Tetrads in General Relativity

   GRT may be viewed as a gauge theory of the Lorentz algebra based on the restricted Lorentz group ${\text{S}}{{\text{O}}^ + }(1,3)$. This is the component of the Lorentz group connected to the identity.  Alternatively, it can be defined as the subgroup of transformations which are orthochronous (preserve the direction of time) and proper (preserve the orientation).

   For this restricted group, the representation ${\Lambda ^\mu }_\nu $ of 4 × 4 real matrices acting on ${\mathbb{R}^{1,3}}$ may be put in the exponentiated form

The anti-symmetric group generators are 4x4 matrices that satisfy the Lorentz algebra commutation relations with the components

The anti-symmetric parameters ${{\varepsilon} _{ab}} = - {\varepsilon _{ab}}$ may be identified with an infinitesimal Lorentz transformation: ${\Lambda _{ab}} = {\eta _{ab}} + {\varepsilon _{ab}}$. Being anti-symmetric, the ${\varepsilon _{ab}}$ have 4x3/2 = 6 independent components, which agrees with the 6 transformations of the Lorentz group: 3 rotations and 3 boosts. [Wikipedia: Dirac Algebra]

   More generally, one may associate six anti-symmetric 4x4 matrices ${J^{ab}}$ with the Lorentz algebra, to generate any 4x4 matrix representation $S(\Lambda )$ of ${\text{S}}{{\text{O}}^ + }(1,3)$ by the exponential form:

   Under local LTs, the parameters ${\varepsilon _{ab}}$ become a function of spacetime: ${\varepsilon _{ab}} \to {\varepsilon _{ab}}(x)$. This step of making a symmetry local is called gauging the global symmetry. As a consequence, derivatives of fields like ${\partial _\mu }\psi (x)$, are no longer homogeneous due to the occurrence of terms like $[{\partial _\mu }{\varepsilon _{ab}}(x)]\psi (x)$, which are non-vanishing unless ${\varepsilon _{ab}}$ is constant. The procedure to go from a locally flat to a curved spacetime involves the generalization of $\partial _\mu $ to a covariant derivative ${\mathcal{D}_\mu }$ to compensate for these extra terms.