Constants of motion for vacuum general relativity

TL;DR

This paper reformulates a sector of vacuum general relativity as an $SL(2,R)$ principal chiral model and introduces a method to generate an infinite set of non-local conserved quantities, enhancing understanding of its integrability.

Contribution

It provides a novel procedure to derive an infinite number of non-local constants of motion for vacuum Einstein equations with symmetries, using a chiral model formulation.

Findings

Infinite non-local conserved quantities are constructed.

Constants of motion are explicitly functional on the phase space.

The approach links Einstein equations to integrable models.

Abstract

The 3+1 Hamiltonian Einstein equations, reduced by imposing two commuting spacelike Killing vector fields, may be written as the equations of the $SL(2,R)$ principal chiral model with certain `source' terms. Using this formulation, we give a procedure for generating an infinite number of non-local constants of motion for this sector of the Einstein equations. The constants of motion arise as explicit functionals on the phase space of Einstein gravity, and are labelled by sl(2,R) indices.