Black holes in supergravity and integrability

TL;DR

This paper demonstrates the integrability of geodesic equations describing stationary black holes in supergravity, enabling a Hamilton-Jacobi approach and a new Lax integration method for solving these equations efficiently.

Contribution

It proves Liouville integrability of black hole geodesic equations in symmetric coset spaces and introduces a one-step Lax integration algorithm for these equations.

Findings

Liouville integrability of geodesic equations established

Hamilton-Jacobi formalism applicable to black hole solutions

New Lax integration algorithm developed and illustrated

Abstract

Stationary black holes of massless supergravity theories are described by certain geodesic curves on the target space that is obtained after dimensional reduction over time. When the target space is a symmetric coset space we make use of the group-theoretical structure to prove that the second order geodesic equations are integrable in the sense of Liouville, by explicitly constructing the correct amount of Hamiltonians in involution. This implies that the Hamilton-Jacobi formalism can be applied, which proves that all such black hole solutions, including non-extremal solutions, possess a description in terms of a (fake) superpotential. Furthermore, we improve the existing integration method by the construction of a Lax integration algorithm that integrates the second order equations in one step instead of the usual two step procedure. We illustrate this technology with a specific example.