# I Help with derivation of linearized Einstein field equations

1. Oct 25, 2016

### epovo

Hi all -
I am trying to follow a derivation of the above. At some point I need to find gαβ for
gαβ = ηαβ + hαβ
with |hαβ|<<1
I am stuck. The text says
gαβ = ηαβ - hαβ
but I cannot figure out why. Can anybody help?

2. Oct 25, 2016

### 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?

3. Oct 25, 2016

### 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?

4. Oct 25, 2016

### epovo

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

5. Oct 25, 2016

### 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.

6. Oct 26, 2016

### epovo

Thank you guys!