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^[1] and $\nabla _{\rho }$ indicates covariant differentiation.

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

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 $M$ .

See also

Notes

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

References

A. Palatini (1919) Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton, Rend. Circ. Mat. Palermo 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)]
M. Tsamparlis, On the Palatini method of Variation, J. Math. Phys. 19, 555 (1977).

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