Ingeneral relativity and tensor calculus, the Palatini identityis
where denotes the variation of Christoffel symbols and
indicates covariant differentiation.[1]
The "same" identity holds for the Lie derivative . In fact, one has
where denotes any vector field on the spacetime manifold
.
The Riemann curvature tensor is defined in terms of the Levi-Civita connection as
Its 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