Gauge aspect of tetrad field in gravity
In general relativity, an inertial frame can only be established in a small region of spacetime, and a locally inertial frame is mathematically represented by a tetrad field in gravity. The tetrad field is not unique due to the freedom to perform a local Lorentz transformation in an inertial frame, and there exists freedom to choose the locally inertial frame at each spacetime. 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.