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Weak gravitational field

  1. Nov 10, 2017 #1
    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. jcsd
  3. Nov 15, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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