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Tetrads in General Relativity

X. Curvature

First Bianchi Identity

   The Riemann (10.2) satisfies the first Bianchi identity, also called the ‘algebraic’ Bianchi identity. It may be obtained by taking the exterior coderivative of the first Cartan structure equation (4.24)

10.9

\[ {\text{D}}{{\mathbf{T}}^a} = {\text{d}}{{\mathbf{T}}^a} + {{\mathbf{\omega }}^a}_b \wedge {{\mathbf{T}}^b} = {{\mathbf{R}}^a}_b \wedge {{\mathbf{e}}^b}\]

where ${{\mathbf{T}}^a}$ is the torsion form as it appears in the first Cartan equation; ${{\mathbf{\omega }}^a}_b$ is the Cartan connection (4.21). The right-hand side is identically equal to the left-hand side, as required of an identity. Proof: just substitute the first Cartan equation into (10.9) and then use the second Cartan equation (10.6). [Wikipedia: Spin Connection]

   If spacetime is torsionless, identity (10.9) reduces to

10.10

\[{{\mathbf{R}}_{\mu \nu }} \wedge {{\mathbf{g}}^\nu } = \frac{1}{2}{R_{\mu \nu \kappa \lambda }}{{\mathbf{g}}^\kappa } \wedge {{\mathbf{g}}^\lambda } \wedge {{\mathbf{g}}^\nu } = 0\]

which is equivalent to the vanishing of the antisymmetric part of the Riemann as in (10.4d). Although known as the ‘first Bianchi identity’, this property was actually discovered by Gregorio Ricci-Curbastro and Tullio Levi–Civita; it is not so much an identity as a theorem, because it is only valid in torsionless spacetimes.

   The first Bianchi identity may also be derived directly from definition (10.1) of the curvature bivector. By using the property ${D_\mu }{{\mathbf{g}}_{_v}} - {D_\nu }{{\mathbf{g}}_{_\mu }} = 0$ of a torsionless spacetime, equation (1.25), one may manipulate the left-hand side of (10.1) into

10.11

\[[{D_\mu },{D_\kappa }]{{\mathbf{g}}_\nu } - [{D_\nu },{D_\kappa }]{{\mathbf{g}}_\mu } = {{\mathbf{R}}_{\mu \nu }} \cdot {{\mathbf{g}}_\kappa }\]

With the notation (10.8), this may be translated into the cyclic form of the first Bianchi identity:

10.12

\[{\text{Riem}}({{\mathbf{g}}_\mu } \wedge {{\mathbf{g}}_\nu }) \cdot {{\mathbf{g}}_\kappa } + {\text{cyclic}} = 0\]