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

DOI10.1098/RSPA.2015.0350zbMATH Open1371.83019arXiv2108.03435OpenAlexW2494613824MaRDI QIDQ5363438

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Publication date: 29 September 2017

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Abstract: Einstein's equations with two commuting Killing vectors and the associated Lax pair are considered. The equations for the connection $A(varsigma,eta,gamma)=Ps{i}_{,gamma}Ps{i}^{-1}$, where $gamma$ the variable spectral parameter are considered. A transition matrix $calT=A(varsigma,eta,gamma)A^{-1}(xi,eta,gamma)$ 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 $varsigma=mboxconstant$ 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 $sigma=lnN$, $N$ 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 $fracpartialpartialt$, $fracpartialpartialz$ and this means that the full Poincare group is at our disposal.

Full work available at URL: https://arxiv.org/abs/2108.03435

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