Paper · Feb 19, 2016

Geometric Flows and Perelman's Thermodynamics for Black Ellipsoids in and Einstein Gravity Theories

Tamara Gheorghiu, Vyacheslav Ruchin, Olivia Vacaru, Sergiu I. Vacaru

gr-qcarXiv:1602.08512
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Abstract

We study geometric relativistic flow and Ricci soliton equations which (for respective nonholonomic constraints and self-similarity conditions) are equivalent to the gravitational field equations of gravity and/or to the Einstein equations with scalar field in general relativity, GR. Perelman's functionals are generalized for modified gravity theories, MGTs, which allows to formulate an analogous statistical thermodynamics for geometric flows and Ricci solitons. There are constructed and analyzed generic off-diagonal black ellipsoid, black hole and solitonic exact solutions in MGTs and GR encoding geometric flow evolution scenarios and nonlinear parametric interactions. Such new classes of solutions in MGTs can be with polarized and/or running constants, nonholonomically deformed horizons and/or imbedded self-consistently into solitonic backgrounds. They exist also in GR as generic off-diagonal vacuum configurations with effective cosmological constant and/or mimicking effective scalar field interactions. Finally, we compute Perelman's energy and entropy for black ellipsoids and evolution solitons in gravity.

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Submitted
Feb 19, 2016
Subjects
gr-qc, hep-th, math-ph
Journal
Annals of Physics NY 369 (2016) 1-35
DOI
10.1016/j.aop.2016.02.008

Comments: latex2e, 12pt, 46 pages; accepted: Annals Phys. NY

Ricci solitonsPerelman entropyPhysics

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  1. The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory

    Sergiu I. Vacaru and Elşen V. Veliev

    Topics Perelman entropy, Physics · Same author Sergiu I. Vacaru · Shared terms “Einstein gravity”, “gravity theories”, “diagonal”

  2. On Supersymmetric Geometric Flows and Inflation From Scale Invariant Supergravity

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    Topics Perelman entropy, Physics, Ricci solitons · Same author Sergiu I. Vacaru · Shared terms “gravity theories”, “geometric flows”, “diagonal”

  3. Nonassicative cosmological solitonic R-flux deformations in gauge gravity and G. Perelman geometric flow thermodynamics

    L. Bubuianu, J. O. Seti, S. Vacaru et al.

    Topics Perelman entropy, Physics · Same author Sergiu I. Vacaru · Shared terms “thermodynamics”, “solitonic”, “gravity theories”

  4. Nonmetric geometric flows and quasicrystalline topological phases for dark energy and dark matter in cosmology

    L. Bubuianu, E. Nurlan, J. O. Seti et al.

    Topics Perelman entropy, Physics, Ricci solitons · Same author Sergiu I. Vacaru · Shared terms “geometric flows”, “gravity theories”, “MGTs”

  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 · Same author Sergiu I. Vacaru · Shared terms “gravity theories”, “solitonic”, “MGTs”

  6. Classical and quantum geometric information flows and entanglement of relativistic mechanical systems

    Sergiu I. Vacaru and Laurenţiu Bubuianu

    Topics Perelman entropy · Same author Sergiu I. Vacaru · Shared terms “geometric flows”, “relativistic”, “statistical”

  7. Trace Anomaly, Perelman's Functionals and the Cosmological Constant

    M. J. Luo

    Topics Perelman entropy, Physics · Shared terms “Perelman functionals”, “cosmological constant”, “effective”

  8. Nonassociative geometric and quantum information flows and R-flux deformations of wormhole solutions in string gravity

    Laurenţiu Bubuianu, Douglas Singleton, Sergiu I. Vacaru et al.

    Topics Physics · Same author Sergiu I. Vacaru · Shared terms “gravity theories”, “respective”, “modified gravity”

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Perelman entropy Perelman found that Ricci flow is a gradient flow once a measure is fixed. His 𝓕-functional and 𝓦-entropy increase along the flow; a lower bound… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6B.16
    Eigenvalue Problems and Perelman’s λBook 6B · Linear PDE: Characteristics, Waves and Regularity

    Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.