The symplectic 2-form and Poisson bracket of null canonical gravity

Abstract

It is well known that free (unconstrained) initial data for the gravitational field in general relativity can be identified on an initial hypersurface consisting of two intersecting null hypersurfaces. Here the phase space of vacuum general relativity associated with such an initial data hypersurface is defined; a Poisson bracket is defined, via Peierls' prescription, on sufficiently regular functions on this phase space, called ``observables''; and a bracket on initial data is defined so that it reproduces the Peierls bracket between observables when these are expressed in terms of the initial data. The brackets between all elements of a free initial data set are calculated explicitly. The bracket on initial data presented here has all the characteristics of a Poisson bracket except that it does not satisfy the Jacobi relations (even though the brackets between the observables do). The initial data set used is closely related to that of Sachs. However, one significant difference is that it includes a ``new'' pair of degrees of freedom on the intersection of the two null hypersurfaces which are present but quite hidden in Sachs' formalism. As a step in the calculation an explicit expression for the symplectic 2-form in terms of these free initial data is obtained.