Palatini identity

In general relativity and tensor calculus, the Palatini identity is:

${\displaystyle \delta R_{\sigma \nu }=\nabla _{\rho }(\delta \Gamma _{\nu \sigma }^{\rho })-\nabla _{\nu }(\delta \Gamma _{\rho \sigma }^{\rho }),}$

where ${\displaystyle \delta \Gamma _{\nu \sigma }^{\rho }}$ denotes the variation of Christoffel symbols and ${\displaystyle \nabla _{\rho }}$ indicates covariant differentiation.[1]

A proof can be found in the entry Einstein–Hilbert action.

The "same" identity holds for the Lie derivative ${\displaystyle {\mathcal {L}}_{\xi }R_{\sigma \nu }}$. In fact, one has:

${\displaystyle {\mathcal {L}}_{\xi }R_{\sigma \nu }=\nabla _{\rho }({\mathcal {L}}_{\xi }\Gamma _{\nu \sigma }^{\rho })-\nabla _{\nu }({\mathcal {L}}_{\xi }\Gamma _{\rho \sigma }^{\rho }),}$

where ${\displaystyle \xi =\xi ^{\rho }\partial _{\rho }}$ denotes any vector field on the spacetime manifold ${\displaystyle M}$.