![]() Let’s use a prime for the derivative with respect to So, for example, we have I’ve been using a dot for the ordinary time derivative, following Newton. To see this, first rewrite conservation of energy using this new notion of time. In fact, with this new kind of time, a planet moves just as fast when it’s farthest from the sun as when it’s closest.Īmazing stuff happens with this new notion of time! This compensates for the planet’s ordinary tendency to move slower when it’s far from the sun. So, measured using this new time, a planet far from the sun might travel in one day what would normally take a week. For a planet very far from the sun, one day of this new time could equal a week of ordinary time. If that seems backwards, just think about it. So, using this new time speeds up the planet’s motion when it’s far from the sun. This new kind of time ticks more slowly as you get farther from the sun. The big idea, apparently due to Moser, is to switch from our ordinary notion of time to a new notion of time! We’ll call this new time and demand that If you prefer an approach that keeps in the units, see Göransson’s paper. ![]() This will reduce the clutter of letters and let us focus on the key ideas. I only want to study orbits of a single fixed energy This frees us to choose units of mass, length and time in which It's a planet moving around the sun, where we treat the sun as so heavy that it remains perfectly fixed at the origin. Let’s consider an attractive force, so and elliptical orbits, so Let's call the particle a 'planet'. From this we can derive the law of conservation of energy, which saysįor some constant that depends on the particle’s orbit, but doesn’t change with time. ![]() Where is its position as a function of time, is its distance from the origin, is its mass, and says how strong the force is. Suppose we have a particle moving in an inverse square force law. But I can’t resist summarizing the key results. Jesper Göransson explains how this works in a terse and elegant way. All elliptical orbits with the same energy are really just circular orbits on the same sphere in 4 dimensions! 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. 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! The interesting part is that we can also do 4-dimensional rotations. Of course we can rotate an elliptical orbit about the sun in the usual 3-dimensional way and get another elliptical orbit. You can take any elliptical orbit, apply a rotation of 4-dimensional space, and get another valid orbit! The best thing about Göransson’s 4-dimensional description of planetary motion is that it gives a clean explanation of an amazing fact. I get a lot of papers by crackpots in my email, but the occasional gem from someone I don’t know makes up for all those. Jesper Göransson, Symmetries of the Kepler problem, 8 March 2015.A lot of papers have been written about it.īut I only realized how simple it is when I got this paper in my email, from someone I’d never heard of before: Physicists have known about this viewpoint at least since 1980, thanks to a paper by the mathematical physicist Jürgen Moser. It’s just a different way of thinking about Newtonian physics! But in ordinary time, its shadow in 3 dimensions moves faster when it’s closer to the sun.Īll this sounds crazy, but it’s not some new physics theory. Relative to this other time, the planet is moving at constant speed around a circle in 4 dimensions. 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. What’s this fourth dimension I’m talking about here? It’s a lot like time. But down here in 3 dimensions, its ‘shadow’ moves in an ellipse! The planet goes around in a circle in 4-dimensional space. The vertical direction is the mysterious fourth dimension. The plane here represents 2 of the 3 space dimensions we live in.
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