The Poisson Bracket on Free Null Initial Data for Gravity

Free initial data for general relativity on a pair of intersecting null hypersurfaces are well known, but the lack of Poisson brackets 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 Poisson brackets on free data are given. On sufficiently regular functions of the solution spacetime geometry these brackets match the Poisson brackets defined on such functions by the Hilbert action via Peierls’ prescription. The symplectic 2-form is also given in terms of free data.

Figure 1

(a) A double null sheet in $2+1$ dimensional spacetime. (b) In $3+1$ dimensional spacetime $\mathcal{N}$ is a 3-manifold consisting of two solid cylinders joined on a disk, here shown without regard to their embedding in spacetime.