Spontaneous breaking of Lorentz symmetry for canonical gravity

TL;DR

This paper introduces a Lorentz covariant formulation of canonical gravity using a local observer field, which spontaneously breaks Lorentz symmetry but remains consistent with Lorentz covariance, and suggests observer space as fundamental.

Contribution

It generalizes the foliation concept to a local observer field, demonstrating spontaneous Lorentz symmetry breaking within a covariant framework and proposing observer space as the fundamental setting.

Findings

Spontaneous Lorentz symmetry breaking from SO(3,1) to SO(3).

A geometric Cartan geometrodynamics interpretation.

Potential applications in gravitational physics using observer space.

Abstract

In the Ashtekar-Barbero formulation of canonical general relativity based on an SU(2) connection, Lorentz covariance is a subtle issue which has been the focus of some debate. Here we present a Lorentz covariant formulation generalising the notion of a foliation of spacetime to a field of local observers which specify a time direction only locally. This field spontaneously breaks the local SO(3,1) symmetry down to a subgroup SO(3); we show that the apparent symmetry breaking to SO(3) is not in conflict with Lorentz covariance. We give a geometric picture of our construction as Cartan geometrodynamics and outline further applications of the formalism of local observers, motivating the idea that observer space, instead of spacetime, should serve as the fundamental arena for gravitational physics.