Jump to content

Palatini identity

From Wikipedia, the free encyclopedia

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]