# Does gravity affect the polarization of electromagnetic waves

## How do gravitational waves affect me and matter?

### 2.2 Sound and electromagnetic waves

Basically, waves transmit energy. In the rope experiment in Chap. 1 the second person who is hit by the rope wave is shaken up properly.

Sound waves apply pressure to our eardrum so that we can hear tones and noises. Our ears are only sensitive in a certain frequency range between 20 and 20,000 Hertz. Lower tones are called infrasound. Elephants in the African savannah communicate via infrasound, which they can hear over many kilometers. At the high-frequency end above 20,000 hertz there is also a "beast". Bats emit high-frequency ultrasound to orient themselves. As with an echo sounder, the ultrasound bounces back against cave walls, trees, houses and other obstacles and is picked up by the ears of the bats. The bat determines the distance to the obstacle from the time between issuing the call and hearing it, and can thus cleverly prevent a collision. With her ultrasound hearing, she can also track down her prey.

Electromagnetic waves affect us very differently depending on the energy. In general, the shorter the wavelength or the higher the frequency, the higher the energy of the electromagnetic wave. In a narrow band between 380 and 780 billionths of a meter (nanometers) wavelength, we perceive these waves as visible light. Depending on the wavelength, our brain interprets this as a color: 380 nanometers appear blue and 780 nanometers appear red. The well-known rainbow colors are arranged in between. Beyond the 780 nanometers at longer wavelengths, the infrared follows. We cannot see these waves, but we can feel them as warmth. At even greater wavelengths, we get into the microwave and radio range. We can warm food with microwaves with a wavelength of a few centimeters. But the oldest form of radiation that mankind can measure is in the microwave range: the cosmic background radiation, which spread shortly after the Big Bang 13.8 billion years ago and was emitted by the first, light, chemical elements. The very long-wave radio waves with wavelengths in the range of meters are emitted by radio stations. They have encoded information onto the radio waves, e.g. B. Music or the weather forecast for tomorrow.

The ultraviolet follows at the short-wave end below 380 nanometers. Some animals, such as bees, can perceive this radiation, so that flowers glow in even more striking colors. At even shorter wavelengths, the X-rays are around a nanometer. It is very energetic and can therefore easily cross a person. The X-ray doctor makes use of this property to photograph bones in the human body. Because much less X-ray radiation goes through bones. However, X-rays are not entirely safe because they can damage the body and cells if you are exposed to them for too long. The more energetic gamma rays, which have shorter wavelengths than X-rays or higher frequencies, are even more dangerous. Gamma radiation usually comes from inside atomic nuclei and is a result of radioactive gamma decay. It is the most dangerous form of radioactivity and was what made the Chernobyl accident in 1986 so life-threatening. In astronomy, sources are known that also generate gamma rays in other ways, e.g. in the form of synchrotron radiation.

### 2.3 Deformed ring made of test masses

Are gravitational waves also dangerous under certain circumstances? What do they do in our body when they cross us? Before we look at this, we need to clarify another property of waves: polarization. Let us remember our mechanical rope wave from Sect. 1.1. The person creating the wave can swing the rope vertically up and down. But you can also swing the rope horizontally to the left and right. The oscillation always takes place in one plane. Such a wave is called linearly polarized.

It is also possible that the oscillation does not take place in a plane, but rotates on a circle. That would happen if you had both waves just described in vertical and horizontal oscillation direction with the same amplitude and one Phase difference would overlay from zero. Phase difference zero means that both partial waves (identical frequency) meet each other in such a way that wave crest fits wave crest. This waveform is called circularly polarized. A distinction is made between left and right circularly polarized, depending on whether the wave rotates counterclockwise or counterclockwise when looking in the direction of propagation. If the two linearly polarized waves with unequal amplitudes and zero phase difference overlap, the wave no longer oscillates on a circle, but on an ellipse. It is called accordingly elliptically polarized.

A precise evaluation of the wave equation for gravitational waves in general relativity shows that there are two forms of oscillation, more precisely polarizations of the waves. A distinction is made between the polarization + ("plus") and the polarization × ("cross"). They are rotated against each other by 45 °. But why exactly 45 °? In Sect. 1.2 we learned that gravity is described by a tensor field. Such fields are mediated by spin-2 particles, as quantum field theory teaches (Box 5.4). However, it is currently not clear whether gravity has to be quantized and thus described as quantum gravity. So far, gravitation has behaved classically in all experiments, i.e. it does not show any quantum behavior and behaves as Einstein's ART intended. If there is a quantum particle that mediates the gravitational force - the graviton (Box 10.2) - then it must have spin S = 2 due to the tensor character of gravitation. In general, the following applies: The radiation field, which is described by a particle with spin S, is symmetrical with rotations through the angle 360 ​​° / S around its direction of propagation [87]. We see this for gravity with S = 2 directly in Fig. 2.2. Because here the angle of rotation is 360 ° / 2 = 180 °. Another rule says: A radiation field with spin S has exactly two independent states of linear polarization, which are mutually rotated by an angle of 90 ° / S. For gravity with S = 2 we get 90 ° / 2 = 45 °. Aha, there is the 45 ° angle! It is different with electromagnetic waves. The exchange quanta of the electromagnetic force are the photons with spin S = 1. The linear polarizations of electromagnetic waves are accordingly rotated by 90 ° / 1 = 90 ° against each other.

The effect of a gravitational wave becomes very clear if we imagine a circular ring on which many small, point-like masses sit. These point masses should be at rest relative to one another at the beginning and no forces other than gravity should act on them. A linearly polarized gravitational wave now hits the surface of the ring perpendicularly. The shaft then squeezes the ring into an ellipse. Then the ellipse swings back to the ring shape and is deformed into an ellipse, which is now perpendicular to the first ellipse. Finally, the initial shape of the circular ring is restored. The areas of the circle or ellipse always remain the same size; they are only deformed. Gravitational waves are transverse waves, i.e. the geometric relationships are only changed perpendicular to the direction of propagation.

The exciting question now is how much the circular ring shape is squeezed together. Let us assume a typical source, e.g. two black holes circling each other, each with ten solar masses, at a distance of one billion light years. The ring should now have the diameter of the earth, around 10,000 kilometers - if we round it off generously. The gravitational wave of the two black holes compresses this earth-sized ring by only 10−14 Meters together; this corresponds to only ten atomic nucleus diameters!

Gravitational wave researchers like to express this as a relative change in length, i.e. as a measured change in length ΔL in meters (the 10−14 Meters) divided by the original length (or diameter) of the object. In the example just mentioned, that was the earth's diameter of 10,000 kilometers. Accordingly, the relative change in length is ΔL / L = 10−21, a dimensionless number. A really amazingly tiny number! However, this number is not universal, but depends on the masses and distances involved, i.e. the properties of the source. We will call it dimensionless gravitational wave amplitude. The number is always extremely small. Gravitational waves stretch and compress reference lengths very, very, very little. Why is that?

### 2.4 The gravitational wave and me

In discussing the quadrupole formula we had the destructive prefactor G /c5 discovered. It flattens all gravitational waves, one could say quite casually. The gravitational wave amplitude is also given a similar prefactor G /c4 strongly suppressed. We can now answer fundamental questions like this: Does it hurt when a gravitational wave penetrates me? Will I even get slim with gravitational waves? Ha, nice try, but we have to clearly deny both. The expansion and compression caused by a gravitational wave are extremely small and move far below the subatomic length scale. We do not feel such tiny deformations, and we are not damaged in the process. But if the effects on the human body are so small, how could a measuring device show such extremely small effects? Obviously you have to have a very subtle measuring method to do this.

But there is also a fascinating property of gravitational waves. We might wonder if we could use a ruler to detect the passage of a gravitational wave. The ruler should be stretched and compressed periodically. Gravitational waves also have an effect on the structure of space-time, i.e. the ruler itself is stretched and compressed so that a reference distance (e.g. one centimeter) itself swings back and forth! As the relativity theorists say: The metric is set in motion.

The dimensions of the body must also fit. Not every gravitational wave makes an effect. If the frequency of the gravitational wave is lower than the basic frequency (natural mode) of the ruler, then there will be no effect of the gravitational wave. This is similar to a beam that is firmly clamped at the ends and is supposed to be brought into resonance with sound so that the beam vibrates. This only happens when the sound wave has a wavelength that is half the size of the bar. This is the fundamental oscillation at which the nodes of the sound wave coincide exactly with the ends of the beam. Furthermore, there are higher frequency harmonics (harmonics) that can excite the beam. Here, too, there are nodes on the ends of the beam, but there are now also nodes between the ends of the beam. However, nothing can be done below the basic frequency. This is also the case with gravitational waves. The corresponding detectors based on this measuring principle are the resonance detectors (Section 3.2). As we saw in Chap. 3, "rulers" have been built from laser light since the early 1970s, which could subsequently be used for extremely precise distance measurements. They are called interferometers, in which mirrors let the light zoom back and forth over kilometers. The speed of light in a vacuum is always the same - regardless of the speed of the light source. The phase shift that a gravitational wave causes in the detector is Not only a spatial effect that affects the (spatial) mirror spacing. If that were the case, you could make light faster than the speed of light cby simply shortening the travel distance of light with a gravitational wave.

Nor is it just a time effect. Rather, it is a Space-time effectcaused by a gravitational wave. The Einstein equations of the theory of relativity are formulated independently of coordinate systems - covariant. Thus the gravitational waves are also independent of the coordinate system. There are different but equivalent ways of looking at the effects of the gravitational wave: In the coordinates that the theorists use (the so-called TT calibration2), the ends of the bar (or mirrors in an interferometer, Chapter 3) do not move at all, but are always at the same positions in the coordinate system. But the coordinate system between them oscillates back and forth. Quite different in the laboratory system used by the experimenter: Here the mirrors move and oscillate back and forth due to the influence of the gravitational wave. Does a gravitational wave stretch a light wave? Or does it make it vibrate more slowly? You can see it either way. It can be shifted to the wavelength of light - the spatial component - or to the frequency - the temporal component. The speed of light c is definitely always a constant. That says Einstein's SRT, and the constancy of the speed of light has proven itself to this day.

### Footnotes

1 Generally applies to the wavelength λ = c/f at the speed of light c and the wave frequency f.
2 Attention, only for theorists of relativity: TT stands for transverse traceless, i.e. transversal and without a trace, and refers to a certain choice of a coordinate system in which the disturbance hμν from Box 1.2 only has spatial components, h = 0, and is therefore transversal. It is also track-free, i.e. hkk = 0, where k only accepts spatial indices.

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