# 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:

*Courtesy Caltech/MIT/LIGO Laboratory*

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.

### Gravitational waves

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.

*Courtesy Caltech/MIT/LIGO Laboratory*

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.

## 18 Comments

Great article Jason. I really liked the explanatory pictures.

However, what statistical analyses has been done on these results? There was a similar announcement last September by LIGO that turned out to be a false indicator. Furthermore the wave results of the two LIGO sites differ significantly in the lower time range. Considering the very tiny scales that they are measuring, how does this affect the significance of the results? Were the wave signals detected at exactly the same time accounting for geographic displacement?

I can’t help feeling that this is a result that everyone in cosmology wants to see as a centenary anniversary of Einstein’s GT of R. It just seems too fortuitous and there are no coincidences in science.

I didn’t hear any official announcements last September, but there were rumors going around. I would note the signal that they showed us was taken September 14, 2015, so you might get an idea where the rumors originated…

How can you do a statistical analysis on a single data point? They have only managed to record a single chirp, https://www.youtube.com/watch?v=TWqhUANNFXw

That wump (as I prefer to call it) does fit with two massive objects accelerating as they are attracted to each other because as their velocity increases they approach the causality speed limit, c, and get evermore massive thus the curvature of space-time increases proportionally, exponentially, until the objects merge and have zero velocity (other than spin). Imagine pushing your finger into a membrane until it breaks through the surface and the membrane returns to the shape it started with. Except there are two objects so two membranes.

I suspect, though I don’t want to speak for Michael, that he is referring to the analysis behind whether the signal is a match. Noting that I never worked in the data-analysis side so I may have some specifics wrong, but there is clearly noise in the data stream and the theoretical models, which you can get a good sense of at http://arxiv.org/pdf/0901.4399.pdf, are much “cleaner looking”. So the analysis that needs to be done is to determine how well we think this signal matches the theoretical models. The analysis is, how much do we think this is a hit, rather than how does these two populations compare to one another. I believe that the actual papers in the field show that this is a “very good match” to the theory.

Actually, there are plenty of coincidences in science, as in life.

To your point about the rigor of the statistical analysis and model-building used to extract black hole masses, distances, and other data that pertain to the black hole merger, the physics and astrophysics communities have been using such techniques for decades.

In these fields, signals are often fleeting and minuscule compared to ambient noise. Confounding instrumental and environmental phenomena are always present and must be properly accounted for.

Particle physicists in the 1960s were the first to make extensive use of sophisticated statistical hypothesis testing and model building. The LIGO analysts are experts in applying these analytical tools and have had years to model and simulate all kinds of scenarios that would generate gravitational waves.

If you read the discovery paper, I believe you will agree that the LIGO analysis team has done an exemplary job of squeezing an extraordinary amount of information from a tiny signal, and clearly documenting all known experimental uncertainties that affect their conclusions.

Thought it would be nice to link a video of the press conference given by the LIGO leadership: https://www.youtube.com/watch?v=aEPIwEJmZyE

Jason Grisby, thankyou so much for this impressively intuitive and compelling explanation. From this, I am in awe of the moment we given with this scientific discovery. Cheers to that.

I’m glad you found the post helpful. I used to work in Numerical Relativity studying binary black holes but I was always better at providing explanations than producing new results. I hope that here I have provided something useful.

Hi Jason

Very fine work, and hot on the tail of the big announcement. I downloaded the B. P. Abbott et al., Phys. Rev. Lett. 116, 061102 (2016) paper this morning, and an extra thrill was to see explicit mention of Bayesian analysis, and the reporting of 90% credible (*not* confidence) intervals. State-of-the-art and expensive data deserves the best analysis. :-)

Do we know of any use of Mathematica in all of this?

Thanks,

Barrie

So I worked in Numerical Relativity on the theoretical side of this problem and I definitely used Mathematica in deriving equations for simulations. The reality is that around a thousand people overall contributed to the efforts if you include theoretical work and the observation and Mathematica has long been a common tool in those communities. Whether or not it was used for a particular task such as data analysis or instrument mechanics, I couldn’t say.

Do you happen to know about the source of the estimates of masses of the binary black hole pair? Is it deduced from the signal itself or by some other observations? In pop sci media it is presented as a well known and established fact.

The sources are being estimated from the signal itself and there are some variations possible. Total mass, mass-ratio, and a few other factors combine to determine frequency and the changes in the amplitude of the signal. Once you have a mass-estimate, distance can be figured by the overall strength of the signal.

I have since read the published paper on the gravitational waves result called Observation of Gravitational Waves from a Binary Black Hole Merger in the journal Physical Review Letters, which is available to the public at http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.116.061102.

All my doubts about the statistical analysis are blown away by the detailed process of testing, model building and verification. The paper itself is only 8 pages long and can be read before going to bed. It is a model of clarity and exposition. It describes a mixture of ideas going back over the last century and in sections IV and V describes the detectors which themselves represent 50 years of refined engineering technology. The references and full author list goes for about 6 pages. The institutions involved are from Europe and America mainly but the collaboration is essentially global. A truly awesome result with profound implications for testing hypotheses about dark matter and dark energy.

re michelson interferometer: the animation shows a beam being split at the 45 degree mirror. Each of the resulting beams are retro-reflected by flat mirrors and re-combined at the beam splitter before being send to the detector, where interference occurs. Your text talks about multiple reflections. Can you explain?

regards

alex

Hi Alex, so the basic interferometer I show here is just illustrative. I do not know the details of what was engineered at LIGO but the story I heard was that effectively, the beam was reflected up and down the arms of the interferometer in what is the equivalent to 75 times the length of the arms. There may be different reasons for this but one is to effectively increase the effect of a passing gravitational wave on the phase change in the beam.

Dear Jason,

on the bottom of this web page there is a computable document of an interferometer. It also allows to experience the influence of the arm length on the output intensity:

http://hagen-lorenz.bplaced.net/Interferenz.html

Best regards

Thanks. I will keep track of that.

W against T, reference required. #conspiracyofscience