Palatini identity

In general relativity and tensor calculus, the Palatini identity is

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

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

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

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

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

Proof

The Riemann curvature tensor is defined in terms of the Levi-Civita connection $\Gamma _{\mu \nu }^{\lambda }$ as

${R^{\rho }}_{\sigma \mu \nu }=\partial _{\mu }\Gamma _{\nu \sigma }^{\rho }-\partial _{\nu }\Gamma _{\mu \sigma }^{\rho }+\Gamma _{\mu \lambda }^{\rho }\Gamma _{\nu \sigma }^{\lambda }-\Gamma _{\nu \lambda }^{\rho }\Gamma _{\mu \sigma }^{\lambda }$.

Its variation is

$\delta {R^{\rho }}_{\sigma \mu \nu }=\partial _{\mu }\delta \Gamma _{\nu \sigma }^{\rho }-\partial _{\nu }\delta \Gamma _{\mu \sigma }^{\rho }+\delta \Gamma _{\mu \lambda }^{\rho }\Gamma _{\nu \sigma }^{\lambda }+\Gamma _{\mu \lambda }^{\rho }\delta \Gamma _{\nu \sigma }^{\lambda }-\delta \Gamma _{\nu \lambda }^{\rho }\Gamma _{\mu \sigma }^{\lambda }-\Gamma _{\nu \lambda }^{\rho }\delta \Gamma _{\mu \sigma }^{\lambda }$.

While the connection $\Gamma _{\nu \sigma }^{\rho }$ is not a tensor, the difference $\delta \Gamma _{\nu \sigma }^{\rho }$ between two connections is, so we can take its covariant derivative

$\nabla _{\mu }\delta \Gamma _{\nu \sigma }^{\rho }=\partial _{\mu }\delta \Gamma _{\nu \sigma }^{\rho }+\Gamma _{\mu \lambda }^{\rho }\delta \Gamma _{\nu \sigma }^{\lambda }-\Gamma _{\mu \nu }^{\lambda }\delta \Gamma _{\lambda \sigma }^{\rho }-\Gamma _{\mu \sigma }^{\lambda }\delta \Gamma _{\nu \lambda }^{\rho }$.

Solving this equation for $\partial _{\mu }\delta \Gamma _{\nu \sigma }^{\rho }$ and substituting the result in $\delta {R^{\rho }}_{\sigma \mu \nu }$, all the $\Gamma \delta \Gamma$-like terms cancel, leaving only

$\delta {R^{\rho }}_{\sigma \mu \nu }=\nabla _{\mu }\delta \Gamma _{\nu \sigma }^{\rho }-\nabla _{\nu }\delta \Gamma _{\mu \sigma }^{\rho }$.

Finally, the variation of the Ricci curvature tensor follows by contracting two indices, proving the identity

$\delta R_{\sigma \nu }=\delta {R^{\rho }}_{\sigma \rho \nu }=\nabla _{\rho }\delta \Gamma _{\nu \sigma }^{\rho }-\nabla _{\nu }\delta \Gamma _{\rho \sigma }^{\rho }$.

See also

• Einstein–Hilbert action
• Palatini variation
• Ricci calculus
• Tensor calculus
• Christoffel symbols
• Riemann curvature tensor

References

• Palatini, Attilio (1919), "Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton" [Invariant deduction of the gravitanional equations from the principle of Hamilton], Rendiconti del Circolo Matematico di Palermo, 1 (in Italian), 43: 203–212, doi:10.1007/BF03014670, S2CID 121043319 [English translation by R. Hojman and C. Mukku in P. G. Bergmann and V. De Sabbata (eds.) Cosmology and Gravitation, Plenum Press, New York (1980)]
• Tsamparlis, Michael (1978), "On the Palatini method of Variation", Journal of Mathematical Physics, 19 (3): 555–557, Bibcode:1978JMP....19..555T, doi:10.1063/1.523699