Observables in terms of connection and curvature variables for Einstein's equations with two commuting Killing vectors

TL;DR

This paper develops a framework to express Einstein's equations with two commuting Killing vectors using connection and curvature variables, revealing integrals of motion and a Klein-Gordon interpretation of the lapse function.

Contribution

It introduces a novel hierarchy of integrals of motion in terms of connection and curvature variables, linking integrable systems theory with Einstein's equations.

Findings

Hierarchy of integrals of motion derived

Connection and curvature variables linked to conserved quantities

Klein-Gordon equation interpretation for lapse function

Abstract

Einstein's equations with two commuting Killing vectors and the associated Lax pair are considered. The equations for the connection $A(\varsigma, \eta, \gamma)=\Psi_{,\gamma}\Psi^{-1}$, where $\gamma$ the variable spectral parameter are considered. A transition matrix ${\cal T}= A(\varsigma, \eta, \gamma)A^{-1}(\xi, \eta, \gamma)$ for $A$ is defined relating $A$ at ingoing and outgoing light cones. It is shown that it satisfies equations familiar from integrable pde's theory. A transition matrix on $\varsigma={\mbox constant}$ is defined in an analogous manner. These transition matrices allow us to obtain a hierarchy of integrals of motion with respect to time, purely in terms of the trace of a function of the connections $g_{,\varsigma}g^{-1}$ and $g_{,\eta}g^{-1}$. Furthermore a hierarchy of integrals of motion in terms of the curvature variable $B=A_{,\gamma}A^{-1}$, involving the commutator $[A(1), A(-1)]$, is obtained. We interpret the inhomogeneous wave equation that governs $\sigma=ln N$, $N$ the lapse, as a Klein-Gordon equation, a dispersion relation relating energy and momentum density, based on the first connection observable and hence this first observable corresponds to mass. The corresponding quantum operators are $\frac{\partial}{\partial t}$, $\frac{\partial}{\partial z}$ and this means that the full Poincare group is at our disposal.