(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that a plane wave with ##A_{xy}=0## (see below) has the metric ##ds^2=-dt^2+(1+h_+)dx^2+(1-h_+)dy^2+dz^2##, where ##h_+=A_{xx}sin[\omega(t-z)]##

2. Relevant equations

##h_{\mu \nu}## is small perturbation of the Minkowski metric i.e. in the space now ##g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} ##. If we go in TT gauge, and consider a plane traveling in the z direction ##k = (\omega,0,0,\omega)##, we have ##h_{\mu \nu}^{TT}=A_{\mu \nu}^{TT}exp(ik_\alpha x^\alpha)##, with ##A_{\mu \nu}^{TT} = \begin{pmatrix}

0 & 0 & 0 & 0 \\

0 & A_{xx} & A_{xy} & 0 \\

0 & A_{xy} & -A_{xx} & 0 \\

0 & 0 & 0 & 0

\end{pmatrix}##

This is from Schutz page 205 Second edition

3. The attempt at a solution

So as ##g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} ## and only ##h_{xx}## and ##h_{yy}## are non-zero, we have ##h_{xx}=A_{xx}e^{i\omega(z-t)}##. For ##dt^2## and ##dz^2## it is obvious that they remain the same but for ##dx^2## for example, the coefficient would be ##1+A_{xx}e^{i\omega(z-t)}##. How do I get from this to ##1+A_{xx}sin[\omega(t-z)]##? Thank you!

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# Homework Help: Weak gravitational field

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