The Poisson bracket on free null initial data for gravity

TL;DR

This paper introduces a Poisson bracket for free initial data in general relativity on null hypersurfaces, addressing previous issues with caustics and generator crossings, and aligns with the Hilbert action's Peierls' bracket.

Contribution

It provides a novel Poisson bracket and symplectic form for null initial data in gravity, avoiding caustics and generator crossings, and matches the Peierls' bracket from the Hilbert action.

Findings

Poisson bracket on free null data is explicitly constructed.

Caustics and generator crossings can be effectively avoided.

The bracket aligns with the Peierls' prescription for the Hilbert action.

Abstract

Free initial data for general relativity on a pair of intersecting null hypersurfaces are well known, but the lack of a Poisson bracket and concerns about caustics have stymied the development of a constraint free canonical theory. Here it is pointed out how caustics and generator crossings can be neatly avoided and a Poisson bracket on free data is given. On sufficiently regular functions of the solution spacetime geometry this bracket matches the Poisson bracket defined on such functions by the Hilbert action via Peierls' prescription. The symplectic form is also given in terms of free data.