- #1

Taha

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## 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?