Palatini identity

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

$\delta R_{{\mu \nu }}{}=(\delta \Gamma ^{{\lambda }}{}_{{\mu \nu }})_{{;\lambda }}-(\delta \Gamma ^{{\lambda }}{}_{{\mu \lambda }})_{{;\nu }}$

where $\delta \Gamma ^{{\lambda }}{}_{{\mu \nu }}$ denotes the variation of Christoffel symbols^[1] and semicolon ";" indicates covariant differentiation.

Proof can be found in the entry Einstein–Hilbert action.

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", Jour. 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).