Local Lorentz tetrad and connection

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the canonical tangent-bundle and tetrad conventions of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model and Local frame fields, coordinate frames and tetrad fields. Let $(M,g,N)$ be a smooth, time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$, let $U\subseteq M$ be nonempty and open, and let $E=(E_0,E_1,E_2,E_3)$ be a smooth future-directed tetrad on $U$. Thus the $E_J$ are a pointwise basis of $TM{\mathrm{\mid }}_U$, $g(E_I,E_J)={\eta }_{IJ}$ for $\eta =\mathrm{diag}(-1,1,1,1)$, and $E_0$ is future-directed. Write $e=(e^0,e^1,e^2,e^3{)}^{\mathsf{T}}$ for the dual coframe, so $e^I(E_J)={\delta }^I{}_J$.

A tetrad connection is a smooth affine connection $\mathrm{\nabla }$ on $TM{\mathrm{\mid }}_U$; its connection matrix $\omega =({\omega }^I{}_J)$ in $E$ is defined by ${\mathrm{\nabla }}_X{E}_J={\sum }_I{\omega }^I{}_J(X)E_I$. It is a Lorentz connection when it is metric-compatible, equivalently when the matrix-valued one-form is $\eta$-skew: ${\omega }_{IJ}:={\eta }_{IK}{\omega }^K{}_J$ satisfies ${\omega }_{IJ}=-{\omega }_{JI}$. A smooth local Lorentz change is a smooth map $\mathrm{\Lambda }:U\to S{O}^{+}(1,3)$. With the convention $e^{\mathrm{\prime }}=\mathrm{\Lambda }e$ (equivalently $E^{\mathrm{\prime }}=E{\mathrm{\Lambda }}^{-1}$), the connection matrix transforms as

$e^{\mathrm{\prime }}=\mathrm{\Lambda }e,{\omega }^{\mathrm{\prime }}=\mathrm{\Lambda }\omega {\mathrm{\Lambda }}^{-1}-d\mathrm{\Lambda }{\mathrm{\Lambda }}^{-1}\mathrm{.}$

Here $d$ is the componentwise exterior derivative, and products of matrix-valued forms use matrix multiplication and the wedge product of their scalar form entries. Define the torsion and curvature forms by

$T^I:=d{e}^I+{\sum }_J{\omega }^I{}_J\wedge e^J,F^I{}_J:=d{\omega }^I{}_J+{\sum }_K{\omega }^I{}_K\wedge {\omega }^K{}_J\mathrm{.}$

Thus $T$ is an ${\mathbb{R}}^4$-valued two-form and $F$ is an $\mathrm{End}({\mathbb{R}}^4)$-valued two-form. With this convention, $F$ is the local curvature two-form of $\mathrm{\nabla }$. A local $\mathrm{\Lambda }$ is a change of frame in a trivialization; it represents a gauge transformation in the sense of Gauge group of a smooth principal bundle only when it extends to a principal-bundle automorphism over the identity. No such extension, global tetrad, or spin structure is part of this definition.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Local frame fields, coordinate frames and tetrad fields

Linear and affine Lorentz maps, and the Lorentz group

Affine connection on a smooth manifold

Smooth principal $G$-bundle

Gauge group of a smooth principal bundle

A smooth differential $k$-form

Finite-dimensional vector-valued forms and their exterior derivative

The wedge product of differential forms

The exterior derivative by the invariant vector-field formula

Curvature two-form structure equation

Second Bianchi identity for a bundle connection

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The change between two future-directed tetrads is a Lorentz matrix

The coframe and connection form are local data on $U$; no global tetrad or global spin structure is asserted. The metric-selected Levi-Civita connection is one particular torsion-free instance of this local connection interface. For any connection on $TM{\mathrm{\mid }}_U$, the curvature $F$ also satisfies the second Bianchi identity supplied by the cited bundle-connection theorem.