Orthogonality of Gravitational Wave Polarizations

[Taha]

Homework Statement

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 $A^{\mu\nu}$.

Homework 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.

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?

 

[Orodruin]

How do rank two tensors transform under rotations?

 

[Taha]

How do rank two tensors transform under rotations?

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?