Symmetries and Ricci Tensor

• By definition, $R_{\sigma \mu \nu }^{\rho }=-R_{\sigma \mu \nu }^{\rho }$
• $R_{\rho \sigma \mu \nu }=-R_{\sigma \rho \mu \nu }$
• $R_{\rho \sigma \mu \nu }=R_{\mu \nu \rho \sigma }$
• $R_{\rho [\sigma \mu \nu ]}=R_{\rho \sigma \mu \nu }+R_{\rho \mu \nu \sigma }+R_{\rho \nu \sigma \mu }=0$
• Bianchi Identity ${\nabla }_{[\lambda }{R}_{\rho \sigma ]\mu \nu }=0$
• Ricci Tensor $R_{\mu \nu }=R_{\mu \lambda \nu }^{\lambda }=R_{\nu \mu }$
• Ricci Scalar $R=g^{\mu \nu }{R}_{\mu \nu }$
• ContractedBianchi Identity ${\nabla }_{\mu }{R}_{\rho }^{\mu }=\frac{1}{2}{\nabla }_{\rho }R$
  • DefiningEinstein-Tensor $G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R$
  • ${\nabla }^{\mu }{G}_{\mu \nu }=0$