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Tetrads in General Relativity

X. Curvature

Curvature Bivector

   For the purpose of relating the curvature bivector to the spin (Lorentz) connection, it is useful to establish a link with the Cartan formalism. In this formalism the curvature of a manifold is characterized by the tetrad curvature 2-form

10.5

\[{{\mathbf{R}}^{ab}}: = {e_\mu }^a{e_\nu }^b{{\mathbf{R}}^{\mu \nu }} = \frac{1}{2} R_{\mu \nu }^{ab}{{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu }\]

Inserting the tensor components as specified in (6.3) and subsequently writing out the right-hand side in terms of the connection 1-form defined in (4.21), one obtains the second Cartan structure equation [Wikipedia: Spin Connection]:

10.6

\[{{\mathbf{R}}^a}_b = {\text{d}}{{\mathbf{\omega }}^a}_b + {{\mathbf{\omega }}^a}_c \wedge {{\mathbf{\omega }}^c}_b \]

   The equivalent GA-representation is obtained by substituting (6.3) into the last member of equation (10.2). The result is

10.7

\[{{\mathbf{R}}_{\mu \nu }}  = {\partial _\mu }{{\mathbf{\omega }}_\nu } - {\partial _\nu } {{\mathbf{\omega }}_\mu } + \left[ {{{\mathbf{\omega }}_\mu }, {{\mathbf{\omega }}_\nu }} \right]\]

defining the curvature bivector in terms of the spin connection bivector (4.4). The derivation shows that the GA curvature bivector (10.7) is completely isomorphic to the Cartan curvature 2-form (10.6), offering an alternative way to describe and calculate curvature.

   The curvature bivector (10.7) may be thought of as a point-wise multi-linear map

10.8

\[{{\mathbf{R}}_{\mu \nu }}: = \operatorname{Riem} ({{\mathbf{g}}_\mu } \wedge {{\mathbf{g}}_\nu })\]

from tangent bivectors to tangent bivectors. i.e. $\operatorname{Riem} ({{\mathbf{g}}_\mu } \wedge {{\mathbf{g}}_\nu })$ is a bivector-valued function of bivectors. In the mixed indices representation at the-right-hand side of (10.2), it connects a bivector area ${{\mathbf{g}}_\mu } \wedge {{\mathbf{g}}_\nu }$ with a bivector ${{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b}$ which is the generator of the rotation that a vector experiences when transported around the area perimeter.