What is varing in the variational principle of GR

[spocchio]

Consider the variational principle used to obtain that in the vacuum the Einstein tensor vanish.
So we set the lagrangian density as $L(g,\partial g)=R$
and asks for the condition
$0 = \delta S =\delta\int{d^4 x \sqrt{-g}L}$

proceeding with the calculus I finally have to vary R such that
$\delta R = \delta{g^{\mu\nu}R_{\mu\nu}}$
but what is $\delta g$??

i think (but not sure) $\delta{g^{\mu\nu}}=g^{\mu\nu}-{g'}^{\mu\nu}$
since g transform as a tensor
$\delta{g^{\mu\nu}}=g^{\mu\nu}-J_{\rho}^{\mu} J_{\sigma}^{\nu} {g}^{\rho\sigma}$

where J is the jacobian of the transformation..am I right?

 

[Bill_K]

Usually the independent variables to be varied are taken as g_μν. The equations you need are:

δg^μν = - g^μσ g^ντ δg_στ

δ√-g = ½√-g g^μν δg_μν

δR = g^μν δR_μν - R^μν δg_μν

δR_μν = ½ g^στ(g_μν;στ + g_στ;μν - g_μσ;ντ - g_ντ;μσ)