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Variation of the Ricci tensor with respect to the metric.
In
general relativity and
tensor calculus, the
Palatini identity is
![{\displaystyle \delta R_{\sigma \nu }=\nabla _{\rho }\delta \Gamma _{\nu \sigma }^{\rho }-\nabla _{\nu }\delta \Gamma _{\rho \sigma }^{\rho },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f693a20c2373be8315b17628d827d9775b97630c)
where
denotes the variation of
Christoffel symbols and
indicates
covariant differentiation.
[1]
The "same" identity holds for the
Lie derivative
. In fact, one has
![{\displaystyle {\mathcal {L}}_{\xi }R_{\sigma \nu }=\nabla _{\rho }({\mathcal {L}}_{\xi }\Gamma _{\nu \sigma }^{\rho })-\nabla _{\nu }({\mathcal {L}}_{\xi }\Gamma _{\rho \sigma }^{\rho }),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad2138c27b21c568671f6a90f85e9edf3e08f3c9)
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
Notes
References
-
Palatini, Attilio (1919),
"Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton" [Invariant deduction of the gravitanional equations from the principle of Hamilton], Rendiconti del Circolo Matematico di Palermo, 1 (in Italian), 43: 203–212,
doi:
10.1007/BF03014670,
S2CID
121043319 [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)]
- Tsamparlis, Michael (1978),
"On the Palatini method of Variation", Journal of Mathematical Physics, 19 (3): 555–557,
Bibcode:
1978JMP....19..555T,
doi:
10.1063/1.523699