Hamiltonian Reduction of Einstein's Equations without Isometries

Jong Hyuk Yoon

doi:10.1088/1742-6596/600/1/012068

Abstract

I apply the Hamiltonian reduction procedure to general spacetimes of 4 dimensions with no isometries in the (2+2) formalism and find privileged spacetime coordinates. Physical time is chosen as the area element of the two dimensional cross-section of null hypersurfaces. The physical spatial coordinates are defined by equipotential surfaces on a given spacelike hypersurface of constant physical time. The physical Hamiltonian is manifestly local and positive-definite in the privileged coordinates. The complete set of Hamilton's equations is presented and it is found that they coincide with the Einstein's equations written in the privileged coordinates. This shows that the Hamiltonian reduction is a self-consistent procedure.

Highlights

It is well-known that the true gravitational degrees of freedom of general relativity reside in the conformal two metric of the spatial cross section of null hypersurfaces[1, 2]
The area element of the spatial cross section of a null hypersurface emerges as the physical time, and the physical radial coordinate is defined by equipotential surfaces on a given spacelike hypersurface of constant physical time
We present the fully reduced physical Hamiltonian in these privileged coordinates[14], which turns out to be local and positive

Summary

Introduction

It is well-known that the true gravitational degrees of freedom of general relativity reside in the conformal two metric of the spatial cross section of null hypersurfaces[1, 2]. The area element of the spatial cross section of a null hypersurface emerges as the physical time, and the physical radial coordinate is defined by equipotential surfaces on a given spacelike hypersurface of constant physical time. R(2) is the scalar curvature of N2, and the diffN2-covariant derivative[13] of a tensor density qab with weight s is defined as. The diffN2-covariant derivatives of the area element τ and the conformal metric ρab, which are a scalar density and a tensor density with weight 1 and −1 with respect to the diffN2 transformations, respectively, are given by. The diffN2-covariant derivatives of the conjugate momenta πτ, πh, πa, and πab, which are tensor densities of weights.

Hamiltonian reduction I: τ as physical time

Hamiltonian reduction II

Main results