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Palatini identity

Variation of the Ricci tensor with respect to the metric. From Wikipedia, the free encyclopedia

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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 .

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Proof

Summarize
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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

.
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See also

Notes

References

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