  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.
