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

Abstract

We derive a general form of first-order flow equations for extremal and nonextremal, static, spherically symmetric black holes in theories with massless scalars and vectors coupled to gravity. By rewriting the action as a sum of squaresà 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.

Black holes and black strings of N = 2, d = 5 supergravity in the H-FGK formalism

“…Since V bh (B ± ) = V bh (A), combining this expression with the Bogomol'nyi bound we get the wellknown formula 10) which also admits the suggestive expression 11) and leads to the suggestive relation…”

“…The derivation of flow equations for extremal black holes presented here differs from their previous presentation [29,30,31] in two aspects: Firstly, no specific form of the relation between the fake and actual charges is assumed here, instead the derivation is based on the assumption of harmonicity of the variables H I . 13 Secondly, expressing the black hole potential directly by harmonic functions makes it possible to determine the fake charges by extremization.…”

“…Observe that the above equations are, mutatis mutandis, identical to flow equations (2.70, 2.71) for the black holes. This means that as long as we are considering the same kind of ansatz for the seed functions, which is the case, we will find that the above flow equations will lead to a solution of the FGK equations of motion as long as 31) and the non-extremality parameter can be obtained from…”

“…Even though not all extremal black holes are supersymmetric [5,6] and all non-extremal black holes break all supersymmetries, it turns out [7] that analogous flow equations may be derived for four- [8] and five-dimensional extremal black holes [9], as well as for non-extremal p-branes [10] and black holes [11]. This fact alone already hints at a possibility that perhaps non-supersymmetric solutions could be obtained in a similar way to the supersymmetric ones.…”

“…where we have defined the black string potential as 11) where in the last step we introduced the (magnetic) string central charge Z m = h I p I ; B is again a non-extremality parameter. In order to obtain the above action in the K-variables, we will need the straightforward identity…”