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]

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 ${\displaystyle M}$.

Proof

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

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

Its variation is

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

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

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

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

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

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

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

See also

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

Notes

1. Christoffel, E.B. (1869), "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades", Journal für die reine und angewandte Mathematik, B. 70: 46–70

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