# Weak gravitational field

1. Nov 10, 2017

### Silviu

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!

2. Nov 15, 2017

### PF_Help_Bot

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