First-order flow equations for extremal and non-extremal black holes

TL;DR

This paper develops a unified first-order flow equation framework for static black holes, applicable to both extremal and non-extremal cases, using a generalized superpotential approach in various dimensions.

Contribution

It introduces a general method to derive first-order flow equations for black holes with scalars and vectors, extending previous supersymmetric techniques to non-supersymmetric cases.

Findings

Derived first-order flow equations for black holes in various theories.

Identified the generalized superpotential linking extremal and non-extremal solutions.

Provided conditions for the existence of superpotentials in symmetric scalar manifolds.

Abstract

We derive a general form of first-order flow equations for extremal and non-extremal, static, spherically symmetric black holes in theories with massless scalars and vectors coupled to gravity. By rewriting the action as a sum of squares a la Bogomol'nyi, we identify the function governing the first-order gradient flow, the `generalised superpotential', which reduces to the `fake superpotential' for non-supersymmetric extremal black holes and to the central charge for supersymmetric black holes. For theories whose scalar manifold is a symmetric space after a timelike dimensional reduction, we present the condition for the existence of a generalised superpotential. We provide examples to illustrate the formalism in four and five spacetime dimensions.