Paper · Jul 8, 2018

Lorentzian Einstein-Ricci Flows

Aditya Dhumuntarao

gr-qcarXiv:1807.02731v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We study the Ricci flow for the Lorentzian Einstein-Hilbert action. We show that Einstein gravity emerges as a fixed point of the Einstein-Ricci flow equations and derive a renormalization group flow in Euclidean signature. By considering linearizations near the fixed point, the dynamics of the metric reveal that curvature deformations flow according to a forward heat equation with the stress-energy tensor acting as a source.

View PDF View HTML arXiv page
Submitted
Jul 8, 2018
Last revised
Oct 1, 2021 (v2)
Subjects
gr-qc, hep-th, math-ph

Comments: 4 pages, comments are welcome. v2: Latex, 0 figures; revised argument concerning the RG flow and updated the argument for the geometric flow as a gradient flow; incorporated comments from Jiaping Wang; typos corrected and several minor changes; references added; footnote added

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 2016, 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 Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory (Mauro Carfora, Claudio Dappiaggi, Nicolò Drago et al., 2018)6Carfora 2018 Cosmology from the two-dimensional renormalization group acting as the Ricci flow (Daniel Friedan, 2019)8Friedan 2019 Strings in Bimetric Spacetimes (Ziqi Yan, 2021)4Yan 2021 Brownian Motion in the Hilbert Space of Quantum States along with the Ricci Flow and Stochastically Emergent Einstein-Hilbert Action: Formulating a Well-Defined Feynman Path-Integral Measure for Quantum Fields in the Presence of Gravity (A. A. Varshovi, 2022)2Varshovi 2022 Quantum Modified Gravity at Low Energy in the Ricci Flow of Quantum Spacetime (M. J. Luo, 2022)7Luo 2022 Stress-energy tensor deformations, Ricci flows and black holes (Nicolò Brizio, Tommaso Morone and Roberto Tateo, 2024)1Brizio 2024 Solutions to the Ricci Flow via Einstein Field Equations (Tommaso Morone and Roberto Tateo, 2024)5Morone 2024 The perturbative Ricci flow in gravity (Robert V. Harlander, Yannick Kluth, Jonas T. Kohnen et al., 2026)3Harlander 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stress-energy tensor deformations, Ricci flows and black holes

    Nicolò Brizio, Tommaso Morone and Roberto Tateo

    Topics Physics · Shared terms “energy tensor”, “stress energy”, “Einstein Ricci”

  2. Brownian Motion in the Hilbert Space of Quantum States along with the Ricci Flow and Stochastically Emergent Einstein-Hilbert Action: Formulating a Well-Defined Feynman Path-Integral Measure for Quantum Fields in the Presence of Gravity

    A. A. Varshovi

    Topics Physics · Shared terms “Hilbert action”, “Einstein Hilbert”

  3. The perturbative Ricci flow in gravity

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

    Topics Physics · Shared terms “fixed point”, “renormalization group”

  4. Strings in Bimetric Spacetimes

    Ziqi Yan

    Topics Physics · Shared terms “Einstein gravity”, “group flow”, “renormalization group”

  5. Solutions to the Ricci Flow via Einstein Field Equations

    Tommaso Morone and Roberto Tateo

    Topics Physics

  6. Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

    Mauro Carfora, Claudio Dappiaggi, Nicolò Drago et al.

    Topics Physics · Shared terms “group flow”, “renormalization group”, “Euclidean”

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

    M. J. Luo

    Topics Physics · Shared terms “Einstein gravity”, “Hilbert action”, “Einstein Hilbert”

  8. Cosmology from the two-dimensional renormalization group acting as the Ricci flow

    Daniel Friedan

    Topics Physics · Shared terms “renormalization group”

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.