We’re more than a week into a discussion of Professor Richard Muller’s claim that “According to the general theory of relativity, the Sun does orbit the Earth. And the Earth orbits the Sun. And they both orbit together around a place in between. And both the Sun and the Earth are orbiting the Moon.” Though many readers have made interesting and compelling attempts to prove the Earth orbits the Sun, none have yet been able to say why Muller is wrong.
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?
Standard Geocentric Coordinates
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.
General Geocentric 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?