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 ${\displaystyle M}$.

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 [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