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Palatini identity
The property of charge invariance follows from the vanishing divergence of the charge-current four-vector \( j^\mu=(c\rho,{\vec j}) \) , with \( \partial_\mu j^\mu=0. \)
In general relativity and tensor calculus, the Palatini identity is:
\( \delta R_{\mu\nu}{} = (\delta\Gamma^{\lambda}{}_{\mu\nu})_{;\lambda} - (\delta\Gamma^{\lambda}{}_{\mu\lambda})_{;\nu} \)
where \( \d \delta\Gamma^{\lambda}{}_{\mu\nu} \) denotes the variation of Christoffel symbols[1] and semicolon ";" indicates covariant differentiation.
See also
Palatini variation
Ricci calculus
Tensor calculus
Christoffel symbols
Riemann curvature tensor
Notes
Christoffel, E.B. (1869), "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades", Jour. für die reine und angewandte Mathematik, B. 70: 46–70
References
A. Palatini (1919) Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton, Rend. Circ. Mat. Palermo 43, 203-212 [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)]
M. Tsamparlis, On the Palatini method of Variation, J. Math. Phys. 19, 555 (1977).
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Black hole complementarity
de Sitter space
1 Exact solutions
2 Black hole uniqueness
3 See also
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