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Gravitational Waves

Many sci-fi movies feature reality-warping, where some fantastical device or ability allows one to bend the space around them. And while it’s true that such power is far beyond what humans can do, this is actually something that can happen in real-life. It is possible for the space between matter to change, so that objects expand and contract. The phenomena that can cause such bizarre behavior are called gravitational waves.


One of the many predictions Einstein made from his theory of general relativity was the existence of something called gravitational waves. First predicted in 1916, they were not confirmed to exist until 2015, almost a century later. The direct observation of these waves seems to be the latest experimental confirmation of Einstein’s theory. Ever since the Eddington Expedition, which confirmed gravitational lensing, countless tests have been done to see if the strange implications of relativity really exist. And time and time again, we see that Einstein’s predictions have been correct. Relativity’s survival in the face of so many attempts to prove it wrong has led it to become the prime example of a successful scientific breakthrough. With this in mind, let us examine the most recent of Einstein’s vindications, the gravitational wave.


One of the most fundamental principles of general relativity is how massive objects bend spacetime. And as these massive objects move, the curvature they generate also changes to match their motion.




This is the concept of the spacetime fabric, which is illustrated in the picture above. The classic analogy involves a blanket and heavy ball. Imagine stretching out this blanket so that it’s taut. Now imagine dropping a heavy ball on this blanket. What would happen? The ball would make a sizable dent in the blanket. This is something you can try at home very easily. But now imagine that the blanket is the very fabric of space and time. And that heavy ball is some massive object, such as the Earth or a star. Again, this is illustrated in the picture above. Massive objects create “dents” in spacetime, and these deformations are what cause the effects of gravity.



In certain circumstances, accelerating objects generate fluctuations in the curvature of spacetime, which propagate at the speed of light as gravitational waves. These gravitational waves also carry energy with them. So if gravitational waves are emitted from a system, that means that the system is losing gravitational potential energy.

Now, what are the “certain circumstances” that can lead to the formation of gravitational waves? By definition, waves are created from the oscillation of something. For example, waves in water come from the oscillation of, well, water itself. Sound waves come from the oscillation of pressure in any material they pass through. Light waves are generated from the fluctuation of electromagnetic fields. Similarly, gravitational waves are created from the fluctuation of gravitational fields. So, the phenomena that generate gravitational waves must involve changing a gravitational field.


We may first discuss what scenarios do not generate gravitational waves. Consider a perfectly spherical star orbiting about its axis. By geometry we know that the gravitational field created by this spherical star is equivalent to all of its mass concentrated at its center of mass. Since the spherical star is perfectly symmetric about its axis of rotation, its center of mass is not changing as it rotates. Thus its gravitational field does not change. Thus it does not generate gravitational waves.




Now consider that this perfectly spherical star is expanding in a perfectly spherical manner. If it is expanding spherically, then the center of mass does not change. And since again, the gravitational field of this spherical star is equivalent to the gravitational field of that star’s entire mass concentrated on the center of mass, the gravitational field isn’t changing. And so again, no gravitational waves are emitted. This can be taken to the extreme. If a supernova were to somehow explode in a perfect sphere, it would not generate gravitational waves (despite all the debris it flings out).



But now consider a star with a “dimple” on one end. Because of this uneven feature, the star is no longer symmetric about its axis of rotation. Thus the center of mass changes as the star rotates, creating a fluctuating gravitational field. This oscillation will generate gravitational waves. In addition, any imperfect explosions or implosions would generate bursts of gravitational waves. This is because the center of mass of the ever-expanding cloud of matter changes in position and thus the overall gravitational field of the debris also fluctuates. Finally, we can focus on binary systems, which have been the source of the only gravitational waves we have detected so far. They are caused by the fluctuating “gravitational quadrupole” of the two bodies. The mathematics are complex, but they ultimately show how the two rotating objects create a fluctuating gravitational field. And so just as before, this constant fluctuation creates gravitational waves. This can be visualized by the following animation:



Now that the causes of gravitational waves have been discussed, we may turn to the question of why they travel at the speed of light. The key is that the speed of light isn’t just for light, it is the general speed limit in the universe. Anything massless would travel at light speed in a vacuum, and gravitational radiation is just that. One can also draw a simple analogy to explain why gravitational waves travel at the speed of light. Light, which is electromagnetic radiation, is analogous to gravitational waves, which is gravitational radiation. Both result from the disturbance of a certain field. And so similarly, both propagate at speed c.



We can now ask the question: what are the effects of a passing gravitational wave? The theory of general relativity states that all matter rests on spacetime. And we know that gravitational waves are like ripples through spacetime, since gravity is a curvature in spacetime. Thus we can reasonably infer that gravitational waves affect matter in some way. In fact, the passing of a gravitational wave causes the space between matter to fluctuate. For example, suppose you had a collection of particles lying flat on a plane. Now suppose a gravitational wave passes perpendicularly through that plane. It would cause the planar collection of particles to distort, as if they were stretched from top to bottom then from side to side. The area enclosed within the particles does not change and the planar set does not move in any way along the direction of the wave’s propagation.


This is the reality-warping effect that I referred to at the beginning of the article. These particles are literally being stretched in one direction and compressed in the other. If you were to stand in the path of a passing gravitational wave, you could behave similarly. You would grow taller and skinnier, and then shorter and wider. You’d keep oscillating like this as long as the waves continued to pass through you.


With all of this theoretical information about gravitational waves, we may now turn to how they were eventually detected. The problem with detecting gravitational waves is that they are tiny by the time they reach Earth. Although the sources that create them are extremely energetic, the distance they must travel to reach Earth means that by the time they arrive they are thousands of billions of times smaller. The amount of spatial distortion they create is less than 1000 times smaller than the nucleus of an atom. This is why the people you see on the street aren’t constantly being stretched and compressed. The effects of any gravitational waves that reach Earth are far too small to see.

The first clues about gravitational waves are from indirect detection. The discovery of pulsars meant that physicists had reliable clocks they could use to infer many properties. In a binary star system where one of the stars is a pulsar, the periodic radio signals emitted from the pulsar can be used to determine the characteristics of its orbit. General relativity states that this binary system will radiate energy through gravitational waves, causing the system to lose energy. This would cause orbits of the stars to decrease and the orbital periods to decrease as well. This means that the pulsar’s signals would be received at an ever-increasing frequency. This is exactly what scientists were detecting on Earth, which led to the inference that gravitational waves do exist.

But eventually, scientists used instruments called laser interferometers to directly detect the first gravitational waves. The idea was to use the wave interference properties of lasers to measure the incredibly small changes in gravitational waves would make. The laser interferometer that detected the first gravitational wave was LIGO.


Here is the layout of LIGO’s laser interferometer. The laser is shot forward and hits a beam splitter. This sends out two separate beams which are perpendicular to each other. One continues going right while the other goes up. Each travels 4 km up an “arm” before hitting a mirror and traveling back toward the beam splitter. Upon return, the two separate beams are recombined and travel down toward a detector.

Under normal circumstances, the system is aligned so that the recombined beams interfere destructively. Thus they “cancel each other out” and the detector does not receive a signal. This is shown by the image below, the red waves travelling in one direction interfere with the blue waves that travel in the opposite direction. Together they cancel out:



However, when a gravitational wave passes through the system, it causes the length of each arm to fluctuate. One will be stretched while the other contracts, and then vice versa. This is similar to the animation of the ring of particles. The dimensions of the lab oscillate. This oscillation disrupts the interference pattern of the lasers. As the distances change, the waves no longer line up so that they cancel out, and the detector will detect light..


The intensity of the light is dependent on how much the arms of the lab elongate or contract which is in turn dependent on the size of the gravitational wave. Thus, by analyzing the light detected during the passing of a gravitational wave, scientists can determine its properties. The first direct detection of a gravitational wave was on September 14th, 2015. The waves detected were from the collision of two black holes. From that first observation, a number of others have been made. We have yet to figure out many aspects of gravitational waves, such as how to detect smaller waves from other sources, so it has become a burgeoning and promising topic and the field of modern physics.


Bibliography

Betz, Eric. “Why Does Gravity Travel at the Speed of Light?” Discover Magazine, Discover Magazine, 17 Apr. 2020, www.discovermagazine.com/the-sciences/why-does-gravity-travel-at-the-speed-of-light.

Goss, Heather. “How LIGO Works.” Air & Space Magazine, Air & Space Magazine, www.airspacemag.com/videos/how-ligo-works/?no-ist.

“Gravitational Wave.” Wikipedia, Wikimedia Foundation, 29 June 2021, https://en.wikipedia.org/wiki/Gravitational_wave.

“LIGO's Interferometer.” Caltech, www.ligo.caltech.edu/page/ligos-ifo.

“What Are Gravitational Waves?” Caltech, www.ligo.caltech.edu/page/what-are-gw.



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