# alternative Bianchi-identity

24 Mar 2021The homogeneous Maxwell-equation can be written with the dual field tensor \(\tilde{F}^{\mu\nu}\) as \(\partial_\mu\tilde{F}^{\mu\nu}=0\). Could you generalise this idea to the (contracted) Bianchi-identity in general relativity?

Expanding the homogeneous Maxwell-equation yields \(\partial_\lambda F_{\mu\nu} + \partial_{\mu} F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0\) which is completely equivalent to the Bianchi-identity for e.g. the Riemann-tensor, \(\nabla_\lambda R_{\mu\nu\alpha\beta} + \nabla_\mu R_{\nu\lambda\alpha\beta} + \nabla_\nu R_{\lambda\mu\alpha\beta} = 0\).