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Palatini identity

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Variation of the Ricci tensor with respect to the metric.

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

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

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