# Weak gravity approximation gauge invariance

Hey guys,

So I have a question about the gauge invariance of the weak field approximation. So if I write the approximation as

$\Box h^{\mu\nu} -\partial_{\alpha}(\partial^{\mu}h^{\nu\alpha}+\partial^{\nu}h^{\mu\alpha})+\partial^{\mu}\partial^{\nu}h=0$

then this is invariant under the gauge transformation

$\delta h^{\mu\nu}=\partial^{\mu}\epsilon^{\nu}+\partial^{\nu}\epsilon^{\mu}+\mathcal{O}(\epsilon, h)$

if you ignore the correction terms. So my question is...how does this variation come about? I mean how would I calculate this variation from first principles, using $g^{\mu\nu}(x)=\eta^{\mu\nu}+h^{\mu\nu}(x)$?

I looked at wikipedia and I didnt understand a word...so can someone please offer a simplified explanation of how to achieve this expression?

Thanks guys!

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