Hamiltonian reduction for Einstein’s equations in the (2+2) formalism

Abstract

I apply the Hamiltonian reduction procedure to general spacetimes of 4 dimensions in the (2+2) formalism and find privileged spacetime coordinates in which the physical Hamiltonian is expressed in terms of the conformal two metric and its conjugate momentum. Physical time is the area element of the spatial cross-section of null hypersurfaces, and the physical radial coordinate is defined by equipotential surfaces on a given spacelike hypersurface of constant physical time. The physical Hamiltonian is local and positive in the privileged coordinates. I present the complete set of Hamilton’s equations and find that they coincide with the Einstein’s equations written in the privileged coordinates. This shows that the Hamiltonian reduction is self-consistent and respects general covariance.