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in my book, it says:

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Beacause of [tex]T^{\mu\nu}{}{}_{;\nu} = 0[/tex] and the symmetry of [tex]T^{\mu\nu}[/tex], it holds that

[tex]\left(T^{\mu\nu}\xi_\mu\right)_{;\nu} = 0[/tex]

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(here, [tex]T^{\mu\nu}[/tex] ist the energy momentum tensor and [tex]\xi_\mu[/tex] a killing vector. The semicolon indicates the covariant derivative, i.e. [tex]()_{;}[/tex] is the generalized divergence)

I don't understand, why from

" [tex]T^{\mu\nu}{}{}_{;\nu} = 0[/tex] and the symmetry of [tex]T^{\mu\nu}[/tex] "

it follows, that

[tex]\left(T^{\mu\nu}\xi_\mu\right)_{;\nu} = 0[/tex]

must hold.

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derivator

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# Divergence of product of killing vector and energy momentum tensor vanishes. Why?

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