0

I have a term of the form

$R_{\mu \nu \rho \sigma}R^{\mu \nu}_{\lambda \kappa} (g^{\rho \sigma}g^{\lambda \kappa} -g^{\rho \lambda}g^{\sigma \kappa})$ which I would like to simplify, but I am obviously doing something completely wrong. My attempt:

$R_{\mu \nu \rho \sigma}R^{\mu \nu}_{\lambda \kappa}g^{\rho \sigma}g^{\lambda \kappa} =R_{\mu \nu \rho \sigma} g^{\rho \sigma}R^{\mu \nu}_{\lambda \kappa}g^{\lambda \kappa}=R_{\mu \nu \sigma}^{\sigma}R^{\mu \nu \kappa}_{\kappa}=R_{\mu \nu}R^{\mu \nu}$

Also

$-R_{\mu \nu \rho \sigma}R^{\mu \nu}_{\lambda \kappa} g^{\rho \lambda}g^{\sigma \kappa}=- R_{\mu \nu \rho \sigma}R_{\kappa}^{\mu \nu \rho}g^{\sigma\kappa}=- R_{\mu \nu \rho \sigma}R^{\mu \nu \rho \sigma}$.

According to the text I am using, the first term vanishes because of antisymmetry of $R_{\mu \nu \rho \sigma}$ in the last two indices?

Henrymerrild
  • 377

1 Answers1

2

Be careful with the placement of the indices when you raise and lower indices of nonsymmetric tensors.

It should be $R_{\mu\nu\rho\sigma} g^{\rho\sigma} = R_{\mu\nu}{}^{\sigma}{}_{\sigma},$ which does not become the Ricci tensor, since that is given by contraction on the first and third indices, not the third and fourth indices. Instead it becomes zero because of the skew symmetry: $$ R_{\mu\nu\rho\sigma} g^{\rho\sigma} = \{ \text{ skew symmetry of $R$ and symmetry of $g$ } \} \\ = (-R_{\mu\nu\sigma\rho}) (g^{\sigma\rho}) = \{ \text{ swap $\rho$ and $\sigma$ as dummy indices} \} \\ = -R_{\mu\nu\rho\sigma} g^{\rho\sigma} $$ which implies $R_{\mu\nu\rho\sigma} g^{\rho\sigma} = 0.$

md2perpe
  • 26,770