Gauge Theory in Riem(M)

Abstract

In the geometrodynamical setting of general relativity in Lagrangian form, the objects of study are the {\it Riemannian} metrics (and their time derivatives) over a given 3-manifold $M$. It is our aim in this paper to study the gauge properties that the space Riem(M) of all metrics over $M$ possesses, specially as they relate to the constraints of geometrodynamics. For instance, the Hamiltonian constraint does not generate a group, and it is thus hard to view its action in Riem(M) in a gauge setting. However, in view of the recent results representing GR as a dual theory, invariant under foliation preserving 3--diffeomorphisms and 3D conformal transformations, but not under refoliations, we are justified in considering the gauge structure pertaining only to the groups $\mathcal{D}$ of diffeomorphisms of $M$, and $\mathcal{C}$, of conformal diffeomorphisms on $M$. For these infinite-dimensional symmetry groups, Riem(M) has a natural principal fiber bundle (PFB) structure, which renders the gravitational field amenable to the full range of gauge-theoretic treatment. We discuss some of these structures and construct explicit formulae for supermetric-induced gauge connections. To apply the formalism, we compute general properties for a specific connection bearing strong resemblance to the one naturally induced by the deWitt supermetric, showing it has desirable relationalist properties. Finally, we find that the group of conformal diffeomorphisms solves the pathologies inherent in the \DD group and also brings it closer to Horava gravity and the dual conformal theory called Shape Dynamics.Comment: Version virtually identical to the one published in J. Math. Phys. Contains corrections to a crucial proof, and new figure