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A topic came up in a lecture the other day about how if certain simplifications are made, then the Einstein equation reduces to a form of the wave equation.

When I look at derivations of how this happens, I get a little confused as to how this happens.

I think I'm posting it in the right place putting it here, as it's not strictly a homework question, since it is only to help my understanding of the course so far...

Looking at the website http://iopscience.iop.org/1367-2630/7/1/204/fulltext/#nj192710s2

I don't understand why you are allowed to take the step made in Equation (2.9) - i.e. why does putting a prime on the LHS simply mean that you can go ahead and "prime" all of the h in the RHS? If this is allowed, then I can see how the rest of that equality works, no problem.

Secondly, taking a step or two back, where it says substituting "h-bar" into (2.6) "and expanding", how does this simplify down to (2.7)? Because I can see how the terms where both indices are "downstairs" are changed, but what about the h's in the form of a (1,1) tensor? That is one upstairs and one downstairs? I tried applying the Minkowski metric to the (1,1) h, to try and get into "downstairs format" for me to work with, but I was left with some delta's.. Specifically (apologies in advance for appalling Latex):

[itex]h^{c}_{a,bc}=\eta^{da}\bar{h}_{ab,bc}-\frac{1}{2}\eta_{ab}\eta^{ad}\bar{h}_{,bc}=\bar{h}^{d}_{b,bc}-\frac{1}{2}\delta\stackrel{d}{b}\bar{h}_{,bc}[/itex]

and I don't understand how to move on from here?

Any observations would be greatly appreciated.

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# Linearised Gravitational Waves Derivation

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