Black holes as generalised Toda molecules

TL;DR

This paper explores the connection between geodesic formalism and black hole effective potentials, revealing a generalization of Toda molecule equations linked to integrable systems in supergravity theories.

Contribution

It demonstrates that the integrability of geodesic equations extends to black hole potentials, generalizing Toda molecule equations within symmetric supergravity models.

Findings

Geodesic formalism simplifies analysis of symmetric black hole solutions.

Integrability of equations of motion is preserved with black hole potentials.

Generalization of Toda molecule equations is established in this context.

Abstract

In this note we compare the geodesic formalism for spherically symmetric black hole solutions with the black hole effective potential approach. The geodesic formalism is beneficial for symmetric supergravity theories since the symmetries of the larger target space leads to a complete set of commuting constants of motion that establish the integrability of the geodesic equations of motion, as shown in arXiv:1007.3209. We point out that the integrability lifts straightforwardly to the integrability of the equations of motion with a black hole potential. This construction turns out to be a generalisation of the connection between Toda molecule equations and geodesic motion on symmetric spaces known in the mathematics literature. We describe in some detail how this generalisation of the Toda molecule equations arises.