We’ve been having some fun recently with Sun-centered and Earth-centered coordinate systems, as related to a provocative claim by certain serious scientists, most recently Berkeley professor Richard Muller. They claim that in general relativity (Einstein’s theory of gravity, the same fantastic mathematical invention which predicted black holes and gravitational waves and gravitational lensing) the statement that “The Sun Orbits the Earth” is just as true as the statement that “The Earth Orbits the Sun”… or that perhaps both statements are equally meaningless.
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.
Is “the Earth is Not Flat” a meaningful statement?
If Muller is right, can’t we choose coordinates in which the Earth is flat? For example, how about these coordinates:
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 are.
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.
Is “the Earth’s a Sphere” a meaningful statement?
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.
Is “The Earth is Rotating” a meaningful statement?
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 yes, unequivocally, the Earth orbits the Sun (and the Sun does not orbit the Earth)?