Palatini identity explained

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.l{L} _{\xi} R _{\sigma\nu} . In fact, one hasl{L} _{\xi} R _{\sigma\nu} = \nabla _{\rho} ( l{L} _{\xi} \Gamma ^{\rho} _{\nu\sigma} ) - \nabla _{\nu} ( l{L} _{\xi} \Gamma ^{\rho} _{\rho\sigma} ),

where

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

Proof

\Gamma ^{λ} _{\mu\nu} as{R ^{\rho}} _{\sigma\mu\nu} = \partial ^{\rho} _{\nu\sigma} - \partial ^{\rho} _{\mu\sigma} + \Gamma ^{\rho} _{\mu λ} \Gamma ^{λ} _{\nu\sigma} - \Gamma ^{\rho} _{\nu λ} \Gamma ^{λ} _{\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 λ} \Gamma ^{λ} _{\nu\sigma} + \Gamma ^{\rho} _{\mu λ} \delta\Gamma ^{λ} _{\nu\sigma} - \delta\Gamma ^{\rho} _{\nu λ} \Gamma ^{λ} _{\mu\sigma} - \Gamma ^{\rho} _{\nu λ} \delta\Gamma ^{λ} _{\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 λ} \delta \Gamma ^{λ} _{\nu\sigma} - \Gamma ^{λ} _{\mu\nu} \delta \Gamma ^{\rho} _{λ \sigma} - \Gamma ^{λ} _{\mu\sigma} \delta \Gamma ^{\rho} _{\nu λ} .

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} .

See also

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

References

• [English translation by R. Hojman and C. Mukku in [[Peter Bergmann|P. G. Bergmann]] and V. De Sabbata (eds.) Cosmology and Gravitation, Plenum Press, New York (1980)]