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 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

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).