Gravitational observables, intrinsic coordinates, and canonical maps

TL;DR

This paper constructs gravitational invariants using canonical transformations and active gauge transformations, clarifying their relation to diffeomorphism symmetry and providing explicit expressions with physical interpretations.

Contribution

It introduces a symmetry-inspired method to explicitly construct invariants in gravitational theories as limits of canonical transformations, linking them to Dirac brackets.

Findings

Invariants can be expressed as Taylor expansions in observer coordinates.

All invariants, including lapse and shift, satisfy Poisson brackets equal to their Dirac brackets.

A geometric proof shows the invariants' consistent algebraic structure.

Abstract

It is well known that in a generally covariant gravitational theory the choice of spacetime scalars as coordinates yields phase-space observables (or "invariants"). However their relation to the symmetry group of diffeomorphism transformations has remained obscure. In a symmetry-inspired approach we construct invariants out of canonically induced active gauge transformations. These invariants may be intepreted as the full set of dynamical variables evaluated in the intrinsic coordinate system. The functional invariants can explicitly be written as a Taylor expansion in the coordinates of any observer, and the coefficients have a physical and geometrical interpretation. Surprisingly, all invariants can be obtained as limits of a family of canonical transformations. This permits a short (again geometric) proof that all invariants, including the lapse and shift, satisfy Poisson brackets that are equal to the invariants of their corresponding Dirac brackets.