(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)?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A ambiguous variation of Einstein-Hilbert action

**Physics Forums | Science Articles, Homework Help, Discussion**