The FGK formalism for black p-branes in d dimensions

Abstract

We present a generalization to an arbitrary number of spacetime (d) and worldvolume (p+1) dimensions of the formalism proposed by Ferrara, Gibbons and Kallosh to study black holes (p = 0) in d = 4 dimensions. We include the special cases in which there can be dyonic and self-or anti-self-dual black branes. Most of the results valid for 4-dimensional black holes (relations between temperature, entropy and non-extremality parameter, and between entropy and black-hole potential on the horizon) are straightforwardly generalized.We apply the formalism to the case of black strings in N = 2, d = 5 supergravity coupled to vector multiplets, in which the black-string potential can be expressed in terms of the dual central charge and work out an explicit example with one vector multiplet, determining supersymmetric and non-supersymmetric attractors and constructing the non-extremal black-string solutions that interpolate between them.

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

“…Following [15], we introduce the basic definitions concerning the theories we deal with, the metric ansatz and the H-variables. We show how we can get the metric that covers the region lying between the inner (Cauchy) horizon and the singularity (not discussed in [2,3]) from the one that covers the exterior of the outer (event) horizon in the present 5-dimensional case. This will allow us to compute the "entropy" and "temperature" associated with the inner horizon 7 .…”

“…Static, spherically symmetric black-hole solutions of N = 2 supergravity can be conveniently studied in the effective black-hole potential formalism originally developed by Ferrara, Gibbons and Kallosh [1] in four dimensions, later extended to arbitrary dimensions [2], as well as to p-branes [3]. This is especially true for supersymmetric extremal solutions, where expressing the effective potential by the central charge leads to the derivation of first-order flow equations for the scalars (implied by the Killing spinor equations), whose attractor fixed points (corresponding to the sets of values of scalars on the event horizon) [4] are determined by critical points of the central charge.…”

“…This fact alone already hints at a possibility that perhaps non-supersymmetric solutions could be obtained in a similar way to the supersymmetric ones. Indeed, at least in a class of four- [12] and five-dimensional [13,2] black holes and p-branes [3], the known supersymmetric solution can be deformed to a unique non-extremal solution, from which both supersymmetric and nonsupersymmetric extremal solutions are recovered in the different limits in which the non-extremality parameter vanishes. This is, as far as we know, the only systematic method for constructing general extremal non-supersymmetric black-hole solutions, in particular when the black-hole potential has flat directions and the values of the scalar fields on the horizon have some dependence on the asymptotic values [12].…”

“…Explicit solutions are given in section 3. The examples that we analyze include the general nonextremal black holes with constant scalars, in section 3.1 (found earlier in [13] in a different way), the ST U model, in section 3.2, which we solve paying particular attention to the possible signs of the charges, and in section 3.3 the models of the reducible Jordan sequence, whose black-hole potential has flat directions and whose values of scalars on the horizon, in some non-supersymmetric cases, are not completely fixed by the charges.…”

“…Finally, in section 4 we generalize this approach to black strings, using the extension, recently constructed in [3], of the FGK formalism to p-branes, and introducing dual H-variables (which we shall denote by K). Our study of this case follows what we did for the black holes in the previous sections: we find the general solutions for non-extremal black strings with constant scalars for any N = 2, d = 5 supergravity theory in section 4.2, derive flow equations for black strings in section 4.1, and construct explicitly the extremal black strings of the pure and the heterotic ST U model in section 4.3.…”