On plane gravitational waves in real connection variables

TL;DR

This paper studies plane gravitational waves using real connection variables to facilitate loop quantization, revealing non-local brackets and simplifying the constraints despite symmetry breaking.

Contribution

It introduces a classical analysis of plane-fronted gravitational waves in real connection variables, deriving a simplified system with non-local Dirac brackets.

Findings

Derived a reduced canonical system with one pair and one constraint.

Established non-local Dirac brackets for triad variables.

Simplified the constraints of plane-fronted gravitational waves.

Abstract

We investigate using plane fronted gravitational wave space-times as model systems to study loop quantization techniques and dispersion relations. In this classical analysis, we start with planar symmetric space-times in the real connection formulation. We reduce via Dirac constraint analysis to a final form with one canonical pair and one constraint, equivalent to the metric and Einstein equations of plane-fronted with parallel rays waves. Due to the symmetries and use of special coordinates general covariance is broken. However, this allows us to simply express the constraints of the consistent system. A recursive construction of Dirac brackets results in non-local brackets, analogous to those of self-dual fields, for the triad variables chosen in this approach.