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

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-disc 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 (Reisenberger 2008 Phys. Rev. Lett. 101 211101) to calculate the Poisson brackets of the free data.