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:
\bar{h^{TT}_{\mu \nu}} = A^{TT}_{\mu \nu} cos(\omega (t-z)) [\tex]<br /> to<br /> h^{TT}_{\mu \nu} = B^{TT}_{\mu \nu} cos(\omega (t-z)) [\tex]&lt;br /&gt; &lt;br /&gt; &lt;br /&gt; &lt;h2&gt;Homework Equations&lt;/h2&gt;&lt;br /&gt; &lt;br /&gt; \bar{h}_{\mu \nu} = h_{\mu \nu} - \frac{1}{2} \eta_{\mu \nu} h[\tex]&amp;lt;br /&amp;gt; h = h^{\alpha}_{\alpha} = \eta^{\alpha \beta} h_{\alpha \beta}[\tex]&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;h2&amp;amp;gt;The Attempt at a Solution&amp;amp;lt;/h2&amp;amp;gt;&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; There&amp;amp;amp;#039;s just one line where I have an issue, i think. &amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; I&amp;amp;amp;#039;ve used the above equations to say:&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; \bar{h} = \eta^{\mu \nu}\bar{h}_{\mu \nu}[\tex]&amp;amp;amp;lt;br /&amp;amp;amp;gt; then substituted the equation for \bar{h}_{\mu \nu}[\tex] to get:&amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; \bar{h} = h - \frac{1}{2} \eta^{\mu \nu} \eta_{\mu \nu} h[\tex]&amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;gt; which is fine, but for it to work this last line must equal -h[\tex], I think and I&amp;amp;amp;amp;amp;amp;amp;#039;m not sure i see where that comes from.&amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;gt; thanks.&amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;gt; ftj

 

[bigplanet401]

Your eta is the Minkowski metric, right?

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

Take it away.