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