# A ambiguous variation of Einstein-Hilbert action

1. May 19, 2010

### archipatelin

A ambiguous variation of Einstein--Hilbert action

Variation of EH action is:
$$\delta S_{EH}=\int_{\Omega}{\delta(R\sqrt{-g})dx^4}= \int_{\Omega}{G_{\mu\nu}\delta{g^{\mu\nu}}\sqrt{-g}dx^4}=0,$$​
where
$$G_{\mu\nu}:=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$$​
is symmetric einstein's tensor.

The action have to be extremal for each volume $$\Omega$$. This implicate
$$G_{\mu\nu}\delta{g^{\mu\nu}}=0.$$​
Becouse variation of metric is arbitrary and $$G_{\mu\nu}$$ is principal independed on $$\delta{g^{\mu\nu}}$$, latest equation is equivalent with
$$G_{\mu\nu}=0.$$​
This are ordinary einstein's vacuum equations.

But variation of metric is symmetric tensor, therefore more general form of vacuum
field equations are
$$S_{\mu\nu}:=G_{(\mu\nu)}+F_{[\mu\nu]}=0,$$​
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)?