Lorentz gauge and Coulomb gauge for tetrad field of gravity

TL;DR

This paper explores gauge fixing conditions for tetrad fields in gravity, specifically proposing Lorentz and Coulomb gauges, and discusses their properties and similarities to electromagnetic gauge conditions.

Contribution

It introduces Lorentz and Coulomb gauge conditions for tetrad fields in gravity, extending gauge fixing concepts from electromagnetism to gravitational tetrad fields.

Findings

Lorentz and Coulomb gauges for tetrad fields are proposed.

Properties of these gauges are analyzed and compared to electromagnetic gauges.

The gauges help fix the local Lorentz freedom in tetrad formulations of gravity.

Abstract

In general relativity, an inertial frame can only be established in a small region of spacetime, and locally inertial frames are mathematically represented by a tetrad field in gravity. The tetrad field is not unique due to the freedom to perform Lorentz transformations in locally inertial frames, and there exists freedom to choose the locally inertial frame at each spacetime point. The local Lorentz transformations are known as non-Abelian gauge transformations for the tetrad field, and to fix the gauge freedom, corresponding to the Lorentz gauge $\partial^\mu\mathcal{A}_\mu=0$ and Coulomb gauge $\partial^i\mathcal{A}_i=0$ in electrodynamics, the Lorentz gauge and Coulomb gauge for the tetrad field are proposed in the present work. Moreover, properties of the Lorentz gauge and Coulomb gauge for tetrad field are discussed, which show the similarities to those in electromagnetic field.