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

TL;DR

This work derives the symplectic 2-form for vacuum General Relativity on a double null initial hypersurface in terms of free null data, clarifying how diffeomorphism gauge allows non-null-preserving variations to be represented within a consistent phase space. It defines a complete set of free data, separating geometric data on the null branches from diffeomorphism data, and proves the regular area-parameter data are equivalent to Sachs data, yielding a free and complete phase space prone to quantization. Using admissible, null-sheet-preserving variations, the authors express the symplectic form entirely in terms of these data and connect it to the Peierls bracket for observables, thereby establishing a robust canonical structure for null initial data. The analysis also clarifies the auxiliary role of diffeomorphism data and addresses boundary/diffeomorphism subtleties, with potential implications for holographic entropy bounds and quantum gravity formulations that respect area discretization.

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.

Figures (2)

• Figure 1: a) A double null sheet in 2+1 dimensional spacetime. b) In 3+1 dimensional spacetime ${\cal N}$ is a 3-manifold consisting of two solid cylinders joined on a disk (shown here without regard to their embedding in spacetime).
• Figure 2: Panel a) shows a simple example of a caustic and intersections of generators in $2 + 1$ Minkowski space: $S_0$ is a spacelike curve having the shape of a half racetrack - a semicircle extended at each end by a tangent straight line. The congruence of null geodesics normal to $S_0$ and directed inward and to the future sweep out ${\cal N}_R$, which takes the form of a ridge roof, terminated by a half cone over the semicircle. The generators from the semicircle form a caustic at the vertex of the cone. There neighbouring generators intersect. On the other hand generators from the two straight segments of $S_0$ cross on a line (the ridge of the roof) starting at the caustic, but the generators that cross there are not neighbours at $S_0$. Clearly the generator segments beyond the crossing points enter the interior of the domain of dependence of ${\cal N}$. In Panel b) the double null sheet defined by $S_0$ in the covering space is shown, with the points that are identified in the original spacetime indicated.