But, uh… sorry. All this fun with coordinates was beside the point. The truth, falsehood, or meaninglessness of “the Earth orbits the Sun” will not be answered with a choice of coordinates. **Coordinates are labels.** In this context, they are simply ways of labeling points in space and time. Changing how you label a system changes only how you describe that system; it does not change anything physically meaningful about that system. So rather than focusing on coordinates and how they can make things appear, **we should spend some time thinking about which things do not depend on our choice of coordinates.**

And so our question really needs to be this: *does the statement *“The Earth Orbits the Sun (and not the other way round)”* have coordinate-independent meaning, and if so, is it true?*

Because we are dealing with the coordinate-independence of a four-dimensional spacetime, which is not the easiest thing to think about, it’s best to build some intuition by looking at a two-dimensional spatial shape first. Let’s look at what’s coordinate-independent and coordinate-dependent about the surface of the Earth.

If Muller is right, can’t we choose coordinates in which the Earth is flat? For example, how about these coordinates:

or these:

or these:

Look familiar? The third choice represents the coordinates beloved of flat Earthers, and indeed, in this view, the entire Earth, excepting the point at the south pole, is flat. In fact, all three maps are flat. **But they represent a flat labeling of the Earth, not a flattening of the Earth itself.**

More generally, there are an infinite number of possible maps that can be made of the Earth that will make it appear flat. (Many of them are gathered here — have fun!) Some of them will make Greenland appear larger than the continent of Africa; others will make it seem that the center of the world is the USA, or China, or Ethiopia, or the South Pole; still others will make it seem that the south tip of South America is farther from Australia than it is from parts of Asia. It doesn’t matter how things seem. Coordinates can make all kinds of things **seem** to be one way or another. But anything that a coordinate choice can change cannot be real. The only thing that matters is how things **are**. A coordinate system is just about how you * describe* those things that

To avoid confusion about what is real and what is not, you need to know how to measure things in such a way that the answers you obtain don’t depend on your coordinates. The surface area of Greenland and Africa, the shortest distance from South America to Australia or Asia, and the lack of any “central point” on our planet are all things that you can determine using whatever coordinates you choose, or in some cases without ever using coordinates at all; and if you do it correctly, you will always get the same answer no matter which coordinates you use. *(For example, to measure the area of Greenland, make yourself a 1 meter by 1 meter square pieces of cardboard at home, and then go to Greenland yourself and draw a grid on it using your cardboard, until you’ve covered the whole thing.) * You have to account for the distortions the map introduces; that takes some math, but if you use the math correctly, it will undo the distortions in just such a way as to assure that all true facts remain true facts. **Do not let yourself be confused by the mere appearance of the map.**

*[… which is to say… just because I’ve chosen to draw the solar system so that it appears as though the Sun orbits the Earth doesn’t automatically mean that the Sun does orbit the Earth.]*

So, how can we determine if a space is or isn’t flat? Here’s one approach: Take a square-shaped walk: walk N paces, turn right 90 degrees, walk N paces, turn right 90 degrees, walk N paces, and turn right 90 degrees, and walk N paces. On any surface, if N is small enough you will come back to your starting point. As N increases, does this remain true? If the space is flat, it will always be true, no matter how long your walks, no matter where you start and no matter which direction you go — as long as your walk doesn’t collide with the edge of the space. But if, as N increases, you find your end point is further and further from your starting point, that tells you the space is not flat.

*Of course the Earth, or any real-world surface isn’t expected to be exactly a sphere or exactly flat; it has wiggles on it, in the form of hills and valleys. So we have to allow for these statements to not be exactly correct. However, the largest wiggles on Earth are only about 10 miles deep or tall, while the planet is 24000 miles around, so we should have no trouble distinguishing a nearly-flat Earth from a nearly-spherical Earth.*

Imagine you start at the north pole, walk 6225 miles (about 10000 km) in any direction, turn right 90 degrees, walk 6225 miles, turn right 90 degrees, walk 6225 miles, turn right 90 degrees, and walk 6225 miles again. If the Earth were flat, you’d have ended up back at the north pole after the fourth leg of your trip. But on the real Earth, the first leg of your trip brings you to the equator; the second is along the equator; the third brings you back to the north pole, and the fourth takes you back to the equator. End of discussion; **the Earth is not flat.**

The coordinates we put on the Earth’s surface play no role in this determination, because in all this, you never needed to know anything about coordinates applied across the whole planet. The distance you walked can be measured by the wear and tear on your shoes; the direction you walk in each segment is a straight line (one foot in front of the other), and you can make a 90 degree turn using a straight-edge that you carry with you.

Even if we accept the Earth isn’t flat, can’t we choose coordinates that make it look like a cucumber, a pear, a peanut, or maybe even a frisbee? Sure. We can label points however we like, and then draw them however we like, so that it appears to be very different from a sphere. In fact we could use standard latitude-longitude coordinates and project them onto a plane using weird lenses to make them look like any shape we want. **But the Earth is still a sphere**.

How do we see that? What’s true for the north pole is true for every point on Earth. Starting in any direction, if you walk in a **triangle**, not a square, whose sides are length 6225 miles (measured by the number of steps you take as you go in a straight line, and not requiring any coordinate system), and whose angles are 90 degrees, you will come back to your starting point.

Since all points and all directions have this feature, the Earth’s surface is (approximately) a “homogeneous isotropic space” (all points and directions are equivalent). The fact that a triangle with 90 degree angles can bring you back to your starting point means it is “positively curved”, and a two-dimensional positively curved homogeneous isotropic space must be a complete sphere.

I used a different approach to prove the Earth’s a sphere using the Tonga eruption’s pressure waves, way back when I started this series. On a sphere, any journey in a straight line, moving in any direction from any point on the surface, will come back to itself after having traveled the same distance (the sphere’s circumference), or equivalently (if the speed of the journey is constant) having taken the same amount of time. You can tell this without coordinates; you just need to observe that all the pressure waves from Tonga (and from Krakatoa’s eruption also) roughly intersected each other halfway around and all the way around the world.

Both of these coordinate-invariant statements involve studying large paths on the surface. A different approach is to study the properties of the surface using relatively short paths, the method of measuring “local curvature.” There are various ways to do this, but the easiest is to take a triangular walk — any one you like — such that on the third leg of the triangle you return to your initial point. At each of the three points on your walk where you changed direction, measure the angle. We all know that on a flat surface, the sum of the three angles will be 180 degrees. On a positively curved surface like a sphere or cucumber, it will be larger than 180 degrees. The amount of excess angle will grow as we take larger and larger triangles, and we can use this to determine how curved the Earth’s surface is… never using a coordinate system.

Notice that all the coordinate systems we’ve talked about so far “rotate with the Earth”, which is to say, they make it appear that the Earth is * not* rotating. Does that mean it doesn’t rotate? or that rotation is meaningless?

Of course not. Foucault pendulums and gyroscopes do what they do, showing the Earth rotates relative to the slowly drifting stars, independent of whether we set longitude to be fixed upon the Earth’s continents, or whether we fix it longitude in the stars and let the Earth rotate underneath them, or choose some other time-dependent coordinate system. In this sense, the Earth’s rotation is coordinate-independent.

Clearly, we need to be very cautious about drawing any conclusions from coordinates. Being cavalier about coordinates will lead to mistakes. The mere fact that I can redraw the solar system in geocentric coordinates has absolutely nothing to say about whether “Sun orbits the Earth” is false, or meaningless, etc.

A critical issue is to identify what is coordinate-independent and what is not; anything that is not truly coordinate-independent is suspect. Sometimes a particular coordinate-dependent viewpoint is useful, but you should always understand what alternative viewpoints would tell you, so that you don’t overinterpret. (Over the coming months we will see just how deeply this issue, in various sophisticated forms, permeates **all **of modern high-energy physics.)

Another lesson: imagine someone told you that even though spherical coordinates (latitude and longitude) are the **simplest** coordinates (because they make the Earth look simple, and also make the equations describing it simpler), they don’t reflect anything meaningful about the Earth — that with a different choice of coordinate system, the Earth could just as well be a pear or a cucumber or a log. Or flat. That all of these things are equally true.

This person would have drawn the wrong conclusion. Spherical coordinates are certainly not the “right” coordinates — coordinates are arbitrary — but **the fact that they are so simple on the Earth reflects something real about the Earth.** Spherical coordinates are simple not because they are right but because the underlying space is a sphere. Had the Earth been a knobby, blobby, spiky shape, then spherical coordinates would have been no simpler than any others.

So simple coordinates, despite their arbitrariness, can ** reflect** something important and meaningful about the underlying physical system. And that raises a question. We all agree, including professor Muller, that heliocentric (Sun-centered) coordinates make the appearance and behavior of the solar system, and the equations describing its behavior, somewhat simpler. We also all agree that coordinates are arbitrary. But should we then conclude that the simplicity of Sun-centered coordinates for the solar system is a pure fluke? Might it not reflect something simple about the underlying space-time geometry — something which could perhaps tell us that

A number of readers suggested, in one way or another, that we go far from the Sun and Earth and use the fact that out there, far from any complications, Newtonian physics should be good. From there, we can look back at the Sun and Earth, and see what’s going on in an unbiased way. Although Muller would say that you could still claim the Sun orbits the Earth by using “geocentric” coordinates centered on the Earth, these readers argued that such coordinates would not make sense in this distant, Newtonian region.

Are they correct about this?

Let’s make that last argument more precise. About a week ago, I offered you some geocentric coordinates; see below, and also the last two figures in that previous post. These are * non-rotating Cartesian coordinates centered on the Earth*. They can be defined in the usual heliocentric (Sun-centered) coordinate system, the one we normally take for granted, by centering a non-rotating grid on the Earth, shown in Figure 1. This figure shows a simplified solar system (the Sun at center, with Mercury, Venus, Earth, Mars and Jupiter in circular orbits), as well as the Earth-centered grid which follows the Earth around in its orbit.

When we now move to the coordinate system defined by the grid in figure 1, the Earth becomes stationary and the Sun starts moving around it, as shown below. The other planets do some strange loops-within-loops — epicycles, they are called.

The argument against such geocentric coordinates is that it’s not just nearby planets like Jupiter that undergo epicycles. **So would all of the distant stars!** Each will move in a little loop, once an Earth-year! Now indeed, that sounds bad; why would we accept a coordinate system in which extremely distant stars like Sirius or Vega or Betelgeuse would travel in loops that somehow know how long it takes for the Earth to go around the Sun?

Such complaints seem reasonable. This kind of geocentric coordinate system implicitly stretches the Earth’s influence across the entire cosmos, and that doesn’t seem to make any physical or causal sense.

That said, coordinates are just **labels**. They don’t have to make physical sense or preserve a notion of causality. Only physical phenomena have to do that. But still, it seems crazy to take coordinates seriously that have this property.

And the claim that readers implicitly made is that if you forbid these coordinates — if you use coordinates in which the distant stars are fixed, or at least traveling not in Earth-year-long loops — then you inevitably will prefer heliocentric coordinates.

But this claim, and any similar one, is wrong. ** No one said that we have to extend the coordinates out from the Earth in a rigid, Cartesian way.** Einstein claimed that physics is unchanged no matter how crazy the coordinate system you might choose to describe it. So let’s take the following coordinate system, which is warped, remains the same as the heliocentric coordinates at very large distances, but is geocentric at and near the Earth.

In this system of coordinates, here’s what the motion of the Sun and planets looks like.

The Sun goes round the Earth. Notice that Mars still moves with a significant epicycle, but the epicycles of Jupiter are almost gone. By the time you get to the distant stars, none of them are doing loops anymore. The stars, in this coordinate system, move completely independently of Earth’s motion. Yet the coordinate system has Earth as its center, with the Sun moving round it.

For those of you who suggested that it’s obvious (or near-obvious) that Earth orbits the Sun, these are the coordinates that Muller can ask you about. The only effect of these geocentric coordinates is near the Earth and Sun. No hint remains, by the time you get to the distant stars, that anything is different from heliocentric coordinates. And so, if you assumed implicitly or explicitly that because the distant stars are in nearly flat space, you could extend good heliocentric coordinates all the way down to the Sun and apply quasi-Newtonian reasoning, ** these curved geocentric coordinates raise challenging questions that you need to answer.** Does your argument, whatever it was, truly survive the use of a coordinate systems like this one? And why can’t Muller use them to show the Sun orbits the Earth?

It’s blurred out in space by imperfections in the telescopic array that is the “Event Horizon Telescope” (EHT) and by dust between us and our galaxy’s center. It’s blurred out in time by the fact that the glowing material around the black hole changes appreciably by the hour, while the EHT’s effective exposure time is a day. There are bright spots in the image that may just be artifacts of exactly where the telescopes are located that are combined together to make up the EHT. The details of the reconstructed image depend on exactly what assumptions are made.

At best, it shows us just a thick ring of radio waves emitted over a day by an ever-changing thick disk of matter around a black hole.

But it’s **our** galaxy’s black hole. And it’s just the first image. There will be many more to come, sharper and more detailed. Movies will follow. A decade or two from now, what we have been shown today will look quaint.

We already knew the mass of this black hole from other measurements, so there was a prediction for the size of the ring to within twenty percent or so. The prediction was verified today, a basic test of Einstein’s gravity equations. Moreover, EHT’s results now provide some indications that the black hole spins (as expected). And (by pure luck) its spin axis points, very roughly, toward Earth (much like M87’s black hole, whose image was provided by EHT in 2019.)

We can explore these and other details in coming days, and there’s much more to learn in the coming years. But for now, let’s appreciate the picture for what it is. It is an achievement that history will always remember.

]]>At the time, there was also hope that the EHT would produce an image of the region around the black hole at the center of our own galaxy, the Milky Way. That black hole is thousands of times smaller, but also thousands of times closer, than the one in M87, and so appears about the same size on the sky (just as the Moon and Sun appear the same size, despite the Sun being much further away.)

However, the measurements of the Milky Way’s black hole proved somewhat more challenging, precisely because it is smaller. EHT takes about a day to gather the information needed for an image. M87’s black hole is so large that it takes days and weeks for it to change substantially — even light takes many days to cross from one side of the accretion disk to the other — so EHT’s image is like a short-exposure photo and the image of M87 is relatively clear. But the Milky Way’s galaxy’s black hole can change on the times scale of minutes and hours, so EHT is making a long-exposure image, somewhat like taking a 1-second exposure of a tree on a windy day. Things get blurred out, and it can be difficult to determine the true shape of what was captured in the image.

Apparently, the EHT scientists have now met these challenges, at least in part. **We will learn new things about our own galaxy’s black hole on Thursday morning**; links to the press conferences are here.

In preparation for Thursday, you might find my non-expert’s guide to a black hole “silhouette” useful. This was written just before the 2019 announcement, when we didn’t yet know what EHT’s first image would show. The title is a double-entendre, because I myself wasn’t entirely expert yet when I wrote it. The vast majority of it, however, is correct, so I still recommend it if you want to be prepared for Thursday’s presentation.

The only thing that’s not correct in the guide (and the offending sections are clearly marked as such) are the statements about the “photon ring”. It took me until my third follow-up post, two months later, to get it straight; that post is accurate, but it is long and very detailed. Most readers probably won’t want to go into that much detail, so what I’ll do here is summarize the correct parts of what I wrote in the weeks following the announcement, repeating a few of the figures that I made at the time, and then tell you about a couple of new things that have been learned since then about M87’s black hole. Hopefully you’ll find this both interesting on its own and useful for Thursday.

A first thing to know about the M87 black hole is that (as we believe to be true for most black holes with matter falling onto them) it has an accretion disk and jets. These happens to be oriented with one of the jets pointing nearly at us; see the figure below. (The picture at left is schematic; the one at right is more to scale, showing the jets more accurately, but may be harder to parse.) The jets presumably point along the axis around which the black hole is spinning.

At the time of the M87 announcement, there were a lot of claims that the image showed the “photon ring” around the black hole, and the dark region between could be used to make a precise estimate of the black hole’s mass. Although I quoted some of these claims in my early posts, they turned out to be badly misleading. I discussed this in the my third follow-up blog post, “A ring of controversy.” The post starts with a relatively short overview, if you just want a brief sketch; then there follows a detailed discussion, with a careful explanation as to where the photon ring comes from, and why, nevertheless, the image that EHT produced doesn’t actually show it. Today I’ll give you a very quick summary of the conclusions.

The photon ring arises from the effect described in the figure below, in the approximation that we are looking straight at one of the poles of the black hole. *[This is almost true for M87 but may not be true at all for our galaxy’s black hole.]*

Roughly speaking, the punchline for the M87 image is summarized in the figure below. The photon ring, which reflects details of the black hole’s geometry, would be dramatic in a perfect image, but with the blur that EHT introduces, it is swamped in the glare of the accretion disk itself.

From the size of the inner dark region in their image (and other information), the EHT folks were able to estimate the mass of the black hole with more accuracy than before.

*[Measuring the mass won’t be EHT’s major goal for the Milky Way’s black hole, since we can already measure its mass precisely in other ways (e.g., by watching stars that orbit close to it, part of Andrea Ghez’s Nobel Prize-winning work). But we don’t know how our own galaxy’s black hole is oriented, or how fast it might be spinning. Naively we might expect that the accretion disk is in the same plane as the galaxy as a whole, and that the black hole rotates in the same direction as the galaxy does. However, this may not be the case. Maybe EHT can answer that this week.]*

Meanwhile, there have been some developments since then that I didn’t cover. I’m not sure I know all of them, but here are a couple of important ones.

- The EHT used its data to extract the degree of polarization in the radio waves from M87’s black hole. This provides some information about the magnetic field around the accretion disk and jets.

- This in turn has given preliminary indications that the material in the accretion disk is pushed around by the rotating magnetic field [“MAD”], rather than the other way round [“SANE”]. If true, this would likely imply (see the figure below) that
- The accretion disk is probably relatively thin; and
- What is seen in the EHT image likely comes directly from the disk, and not from the regions where the the disk and jet intersect/interact (informally called the “funnel”).

If there’s more I should have mentioned, EHT experts should feel free to let me know.

I hope these remarks are useful to you in the run-up to Thursday! You can expect a post to follow the announcement, after I’ve had a chance to absorb it and look at the accompanying papers.

]]>Today I’m continuing the reader-requested explanation of the “triplet model,” (a classic and simple variation on the Standard Model of particle physics, in which the W boson mass can be raised slightly relative to Standard Model predictions without affecting other current experiments.) The math required is pre-university level, just algebra this time.

The third webpage, showing how to combine knowledge from the first page and second page of the series into a more complete cartoon of the triplet model, is ready. It illustrates, in rough form, how a small modification of the Higgs mechanism of the Standard Model can shift a “W” particle’s mass upward.

Future pages will seek to explain why the triplet model resembles this cartoon closely, and also to explore the implications for the Higgs boson.

Please send your comments and suggestions!

]]>When we’re trying to figure out whether a confusing statement is really true or not, we have to speak precisely. Up to this stage, I haven’t been careful enough, and in this post, I’m going to try to improve upon that. There are a few small but significant points of clarification to make first. Then we’ll look in detail at what it means to “change coordinates” in such a way that would put the Sun in orbit around the Earth, instead of the other way round.

Although this is irrelevant to Muller’s point, I want to be clear about what we mean, in the standard pre-Einstein line of thinking, when we argue that Earth orbits the Sun. We don’t mean it precisely. You can see there’s an issue if you consider two identical objects, pulling on each other via gravity and going round each other. Let’s color them red and blue. Would you expect the red object goes round the blue object? Or vice versa?

Well, if they’re identical (except for their paint), there’s no reason that one of them should be privileged over the other. So what actually happens? The answer is that they go round the point exactly between them.

Suppose now that the red object has 10% less mass than the blue object. Does that mean that the blue object will now be stationary while the red object orbits round it? No; things have only changed a little bit. Instead, as you can see in the second figure, the two objects will now orbit a point that, though between them, is slightly closer to the blue object than the red object. Thus the blue object’s orbit has a slightly smaller radius than that of the red object, and it moves more slowly than the red object.

Finally, suppose that the red object’s mass was a thousand times smaller than that of the blue object. They will still orbit a common point, but that point will be much, much closer to the blue object than the red object. The blue object won’t be stationary, but its orbit will be so tiny that it will be *almost* stationary, as you can see in the third figure.

And so although the red object orbits the common point, the blue object is so close to the common point that to say “the red object orbits the blue object” is very close to true — close enough that we do say it, knowing it’s almost but not exactly right.

It’s in this sense that we say (pre-Einstein) that Earth orbits the Sun. The Sun and all the planets orbit a common point (called the “barycenter”), which tends to lie in the line between the Sun and Jupiter, the planet with the most mass. **But the Sun lies close to that common point, and so the Earth, as it orbits that common point, essentially orbits the Sun.**

This is really what we’re really being taught in school. But everything I’ve said here reflects a Newtonian way of thinking. Einstein’s equations for general relativity are more flexible, and Muller is arguing that they imply that the statements

- “the blue object orbits the red one”
- “the red object orbits the blue one”
- “the two objects orbit their barycenter”

are equivalent, indistinguishable, and thus equally true.

Now, before we return to the issues Muller raises for us, let’s look at a few questions that one might think are on the table, but are not.

First, we’re not asking **whether the Greeks**, in particular Ptolemy and others who thought the Earth lies at the center of the universe, **were correct after all.** They were not, period. Although professor Muller stated in his article that *“the revolution of Copernicus was actually a revolution in finding a simpler way to depict the motion, not a more correct way,“* this is inaccurate. First, the Greek astronomers put the orbits of Mercury and Venus in entirely the wrong place; **the two planets could never pass behind the Sun** from Earth’s perspective. The Copernican revolution included putting these two planets in their correct relative location, and in this sense it * was* indeed a more correct way, not merely a simpler way, of depicting the motion. It was a major step forward in accurate astronomy, not merely a change of perspective.

Moreover, the Greek astronomers (and many post-Copernican astronomers, including Tycho Brahe, mentor to Johannes Kepler) believed the Earth does not spin. Copernicus, by contrast, argued it does spin daily *(and Muller has not gone so far as to claim that even this statement is debatable in general relativity… although he could have.)* So in these two senses, Copernicus was correct and the late Greek astronomers were wrong; it is not merely a matter of the coordinates one chooses.

Second, we are not asking about **rotating coordinate systems**. (This point confused many readers yesterday on the blog and on twitter.) We will indeed compare the usual Sun-centered *(heliocentric)* coordinate system with an Earth-centered *(geocentric)* coordinate system, but the geocentric coordinate system of greatest interest for our initial discussion is one **whose origin moves in a circle but whose axes do not rotate** relative to the heliocentric system. (You’ll see what I mean in a moment.) The stars have nearly-fixed coordinates in this coordinate system, so we need not attribute dramatic coordinate motions to the stars and distant galaxies, as would occur in a rotating coordinate system. This makes the ensuing discussion much simpler and clearer.

Now, let’s look at the solar system in these two sets of coordinates. First, here is a cartoon of the Solar System, showing the planets from Mercury out to Jupiter in the usual heliocentric coordinate system. It’s a cartoon because I’ve given the planets perfectly circular orbits, made the Sun exactly stationary, and removed the planets’ moons. You can see that each planet orbits at a different rate, but each one’s motion is simple.

Now I want to change coordinates to geocentric ones. The origin of the coordinates is the Earth, but the axes are the same as before, so there is **no rotation of the coordinates**, just a shift that moves in a circle. This is illustrated in the next figure

What does the solar system look like in these coordinates? Here you go: in these coordinates, the Earth is stationary, and the Sun orbits the Earth in a circle.

Meanwhile the other planets orbit in complicated paths that involve loops within loops; these are the famous “epicycles” that are characteristic of geocentric coordinate systems. *[In fact, this viewpoint has some practical value, as it shows us what Earthly astronomers see as they watch the Sun and planets move relative to the fixed stars.] *

It’s clearly true that the planets’ orbits are simpler in the heliocentric coordinate system. But… not so fast. I didn’t put the Moon into my animations. If I had, you’d see the Moon’s orbit is very simple in geocentric coordinates, but quite complicated in heliocentric coordinates. The same would be true for the International Space Station, and for other artificial satellites. So is it really true that heliocentric coordinates make the solar system simpler than geocentric ones? Or does it just look that way because the Earth has only one moon, rather than, say, twelve? Imagine if switching from geocentric to heliocentric coordinates simplified the motion of eight planets and complicated the motion of twelve moons; would you be so sure that heliocentric coordinates were simplest? Let’s not confuse luck with principle.

I hope this post helps provide some intuition and clarification for what we need to understand here. As far as I can see, these are the main questions still to address:

- Is the heliocentric coordinate system actually better than others, and if so, why exactly?
- Is it true (or at least truer) that the Earth orbits the Sun rather than the other way round?
- How are questions (1) and (2) related to each other?
- Given the answer to (3), how much merit is there to Professor Muller’s argument?

Your thoughts? Comments are open.

Here, for example, is this statement as written in *Forbes* by professor Richard Muller at the University of California, Berkeley. It opens as follows: “**According to the general theory of relativity, the Sun does orbit the Earth. And the Earth orbits the Sun.**” I invite you to read the rest of it; it’s not long.

What’s his point? In Einstein’s theory of gravity (“general relativity”), time and three-dimensional space combine together to form a four-dimensional shape, called “space-time”, which is complex and curved. And in general relativity, **you can choose whatever coordinates you want on this space-time.**

So you are perfectly free to choose a set of coordinates, according to this point of view, in which the Earth is at the center of the solar system. In these coordinates, the Earth does not move, and the Sun goes round the Earth. The heliocentric picture of the planets and the Sun merely represents the ** simplest** choice of coordinates; but there’s nothing wrong with choosing something else, as you like.

This is very much like saying that to use latitude and longitude on the Earth is just a choice. I could use whatever coordinates I want. The equator is special in the latitude-longitude system, since it lies at latitude=0; the poles are special too, at latitude +90 degrees and -90 degrees. But I could just as well choose a coordinate system in which the equator and poles don’t look special at all.

And so, after Einstein, the whole Copernican question — “is the solar system geocentric or heliocentric?” — is a complete red herring… much ado about nothing. As Muller argues in his article, “**the revolution of Copernicus was actually a revolution in finding a simpler way to depict the motion, not a more correct way.**“

Well? Is this true? If not, why not? Comments are open.

I’m continuing the reader-requested explanation of the “triplet model,” a classic and simple variation on the Standard Model of particle physics, in which the W boson mass can be raised slightly relative to Standard Model predictions without affecting other current experiments.

The math required is pre-university level, mostly algebra and graphing.

The second webpage, describing what particles are in field theory, and how the particles of one field can obtain mass from a second field, is ready now. In other words, the so-called “Higgs mechanism” for mass generation is sketched on the new page.

Meanwhile the first page (describing what the vacuum of a field theory is and how to find it in simple examples) is here.

Please send your comments and suggestions, as I will continue to revise the pages in order to improve their clarity.

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