(adsbygoogle = window.adsbygoogle || []).push({}); A ambiguous variation of Einstein--Hilbert action

Variation of EH action is:

[tex]where

\delta S_{EH}=\int_{\Omega}{\delta(R\sqrt{-g})dx^4}=

\int_{\Omega}{G_{\mu\nu}\delta{g^{\mu\nu}}\sqrt{-g}dx^4}=0,

[/tex]

[tex]is symmetric einstein's tensor.

G_{\mu\nu}:=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R

[/tex]

The action have to be extremal for each volume [tex]\Omega[/tex]. This implicate

[tex]Becouse variation of metric is arbitrary and [tex]G_{\mu\nu}[/tex] is principal independed on [tex]\delta{g^{\mu\nu}}[/tex], latest equation is equivalent with

G_{\mu\nu}\delta{g^{\mu\nu}}=0.

[/tex]

[tex]This are ordinary einstein's vacuum equations.

G_{\mu\nu}=0.

[/tex]

But variation of metric is symmetric tensor, therefore more general form of vacuum

field equations are

[tex]where [tex]F_{\mu\nu}[/tex] is whatever antisymmetric tensor build it from metric and its derivations.

S_{\mu\nu}:=G_{(\mu\nu)}+F_{[\mu\nu]}=0,

[/tex]

Why we can ignore this tensor (it is proven that there is not exist)?

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# A ambiguous variation of Einstein-Hilbert action

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