On the role of conformal three-geometries in the dynamics of General Relativity

TL;DR

This paper reveals that the Chern-Simons functional's invariance under conformal rescalings characterizes vacuum Einstein equations and links the Hamiltonian constraint to the Poisson bracket involving this functional and Misner's time.

Contribution

It establishes a novel connection between the Chern-Simons functional, conformal invariance, and the Hamiltonian and momentum constraints in vacuum general relativity.

Findings

The Chern-Simons functional is conformally invariant iff Einstein vacuum equations hold.

The Hamiltonian constraint is expressed as a Poisson bracket involving the Chern-Simons functional and volume time.

The momentum constraint relates to the change of the Chern-Simons functional along York's time flow.

Abstract

It is shown that the Chern-Simons functional, built in the spinor representation from the initial data on spacelike hypersurfaces, is invariant with respect to infinitesimal conformal rescalings if and only if the vacuum Einstein equations are satisfied. As a consequence, we show that in the phase space the Hamiltonian constraint of vacuum general relativity is the Poisson bracket of the imaginary part of this Chern-Simons functional and Misner's time (essentially the 3-volume). Hence the vacuum Hamiltonian constraint is the condition on the canonical variables that the imaginary part of the Chern- Simons functional be constant along the volume flow. The vacuum momentum constraint can also be reformulated in a similar way as a (more complicated) condition on the change of the imaginary part of the Chern-Simons functional along the flow of York's time.