics

  5. Dark energy and dark matter configurations for wormholes and solitionic hierarchies of nonmetric Ricci flows and gravity

    Laurenţiu Bubuianu, Sergiu I. Vacaru, Elşen Veli Veliev et al.

    Topics Physics · Shared terms “relativity”

  6. Quantum Modified Gravity at Low Energy in the Ricci Flow of Quantum Spacetime

    M. J. Luo

    Topics Physics

  7. Solutions to the Ricci Flow via Einstein Field Equations

    Tommaso Morone and Roberto Tateo

    Topics Physics

  8. Aspects of the Classical Double Copy

    Rashid Alawadhi

    Topics Physics · Shared terms “relativity”

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) 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.