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Tetrads in General Relativity

X. Curvature

Second Bianchi Identity

   The second Bianchi identity is a differential identity obtained by taking the exterior coderivative of the curvature 2-form:

10.13

\[{\text{D}}{{\mathbf{R}}^a}_b = {\text{d}}{{\mathbf{R}}^a}_b + {{\mathbf{\omega }}^a}_c \wedge {{\mathbf{R}}^c}_b - {{\mathbf{\omega }}^c}_b \wedge {{\mathbf{R}}^a}_c = 0\]

Proof: Substitute the second Cartan equation (10.6) and use the property ${\text{dd}}{\mathbf{A}} = 0$ for general ${\mathbf{A}}$; see definition (1.7). After some algebra, taking into account the (anti-)commutation rule ${( - 1)^p}$, one obtains (10.13). [Wikipedia: Spin Connection]

   For the case of no torsion, the second Bianchi identity may be written in the component form

10.14

\[{D_\rho }{R_{\mu \nu \kappa \lambda }} + {D_\mu }{R_{\nu \rho \kappa \lambda }} + {D_\nu }{R_{\rho \mu \kappa \lambda }} = 3{D_{[\rho }}{R_{\mu \nu ]\kappa \lambda }} = 0\]

This equation may be also proven on the basis of the Jacobi identity, since it basically expresses

10.15

\[[[{D_{[\mu }},{D_\nu }],{D_{\kappa ]}}] = 0\]

   By operating with the Jacobi operator on an arbitrary vector and using definitions (10.1) and (10.8), one derives the second Bianchi identity in the cyclic form

10.16

\[{D_\rho }\operatorname{Riem} ({{\mathbf{g}}_\mu } \wedge {{\mathbf{g}}_\nu }) + {\text{cyclic}} = 0\]

   The above version of the second Bianchi identity is true if and only if the connection is symmetric and the torsion is zero.