Components of metric perturbation in TT gauge

[fasterthanjoao]

Homework Statement

I'm just working through a textbook and there's a line in which I'm clearly missing something. What I want to do is show that from:
[tex]\bar{h^{TT}_{\mu \nu}} = A^{TT}_{\mu \nu} cos(\omega (t-z)) [\tex]<br /> to<br /> [tex]h^{TT}_{\mu \nu} = B^{TT}_{\mu \nu} cos(\omega (t-z)) [\tex]<br /> <br /> <br /> <h2>Homework Equations</h2><br /> <br /> [tex]\bar{h}_{\mu \nu} = h_{\mu \nu} - \frac{1}{2} \eta_{\mu \nu} h[\tex]<br /> [tex]h = h^{\alpha}_{\alpha} = \eta^{\alpha \beta} h_{\alpha \beta}[\tex]<br /> <br /> <br /> <h2>The Attempt at a Solution</h2><br /> <br /> There's just one line where I have an issue, i think. <br /> <br /> I've used the above equations to say:<br /> <br /> [tex]\bar{h} = \eta^{\mu \nu}\bar{h}_{\mu \nu}[\tex]<br /> then substituted the equation for [tex]\bar{h}_{\mu \nu}[\tex] to get:<br /> <br /> [tex]\bar{h} = h - \frac{1}{2} \eta^{\mu \nu} \eta_{\mu \nu} h[\tex]<br /> <br /> which is fine, but for it to work this last line must equal $$-h[\tex], I think and I'm not sure i see where that comes from.<br /> <br /> thanks.<br /> <br /> ftj$$[/tex][/tex][/tex][/tex][/tex][/tex][/tex]

 

[bigplanet401]

Your eta is the Minkowski metric, right?

eta_{alpha beta} eta^{mu nu} = delta_{alpha}^{mu} delta_{beta}^{nu}

Take it away.