Palatini identity

{{short description|Variation of the Ricci tensor with respect to the metric.}}

In general relativity and tensor calculus, the Palatini identity is

: $\delta R_{\sigma\nu} = \nabla_\rho \delta \Gamma^\rho_{\nu\sigma} - \nabla_\nu \delta \Gamma^\rho_{\rho\sigma},$

where $\delta \Gamma^\rho_{\nu\sigma}$ denotes the variation of Christoffel symbols and $\nabla_\rho$ indicates covariant differentiation.{{citation|title=Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades|last=Christoffel|first=E.B.|author-link=Elwin Bruno Christoffel|journal=Journal für die reine und angewandte Mathematik|volume=B. 70|pages=46–70|year=1869|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002153882&IDDOC=266356}}

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

: $\mathcal{L}_{\xi} R_{\sigma\nu} = \nabla_\rho (\mathcal{L}_{\xi} \Gamma^\rho_{\nu\sigma}) - \nabla_\nu (\mathcal{L}_{\xi} \Gamma^\rho_{\rho\sigma}),$

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 $\Gamma^\lambda_{\mu\nu}$ as

: ${R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}$ .

Its variation is

: $\delta{R^\rho}_{\sigma\mu\nu} =
\partial_\mu \delta\Gamma^\rho_{\nu\sigma} - \partial_\nu \delta\Gamma^\rho_{\mu\sigma} + \delta\Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} + \Gamma^\rho_{\mu\lambda} \delta\Gamma^\lambda_{\nu\sigma} - \delta\Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma} - \Gamma^\rho_{\nu\lambda} \delta\Gamma^\lambda_{\mu\sigma}$ .

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

: $\nabla_\mu \delta \Gamma^\rho_{\nu\sigma} =
\partial_\mu \delta \Gamma^\rho_{\nu\sigma} + \Gamma^\rho_{\mu\lambda} \delta \Gamma^\lambda_{\nu\sigma} - \Gamma^\lambda_{\mu\nu} \delta \Gamma^\rho_{\lambda\sigma} - \Gamma^\lambda_{\mu\sigma} \delta \Gamma^\rho_{\nu\lambda}$ .

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

: $\delta{R^\rho}_{\sigma\mu\nu} =
\nabla_\mu \delta\Gamma^\rho_{\nu\sigma} - \nabla_\nu \delta\Gamma^\rho_{\mu\sigma}$ .

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

: $\delta R_{\sigma\nu} = \delta {R^\rho}_{\sigma\rho\nu} =
\nabla_\rho \delta \Gamma^\rho_{\nu\sigma} - \nabla_\nu \delta \Gamma^\rho_{\rho\sigma}$ .

Notes

References

{{citation

|author-first=Attilio

|author-last=Palatini

|author-link =Attilio Palatini

|year=1919

|title=Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton

|trans-title=Invariant deduction of the gravitanional equations from the principle of Hamilton

|journal=Rendiconti del Circolo Matematico di Palermo

|series=1

|language=Italian

|volume=43

|issue=

|pages=203–212

|url=https://link.springer.com/article/10.1007/BF03014670

|doi=10.1007/BF03014670

|jfm=

|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)]

{{citation

|author-first=Michael

|author-last=Tsamparlis

|year=1978

|title=On the Palatini method of Variation

|journal=Journal of Mathematical Physics

|series=

|volume=19

|issue=3

|pages=555–557

|url=https://aip.scitation.org/doi/10.1063/1.523699

|doi=10.1063/1.523699

|bibcode = 1978JMP....19..555T

|jfm=

}}

Category:Equations of physics

Category:Tensors

Category:General relativity

Palatini identity

Proof

See also

Notes

References