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Tetrads in General Relativity

VI. Physical Geometry

Gauge Principle

   A tetrad vector (covector) base $\{ {{\mathbf{e}}_a},{{\mathbf{e}}^a}\} $ of a local tangent space is not unique, since it is always possible to find an infinity of other bases by local Lorentz transformations

6.1

\[{{\mathbf{e'}}_a}(x) := {\Lambda _a}^b(x){{\mathbf{e}}_b}(x)\]

The equivalence of these tangent bases may be seen as a gauge principle, where the local gauge group is the symmetry group $\text{SO}(1,3)$ of position dependent Lorentz rotations (6.1). Gauge invariance is then the statement that physical laws cannot depend on such transformations. [R. Utiyama, 1956]

   From a given local frame, different classes of non-inertial frames may be obtained by performing local (point-dependent) Lorentz transformations. This includes transformations that boost away local gravity. The gauge principle of local Lorentz symmetry thus encompasses the essence of Einstein’s equivalence principle that gravitation may be annulled by acceleration.

To derive the gauge invariant equations implied by the gauge principle, one needs a gauge covariant derivative. To ascertain that the coderivative ${D_\mu }$ fullfills the necessary and sufficient conditions, one may differentiate (6.1) to get

6.2

\[{D_\mu }{{\mathbf{e'}}_a}(x) = {\Sigma '}{{_\mu}{^b}}{_a}(x){{\mathbf{e'}}_b}(x)\]

Here $\Sigma '$ is the locally transformed spin connection (4.9). This proves that the coderivative ${D_\mu }$ is invariant under a change of gauge, the spinor (Lorentz) connection $\Sigma _\mu ^{ab}$, with one spacetime index and two tetrad indices, being the relevant gauge field.

   The gauge field strength, in gauge theory defined by the coderivative of the gauge field

6.3

\[R_{\mu \nu }^{ab} : = 2{D_{[\mu }}\Sigma _{\nu ]}^{ab} = {\partial _\mu }\Sigma _\nu ^{ab} - {\partial _\nu }\Sigma _\mu ^{ab} + \Sigma _{\mu c}^a\Sigma _\nu ^{cb} - \Sigma _{\nu c}^a\Sigma _\mu ^{cb}\]

turns out to be the curvature tensor associated with the spin connection; see equation (10.2). This field is a tensor under MDs owing to the antisymmetric derivatives, and transforms homogeneously under LLTs.