Gradient flow of Einstein-Maxwell theory and Reissner-Nordstr\"om black holes

TL;DR

This paper explores the Ricci flow analog for Einstein-Maxwell theory, analyzing the stability and evolution of Reissner-Nordström black holes, revealing stability for magnetic charge and instability for electric charge under the flow.

Contribution

It introduces a Ricci flow for Einstein-Maxwell theory, studies its effects on black hole horizons, and identifies stability conditions for magnetic versus electric Reissner-Nordström black holes.

Findings

Magnetic RN solutions become stable fixed points under the flow.

Electric RN black holes are always unstable in the flow.

Flow behavior includes horizon shrinking to singularities or expansion to stable solutions.

Abstract

Ricci flow is a natural gradient flow of the Einstein-Hilbert action. Here we consider the analog for the Einstein-Maxwell action, which gives Ricci flow with a stress tensor contribution coupled to a Yang-Mills flow for the Maxwell field. We argue that this flow is well-posed for static spacetimes with pure electric or magnetic potentialsand show it preserves both non-extremal and extremal black hole horizons. In the latter case we find the flow of the near horizon geometry decouples from that of the exterior. The Schwarzschild black hole is an unstable fixed point of Ricci flow for static spacetimes. Here we consider flows of the Reissner-Nordstr\"om (RN) fixed point. The magnetic RN solution becomes a stable fixed point of the flow for sufficient charge. However we find that the electric RN black hole is always unstable. Numerically solving the flow starting with a spherically symmetric perturbation of a non-extremal RN solution, we find similar behaviour in the electric case to the Ricci flows of perturbed Schwarzschild, namely the horizon shrinks to a singularity in finite time or expands forever. In the magnetic case, a perturbed unstable RN solution has a similar expanding behaviour, but a perturbation that decreases the horizon size flows to a stable black hole solution rather than a singularity. For extremal RN we solve the near horizon flow for spherical symmetry exactly, and see in the electric case two unstable directions which flow to singularities in finite flow time. However, even turning these off, and fixing the near horizon geometry to be that of RN, we numerically show that the flows appear to become singular in the vicinity of its horizon.