Hamiltonian reduction of Einstein’s equations without isometries

Jong Hyuk Yoon,B Gwak,W Lee,H.-C Kim,J Lee,C.-H Lee,C Kim,S Lee,G Kang

doi:10.1051/epjconf/201816802001

Abstract

I apply the Hamiltonian reduction procedure to 4-dimensional spacetimes without isometries 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 cross section of null hypersurface, 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. Einstein’s equations in the privileged coordinates are presented as Hamilton’s equations of motions obtained from the physical Hamiltonian.

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