Gauging Geometry: A Didactic Lecture

TL;DR

This paper presents a didactic overview of physical geometry based on local inertial frame invariance, describing it as a gauge theory of the vierbein that aligns with Einstein's theory of gravity.

Contribution

It introduces a gauge-theoretic formulation of physical geometry using vierbeins, connecting local Lorentz invariance to Einstein's relativistic gravitation.

Findings

Verbein invariance under Lorentz transformations

Gauge theory formulation of gravity

Equivalence to Einstein's tensor form

Abstract

Local inertial frame invariance is taken as the fundamental principle of physical geometry, where a local inertial frame is represented by a verbein. Invariance of the vierbein with respect to local Lorentz transformations then expresses local inertial frame invariance. The dynamics of physical geometry develops as a gauge theory of the verbein that is closely analogous to the Yang-Mills field provided the verbein connection and curvature correspond to the geometric potential and field respectively. The resulting theory is shown to be equivalent to Einstein's tensor form of relativistic gravitation.