Linking Covariant and Canonical General Relativity via Local Observers

TL;DR

This paper introduces a covariant formulation of Hamiltonian gravity using a field of observers, bridging Ashtekar variables with covariant phase space and suggesting observer-dependent spacetime geometry.

Contribution

It presents a spacetime covariant approach that incorporates local observers, connecting Hamiltonian and covariant formulations of gravity in a novel way.

Findings

Fields are covariant under spacetime symmetries.

When the observer field is normal to a foliation, fields adopt Hamiltonian form.

Framework suggests observer-dependent spacetime geometry.

Abstract

Hamiltonian gravity, relying on arbitrary choices of "space," can obscure spacetime symmetries. We present an alternative, manifestly spacetime covariant formulation that nonetheless distinguishes between "spatial" and "temporal" variables. The key is viewing dynamical fields from the perspective of a field of observers -- a unit timelike vector field that also transforms under local Lorentz transformations. On one hand, all fields are spacetime fields, covariant under spacetime symmeties. On the other, when the observer field is normal to a spatial foliation, the fields automatically fall into Hamiltonian form, recovering the Ashtekar formulation. We argue this provides a bridge between Ashtekar variables and covariant phase space methods. We also outline a framework where the 'space of observers' is fundamental, and spacetime geometry itself may be observer-dependent.