spocchio
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Consider the variational principle used to obtain that in the vacuum the Einstein tensor vanish.
So we set the lagrangian density as [itex]L(g,\partial g)=R[/itex]
and asks for the condition
[itex]0 = \delta S =\delta\int{d^4 x \sqrt{-g}L}[/itex]
proceeding with the calculus I finally have to vary R such that
[itex]\delta R = \delta{g^{\mu\nu}R_{\mu\nu}}[/itex]
but what is [itex]\delta g[/itex]??
i think (but not sure) [itex]\delta{g^{\mu\nu}}=g^{\mu\nu}-{g'}^{\mu\nu}[/itex]
since g transform as a tensor
[itex]\delta{g^{\mu\nu}}=g^{\mu\nu}-J_{\rho}^{\mu} J_{\sigma}^{\nu} {g}^{\rho\sigma}[/itex]
where J is the jacobian of the transformation..am I right?
So we set the lagrangian density as [itex]L(g,\partial g)=R[/itex]
and asks for the condition
[itex]0 = \delta S =\delta\int{d^4 x \sqrt{-g}L}[/itex]
proceeding with the calculus I finally have to vary R such that
[itex]\delta R = \delta{g^{\mu\nu}R_{\mu\nu}}[/itex]
but what is [itex]\delta g[/itex]??
i think (but not sure) [itex]\delta{g^{\mu\nu}}=g^{\mu\nu}-{g'}^{\mu\nu}[/itex]
since g transform as a tensor
[itex]\delta{g^{\mu\nu}}=g^{\mu\nu}-J_{\rho}^{\mu} J_{\sigma}^{\nu} {g}^{\rho\sigma}[/itex]
where J is the jacobian of the transformation..am I right?