Quantum gravitational measure for three-geometries

TL;DR

This paper constructs a gravitational measure for three-manifolds, revealing how it depends on the conformal factor, and links it to the Liouville and Einstein-Hilbert actions, with implications for quantum gravity and the Hartle-Hawking wave function.

Contribution

It introduces a new measure for three-geometries that incorporates conformal factor dependence and connects it to known gravitational actions and quantum wave functions.

Findings

Measure depends on conformal factor only for manifolds with boundary.

Liouville action appears in the measure for boundary manifolds.

Divergent Jacobian part generates Einstein-Hilbert action.

Abstract

The gravitational measure on an arbitrary topological three-manifold is constructed. The nontrivial dependence of the measure on the conformal factor is discussed. We show that only in the case of a compact manifold with boundary the measure acquires a nontrivial dependence on the conformal factor which is given by the Liouville action. A nontrivial Jacobian (the divergent part of it) generates the Einstein-Hilbert action. The Hartle-Hawking wave function of Universe is given in terms of the Liouville action. In the gaussian approximation to the Wheeler-DeWitt equation this result was earlier derived by Banks et al. Possible connection with the Chern-Simons gravity is also discussed.