Help with Derivation of Linearized Einstein Field Eqns

[epovo]

Hi all -
I am trying to follow a derivation of the above. At some point I need to find g^αβ for
g_αβ = η_αβ + h_αβ
with |h_αβ|<<1
I am stuck. The text says
g^αβ = η^αβ - h^αβ
but I cannot figure out why. Can anybody help?

 

[haushofer]

Well, you want both expressions to be each-other inverses, up to linear order in the metric perturbation h. So what do you get?

 

[Matterwave]

Take $g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}$ and $g^{\alpha\beta}=\eta^{\alpha\beta}-h^{\alpha\beta}$ and multiply them together and what do you get?

 

[epovo]

I get an identity as long as $h_{\alpha\beta} h^{\alpha\beta}$ is <<1

 

[dextercioby]

Voila!. The quadratic term is discarded. Depending on the source, $h_{\alpha\beta}$ is called the Pauli-Fierz field. It was discovered by Pauli and Fierz as far back as 1939 that the only Lagrange action (hence field equations) describing a spin 2 field is necessarily the linearized Hilbert-Einstein action.

 

[epovo]

Thank you guys!