Of Particular Significance

# Chapter 2, Figure 2

From Chapter 2:

Now in what sense are people in Miami going round in a circle from the point of view of people in Boston?

## One City Circles Another

In the figure below (click to animate it), I’ve drawn a rotating Earth and drawn an arrow between two cities. (I didn’t use Boston and Miami because they’re a bit too close together and the arrow would be rather short; instead I used two cities that are about 3000 miles apart, so that the arrow is easier to see. In any case, the conceptual point is not dependent on the specifics, as we’ll see.)

Notice that the arrow points in different directions as the Earth rotates. For instance, compare these two images showing the Earth twelve hours apart (i.e. from one moment to another half a rotation later.) You can see that the arrow from the northern city to the southern city points leftward at the earlier time and rightward at the later time.

In fact, if we look at the arrow from the point of view of the northern city, the length of the arrow always stays the same, but the orientation of the arrow changes continuously, as shown below.

And thus. from the point of view of people in the northern city, who feel themselves stationary (just as we all do), the southern city is going round in a circle, one that lies to their south and is always at the same distance from the northern city. This is true for any two such cities, including Boston and Miami. The Earth’s equator moves, relative to its poles, at over 1000 miles (over 1500 km) per hour, so it’s no surprise that the relative speeds of most pairs of cities is at hundreds of miles (hundreds of km) per hour.

Miami residents view this in a similar way, except that they point the arrow in Fig. 1 in the opposite direction. For them, Boston goes in a circle that lies to their north.

## Cities and Circles

This isn’t limited to cities that are oriented to each others’ north or south. It’s true for most pairs of cities. In Fig. 4, I add two more cities into Fig. 1, giving three rotating arrows joining three cities to a fourth one. (Click to animate it.)

Again, from the point of view of the city from which the arrows emanate, all three arrows move in circles, as shown in Fig. 5. Thus, from the point of view of those living in that city, the residents of the other cities all move in daily circles of different diameters and locations.

All this motion of circles and circles goes undetected and unrecognized because of Galileo’s principle of relativity — the oldest law of physics never to have been modified. This principle, long predating Einstein. states that straight-ahead steady motion cannot be detected at all — and as a consequence, motion in a very large circle is almost impossible to detect. It’s this same principle that makes us unaware of the Earth’s rotation, and can trick us into imagining both ourselves and our cousins in other cities as stationary relative to each other.

And it’s this principle, more than any other, that makes our universe an “Impossible Sea”.

### 13 Responses

1. RichardF says:

My copy of your book arrived on the weekend and it’s truly an excellent read – thank you! (about halfway through so far).
One thing that troubles me is the definition of “moving relative to” in rotating references frames. My naive intuition is that if you draw a line segment L between two cities (points) on a sphere A and B and then define this line segment as your frame of reference, then they aren’t moving relative to each other. To visualise, make the sphere transparent and have the line goes through the sphere itself (chord).
In the reference frame L, from time T1 to time T2 while the sphere is spinning, neither the distance from A to B changes, nor does the angle from A to B change. i.e. an observer at A looking bore sighted at B doesn’t have to rotate their view to keep looking straight at B. So the vector AB at times T1 and T2 hasn’t changed (in the reference frame L).
If you anchor A at a fixed point (like you do in the animations) and then then measure again, then sure, an observer at A will keep having to rotate their view to keep looking at B as their segment rotates in space, but that’s akin to having an outside observer in a separate, non-rotating reference frame.
When I spin an umbrella in my hand, then yes, the tip of the umbrella is moving relative to my hand (because the handle keeps sliding in the pivot point of my hand), but in the rotating reference frame of the umbrella, the handle and tip aren’t moving relative to each other.
A and B could both feel they’re accelerating radially*, but that’s not the question – the question is do they consider themselves moving relative to each other?
Pick another point C on the sphere. If you ask observers at points A and B what’s the angle and distance to C at time T1 and again at T2, their answers won’t change. But if you ask the same question about some star S outside the sphere, then of course their answers will change.

The circles you trace out only seem to exist if you use the stars or some other reference frame outside the sphere to define them. i.e. they are all projections from the frame of the sphere to an outside space.

I’m sure my questions have been asked ad nauseum by undergraduates, but my tired old brain hasn’t found a succinct response yet. Am I trying to find a “rotating” analogy to Einstein’s man floating in a falling elevator, when there really isn’t one?

* or do they – my murky memory of university recalls some conversation between Einstein and Mach on the subject…

1. These are very good, subtle questions, and it’s true that I haven’t fully addressed them here.

Before I try to answer, however, may I suggest that you give the following posts a read? At least the first one, and perhaps the second. They do address some of these issues. Once you’ve done that, swing back to me here, and let me know which of your questions still need answering.

1. The most thorough: https://profmattstrassler.com/2022/02/07/the-best-proof-that-the-earth-spins/

2. An intro you might enjoy starting with: https://profmattstrassler.com/2022/02/02/how-can-you-check-that-the-earth-spins/

3. A brief summary of the first two posts, with a few other side comments: https://profmattstrassler.com/2023/09/21/how-to-tell-that-the-earth-spins/

1. RichardF says:

Thanks Matt, these are really clear and clever explanations.
I guess what they don’t address is the relative motion of two observers fixed in the same rotating reference frame. (All the talk about cannon balls, satellites, hurricanes etc is about objects which have
– at least partially – left the rigid reference frame of their departure points. Even the magic wand moves relative to the Earth because the end on the ground pivots like a thigh bone in a hip joint.)

Like I said in the question above, A and B both know they’re rotating, that’s not in doubt. But what definition of “moving relative to each other” works for A and B just using measurements they can perform in that frame? (The curling cannon ball just confirms they’re rotating, not that they’re moving relative to each other).
In a non-rotating frame (from memory, an inertial frame?), we just say either the distance or angle between them changes. Two joggers, running in a straight line next to each other, both agree they’re moving (relative to the track) but not moving relative to each other.

Imagine two people standing in a long train carriage, facing each other from either end of the carriage. As the train goes (rotates) around a bend, they don’t have to do anything to keep staring straight at each other* and their distance doesn’t change. But for an observer on the ground near the start of the bend, the leading and trailing person would exhibit different relative motions to them at any given time. Everyone agrees on this.

In your example above, make the train carriage stretch from Miami to Boston (it would have to go through the Earth). It’s not moving on tracks, just sitting there, wedged in the crust between those two cities. As the Earth turns, the two people keep staring at each other (they have very good eyesight). Or they might have two mirrors and a torch and both agree that the return light time between them doesn’t change and they don’t have to change the tilt of the mirrors to keep the light bouncing back and forth.
They even both have their perfect gyroscopes and both acknowledge they are rotating (relative to the stars they cannot see from inside the carriage). But they insist neither has moved relative to the train carriage or to each other. How do we prove them wrong?
——
* Thinking about this more deeply, perhaps in a real physical world, A and B would experience different torque at either ends of the carriage and as this force travelled along the carriage and up through their legs and torso to their eyes, they would indeed rotate a tiny amount at a molecular level to keep bore-sighted at each other. — this paragraph might be nonsense, but I’m just trying to ground the thought experiment in reality. I did a pure maths degree (many moons ago) rather than physics, so I think in terms of lines of zero width with points of zero size facing each other without having to “turn”.

2. Excellent! The law of inertia (linear for stones and arrows, or “circular”, for planets) was discovered by Buridan in Paris circa 1340 CE. Buridan also discovered impetus ( = mv, momentum): Buridan’s Quaestiones super libros De generatione et corruptione Aristotelis explained that Aristotle’s physics was false because he had neglected air resistance and other forms of friction. Here is Buridan:

“When a mover sets a body in motion he implants into it a certain impetus, that is, a certain force enabling a body to move in the direction in which the mover starts it, be it upwards, downwards, sidewards, or in a circle. The implanted impetus increases in the same ratio as the velocity. It is because of this impetus that a stone moves on after the thrower has ceased moving it…”

Buridan’s students (Oresme, Albert of Saxony, Oxford Computing school, etc.) developed calculus, giving geometrical proofs, for example of the mean value theorem… Newton used the same geometrical calculus, by the way… Buridan explained that circular inertia explained how planets rotated around the Sun. But, because of relativity, we could not tell one way or another. Thus we may as well follow “scripture which says that the Sun rotates around the Earth”. Buridan, rector of the university of Paris, was adviser to four kings and lover of one queen. He was untouchable, and his astounding career marked the launch of the scientific renaissance…

However, a century later, the Inquisition (propped by Ockham’s partisans) condemned all the works of Buridan… Except in Krakow, Poland, where Copernicus had to study them…

1. Buridan was a very intelligent and insightful proto-scientist. Still, we should not overstate his abilities or his originality. There were earlier thinkers who knew some of these things, too. And there was a lot that he was missing. [I do mention him in the book, by the way, though the book isn’t intended to be a history.]

The problems that all these proto-scientists had were (a) they did not have a completely coherent view of how the universe worked, having found only some of the main principles, and therefore they were still open to criticism, and (b) they were surrounded by a wide range of incoherent views, which made it hard for theirs to stand out at the time; we only pick them out of the heap because of hindsight, since we know they had the right ideas, while their contemporaries did not know that.

What makes Newton exceptional is that he had all the right ideas and almost none of the wrong ones, and so he could put them together into a completely coherent story which had been impossible for anyone to do prior to him. This was possible because of an interplay of data and theory. So successful were the observations of Brahe combined with the mathematical insights of Kepler, along with Galileo’s experiments and theory and those of people who followed, such as Huygens, that they began clearing the decks of all the bad ideas, and made more and more of the universe’s workings explicable. That set the stage for Newton.

So Buridan wasn’t the only man ahead of his time. He was, however, a light in the wilderness, a light whose brightness we can only see today, and which was not so visible when he was alive. Copernicus is similar; his vision was not accepted widely for 100 years. Also right but too early were Aristarchus and Aryabhata; there are also Arab, Turkish and Persian scholars from the Muslim golden age.

The lesson here is that it is not merely enough to have some correct ideas when untangling a mystery. In the end, you need all of them, and in putting them all together, you will probably need to borrow some of them from earlier scholars, and rediscover others that were already known to someone else whom you may never have heard of. Einstein, too, relied on earlier work; his key achievement was a huge conceptual insight that made it possible to put several key ideas together and take them to a new level.

This is also why it’s easy to spot a crackpot. Crackpot scientists invent their own theories of the world without any knowledge of or reliance on any scientific work done in the past 100 years. No one, not even Newton or Einstein, has ever successfully done something like that.

I’m not a historian and neither was Carl Sagan but my impression was that in his Cosmos he and cowriters pushed the notion of mercantilism with its need for empirics and measures that ushered in brief periods of (pre)science.

First in Greece and then in the Muslim world before organized superstition could suppress them and lastly in Europe where the suppression failed. When I check the timing in Newton’s England it fits, and then it is a matter of opinion if that is merely correlation from biased picking of observation or causality from longitudinal study of the process.

1. It’s an argument one can make. But I suspect that it’s too simple. I suspect there can be many reasons for a society to want to keep track of seasons, stars, and/or commerce with math; and there can be many reasons that the world might be assumed to be substantially predictable or substantially unpredictable.

3. “… two cities that are about 3000 thousand miles apart …”

Rather big planet 😉

4. Bill says:

Um, are you making a big deal out of nothing?
Your last sentence “… leads us to imagine both ourselves and our cousins sitting in other cities as stationary.” Not me. I imagine no such thing. I think we are simply sitting on a large sphere and it’s rotating and we’re both moving by way of angular motion.
If my knowledge was limited to maybe 50,000 years ago, I would probably “imagine “ we were both sitting on a flat plane and not moving relative to each other. Understandably so.
I very much appreciate and enjoy you and your writings.

1. Your understanding of astronomy is commendable. Your knowledge of history and psychology, less so.

First, “50,000 years” is a massive overestimate; the real number is much less than 1% of this. As recently as 424 years ago — 50 years after Copernicus’ had died — Tycho Brahe, the world’s greatest astronomer at the time, insisted that the Earth could not possible be rotating. Yes, people knew it was a sphere since classical Greece, but not that it rotates. I’m sure that in 1800, and probably even 1900, the majority of humans on the planet were still unaware (or at least unconvinced) that the Earth is a rotating sphere.

Psychologically, it’s well known that making consistent sense of motion is challenging for most people who haven’t taken university physics in college, and even for many who have. Any physics teacher will confirm this, but you can check this for yourself. (1) Ask a friend how fast they think they are currently moving, (2) point out the Earth is going round the Sun at 20 miles per second, (3) repeat step 1, (4) point out the Earth and Sun are going round the galaxy at 150 miles per second, (5) repeat step 1. By step 5, most people struggle to give a sensible answer to the question — for good reason. To give a sensible, coherent answer is hard. It goes against basic psychology, and against the English language, which reflects that psychology.

2. Note that rotation is not inertial, but involves detectable accelerations, and so can be measured in a laboratory, for example using what is called the Sagnac effect (the difference in delay, and thus in phase, for photons going with and against the rotation of a closed path). This is the basis of ring laser gyros and fiber optic gyros, which are used routinely in aircraft and spacecraft. And, yes, in principle you can use this to tell absolute, not just relative, rotations.

So, in your closed laboratory, you absolutely could tell you were moving in a daily circle, and that is fully compliant with Galileo’s / Einstein’s principle.

1. Indeed. Nevertheless, the Earth gives us a surprising example… when we travel at 1000 miles per hour and traverse a loop every 24 hours, the rotation is too slow, despite the high speed, for us to easily detect it. We don’t feel it with our senses, obviously. It took humans tens of thousands of years to invent Foucault’s pendulum, and we still don’t have mechanical gyroscopes stable enough for us to measure the Earth’s rotation easily [though we have electronic gyroscopes that can do it.] We see its effects in the Coriolis effect, but that effect was small enough and confusing enough that people actually argued over it for a long time when first trying to determine whether the Earth spins or not. So yes, you absolutely can tell — but it’s very difficult for humans to do it, and therefore, historically, it was not without controversy.

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