First Order Description of Black Holes in Moduli Space

TL;DR

This paper demonstrates that extremal black hole solutions in moduli space can be described by first order equations derived from a prepotential, linking their properties to a c-function and providing explicit formulas in supergravity theories.

Contribution

It generalizes previous results by showing that second order equations are implied by first order equations involving a prepotential, applicable to various supergravity black hole solutions.

Findings

The squared prepotential acts as a c-function interpolating between ADM and near-horizon masses.

Explicit prepotential expressions are found for extended supergravity black holes at any radius.

Constraints on U-duality invariants for non-BPS solutions are identified.

Abstract

We show that the second order field equations characterizing extremal solutions for spherically symmetric, stationary black holes are in fact implied by a system of first order equations given in terms of a prepotential W. This confirms and generalizes the results in [14]. Moreover we prove that the squared prepotential function shares the same properties of a c-function and that it interpolates between M^2_{ADM} and M^2_{BR}, the parameter of the near-horizon Bertotti-Robinson geometry. When the black holes are solutions of extended supergravities we are able to find an explicit expression for the prepotentials, valid at any radial distance from the horizon, which reproduces all the attractors of the four dimensional N>2 theories. Far from the horizon, however, for N-even our ansatz poses a constraint on one of the U-duality invariants for the non-BPS solutions with Z \neq 0. We discuss a possible extension of our considerations to the non extremal case.