Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2011-02-27
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
Scientific paper
The notion of Superspace, the configuration space of General Relativity, has been constantly refined over the last fifty years. In Wheeler's Geometrodynamics, it is currently identified with the higher Teichm\"uller Space of conformal structures on a spatial hypersurface. In Ashtekar's Connection-dynamics however, Superspace has considerably less geometric structure - being identified with the space of suitably regular soldering forms modulo diffeomorphisms. Under York's assumption that the spatial hypersurface has the structure of a conformal 3-manifold, we use conformal spin geometry to prove that the soldering forms are naturally partitioned into conformal equivalence classes. This modifies the structure of Connection-dynamic Superspace, and the two configuration spaces are seen - as they were at the introduction of Ashtekar's new variables - to be 2-1 homomorphic. Lastly, we show that the enhanced structure on Connection-dynamic Superspace removes the (debatably unphysical) Barbero-Immirzi parameter of Loop Quantum Gravity at the expense of a non-dynamical scalar field: a geometric analogue of a result obtained by Wang.
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