Palatini Identity

In general relativity and tensor calculus, the Palatini identity is

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

where $\delta \Gamma^\rho_$ denotes the variation of Christoffel symbols and $\nabla_\rho$ indicates covariant differentiation.

The "same" identity holds for the Lie derivative $\mathcal_ R_$. In fact, one has

:$\mathcal_ R_ = \nabla_\rho (\mathcal_ \Gamma^\rho_) - \nabla_\nu (\mathcal_ \Gamma^\rho_),$

where $\xi = \xi^\partial_$ denotes any vector field on the spacetime manifold $M$.

Proof

The Riemann curvature tensor is defined in terms of the Levi-Civita connection $\Gamma^\lambda_$ as

: $_ = \partial_\mu\Gamma^\rho_ - \partial_\nu\Gamma^\rho_ + \Gamma^\rho_ \Gamma^\lambda_ - \Gamma^\rho_\Gamma^\lambda_$.

Its variation is

: $\delta_ = \partial_\mu \delta\Gamma^\rho_ - \partial_\nu \delta\Gamma^\rho_ + \delta\Gamma^\rho_ \Gamma^\lambda_ + \Gamma^\rho_ \delta\Gamma^\lambda_ - \delta\Gamma^\rho_ \Gamma^\lambda_ - \Gamma^\rho_ \delta\Gamma^\lambda_$.

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

: $\nabla_\mu \delta \Gamma^\rho_ = \partial_\mu \delta \Gamma^\rho_ + \Gamma^\rho_ \delta \Gamma^\lambda_ - \Gamma^\lambda_ \delta \Gamma^\rho_ - \Gamma^\lambda_ \delta \Gamma^\rho_$.

Solving this equation for $\partial_\mu \delta \Gamma^\rho_$ and substituting the result in $\delta_$, all the $\Gamma \delta \Gamma$-like terms cancel, leaving only

: $\delta_ = \nabla_\mu \delta\Gamma^\rho_ - \nabla_\nu \delta\Gamma^\rho_$.

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

: $\delta R_ = \delta _ = \nabla_\rho \delta \Gamma^\rho_ - \nabla_\nu \delta \Gamma^\rho_$.

See also

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

Notes

References

* P. G. Bergmann and V. De Sabbata (eds.) Cosmology and Gravitation, Plenum Press, New York (1980)">Peter_Bergmann.html" ;"title="nglish 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)*{{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=
, url-access=subscription

Equations of physics
Tensors
General relativity