A construction principle for ADM-type theories in maximal slicing gauge

TL;DR

This paper constructs ADM-type theories in a specific time gauge from first principles, providing an independent derivation of ADM gravity and addressing conceptual issues in quantum gravity.

Contribution

It introduces a new construction principle for ADM-type theories in maximal slicing gauge, independent of Shape Dynamics, based on phase space constraints and symmetry considerations.

Findings

Derivation of ADM gravity from first principles.

Theories can be represented as either Shape Dynamics or ADM.

Addresses criticisms of Shape Dynamics by starting from ADM Hamiltonian.

Abstract

The differing concepts of time in general relativity and quantum mechanics are widely accused as the main culprits in our persistent failure in finding a complete theory of quantum gravity. Here we address this issue by constructing ADM-type theories \emph{in a particular time gauge} directly from first principles. The principles are expressed as conditions on phase space constraints: we search for two sets of spatially covariant constraints, which generate symmetries (are first class) and gauge-fix each other leaving two propagating degrees of freedom. One of the sets is the Weyl generator tr$(\pi)$, and the other is a one-parameter family containing the ADM scalar constraint $\lambda R- \beta(\pi^{ab}\pi_{ab}+(\mbox{tr}(\pi))^2/2))$. The two sets of constraints can be seen as defining ADM-type theories with a maximal slicing gauge-fixing. This work provides an independent, first principles derivation of ADM gravity. The principles above are motivated by a heuristic argument relying in the relation between symmetry doubling and exact renormalization arguments for quantum gravity, aside from compatibility with the spatial diffeomorphisms. As a by-product, these results address one of the most popular criticisms of Shape Dynamics: its construction starts off from the ADM Hamiltonian formulation. The present work severs this dependence: the set of constraints yield reduced phase space theories that can be naturally represented by either Shape Dynamics or ADM. More precisely, the resulting theories can be naturally "unfixed" to encompass either spatial Weyl invariance (the symmetry of Shape Dynamics) or refoliation symmetry (ADM).