The symplectic 2-form for gravity in terms of free null initial data

TL;DR

This paper derives an expression for the symplectic 2-form in vacuum general relativity using free null initial data on a hypersurface formed by two light fronts, facilitating analysis of gravitational degrees of freedom.

Contribution

It provides a novel explicit formula for the symplectic 2-form in terms of free null initial data, accounting for gauge invariance despite non-preservation of nullness under variations.

Findings

Expression for symplectic 2-form in terms of free null data

Application to calculate Poisson brackets of free data

Addresses gauge invariance in null hypersurface variations

Abstract

A hypersurface formed of two null sheets, or "light fronts", swept out by the future null normal geodesics emerging from a common spacelike 2-disk can serve as a Cauchy surface for a region of spacetime. Already in the 1960s free (unconstrained) initial data for general relativity were found for such hypersurfaces. Here an expression is obtained for the symplectic 2-form of vacuum general relativity in terms of such free data. This can be done, even though variations of the geometry do not in general preserve the nullness of the initial hypersurface, because of the diffeomorphism gauge invariance of general relativity. The present expression for the symplectic 2-form has been used previously to calculate the Poisson brackets of the free data.