Ricci flow and black holes

TL;DR

This work analyzes Ricci flow as the gradient flow of the Euclidean Einstein–Hilbert action for 4D gravity in a cavity, revealing how the unstable small black hole (with a GPY-like negative mode) evolves under the flow to either hot flat space or the large black hole, including a topology-changing singularity that mediates passage to hot flat space. By interpreting the flow as a world-sheet RG trajectory in string theory, the authors connect gravity in a box to KK reductions and study the potential end-states of the flow via numerical simulations, including a surgery procedure that preserves symmetry and continuity of the action. A key outcome is the construction of a novel off-shell free-energy diagram derived from the gradient-flow structure, offering a geometric perspective on thermodynamics and RG flow in gravitational systems. The results suggest qualitative similarity of the flows in higher dimensions and among various boundary conditions, with potential implications for AdS/CFT and tachyon condensation analyses.

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.

Figures (14)

• Figure 1: Action versus inverse temperature parameter $b\equiv\beta/(2\pi R)$ for the three saddle points. For $b>b_{\rm crit}$ only hot flat space is allowed, while for $b<b_{\rm crit}$ there are also small and large black hole solutions. At $b_{\rm HP}$ there is a first-order phase transition, the Hawking-Page transition: for $b>b_{\rm HP}$ flat space has the lowest action and therefore dominates thermodynamically, while for $b<b_{\rm HP}$ the large black hole dominates. The dashed lines indicate the three values of $b$ which will be used in later figures to display the qualitative behavior of the simulated Ricci flows. For this and all other figures we have chosen units such that $R=1$.
• Figure 2: Lowest eigenvalue $\mu$ of the Lichnerowicz operator $\Delta_{\rm L}$ on the black hole background (3.1). The small and large black holes are separated by the value $r_0/R=\frac{2}{3}$, where the eigenvalue passes from negative to positive. At this point the mode becomes tangent to this one-parameter family of solutions.
• Figure 3: Euclidean time radius $T$ (red) and sphere radius $S$ (purple) against proper radial coordinate $\rho$, for the flow of the small black hole perturbed by $+h_{\mu\nu}^{\rm GPY}$, at $b = b_{\rm HP}\approx0.59$. The boundary is at $\rho=0$. Snapshots are drawn at intervals of $\lambda$ of $0.05$. The heavy blue curves show those functions for the unperturbed small black hole. The heavy black curves show them for the corresponding large black hole, to which the flow clearly asymptotes at late times.
• Figure 4: Horizon radius, $S(\rho_{\rm H})$, against $\lambda$ for small black hole perturbed by $+h_{\mu\nu}^{\rm GPY}$, at $b = 0.40, 0.59, 0.70$ (the middle value equals $b_{\rm HP}$). The horizontal (red) dashed lines show the horizon radius for the corresponding large black hole, which the metric asymptotically converges to in each case. The curved (orange) dashed lines give the initial exponential growth of the negative mode.
• Figure 5: (Top) $T(\rho)$ (red) and $S(\rho)$ (purple) for the flow of the small black hole perturbed by $-h_{\mu\nu}^{\rm GPY}$, at $b = b_{\rm HP}$. Snapshots are drawn at intervals of $\lambda$ of $0.01$. The metric flows to a singularity, where the horizon shrinks to zero size. (Bottom) The metric after surgery, and the continuation of the Ricci flow. The metric asymptotically tends to flat space.