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]
to
[tex] h^{TT}_{\mu \nu} = B^{TT}_{\mu \nu} cos(\omega (t-z)) [\tex]

Homework Equations

[tex]\bar{h}_{\mu \nu} = h_{\mu \nu} - \frac{1}{2} \eta_{\mu \nu} h[\tex]
[tex]h = h^{\alpha}_{\alpha} = \eta^{\alpha \beta} h_{\alpha \beta}[\tex]

The Attempt at a Solution

There's just one line where I have an issue, i think.

I've used the above equations to say:

[tex] \bar{h} = \eta^{\mu \nu}\bar{h}_{\mu \nu}[\tex]
then substituted the equation for [tex]\bar{h}_{\mu \nu}[\tex] to get:

[tex] \bar{h} = h - \frac{1}{2} \eta^{\mu \nu} \eta_{\mu \nu} h[\tex]

which is fine, but for it to work this last line must equal [tex]-h[\tex], I think and I'm not sure i see where that comes from.

thanks.

ftj

 

[bigplanet401]

Your eta is the Minkowski metric, right?

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

Take it away.