Of Particular Significance

Author: Matt Strassler

It’s a lot easier to map the Earth than it used to be.  Before satellites, you had to do many careful measurements of distances and directions, at many different locations around the world, and combine them all to build a picture of a world you couldn’t see.  That’s part of why maps and globes made in past centuries had so many inaccuracies and distortions; it was a tough business.  

How that changed in the 1960s!   The first full photograph of the Earth that I’m aware of was made in 1967 by the ATS-3 satellite (were there earlier ones?)  So much simpler… the whole planet laid out in front of you.  You just need a few photographs like this, and the era of measuring from one point on the ground to another is mostly over.

But the challenge of trying to measure things when you’re stuck within them, and can’t step outside them, hasn’t gone away.  Just as we could see in telescopes that the Moon and Mars are ball-shaped, long before we could observe the Earth itself, today we can see other galaxies in great detail, but we still struggle to build a complete picture of our own, the Milky Way. The Gaia satellite is trying hard.

To determine the Earth’s two-dimensional surface is really round took some clever thinking. Aristotle, in ancient Greek times, noted that the Earth’s shadow on the Moon during a lunar eclipse is always curved in the same way — it doesn’t matter what time of day or year the eclipse occurs, or whether the shadow is on the north, east, west or south side of the Moon.  This feature is to be expected if the Earth’s a ball, like the Moon and Sun, and very difficult to explain otherwise.  [Try to figure out what you might see if it were cylinder-shaped!]

But there are other tricks you can use if you have a hunch that the place you live on, or in, is of finite size.

One Dimension: the Possibly Circular Canal

Suppose you live on the banks of a canal, a long thin channel extending off to the horizon, like a river without any flow.  And suppose you suspect that this canal forms a loop, surrounding a large island.  How could you check?   Well, if you had a boat, you could row yourself down the canal; or you could walk along the shore. If the canal is really in the shape of a loop, you’ll eventually come back to your starting point.  But maybe you’re worried such a journey would be too long, difficult, risky, expensive. Do you have other options?  

Here’s one: suppose you could make a big wave moving in the clockwise direction around the canal.  The wave, unlike you, wouldn’t need any food and drink or fuel for the journey — so time and money would not be a problem. The wave would move down the canal at a definite speed [I’m assuming here that it maintains its height], and if the canal were really a loop, then after some time T you’d see the wave return, still moving clockwise, and pass by you.  If you waited the same amount of time T again, you’d see the same wave a second time, again clockwise.  After the same amount of time T, you’d see it a third time. 

If instead the canal were a finite strip, then the wave would reflect off the end, and so the wave would return from the opposite direction. If it were infinite in length, it would never return. And if it had a complicated shape — perhaps a P or an R or a B instead of an O — you would get multiple waves in a complex pattern. But the simple pattern in which the waves return again and again, from the same direction, after a time T, is consistent with the canal being a simple loop.

You could try sending a wave counterclockwise too, and you’d expect the same pattern if the canal’s a loop.

As the wave passes you, you can also estimate its speed v. Having also measured T, you can now determine the length L of the canal. It’s the wave speed times the time T for the wave to go round once:

  • L = v T
Figure 1: You live on the shore of a canal, which you suspect is circular. You could find out how big it is by sending a large wave in either direction, and measuring the time T that it takes to return.

Perhaps making such a wave is too difficult for you, but if you’re lucky, someone or something down the canal may make a giant splash. Then you’ll see the ripples from the splash come by in a similar pattern. Now waves will travel both counterclockwise and clockwise around the canal, and they probably won’t arrive at the same time. That doesn’t matter, though. You’ll see the clockwise waves repeat after a time T, and you’ll see the same for the counterclockwise waves. Seeing both of them repeat after the same time T will give you confidence that the canal’s really a simple loop

To be specific, let’s call t1 the time you measure the first wave, t2 the second wave, t3 the third, t4 the fourth, and so on; if the first wave is counterclockwise, then the second is clockwise (see Figure 2), the third counterclockwise, and so on. (This won’t be true if instead of a loop the canal is in the form of a line segment! A reflection off the end could make the first two waves come from the same direction.) As the clockwise waves will repeat after a time T, and the same for the counterclockwise waves, it will be the case, if the canal’s a loop, that

  • t3 – t1 = t4 – t2 = T
  • L = v T

There’s more; if you know the time ts when the splash happened and you know the wave speed, then you can learn how far away the splash was from you:

  • D = v ( t1 – ts )

But even if you don’t know what time the splash happened, you can figure it out; see Figure 2. The distance traveled by the counterclockwise wave to get to you, plus the distance traveled by the clockwise wave to do the same, equals the full distance round the circle (Figure 2), so the time that the counterclockwise wave required to reach you ( t1 – ts ) plus the corresponding time for the clockwise wave ( t2 – ts ) must be equal to T.

  • T = ( t1 – ts ) + ( t2 – ts ) = t1 + t2 – 2ts , which implies ts = 1/2 (t1 + t2 – T)

If you look closely at these four bold-faced equations, they tell you that you can determine T, L, D and ts , properties of the loop and the splash, if you know t1, t2 and t3 and v, which are all things that you can measure without going anywhere. From this point of view t4 is a bonus, a nice check that things are working as expected.

Even better, if you have a friend down the canal who makes the same measurements, that friend won’t get the same answers for t1, t2, t3 and t4 ; the waves arrive at different times for your friend than for you. But when you obtain T and L and ts from the waves you see, and your friend does the same, you’d better get the same answer — because these are properties of the loop and splash, and don’t care where either you or your friend is located.

Figure 2: A large splash occurs at time ts, and waves travel both counterclockwise (green), in which case they reach you at time t1, and clockwise (red), reaching you at time t2.

By themselves, these equations do not prove the canal is round, though they are consistent with it. They only tell you that it’s a loop of length L, with no kinks which could cause extra reflections. Still, it’s a lot of information for a very low price, without taking a boat around the loop, walking all around it, or sending up a drone to take a photograph. The waves have done all the work for you.

Figure 3: After the counterclockwise wave passes you at time t1, it continues round the canal, and passes you again at time t3 = t1 + T.

Two Dimensions: the Possibly Round Surface of the Earth

What would be different if you lived on a sphere?  (A subtlety of language: by “sphere,” I do not mean “ball”, which is three-dimensional; I mean the surface of the ball, which is two-dimensional.  In this terminology, the Earth is a ball, while its surface is a sphere, approximately.)  Again, of course, you always have the option of traveling round the sphere yourself and exploring it, checking that no matter what direction you go in, if you walk in a perfectly straight line, you will always come back to your starting point after you travel the circumference of the sphere.  But that’s expensive and time-consuming and not very practical.  What other options do you have?

You could wait for a big splash in the atmosphere — a natural one like a volcanic eruption, or an artificial one of similar size (fortunately now forbidden by nuclear testing treaties).   This opportunity, if you want to call it that, came this past week, unfortunately near an inhabited area and at the ocean’s surface within the Kingdom of Tonga, with ensuing loss of life, as well as the destruction of crops and homes; the resulting tsunami even took lives far across the Pacific ocean.  It’s not an experiment we would happily have chosen. But nature has carried it out without asking us; we may as well learn what we can from it.

When water hits hot magma and turns to steam, there’s an immense release of energy, especially if the magma is itself packed with compressed gasses. This is partly why some of the largest explosions in the last two hundred years have occurred when volcanic islands self-destructed; Krakatoa is the most famous.  The latest estimate as of the time of writing is that the one in Tonga last week was overall perhaps only 1/20 times as powerful as Krakatoa, but its plume was enormous, and its shock waves were strong enough to be detected multiple times, in many places, as they traveled round and round the Earth.

The shock wave emanated from the explosion in all directions, moving outward as an ever expanding circle, as you can guess by pure reasoning but also as confirmed by satellite.  After traveling 1/4 of the way around the Earth, the wave front reached a maximum extent — the same size and shape as the equator, though with a different orientation — and then shrank again, converging to a point in Africa exactly halfway around the Earth from the explosion’s location. (A nice visualization of this, and of what I’ll say next, can be found here.) Then the shockwave continued onward, again expanding to the Earth’s full extent, and then shrinking and converging on the very spot where it was created in the first place.   And this process repeated, until the shock wave, gradually losing its energy, faded beyond the point of detectability.

This pattern of outward expansion, convergence to the opposite point, return-ward expansion, and convergence to the original point, means that the waves from the explosion passed every point on Earth multiple times, and did so first moving away from the explosion, then returning, then again moving away, and again returning, until finally they were too small to observe.  That this pattern was seen everywhere, in countries widely spread around the globe, by both professional and civilian weather stations, gives some qualitative evidence that the Earth’s a smooth object with a rounded surface of some type.  For example, here is the pattern of multiple waves crossing, returning, re-crossing and re-returning as measured by weather stations in China; we can see three wave passages clearly (the fourth is too dim to measure well).  And here is a similar pattern in the Netherlands; though it’s only at one location, and only the main shock wave is detected, the shock is seen six times. 

What’s nice is that for a sphere — and only for a sphere [see caveat below] — the equations I wrote earlier for a circular canal still hold, and importantly, they hold everywhere, and have to give the same circumnavigation time T and the same splash time ts. That’s because if you are on a sphere, motion away from the volcano (or indeed any point), in any direction, will take you on a circular path of length equal to the sphere’s circumference. On any other shape, this won’t be true.

[To be fair, I am making a couple of assumptions: for instance, that the volcano was located on a random, not special, point on the Earth. (For example, if the Earth’s surface was oblong instead of circular, then the two points at either end of the oblong are special.) To make a long story short, there are still loopholes to the argument I’m giving here, but they are only relevant if there are very special and unlikely coincidences. Additional volcanoes, would quickly close the loopholes.]

In particular, the equations I introduced earlier should hold in China, about 1/4 around the Earth from Tonga. And they should also hold in the Netherlands, much further from Tonga, in a quite different direction. If the Earth had an uneven shape, then the time to go round the Earth in the direction from Tonga to Beijing would be different from the time to go round it from Tonga to the Netherlands; you wouldn’t get the same T. And if the Earth had edges (as in the absurd flat-earth map), you would see reflection waves; you wouldn’t get the same T or the same ts, and the second big wave across China wouldn’t look like the original one retracing its steps (a fact which already gives qualitative evidence for a round Earth.)

Using publicly available data from anywhere in the world, including what I’ve shown you from China and from the Netherlands, we can check ourselves that the Earth’s a ball and measure its circumference. Let’s do it.

So as not to spoil the fun, I’m going to wait until after the weekend to post the results. You are all encouraged to gather your children together and to try to measure:

  • T, the time it took for the waves to travel around the Earth; do this both with the data from the Netherlands and that from China; do you get similar answers?
  • ts, the time when the eruption occurred; use both the data from the Netherlands and the data from China (make sure you’re using UTC time, so you don’t get confused by time zones). Do you get similar and roughly accurate answers? Is it close to the time reported in this article?
  • v, the speed of the waves, which you can determine by watching how long it takes them to cross a part of China and comparing that time with the distance of that path; caution, make sure you trace a path perpendicular to the wave front.
  • C = T v , the circumference of the Earth, equal to the time it took for the waves to circumnavigate the Earth times their speed. Can you get fairly close?

Caution: You’re not going to get exactly the precise scientifically-known answers, nor will your answers be perfectly consistent, because the data I’ve linked to was neither taken nor presented with scientific levels of precision. But you should be able to get within 10-20%, enough to convince you the Earth’s surface pretty darn close to a sphere. If you want more precision, I’m sure precision data is available (anybody have a good link?) [Also note that there are some extra waves seen in the China map, some of them reverberations from the original explosion, and some due to later, smaller explosions; they travel in the same directions as the original ones, showing they come from the same place. For our purpose here, just keep your focus on the biggest waves.]

The point is that we can learn the Earth is ball-shaped without ever stepping off the Earth, and in fact without even traveling; and we can even learn, from the timing, how big the Earth is.  All it takes is a natural explosion, measurements from a few places, some logic, and simple algebra.   The data is now publicly available, and every science teacher in the world ought to encourage their teenage students to do this exercise!  Not only does it confirm we live on a sphere, it shows that one needs neither a photograph taken from outer space, nor a flight around the world, nor specialized map-making skills, to obtain that proof.

Three Dimensions: The Universe

Now what about the universe as whole?  The Earth and Sun are carrying us along as they travel within a three-dimensional surface.  What is its shape?  How can we know?  [There is also the question of the four-dimensional surface that makes up the space and time of the universe.  I’m not addressing that here, that’s even more complex.]

A circle is a one-dimensional sphere; the surface of the Earth (not its interior) is a two-dimensional sphere. Could the universe be a three-dimensional sphere?   We can’t stand outside it to find out.  In fact it’s far from clear there is meaning to “outside” since, after all, it’s the universe, and might be everything there is. Nevertheless, we can imagine, at least, trying to do a similar experiment.  If there were a huge supernova explosion, or a tremendous flare from a distant black hole as it ripped apart a star, maybe we would see the light arrive from one side of us, and then later see it arrive from the other side, and yet again from the first direction, and so on.

Back before we knew the huge scale of the universe and the tiny speed of light, that might have seemed plausible.  We can’t hope to do anything like this, unfortunately.   But it’s not because the question makes no sense.  The natural Tonga volcano experiment worked thanks to the fact that it’s a small world (after all) and the speed of sound is relatively fast, so it all took less than a day or two.  In the universe, it’s the reverse; it’s a big place and the speed of light is relatively slow.  Our own galaxy, the Milky Way, is itself 100,000 light-years across [i.e. it’s so big that it takes 100,000 years for light, traveling at the fastest speed our universe allows, to cross it], so even if our galaxy were the entire cosmos, as was thought until the 1920s, it would take at least 100,000 years to do this experiment.  And of course we now know the universe is immensely larger than our own galaxy; indeed the most recent map of galaxies extends out, for the brightest galaxies, as far as 10,000,000,000 light years.  Hopeless.

Nevertheless, the possibility that the universe has an interesting shape, and though huge might be small enough that we could see some evidence of its shape, remains a topic of research.  The light from events in the distant past might give us clues.  While a blast wave isn’t something we’d be able to see from multiple perspectives, a long-lasting bright spot on the sky could potentially be seen reaching us from different paths around a complex universe.  The fact that the universe has been expanding over the billions of years since the Hot Big Bang began complicates the thinking, but also provides opportunities.

To give insight into how this could be done is beyond the scope of this blog post, but if you’re curious about it, you might try this long-form article from Quanta Magazine (a highly recommended source for interesting articles.)  

The Lesson for Humankind

The big lesson here: geometry can be learned from the inside.  You don’t need to be outside an object to map it and learn its shape and size. That this is possible explains how mapmakers knew the shapes of continents long before satellites, and how one can determine that the universe is expanding while remaining within it (though the story of how scientists did this, without using the methods described in this post, is for another day.) And if the object is finite, so that no wave can travel forever without eventually returning to you, then it’s possible to infer its shape just by learning how waves travel and bounce around the object. That’s how the depth of the ocean’s deepest point was recently measured, as I described in my last post; and that’s how children (of all ages) should prove for themselves, using publicly available data from last weekend and simple algebra, that the Earth is indeed round.

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POSTED BY Matt Strassler

ON January 21, 2022

I took a short break from other projects this weekend. Poking around, I found three particularly lovely science stories, which I thought some readers might enjoy as well. They all involve mapping, but the distances involved are amazingly different.

Starbirth Near to Home

The first one has been widely covered in the media (although a lot of the articles were confused as to what is new news and what is old news.) Our galaxy is full of stars, but also of gas (mainly hydrogen and helium) and dust (tiny grains of material made mainly from heavier elements forged in stars, such as silicon). That gas and dust, referred to as the “interstellar medium”, is by no means uniform; it is particularly thin in certain regions which are roughly in the shape of bubbles. These bubbles were presumably “blown” by the force of large stellar explosions, i.e. supernovas, whose blast waves cleared out the gas and dust nearby.

It’s been known for several decades that the Sun sits near the middle of such a bubble. The bubble and the Sun are moving relative to one another, so the Sun’s probably only been inside the bubble for a few million years; since we’re just passing through, it’s an accident that right now we’re near its center. Called simply the “Local Bubble”, it’s an irregularly shaped region where the density of gas and dust is 1% of its average across the galaxy. If you orient the galaxy in your mind so that its disk, where most of the stars lie, is horizontal, then the bubble stretches several hundred light-years across in the horizontal direction, and is elongated vertically. [For scale: a light-second is 186,000 miles or 300,000 km; the Sun is 8 light-minutes from Earth; the next-nearest star is 4 light-years away; and our Milky Way galaxy is about 100,000 light years across.] It’s been thought for some time that this bubble was created some ten to twenty million years ago by the explosion of one or more stars, probably siblings that were born close by in time and in space, and which at their deaths were hundreds of light-years from the Sun, far enough away to do no harm to Earthly life. [For scale: recall the Sun and Earth are about 5 billion years old.]

Meanwhile, it’s long been suggested that explosion debris from such supernovas can sweep up gas and dust like a snowplow, and that the compression of the gas can lead it to start forming stars. It’s a beautiful story; large stars live fast and hot, and die young, but perhaps as they expire they create the conditions for the next generation. [A star with 40 times the raw material as the Sun has will burn so hot that it will last only a million years, and most such stars die by explosion, unlike Sun-like stars.]

If this is true, then in the region near the Sun, most if not all the gas clouds where stars are currently forming should lie on the current edge of this bubble, and moreover, all relatively young stars, less than 10 million years old, should have formed on the past edge of the bubble. Unfortunately, this has been hard to prove, because measuring the locations, motions and ages of all the stars and clouds isn’t easy. But the extraordinary Gaia satellite has made this possible. Using Gaia’s data as well as other observations, a team of researchers (Catherine Zucker, Alyssa A. Goodman, João Alves, Shmuel Bialy, Michael Foley, Joshua S. Speagle, Josefa Groβschedl, Douglas P. Finkbeiner, Andreas Burkert, Diana Khimey & Cameren Swiggum) has claimed here that indeed the star-forming regions lying within a few hundred light-years of the Sun all lie on the Local Bubble’s surface, and that nearby stars younger than ten million years were born on the then-smaller shell of the expanding bubble. Moreover they claim that the bubble probably formed 14-15 million years ago, at a time when the Sun was about 500 light-years distant.

(more…)
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POSTED BY Matt Strassler

ON January 17, 2022

There’s a plot afoot. It’s a plot that involves a grid of earthquake locations, under the island of La Palma.

Conspiracy theory would be hysterically funny if it weren’t so widespread and so incredibly dangerous. Today it threatens democracy, human health, and world peace, among many other things. In the internet age, scientists and rational bloggers will have no choice but to take up arms against it on a regular basis.

The latest conspiracy theory involves the ongoing eruption of the Cumbre Vieja volcanic system on the island of La Palma. This eruption, unlike the recent one in Iceland, is no fun and no joke; it is occurring above a populated area. Over the past month, thousands of homes have been destroyed by incessant lava flows, and many more are threatened. The only good news is that, because the eruption is relatively predictable and not overly explosive, no one has yet been injured.

The source of the latest conspiracy theory is a graph of earthquakes associated with the eruption. You can check this yourself by going to www.emsc-csem.org and zooming in on the island of La Palma. You’ll see something like the plot below, which claims to show earthquake locations. You can see something is strange about it: the earthquakes are shown as occurring on a grid.

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Picture of POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON October 26, 2021

Last Thursday, an experiment reported that the magnetic properties of the muon, the electron’s middleweight cousin, are a tiny bit different from what particle physics equations say they should be. All around the world, the headlines screamed: PHYSICS IS BROKEN!!! And indeed, it’s been pretty shocking to physicists everywhere. For instance, my equations are working erratically; many of the calculations I tried this weekend came out upside-down or backwards. Even worse, my stove froze my coffee instead of heating it, I just barely prevented my car from floating out of my garage into the trees, and my desk clock broke and spilled time all over the floor. What a mess!

Broken, eh? When we say a coffee machine or a computer is broken, it means it doesn’t work. It’s unavailable until it’s fixed. When a glass is broken, it’s shattered into pieces. We need a new one. I know it’s cute to say that so-and-so’s video “broke the internet.” But aren’t we going a little too far now? Nothing’s broken about physics; it works just as well today as it did a month ago.

More reasonable headlines have suggested that “the laws of physics have been broken”. That’s better; I know what it means to break a law. (Though the metaphor is imperfect, since if I were to break a state law, I’d be punished, whereas if an object were to break a fundamental law of physics, that law would have to be revised!) But as is true in the legal system, not all physics laws, and not all violations of law, are equally significant.

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POSTED BY Matt Strassler

ON April 12, 2021

There have been dramatic articles in the news media suggesting that a Nobel Prize has essentially already been awarded for the amazing discovery of a “fifth force.” I thought I’d better throw some cold water on that fire; it’s fine for it to smoulder, but we shouldn’t let it overheat.

There could certainly be as-yet unknown forces waiting to be discovered — dozens of them, perhaps.   So far, there are four well-studied forces: gravity, electricity/magnetism, the strong nuclear force, and the weak nuclear force.  Moreover, scientists are already fully confident there is a fifth force, predicted but not yet measured, that is generated by the Higgs field. So the current story would really be about a sixth force.

Roughly speaking, any new force comes with at least one new particle.  That’s because

  • every force arises from a type of field (for instance, the electric force comes from the electromagnetic field, and the predicted Higgs force comes from the Higgs field)
  • and ripples in that type of field are a type of particle (for instance, a minimal ripple in the electromagnetic field is a photon — a particle of light — and a minimal ripple in the Higgs field is the particle known as the Higgs boson.)

The current excitement, such as it is, arises because someone claims to have evidence for a new particle, whose properties would imply a previously unknown force exists in nature.  The force itself has not been looked for, much less discovered.

The new particle, if it really exists, would have a rest mass about 34 times larger than that of an electron — about 1/50th of a proton’s rest mass. In technical terms that means its E=mc² energy is about 17 million electron volts (MeV), and that’s why physicists are referring to it as the X17.  But the question is whether the two experiments that find evidence for it are correct. (more…)

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POSTED BY Matt Strassler

ON November 25, 2019

The untimely and sudden deaths of Steve Gubser and Ann Nelson, two of the United States’ greatest talents in the theoretical physics of particles, fields and strings, has cast a pall over my summer and that of many of my colleagues.

I have not been finding it easy to write a proper memorial post for Ann, who was by turns my teacher, mentor, co-author, and faculty colleague.  I would hope to convey to those who never met her what an extraordinary scientist and person she was, but my spotty memory banks aren’t helping. Eventually I’ll get it done, I’m sure.

(Meanwhile I am afraid I cannot write something similar for Steve, as I really didn’t know him all that well. I hope someone who knew him better will write about his astonishing capabilities and his unique personality, and I’d be more than happy to link to it from here.)

In this context, I’m gratified to see that the New York Times has given Ann a substantive obituary, https://www.nytimes.com/2019/08/26/science/ann-nelson-dies.html, and appearing in the August 28th print edition, I’m told. It contains a striking (but, to those of us who knew her, not surprising) quotation from Howard Georgi.  Georgi is a professor at Harvard who is justifiably famous as the co-inventor, with Nobel-winner Sheldon Glashow, of Grand Unified Theories (in which the electromagnetic, weak nuclear, and strong nuclear force all emerge from a single force.) He describes Ann, his former student, as being able to best him at his own game.

  • “I have had many fabulous students who are better than I am at many things. Ann was the only student I ever had who was better than I am at what I do best, and I learned more from her than she learned from me.”

He’s being a little modest, perhaps. But not much. There’s no question that Ann was an all-star.

And for that reason, I do have to complain about one thing in the Times obituary. It says “Dr. Nelson stood out in the world of physics not only because she was a woman, but also because of her brilliance.”

Really, NYTimes, really?!?

Any scientist who knew Ann would have said this instead: that Professor Nelson stood out in the world of physics for exceptional brilliance — lightning-fast, sharp, creative and careful, in the same league as humanity’s finest thinkers — and for remarkable character — kind, thoughtful, even-keeled, rigorous, funny, quirky, dogged, supportive, generous. Like most of us, Professor Nelson had a gender, too, which was female. There are dozens of female theoretical physicists in the United States; they are a too-small minority, but they aren’t rare. By contrast, a physicist and person like Ann Nelson, of any gender? They are extremely few in number across the entire planet, and they certainly do stand out.

But with that off my chest, I have no other complaints. (Well, admittedly the physics in the obit is rather garbled, but we can get that straight another time.) Mainly I am grateful that the Times gave Ann fitting public recognition, something that she did not actively seek in life. Her death is an enormous loss for theoretical physics, for many theoretical physicists, and of course for many other people. I join all my colleagues in extending my condolences to her husband, our friend and colleague David B. Kaplan, and to the rest of her family.

Picture of POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON August 28, 2019

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