I bet you know that planets go around the sun in elliptical orbits. But do you know why?
In fact, they’re moving in circles in 4 dimensions. But when these circles are projected down to 3-dimensional space, they become ellipses! This animation by Greg Egan shows the idea.
The plane here represents 2 of the 3 space dimensions we live in. The vertical direction is the mysterious fourth dimension. The planet goes around in a circle in 4-dimensional space. But down here in 3 dimensions, its ‘shadow’ moves in an ellipse!
What’s this fourth dimension I’m talking about here? It’s a lot like time. But it’s not exactly time. It’s the difference between ordinary time and another sort of time, which flows at a rate inversely proportional to the distance between the planet and the sun.
Egan's animation uses this other sort of time. Relative to this other time, the planet is moving at constant speed around a circle in 4 dimensions. But in ordinary time, its shadow in 3 dimensions moves faster when it’s closer to the sun.
All this sounds crazy, but it’s not some new physics theory. It’s just a different way of thinking about Newtonian physics! Of course you can see that planets move in elliptical orbits without resorting to the 4th dimension. But it becomes a lot more obvious if you do!
Physicists have known about this viewpoint at least since 1980, thanks to a paper by the mathematical physicist Jürgen Moser. Some parts of the story are much older. A lot of papers have been written about it.
But I only realized how simple it is when I got a paper in my email from someone I didn't know: an amateur mathematician named Jesper Göransson. I get a lot of papers by crackpots, but the occasional gem like this makes up for all those.
The best thing about Göransson’s 4-dimensional description of planetary motion is that it gives a clean explanation of an amazing fact. You can take any elliptical orbit, apply a rotation of 4-dimensional space, and get another valid orbit!
Of course we can rotate an elliptical orbit about the sun in the usual 3-dimensional way and get another elliptical orbit. The interesting part is that we can also do 4-dimensional rotations. This can make a round ellipse look skinny: when we tilt a circle into the fourth dimension, its ‘shadow’ in 3-dimensional space becomes thinner!
In fact, you can turn any elliptical orbit into any other elliptical orbit with the same energy by a 4-dimensional rotation of this sort. All elliptical orbits with the same energy are really just circular orbits on the same sphere in 4 dimensions!
For the details, see the Azimuth blog:
I had to go through Göransson’s calculations to convince myself that they were right.
And here is his paper:
• Jesper Göransson, Symmetries of the Kepler problem, 8 March 2015, http://math.ucr.edu/home/baez/mathematical/Goransson_Kepler.pdf.
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