Higher Conservation Law for the Multi-Centre Metrics

TL;DR

This paper identifies specific multi-centre metrics with an additional quadratic conserved quantity, revealing integrable systems like Eguchi-Hanson and Taub-NUT within the class of solutions to Einstein's equations.

Contribution

It determines which multi-centre metrics possess a quadratic conserved quantity, expanding understanding of integrable structures in Einstein's solutions.

Findings

Several metrics exhibit extra conserved quantities.

Includes Eguchi-Hanson and Taub-NUT as special cases.

Highlights integrability in certain Einstein solutions.

Abstract

The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. For this full class, we determine which metrics do exhibit an extra conserved quantity quadraic in the momenta, induced by a Killing-Stackel tensor. Our results bring to light several metrics which correspond to classically integrable dynamical systems. They include, as particular cases, the Eguchi-Hanson and Taub-NUT metrics.