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in Linearized theory we know that :

g_{αβ}= η_{αβ}+ h_{αβ}

If I multiply out both terms by η^{θα}η^{λβ}, wouldn't one get :

g^{θλ}= η^{θλ}+ h^{θλ}?EQ1

But we already know g^{θλ}= η^{θλ}- h^{θλ}+ O(h^{2})EQ2

How can we reconciliate EQ1 and EQ2 ? Was it an error to have raised the perturbation h with the η ?

Can we also say that g^{θλ}= g_{θλ}?

I'm a bit confused.

Thanks,

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# Raising and lowering indices in Linearized theory

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