does the Earth really go around the Sun?

tl;dr: Executive summary: It is not fundamentally true that the Earth goes around the Sun; it is just easier to calculate things that way.

We like to say that the critical event that started the scientific revolution is the discovery that the Earth goes around the Sun, and not the other way around. This was incredibly important; the hypothesis by Copernicus led to the immensely important data-taking by Tycho Brahe and the quantitative, theoretical explanation of it by Kepler. Galileo's discovery of moons of Jupiter bolstered the case in important ways, and Newton's quantitative description of it all in terms of the inverse-square law solidified it all into an edifice of great importance, that is just as important and valuable today as it was then. It is also a great example of how a scientific discovery requires both observational and theoretical backing to become confidently adopted by the community.

In the 20th Century, Einstein brought us General Relativity, with the eponymous generality granting us immense coordinate freedom. That is, there are (infinitely) many ways we can make decisions about what is stationary and what is moving, and what we choose as reference points. In some choices, calculations are harder. In other choices, calculations are easier. In yet others, certain symmetries become more obvious or more valuable for making predictions. That is, GR delivers to us lots of choices about how to think about what's moving and how.

So the crazy insane thing is this: In GR, there is no answer to the question of whether the Earth goes around the Sun or whether the Sun goes around the Earth. There is literally no observational answer to the question, and no theoretical answer. All observations can be incorporated to an analysis from either perspective. The question of which goes around which is not a question you can ask in the theory.

That said, it really is far, far easier to do calculations in the Copernican frame. Indeed, absolutely all calculations of Solar System dynamics are done in this frame with post-Newtonian code. The way I see it (with modern eyes) is that Copernicus's hypothesis was based on parsimony or simplicity and was adopted for that reason. Brahe and Kepler confirmed that the data are consistent with Copernicus's simple model (though with the eccentricities added). After Brahe and Kepler it was still possible to understand the observations in an Earth-centered (or even stranger) coordinate system, but was far, far easier to do calculations in the heliocentric frame.

Even today, now that GR is our model of gravity, we still calculate the Solar System with Newtonian codes (with adjustments to approximate GR corrections). And even today, now that we have this amazingly accurate model of the Solar System, we still often calculate the positions of celestial bodies by looking at paths on the celestial sphere, as did Ptolemy. How we calculate something is incredibly context-dependent, and doesn't always respect our most fundamental ideas. And the truth of Copernicus's hypothesis really just represents the pragmatism of the present-day mathematical tools. All these thoughts bolster my rejection of scientific realism and play into questions of social construction and so on. It also bolsters my view that Ockham's Razor should be thought of as a statement about calculation, not truth.

Sure the Earth goes around the Sun! But let's remember that this is a statement about calculation and pragmatism, not the fact of the matter.

oscillations and the metric

In class, I was solving the normal-mode problem for a solid object near equilibrium, using generalized coordinates, in the usual manner. This starts by orthogonalizing the coordinates to make (what I call) the "mass tensor" (the tensor that comes in to the quadratic kinetic-energy term) proportional to (or identical to) the identity. This operation was annoying me: Why do we have to get explicit about the coordinates? The whole point is that the coordinates are general and we don't have to get specific about their form!

In my anger, I solved the problem without this orthogonalization. It turns out that this solution is easier! Of course it is: I can do everything with pure matrix operations.

I had two other in-class epiphanies about the problem. The first is that the solution you get when you don't do the orthogonalization is more analogous to the simple one-dimensional problem in every way. The second is that, in a D-dimensional problem with D generalized coordinates, the tensor that goes in to the kinetic energy term is some kind of spatial metric for a D-dimensional dynamical problem. (Or proportional to it, anyway.) That is simultaneously obvious and deep.

vectors and their derivatives

The time derivative of velocity is acceleration, both vectors of course. But I was reminded in office hours today of just how hard it is to get across the idea that the velocity vector and the acceleration vector can point in totally different directions. And some students have trouble seeing this when a ballistic stone is going upwards along some (parabolic) arc, some have trouble seeing it when it is going down, and some have trouble seeing it at the top. That is, different students have very different problems visualizing the differences of the vectors over time.

I said in lecture that this issue was deep but I didn't emphasize it enough. I feel like it is so big it almost needs its own week!

vector subleties

It is vectors all week this week, in my class, and in two classes I have taught for others. It is understandable that they confuse students, even physics majors with good backgrounds. Here are some subleties that I like to point out:

• Vectors have a magnitude and a direction, but that is not sufficient. They also have a coordinate-free existence or description, and they form a linear space (with the usual linear operators). In this sense, despite what every textbook says, the unit vectors that define the coordinate system are not vectors!
• Although vectors carry around all this geometric baggage, they have a magnitude and a direction and nothing else. I can still confuse the physics majors by sliding around vectors on the board. There is no position associated with a velocity vector, and we confuse the students by always drawing the velocity as coming from the object that is moving.
• Multiplication of a vector by a scalar is usually conceived as changing the magnitude of the vector, which it does, but it also changes the units, in many cases of interest (for example when a displacement is multiplied by an inverse time to make a velocity). So it often produces a new vector that is not longer than the original vector, nor shorter, but really incomparable.
• There is a perfect symmetry between the relationship between velocity and position and the relationship between acceleration and velocity. However, it is far harder for students to understand that the acceleration vector can point perpendicular to the velocity vector than it is to understand that the velocity vector can point perpendicular to the position vector. No amount of class time spent on this point is wasted, in my experience.

everything is an approximation

One of the main things I emphasized in today's class (computing a trajectory in gravity near the surface of the Earth with no air forces) is that every calculation in physics is an approximation. The parabolic trajectory near the surface of the Earth is an approximation to the tip of a very eccentric ellipse, and the eccentric ellipse comes out only in the Newtonian approximation to GR, and even that only holds if there are no other forces acting (and there always are). There are also small adjustments for reduced mass, and if the object has non-trivial extension. Crazy! And in high school this is all taught like it is exact: Just plug numbers into the equations!

weight, gravity, and contact force

On the final exam, I asked the following:

Explain why the astronauts in the Space Shuttle are weightless.

I was lenient in grading. But my position is actually at odds with most of the textbooks. Here's why.

The standard textbook answer is something like Actually, the astronauts on the Shuttle still have weight, since there are still gravitational forces acting on them. However, they feel like they are weightless because they are in an accelerating reference frame that is accelerating at the acceleration that the gravitational force is providing. This will be followed with various things about equivalence and plummeting elevators and non-inertial forces and so on.

My explanation is that the gravitational force on an object is not the weight of the object, when the word weight is properly understood. The weight of an object is not the gravitational force but rather the contact force that holds the object up against gravity (and non-inertial forces). It is this contact force, after all, that a spring scale measures, because a spring scale does its job by providing a contact force. It is also this contact force, after all, that you perceive by having your feet pressed onto the floor or behind pressed into the seat of your chair. Indeed, gravitational forces can never be measured locally or internally (that's equivalence!), all you can measure is the stresses and strains required to oppose them in our non-inertial (by GR standards) frame.

My view makes the astronauts not misled but truly weightless. It also makes it true, not apparent, that one is lighter at the top of a hill and heavier at the bottom of a hill on a roller-coaster, and same for the related changes you experience in an elevator.

rolling down planes

Sanjoy Mahajan (MIT) and I have spent a lot of time talking about balls rolling down planes, in part because it is a very rich physics problem, and in part because it was the experiment that allowed Galileo to infer the constant acceleration behavior and galilean relativity. I started on this problem in class yesterday, but considering only the three energies: potential, linear kinetic, and rotational kinetic. When I asked the class to predict the outcome, I was surprised that I could get all three answers to the question "will the tube roll down the plane faster, slower, or at the same speed as a block sliding with little friction?" Even when we were done, not everyone got the inference, but I have to admit, the issue is subtle. Next time I will start to look at the problem from the point of view of forces; unfortunately, the class isn't quite ready for torques yet.

energy misconception

Here's an energy example problem I discuss briefly every year, because it brings up a serious student misconception. I did it today in class, and it worked as usual, although about one fifth of the class got it right straight off the bat.

A block slides from rest down a long, slanted ramp that ends with a small, up-turned ski jump (I usually draw the end of the ski jump at about 45 degrees elevation above the horizontal). Air resistance and friction are negligible. After sliding down the ramp and leaving the jump, the block will fly on a parabolic trajectory. Will the peak of that parabolic trajectory come up above the vertical height of the starting point, exactly to the height of the starting point, very slightly below the height of the starting point, or well below the height of the starting point?

The students want to go with to or slightly below. The correct answer is well below, because the trajectory in gravity never brings the horizontal component of velocity to zero, and therefore never brings the kinetic energy to zero, or even close to zero. This leads to a nice discussion and an instructive comparison with the typical roller-coaster problems out there.

forces and formality

Yesterday in class I worked through the problem of a bouncing ball, concentrating on estimating the magnitude of the force from the floor at bounce. Not a single student was even close to getting the magnitude of that force correct, even after many minutes of discussion, a few minutes of working in small groups, and more discussion. Eventually two students got it and understood after my demonstration in which I prepare to drop a book on a student's hands (comparing with the case in which the student is just holding the book).

Before, during, and after the class, students asked me if the class is going to be more formal soon or ever. I said yes. But what disturbs me is that if we go and do formal problems with vectors and calculus before the class can see even roughly the magnitude of the normal force on a bouncing ball, we are teaching math, not physics. I understand where the students are coming from: They like physics in part because it is formal. But there is no point in calculating forces when you don't understand what forces are.