The full integration of black hole solutions to symmetric supergravity theories

TL;DR

This paper demonstrates that all stationary, spherically symmetric black hole solutions in theories with symmetric target spaces are integrable, providing an explicit method to solve their equations via geodesic and Lax pair formulations.

Contribution

It introduces a comprehensive integration method for black hole solutions in symmetric supergravity theories, including extremal and non-extremal cases, using geodesic and Lax pair techniques.

Findings

All solutions are integrable via geodesic equations.

Explicit integration method based on Lax pair formulation.

Explicit Wick rotation connecting BPS and non-BPS solutions.

Abstract

We prove that all stationary and spherical symmetric black hole solutions to theories with symmetric target spaces are integrable and we provide an explicit integration method. This exact integration is based on the description of black hole solutions as geodesic curves on the moduli space of the theory when reduced over the time-like direction. These geodesic equations of motion can be rewritten as a specific Lax pair equation for which mathematicians have provided the integration algorithms when the initial conditions are described by a diagonalizable Lax matrix. On the other hand, solutions described by nilpotent Lax matrices, which originate from extremal regular (small) D = 4 black holes can be obtained as suitable limits of solutions obtained in the diagonalizable case, as we show on the generating geodesic (i.e. most general geodesic modulo global symmetries of the D = 3 model) corresponding to regular (and small) D = 4 black holes. As a byproduct of our analysis we give the explicit form of the Wick rotation connecting the orbits of BPS and non-BPS solutions in maximally supersymmetric supergravity and its STU truncation.