# Orthogonality of Gravitational Wave Polarizations

• Taha
In summary, two plane gravitational waves with TT amplitudes, ##A^{\mu\nu}## and ##B^{\mu\nu}##, have orthogonal polarizations if ##(A^{\mu\nu})^*B_{\mu\nu}=0##. After applying a 45 degree rotation to ##B^{\mu\nu}##, it becomes proportional to ##A^{\mu\nu}##, showing that rank two tensors transform under rotations.
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?

How do rank two tensors transform under rotations?

Orodruin said:
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?

## 1. What is the orthogonality of gravitational wave polarizations?

The orthogonality of gravitational wave polarizations refers to the perpendicularity of the two types of polarizations that occur in gravitational waves. These polarizations are known as the plus (+) and cross (x) polarizations and are perpendicular to each other, meaning they are at 90-degree angles.

## 2. How are gravitational wave polarizations measured?

Gravitational wave polarizations are measured using interferometers, which are sensitive instruments that can detect tiny changes in the shape of space caused by passing gravitational waves. The interferometers measure the stretching and squeezing of space in the directions of the two polarizations.

## 3. What is the significance of the orthogonality of gravitational wave polarizations?

The orthogonality of gravitational wave polarizations is significant because it allows us to accurately measure and analyze the properties of gravitational waves. By measuring the amplitudes and phases of the plus and cross polarizations, we can determine the direction, distance, and strength of the source of the gravitational waves.

## 4. Can gravitational wave polarizations change over time?

Yes, gravitational wave polarizations can change over time as the waves propagate through space. The amplitudes and phases of the polarizations can change as the waves interact with different objects and structures in their path, such as stars, planets, and black holes.

## 5. How does the orthogonality of gravitational wave polarizations support the theory of general relativity?

The orthogonality of gravitational wave polarizations is a key prediction of Albert Einstein's theory of general relativity. This theory states that gravitational waves travel at the speed of light and have two perpendicular polarizations, which has been confirmed by numerous gravitational wave detections. This supports the validity of the theory of general relativity and its predictions about the behavior of gravitational waves.

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