Physics:Palatini identity

Short description: Variation of the Ricci tensor with respect to the metric.

In general relativity and tensor calculus, the Palatini identity is

$\delta R_{\sigma \nu }={\mathrm{\nabla }}_{\rho }\delta {\mathrm{\Gamma }}_{\nu \sigma }^{\rho }-{\mathrm{\nabla }}_{\nu }\delta {\mathrm{\Gamma }}_{\rho \sigma }^{\rho },$

where $\delta {\mathrm{\Gamma }}_{\nu \sigma }^{\rho }$ denotes the variation of Christoffel symbols and ${\mathrm{\nabla }}_{\rho }$ indicates covariant differentiation.

The "same" identity holds for the Lie derivative ${{ℒ}}_{\xi }{R}_{\sigma \nu }$. In fact, one has

${{ℒ}}_{\xi }{R}_{\sigma \nu }={\mathrm{\nabla }}_{\rho }({{ℒ}}_{\xi }{\mathrm{\Gamma }}_{\nu \sigma }^{\rho })-{\mathrm{\nabla }}_{\nu }({{ℒ}}_{\xi }{\mathrm{\Gamma }}_{\rho \sigma }^{\rho }),$

where $\xi ={\xi }^{\rho }{\partial }_{\rho }$ denotes any vector field on the spacetime manifold $M$.

Proof

The Riemann curvature tensor is defined in terms of the Levi-Civita connection ${\mathrm{\Gamma }}_{\mu \nu }^{\lambda }$ as

${R^{\rho }}_{\sigma \mu \nu }={\partial }_{\mu }{\mathrm{\Gamma }}_{\nu \sigma }^{\rho }-{\partial }_{\nu }{\mathrm{\Gamma }}_{\mu \sigma }^{\rho }+{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }{\mathrm{\Gamma }}_{\nu \sigma }^{\lambda }-{\mathrm{\Gamma }}_{\nu \lambda }^{\rho }{\mathrm{\Gamma }}_{\mu \sigma }^{\lambda }$.

Its variation is

$\delta {R^{\rho }}_{\sigma \mu \nu }={\partial }_{\mu }\delta {\mathrm{\Gamma }}_{\nu \sigma }^{\rho }-{\partial }_{\nu }\delta {\mathrm{\Gamma }}_{\mu \sigma }^{\rho }+\delta {\mathrm{\Gamma }}_{\mu \lambda }^{\rho }{\mathrm{\Gamma }}_{\nu \sigma }^{\lambda }+{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }\delta {\mathrm{\Gamma }}_{\nu \sigma }^{\lambda }-\delta {\mathrm{\Gamma }}_{\nu \lambda }^{\rho }{\mathrm{\Gamma }}_{\mu \sigma }^{\lambda }-{\mathrm{\Gamma }}_{\nu \lambda }^{\rho }\delta {\mathrm{\Gamma }}_{\mu \sigma }^{\lambda }$.

While the connection ${\mathrm{\Gamma }}_{\nu \sigma }^{\rho }$ is not a tensor, the difference $\delta {\mathrm{\Gamma }}_{\nu \sigma }^{\rho }$ between two connections is, so we can take its covariant derivative

${\mathrm{\nabla }}_{\mu }\delta {\mathrm{\Gamma }}_{\nu \sigma }^{\rho }={\partial }_{\mu }\delta {\mathrm{\Gamma }}_{\nu \sigma }^{\rho }+{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }\delta {\mathrm{\Gamma }}_{\nu \sigma }^{\lambda }-{\mathrm{\Gamma }}_{\mu \nu }^{\lambda }\delta {\mathrm{\Gamma }}_{\lambda \sigma }^{\rho }-{\mathrm{\Gamma }}_{\mu \sigma }^{\lambda }\delta {\mathrm{\Gamma }}_{\nu \lambda }^{\rho }$.

Solving this equation for ${\partial }_{\mu }\delta {\mathrm{\Gamma }}_{\nu \sigma }^{\rho }$ and substituting the result in $\delta {R^{\rho }}_{\sigma \mu \nu }$, all the $\mathrm{\Gamma }\delta \mathrm{\Gamma }$-like terms cancel, leaving only

$\delta {R^{\rho }}_{\sigma \mu \nu }={\mathrm{\nabla }}_{\mu }\delta {\mathrm{\Gamma }}_{\nu \sigma }^{\rho }-{\mathrm{\nabla }}_{\nu }\delta {\mathrm{\Gamma }}_{\mu \sigma }^{\rho }$.

Finally, the variation of the Ricci curvature tensor follows by contracting two indices, proving the identity

$\delta R_{\sigma \nu }=\delta {R^{\rho }}_{\sigma \rho \nu }={\mathrm{\nabla }}_{\rho }\delta {\mathrm{\Gamma }}_{\nu \sigma }^{\rho }-{\mathrm{\nabla }}_{\nu }\delta {\mathrm{\Gamma }}_{\rho \sigma }^{\rho }$.

See also

• Einstein–Hilbert action
• Palatini variation
• Ricci calculus
• Tensor calculus
• Christoffel symbols
• Riemann curvature tensor

References

• "Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton" (in Italian), Rendiconti del Circolo Matematico di Palermo, 1 43: 203–212, 1919, doi:10.1007/BF03014670, https://link.springer.com/article/10.1007/BF03014670  [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)]
• "On the Palatini method of Variation", Journal of Mathematical Physics 19 (3): 555–557, 1978, doi:10.1063/1.523699, Bibcode: 1978JMP....19..555T, https://aip.scitation.org/doi/10.1063/1.523699