Even in Einstein’s General Relativity, the Earth Orbits the Sun (& the Sun Does Not Orbit the Earth)

Back before we encountered Professor Richard Muller’s claim that “According to [Einstein’s] general theory of relativity, the Sun does orbit the Earth. And the Earth orbits the Sun,” I was creating a series of do-it-yourself astronomy posts. (A list of the links is here.) Along the way, we rediscovered for ourselves one of the key laws of the planets: Kepler’s third law, which relates the time T it takes for a planet to orbit the Sun to its distance R from the Sun. Because we’ll be referring to this law and its variants so often, let me call it the “T|R law”. [For elliptical orbits, the correct choice of R is half the longest distance across the ellipse.] From this law we figured out how much acceleration is created by the Sun’s gravity, and concluded that it varies as 1/R².

That wasn’t all. We also saw that objects that orbit the Earth — the Moon and the vast array of human-built satellites — satisfy their own T|R law, with the same general relationship. The only difference is that the acceleration created by the Earth’s gravity is less at the same distance than is the Sun’s. (We all secretly know that this is because the Earth has a smaller mass, though as avid do-it-yourselfers we admit we didn’t actually prove this yet.)

T|R laws are indeed found among any objects that (in the Newtonian sense) orbit a common planet. For example, this is true of the moons of Jupiter, as well as the rocks that make up Jupiter’s thin ring.

Along the way, we made a very important observation. We hadn’t (and still haven’t) succeeded in figuring out if the Earth goes round the Sun or the Sun goes round the Earth. But we did notice this:

• If the Earth goes round the Sun, then its path satisfies the Sun’s T|R law, just like the other planets do.
• If the Sun goes round the Earth, then its path does not satisfy the Earth’s T|R law, although the Moon and various human-built satellites do so.

This was all in a pre-Einsteinian context. But now Professor Muller comes along, and tells us Einstein’s conception of gravity implies that the Sun goes round the Earth just as much (or just as little) as the Earth goes round the Sun. And we have to decide whether to believe him.

Muller versus Kepler

What is Muller saying? He’s certainly right that in Einstein’s approach to gravity and motion, Sun-centered (heliocentric) coordinates and Earth-centered (geocentric) coordinates are equally good. Equal, that is, in the sense that Einstein’s concepts of gravity and motion work equally well in either one. Because of this, in Einstein’s theory, we cannot say whether the Earth goes round the Sun or the Sun goes round the Earth. It can appear either way, or some other way, depending on what coordinates we choose and how we visualize them.

But then Muller goes a step too far. He says: “The Sun orbits the Earth. And the Earth orbits the Sun.” This is where, in my opinion, he makes an error. He has forgotten that gravitational orbits have special properties that general looping trajectories do not have. They have T|R laws.

[More precisely, in any context where a Newtonian would feel it reasonable to say that “X orbits Y due to gravity”, then X’s path satisfies Y’s T|R law to a reasonable approximation.]

In considering Muller’s claim, the fact that the Earth satisfies the Sun’s T|R law very well, but the Sun doesn’t even come close to satisfying the Earth’s, is intriguing. But still, isn’t it just a matter of choosing the appropriate coordinates, geocentric instead of heliocentric?

No it is not! Kepler’s third law, and indeed any T|R law, is coordinate-independent!

[I’ll try to demonstrate carefully that this is true in my next post. But in short, velocities and curvatures in the solar system are small, so neither T, which involves an astronomical year as measured by a local clock on either Sun or Earth, nor R, which is a relative distance that can be estimated by light-travel time, is ambiguous; and neither cares what coordinate system you are using when you measure them.]

This is the crucial observation. You see, it’s not as though the Sun’s path relative to the Earth fails to satisfy Earth’s T|R law in heliocentric coordinates, but succeeds in satisfying it in geocentric coordinates. It fails to satisfy Earth’s T|R law in all coordinate systems.

And why does this tell us that the Sun does not orbit the Earth? Any T|R law, including Kepler’s third law for the Sun and the similar laws for all compact objects including planets, moons and asteroids, provides a quintessential test for diagnosing whether a trajectory is a gravitational orbit. After all, the Sun’s T|R law applies to more than the eight planets. It applies to all the Sun-orbiting rocks and ice-balls and human-made satellites, as well as all imaginable objects that could potentially orbit the Sun gravitationally. [We’ll discuss some of the minor fine print to this strong statement next time.] In one simple equation, it provides a basic rule that all such orbits of the Sun must satisfy. A path which does not meet this rule, at least in some approximate way, cannot be said to be a gravitational orbit of the Sun.

And a path which does not satisfy the T|R law for the Earth, to a reasonable approximation, cannot be said to be a gravitational orbit of the Earth.

Thus, the question of whether the Earth orbits the Sun, or the Sun orbits the Earth, can be addressed using their respective T|R laws, independently of whether we use heliocentric or geocentric coordinates, or any other choice of coordinates.

What is Muller’s mistake?

Muller is correct that if two paths (the Sun’s and the Earth’s, in this case) intertwine, you cannot say which one goes round the other. That’s coordinate-dependent. It is also true that, in Einstein’s gravity, the Sun’s motion around the Earth in geocentric coordinates can be interpreted in a purely gravitational language, which is not true (naively) in Newton’s gravity, where we would normally invoke “fictitious” non-gravitational forces.

But these points, though correct, are irrelevant to the question of gravitational orbits.

A gravitating system, even in Newton’s language, consists of an elaborate structure of both actual and potential orbits, characterized by a T|R law. In Einstein’s language, the system isn’t described just by the trajectories of the massive objects within it. It has an extended four-dimensional space-time geometry, which we can’t and shouldn’t ignore. What Muller has done is focus his (and our) attention on the properties of two or three geodesics in isolation, while ignoring the rest of the geometry. If we look at the spacetime as a whole, and probe its properties in a coordinate-invariant way, it is easy to see that

• The Sun’s T|R law applies approximately to the Earth’s trajectory, and applies approximately to classes of gravitational orbits nearby to the Earth’s trajectory.
• The Earth’s T|R law does not apply (even approximately) for the Sun’s trajectory, nor does it apply to any nearby trajectories.

These are coordinate-invariant statements that care not a whit whether, in some set of coordinates, the Sun’s path goes around a stationary Earth, or for that matter whether both the Sun’s and Earth’s paths go around a stationary Moon, a stationary Venus, or some arbitrary point in space.

So in my opinion, Muller is wrong. The Sun does not orbit the Earth, or the Moon; the Earth does not orbit the international space station or any of the GPS satellites; and the planet Saturn does not orbit any of the tiny rocks that make up its rings. To say otherwise is to misunderstand and misapply the lessons of Einstein’s theory.

During the discussion last week a number of readers suggested other methods for arguing against Muller. Maybe next week we can look at the strengths and weaknesses of these other methods, and discuss them in more detail. But first I’ll write a post putting more meat on today’s bare-bones argument.