In general relativity and tensor calculus, the Palatini identity is

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 .

Proof edit

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 edit

Notes edit

  1. ^ Christoffel, E.B. (1869), "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades", Journal für die reine und angewandte Mathematik, B. 70: 46–70

References edit