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Orthogonality of Gravitational Wave Polarizations

  1. Mar 30, 2017 #1
    1. The problem statement, all variables and given/known data
    Two plane gravitational waves with TT (transverse-traceless) amplitudes, ##A^{\mu\nu}## and ##B^{\mu\nu}##, are said to have orthogonal polarizations if ##(A^{\mu\nu})^*B_{\mu\nu}=0##, where ##(A^{\mu\nu})^*## is the complex conjugate of ##A^{\mu\nu}##. Show that a 45 degree rotation of ##B^{\mu\nu}## makes it proportional to ##A^{\mu\nu}##.

    2. Relevant equations
    For waves propagating in the z direction under the TT gauge, ##A^{xx}##, ##A^{xy}##, and ##A^{yy}=-A^{xx}## are the only non-zero components.

    3. The attempt at a solution
    $$(A^{\mu\nu})^*B_{\mu\nu}=2(A^{xx})^*B_{xx}+2(A^{xy})^*B_{xy}=0$$
    $$B_{xy} = -B_{xx}\frac{(A^{xx})^*}{(A^{xy})^*}$$
    $$ (B_{\mu\nu})=B_{xx}\begin{pmatrix}
    1 & -(A^{xx})^*/(A^{xy})^*\\
    -(A^{xx})^*/(A^{xy})^* & -1\\
    \end{pmatrix} = \frac{B_{xx}}{(A^{xy})^*}\begin{pmatrix}
    (A^{xy})^* & -(A^{xx})^*\\
    -(A^{xx})^* & -(A^{xy})^*\\
    \end{pmatrix}$$

    At this point, I'm not really sure what to do. I don't know what it means to "rotate" a matrix so I assume the correct thing to do is complex rotate each component, but I'm not entirely sure what this entails either. I tried multiplying each component by ##e^{i\pi/4}=(1+i)/\sqrt{2}## but I'm not sure how this helps. Any advice?
     
  2. jcsd
  3. Mar 30, 2017 #2

    Orodruin

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    How do rank two tensors transform under rotations?
     
  4. Mar 30, 2017 #3
    Ahh of course, use the rotation matrix twice. This gets me B proportional to A* if I use the normal (real valued) rotation matrix. Is there a complex version that will get me B proportional A?
     
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