Ricci flow and black holes

Abstract

Gradient flow in a potential energy (or Euclidean action) landscape provides a natural set of paths connecting different saddle points. We apply this method to General Relativity, where gradient flow is Ricci flow, and focus on the example of 4-dimensional Euclidean gravity with boundary S^1 x S^2, representing the canonical ensemble for gravity in a box. At high temperature the action has three saddle points: hot flat space and a large and small black hole. Adding a time direction, these also give static 5-dimensional Kaluza-Klein solutions, whose potential energy equals the 4-dimensional action. The small black hole has a Gross-Perry-Yaffe-type negative mode, and is therefore unstable under Ricci flow. We numerically simulate the two flows seeded by this mode, finding that they lead to the large black hole and to hot flat space respectively, in the latter case via a topology-changing singularity. In the context of string theory these flows are world-sheet renormalization group trajectories. We also use them to construct a novel free energy diagram for the canonical ensemble.

31 pages, 14 color figures. v2: Discussion of the metric on the space of metrics corrected and expanded, references added

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• A numerical approach to finding general stationary vacuum black holes
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• Some Applications of Ricci Flow in Physics
• Numerical Kaehler-Einstein metric on the third del Pezzo
• Negative modes and the thermodynamics of Reissner-Nordström black holes
• Cotton flow
• The Ricci flow on noncommutative two-tori
• Geometric Flow Equations for Schwarzschild-AdS Space-time and Hawking-Page Phase Transition
• The Simplicial Ricci Tensor
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• Thermal AdS(3), BTZ and competing winding modes condensation
• Gravity localization in a string-cigar braneworld
• CFTs in rotating black hole backgrounds
• Geometric flows and black hole entropy
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• Gauge fields in a string-cigar braneworld
• Stability of the microcanonical ensemble in Euclidean Quantum Gravity
• Topologically Massive Gravity and Ricci-Cotton Flow
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• Hedgehog black holes and the Polyakov loop at strong coupling
• Gravitational Kaluza-Klein modes in the String-Cigar Braneworld
• Ricci flows and expansion in axion-dilaton cosmology
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• Emergent dual holographic description as a non-perturbative generalization of the Wilsonian renormalization group
• Investigating Off-shell Stability of Anti-de Sitter Space in String Theory
• Ricci Flow Approach to The Cosmological Constant Problem
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• Entanglement transfer from quantum matter to classical geometry in an emergent holographic dual description of a scalar field theory
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• Local Conformal Instability and Local Non-Collapsing in the Ricci flow of Quantum Spacetime
• A Statistical Fields Theory underlying the Thermodynamics of Ricci Flow and Gravity
• Higher order geometric flows on three dimensional locally homogeneous spaces
• Robinson-Trautman solutions with scalar hair and Ricci flow
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• The sausage sigma model revisited
• Euclidean and complex geometries from real-time computations of gravitational Rényi entropies
• Gravitational thermodynamics without the conformal factor problem: Partition functions and Euclidean saddles from Lorentzian Path Integrals
• A proposal for studying off-shell stability of vacuum geometries in string theory
• Complex Saddles of Charged-AdS Gravitational partition function
• Stochastic Einstein equations
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• Modified Ricci flow and asymptotically non-flat spaces
• A New Geometric Flow on 3-Manifolds: the $K$-Flow
• The quantum non-linear $\sigma$-model RG flow and integrability in wormhole geometries
• The Higgs mechanism and geometrical flows for two-manifolds
• The Canonical Ensemble Reloaded: The Complex-Stability of Euclidean quantum gravity for Black Holes in a Box
• Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow
• Riemann-Christoffel flows