# antisymmetric tensors

17 Feb 2021The Bianchi-identity applies to the Riemann-curvature because of its antisymmetry. would it equally apply to the Weyl-tensor as well? What would this imply for gravitational waves? Would a similar argument apply to the field tensor in electrodynamics?

The Bianchi-identity \(\nabla_\lambda R_{\mu\nu\alpha\beta} + \nabla_\mu R_{\nu\lambda\alpha\beta} + \nabla_\nu R_{\lambda\mu\alpha\beta} = 0\) applies to the Weyl-curvature \(C_{\mu\nu\alpha\beta}\) as well, and there is an analogous statement for the Faraday-tensor, \(\nabla_\lambda F_{\mu\nu} + \nabla_\mu F_{\nu\lambda} + \nabla_\nu F_{\lambda\mu} = 0\).