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A ambiguous variation of Einstein-Hilbert action

  1. May 19, 2010 #1
    A ambiguous variation of Einstein--Hilbert action

    Variation of EH action is:
    [tex]
    \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]​
    where
    [tex]
    G_{\mu\nu}:=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R
    [/tex]​
    is symmetric einstein's tensor.

    The action have to be extremal for each volume [tex]\Omega[/tex]. This implicate
    [tex]
    G_{\mu\nu}\delta{g^{\mu\nu}}=0.
    [/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
    [tex]
    G_{\mu\nu}=0.
    [/tex]​
    This are ordinary einstein's vacuum equations.

    But variation of metric is symmetric tensor, therefore more general form of vacuum
    field equations are
    [tex]
    S_{\mu\nu}:=G_{(\mu\nu)}+F_{[\mu\nu]}=0,
    [/tex]​
    where $F_{\mu\nu}$ is whatever antisymmetric tensor build it from metric and its derivations.

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