The geometric role of symmetry breaking in gravity

TL;DR

This paper explores how symmetry breaking in gravity relates to geometric structures, using Cartan's method to connect metrics and connections, and proposes a new Hamiltonian gravity approach involving spontaneous Lorentz symmetry breaking.

Contribution

It introduces a geometric perspective on symmetry breaking in gravity and develops a novel Hamiltonian formulation with spontaneous Lorentz symmetry breaking.

Findings

Symmetry breaking relates to geometry via Cartan's method.

Broken symmetry is inherent in gravity theories due to geometric reasons.

A new Hamiltonian gravity approach with spontaneous Lorentz symmetry breaking is proposed.

Abstract

In gravity, breaking symmetry from a group G to a group H plays the role of describing geometry in relation to the geometry the homogeneous space G/H. The deep reason for this is Cartan's "method of equivalence," giving, in particular, an exact correspondence between metrics and Cartan connections. I argue that broken symmetry is thus implicit in any gravity theory, for purely geometric reasons. As an application, I explain how this kind of thinking gives a new approach to Hamiltonian gravity in which an observer field spontaneously breaks Lorentz symmetry and gives a Cartan connection on space.