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 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
.
See also
Einstein–Hilbert action
Palatini variation
Ricci calculus
Tensor calculus
Christoffel symbols
Riemann curvature tensor
Notes
References
[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)]