Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2001-10-24
Class.Quant.Grav. 19 (2002) 2375-2392
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
15 pages, plain Tex
Scientific paper
10.1088/0264-9381/19/9/304
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.
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