  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.
