Linearized Gravity and the Transverse-Traceless Gauge Conditions

[Alexrey]

Homework Statement

I'm working on some things to do with linearized gravitational radiation and I'm trying to justify the claim that in the Lorenz gauge, where $$\partial_{\nu}\bar{h}^{\mu\nu}=0 (1.1),$$ we are able to impose the additional conditions $$A_{\alpha}^{\alpha}=0 (1.2)$$ and $$A_{\alpha\beta}u^{\beta}=0 (1.3)$$ in order to find the two physical polarization states of a gravitational wave. All of the books that I have looked at so far have just stated that we are able to impose (1.1) and (1.2) without any workings of how they achieved this claim.

Homework Equations

Equations (1.1), (1.2), (1.3) as well as the vacuum Einstein field equation $$\square\overline{h}_{\mu\nu}=0$$ (where the bar denotes the use of the trace reverse metric perturbation) which leads to the vacuum wave equation $$\overline{h}_{\mu\nu}=\Re(A_{\mu\nu}e^{ik_{\sigma}x^{\sigma}}).$$ In addition to this, it might be helpful to know that the wave amplitude $$A_{\mu\nu}$$ is orthogonal to the wave vector $$k_{\nu},$$ that is, $$k_{\nu}A^{\mu\nu}=0.$$ which removes 4 degrees of freedom from the metric perturbation.

The Attempt at a Solution

As it stands I am quite confused and do not know really know where to start with proving that equations (1.1) and (1.2) are possible. In Schutz book "A First Course in General Relativity" after some calculations (on page 205 if you have the book) he does show that under an infinitesimal coordinate transformation we get $$A_{\alpha\beta}^{'}=A_{\alpha\beta}-ik_{\beta}B_{\alpha}-ik_{\alpha}B_{\beta}+i\eta_{\alpha\beta}k_{\mu}B^{\mu}.$$ where we can choose the $$B_{\alpha}$$ to impose (1.1) and (1.2).

 

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However, I do not really understand how this implies that we can impose conditions (1.1) and (1.2) as it doesn't seem to be explicitly stated. Any help would be much appreciated!