Problem of time and Hamiltonian reduction in the (2+2) formalism

TL;DR

This paper applies Hamiltonian reduction in the (2+2) formalism to 4D spacetimes, deriving a constraint-free, positive-definite Hamiltonian expressed in true degrees of freedom, consistent with Einstein's equations in privileged coordinates.

Contribution

It generalizes ADM Hamiltonian reduction to 4D spacetimes without isometries using the (2+2) formalism, identifying privileged coordinates and true degrees of freedom.

Findings

Hamiltonian is constraint-free and positive-definite in privileged coordinates.

Hamilton's equations match Einstein's equations in these coordinates.

The reduction respects general covariance and is self-consistent.

Abstract

We 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 true degrees of freedom only, namely, the conformal two-metric on the cross section of null hypersurfaces and its conjugate momentum. The physical time is the area element of the cross section of null hypersurface, and the physical radial coordinate is defined by {\it equipotential} surfaces on a given spacelike hypersurface of constant physical time. The physical Hamiltonian is {\it constraint-free} and manifestly {\it positive-definite} in the privileged coordinates. We present the complete set of the Hamilton's equations, and find that they coincide with the Einstein's equations written in the privileged coordinates. This shows that our Hamiltonian reduction is self-consistent and respects the general covariance. This work is a generalization of ADM Hamiltonian reduction of midi-superspace to 4-dimensional spacetimes with no isometries.