In general relativity and tensor calculus, the Palatini identity is
δ R σ ν = ∇ ρ δ Γ ν σ ρ − ∇ ν δ Γ ρ σ ρ , {\displaystyle \delta R_{\sigma \nu }=\nabla _{\rho }\delta \Gamma _{\nu \sigma }^{\rho }-\nabla _{\nu }\delta \Gamma _{\rho \sigma }^{\rho },}
where δ Γ ν σ ρ {\displaystyle \delta \Gamma _{\nu \sigma }^{\rho }} denotes the variation of Christoffel symbols and ∇ ρ {\displaystyle \nabla _{\rho }} indicates covariant differentiation. The "same" identity holds for the Lie derivative L ξ R σ ν {\displaystyle {\mathcal {L}}_{\xi }R_{\sigma \nu }} . In fact, one has
L ξ R σ ν = ∇ ρ ( L ξ Γ ν σ ρ ) − ∇ ν ( L ξ Γ ρ σ ρ ) , {\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 ξ = ξ ρ ∂ ρ {\displaystyle \xi =\xi ^{\rho }\partial _{\rho }} denotes any vector field on the spacetime manifold M {\displaystyle M} .
Proof The Riemann curvature tensor is defined in terms of the Levi-Civita connection Γ μ ν λ {\displaystyle \Gamma _{\mu \nu }^{\lambda }} as
R ρ σ μ ν = ∂ μ Γ ν σ ρ − ∂ ν Γ μ σ ρ + Γ μ λ ρ Γ ν σ λ − Γ ν λ ρ Γ μ σ λ {\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
δ R ρ σ μ ν = ∂ μ δ Γ ν σ ρ − ∂ ν δ Γ μ σ ρ + δ Γ μ λ ρ Γ ν σ λ + Γ μ λ ρ δ Γ ν σ λ − δ Γ ν λ ρ Γ μ σ λ − Γ ν λ ρ δ Γ μ σ λ {\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 Γ ν σ ρ {\displaystyle \Gamma _{\nu \sigma }^{\rho }} is not a tensor, the difference δ Γ ν σ ρ {\displaystyle \delta \Gamma _{\nu \sigma }^{\rho }} between two connections is, so we can take its covariant derivative
∇ μ δ Γ ν σ ρ = ∂ μ δ Γ ν σ ρ + Γ μ λ ρ δ Γ ν σ λ − Γ μ ν λ δ Γ λ σ ρ − Γ μ σ λ δ Γ ν λ ρ {\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 ∂ μ δ Γ ν σ ρ {\displaystyle \partial _{\mu }\delta \Gamma _{\nu \sigma }^{\rho }} and substituting the result in δ R ρ σ μ ν {\displaystyle \delta {R^{\rho }}_{\sigma \mu \nu }} , all the Γ δ Γ {\displaystyle \Gamma \delta \Gamma } -like terms cancel, leaving only
δ R ρ σ μ ν = ∇ μ δ Γ ν σ ρ − ∇ ν δ Γ μ σ ρ {\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
δ R σ ν = δ R ρ σ ρ ν = ∇ ρ δ Γ ν σ ρ − ∇ ν δ Γ ρ σ ρ {\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
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