On the Detection of Gravitational Waves by LIGO
February 11, 2016 — Jason Grigsby, Wolfram Technology Engineer, Wolfram Technology Group
Earlier today at a press conference held at the National Science Foundation headquarters in Washington, DC, it was announced that the Laser Interferometer Gravitational-Wave Observatory (LIGO) confirmed the first detection of a gravitational wave. The image reproduced below shows the signal read off from the Hanford, Washington, LIGO installation. The same signal could be seen in the data from the Livingston, Louisiana, site as well. While this signal may not seem like much, it is one of the most important scientific discoveries of our lifetime.
B. P. Abbott et al., Phys. Rev. Lett. 116, 061102 (2016)
A hundred years ago, Einstein’s theory of general relativity predicted the existence of gravitational waves—little ripples in spacetime that carry energy and information. But it has taken a century of technological progress to provide us the practical means to confirm the theory. LIGO’s historic discovery has not just confirmed Einstein’s theory—it also provides us with a first peek into an entirely new way of conducting astronomy. So what are gravitational waves and how does LIGO measure them? To understand gravitational waves, let’s first take a look at waves we are all familiar with: the electromagnetic spectrum.
The electromagnetic spectrum
Astronomy has relied on various kinds of electromagnetic radiation—light, radio waves, X-rays, microwaves—to see into space and learn new things. Early in recorded history, people would watch the movement of the stars and planets at night. Much later, the first optical telescopes were invented and we could magnify images enough to see that the planets had their own moons. We started to build telescopes that could see into different regions of the electromagnetic spectrum, and we started learning more and more about stars, galaxies, pulsars, quasars, the distribution of dark matter, the expansion of the universe, and much more beyond. All this was done with the electromagnetic spectrum, one rainbow of waves that we have carefully searched for new information. The iconic Hubble Space Telescope is a shining example of how much we have learned by improving how we observe in the electromagnetic spectrum:
Today, we have unlocked access to a completely different spectrum, one dependent on the force of gravity rather than the electromagnetic force, that can provide us a new window into the universe. The key to that access is LIGO, with its two installations at Livingston, Louisiana, and Hanford, Washington.
If these locations seem isolated, that’s on purpose. It turns out that you can “listen” to a great many things with a sensitive-enough interferometer. But many forms of vibration in the Earth show up as noise in the data that is produced. In particular, when I had conversations with people who worked at LIGO, they seemed very frustrated at the regular timing of trucks entering and leaving a logging facility near the Livingston site. To try to reduce local sources of noise, the locations were picked to be as isolated as possible. This is LIGO’s Hanford, Washington, site:
This does not look like a typical observatory. It looks far more like a particle accelerator, but by sending a split laser beam many times down the four-kilometer arms you see here, scientists have captured a change in the length of the arm that is equivalent to a fraction of the diameter of an atom, and thus detected gravitational waves. To understand how this works, it’s best to look into the nature of general relativity itself and see what gravitational waves actually are. Let’s start by posing a question for which general relativity gives the answer.
What would happen if the Sun disappeared?
What if the Sun magically disappeared? In this hypothetical case, I am talking about the Sun being replaced with empty space. This would change the gravitational field in the solar system dramatically. This might seem like a far cry from gravitational waves, but by looking at how different scientific theories treat this scenario, we can get to the motivation behind general relativity and gravitational waves.
First, let’s examine the behavior of light emitted by the Sun. We know that light travels at about 300 million meters per second, and the Earth is about 150 million kilometers from the Sun.
By dividing the distance by the speed of light, we see that it takes a little over eight minutes for light to get from the Sun to the Earth.
That means that when the Sun disappears, there are eight minutes’ worth of light still streaming toward the Earth.
So it would take eight minutes after the disappearance of the Sun for the Earth to go dark. Now let’s see what happens with the Earth’s orbit. The Earth is orbiting the Sun and is bound to it through gravity. If the Sun suddenly ceased to exist, how soon would the path the Earth is traveling on change? Let’s look at Newton’s law of gravity, which governs how the Earth moves around the Sun:
In this treatment of gravity, there is no accounting for time. If the Sun were to suddenly disappear, one of the masses in the formula would go to zero, meaning the force would go to zero instantly, and the Earth would cease orbiting and shoot off into space.
Here’s an animation showing what would happen according to Newton’s law. The blue Earth is shown orbiting the Sun. Yellow circles are used to represent light being emitted by the Sun. When the Sun disappears, the light that was already emitted by the Sun is still hitting the Earth for several more minutes. However, according to Newton’s theory of gravity, the Earth instantly stops orbiting where the Sun was. So here, light waves take time to carry the information of the now missing Sun, but gravitationally, that information is available instantaneously.
At the time that Einstein was looking into related questions, this feature was unique to the theory of gravity. This leads to the question: why do changes in everything else take time to propagate from one location to another, but with gravity propagation is instantaneous? What makes gravity unique?
Einstein’s answer is that gravity is not unique, but the underlying theory needs to be changed. He postulated his theory of general relativity, where gravitational information also propagates at the speed of light via gravitational waves. We explore that in the next section on Einstein’s theory of general relativity.
Einstein’s theory of general relativity
Einstein’s theory of relativity has two parts:
- Gravity is the effect of a curved spacetime on the motion of matter and energy.
- The distribution of matter and energy impacts the shape of spacetime.
Let’s take a look at both points.
Gravity is curvature
The first point is about gravity not being thought of as a force, but the natural outcome of objects moving in a curved spacetime. Imagine a large ball with two ants sitting at the equator. The black line is the equator, the two red points represent the ants, and the arrows point in the direction the ants are headed. Note that the arrows are parallel to each other at this point:
If the two ants travel north on the ball, they are initially moving in parallel. But as they approach the North Pole, they converge. By the time they reach the top, the ants are in the same location.
This illustrates some of the ideas behind what general relativity refers to as “parallel transport”. If you think of north as the direction of time, you can see how curvature can bring two objects together. Similarly, the curvature of spacetime is what draws us to the Earth and keeps the Earth orbiting the Sun.
Matter determines the shape of spacetime
For the second point, imagine spacetime as a sheet of tightly stretched fabric.
If you place a ball onto the middle of that sheet, the sheet will take time to deform and find a settled state. In this case, the ball represents the presence of matter and the sheet is spacetime being curved by the ball’s placement.
The consequences of general relativity have been confirmed repeatedly over the last hundred years. When general relativity came out, it explained a precession of Mercury’s orbit that could not previously be accounted for. In 1919 during a solar eclipse, Arthur Eddington measured how the Sun deflects the light of distant stars, a key prediction of general relativity. Still, until this announcement, there had been no direct confirmation of gravitational waves themselves.
To understand how LIGO detects gravitational waves, let’s step back and consider an example using Newtonian physics. In the graphic below, imagine the two red spheres are distant stars orbiting each other and the rabbit is an observer where you and I are. According to Newton’s theory of gravity, each distant star will pull on the rabbit, as shown by the blue arrows. The forces will sum to the red arrow, indicating the rabbit is drawn to the center of mass of the distant orbiting stars.
The above analysis is treating the rabbit as a point rather than an extended object. In reality, objects have height, depth, and width. When aligned as shown below, the top of the rabbit is pulled a little more toward the upper star and the bottom is pulled a little more toward the lower star. Thus a stretching will occur.
As the distant stars orbit each other, so too will the direction of the stretching change. When the two stars are aligned horizontally, the rabbit is stretched horizontally.
So as these stars orbit, the rabbit is stretched according to the orientation of the stars. In this Newtonian model, though, the stretching is perfectly aligned with the orientation of the stars, because according to Newtonian gravity, it takes zero time for a change in gravity to get to another location.
With general relativity, however, the motion of the stars puts ripples into spacetime itself, and those ripples take time to propagate out to the rabbit. Still, when the gravitational waves interact with an object, they have a similar stretching effect. This animation, based on a common LISA image, illustrates the gravitational waves produced by two orbiting objects.
The source code for the above animation can be found here.
To get back to our previous question, “What if the Sun disappeared?”, under general relativity a gravitational wave would be generated that propagates the new information of the now empty space out to where the Earth is. In other words, the path of the Earth’s orbit would continue to circle the now missing Sun until the last rays of light got to the Earth and the gravitational wave with that new information had passed.
Detecting gravitational waves with LIGO
With our understanding that passing gravitational waves will stretch objects in a rotating manner, we now move on to detecting those gravitational waves. To do that, LIGO uses an interferometer. Below, I have provided an animation of an interferometer based on a Wolfram Demonstration.
On the left, a coherent light source sends a beam of light to a half-silvered mirror. The split beam then travels down two different arms and is reflected by mirrors at the end of each path. The beam is then recombined and sent to a detector. If the two paths are different in length, the two beams will be out of phase when they are recombined, decreasing the overall intensity of the beam. Thus you can measure a change in intensity of the beam to determine a change in distance. What you see at the Hanford LIGO site are four-kilometer arms of an interferometer. A beam is passed back and forth several times before being recombined to measure a change in the distance traveled smaller than the nucleus of an atom.
Given the vanishingly small earthly effects of gravitational waves, it takes some of the most energetic events in the universe to generate gravitational waves that are detectable by LIGO. The most likely to be detected are generated by binary black holes with a total mass of about 10–100 times that of the Sun. Indeed, we heard at the LIGO press conference earlier today that the detected waves were from the merger of two black holes of approximately 65 solar masses total. During the course of spiraling together and merging, three solar masses’ worth of energy were radiated out in a fraction of a second. The actual merger of these two black holes happened approximately 1.3 billion light years away, meaning that these two black holes merged before multicellular life came about on Earth.
So a hundred years after Einstein formulated general relativity, one of its last fundamental predictions has been confirmed. However, as much as this detection is a success for LIGO, the LIGO Scientific Collaboration, and the physics community in general, it is not just a conclusion for theoretical physics. It’s the beginning of a new era in astronomy. As the tools and methods at LIGO improve, more information about sources of gravitational waves, their locations, and their physics will become available. Possible future projects such as the Laser Interferometer Space Antenna will extend the range of detection as well as the range of frequencies available for gravitational wave observations, possibly allowing us to see the results of mergers of supermassive black holes that occur when galaxies collide.