Palatini identity
In general relativity and tensor calculus, the Palatini identity is
where denotes the variation of Christoffel symbols and indicates covariant differentiation.
The "same" identity holds for the Lie derivative. In fact, one has
where denotes any vector field on the spacetime manifold.
Proof
The Riemann curvature tensor is defined in terms of the Levi-Civita connection asIts variation is
While the connection is not a tensor, the difference between two connections is, so we can take its covariant derivative
Solving this equation for and substituting the result in, all the -like terms cancel, leaving only
Finally, the variation of the Ricci curvature tensor follows by contracting two indices, proving the identity