Planets and the fourth dimension

Surely you know that planets move around the sun in elliptical orbits. But why? In fact, they move in circles in four-dimensional space. And if you project these circles on three-dimensional space, they turn into ellipses.



In the figure, the plane represents 2 out of 3 dimensions of our space. The vertical direction is the fourth dimension. The planet moves in a circle in four-dimensional space, and its “shadow” in three-dimensional moves in an ellipse.

What is this 4th dimension? It is like time, but it is not quite time. This is such a special time that flows at a speed inversely proportional to the distance between the planet and the sun. And relative to this time, the planet moves at a constant speed in a circle in 4 dimensions. And in ordinary time, his shadow in three dimensions moves faster when it is closer to the sun.

It sounds strange - but it's just an unusual way of representing ordinary Newtonian physics. This method has been known since at least 1980 thanks to the work of the mathematical physicist Jürgen Moser. And I found out about this by receiving an email job from authorship by Jesper Goranson entitled “Symmetries in the Kepler Problem” (March 8, 2015).

The most interesting thing in this work is that this approach explains one interesting fact. If we take any elliptical orbit, and rotate it in 4-dimensional space, then we get another valid orbit.

Of course, you can rotate an elliptical orbit around the sun and in ordinary space, getting an acceptable orbit. It is interesting that this can be done in 4-dimensional space, for example, narrowing or expanding an ellipse.

In the general case, any elliptical orbit can be turned into any other. All orbits with the same energy are circular orbits on the same sphere in 4-dimensional space.

Kepler's problem

Suppose we have a particle that moves according to the inverse square law. The equation of its motion will be



where r is the position as a function of time,r is the distance from the center, m is the mass, and k determines the force. From here we can derive the law of conservation of energy



for a certain constant E, depending on the orbit, but not changing with time. If this force is attraction, then k> 0, and in an elliptical orbit E <0. We will call the particle a planet. The planet moves around the sun, which is so heavy that its vibrations can be neglected.

We will study the orbits with the same energy E. Therefore, the units of mass, length and time can be accepted as any. Put

m = 1, k = 1, E = -1/2

This will save us from unnecessary letters. Now the equation of motion looks like



a conservation law says



Now, following the idea of ​​Moser, we move from ordinary time to new. We call it s and require that



This time is slower as you move away from the sun. Therefore, the speed of the planet away from the sun increases. This compensates for the tendency of the planets to move as they move away from the sun more slowly in normal time.

Now we rewrite the law of conservation with the help of new time. Since I used the dot for the derivatives of normal time, let's use the stroke for the derivatives of time s. Then for example:



and



Using such a derivative, Goranson shows that the conservation of energy can be written in the form



And this is nothing but the equation of a four-dimensional sphere. The proof will come later. Now let's talk about what this means for us. To do this, we need to combine the coordinate of ordinary time t and the spatial coordinates (x, y, z). Point

(t, x, y, z)

moves in four-dimensional space as parameter s changes. That is, the speed of this point, namely,



moves along a four-dimensional sphere. This is a sphere of radius 1 centered at the point

(1,0,0,0)

Additional calculations show other interesting facts:



and

t '' '= - (t' - 1)

These are the usual equations of a harmonic oscillator, but with an additional derivative. The proof will be later, but for now let's think about what this means. In words, this can be described as follows: 4-dimensional velocity v performs simple harmonic oscillations around a point (1,0,0,0).

But since vat the same time remains on the sphere centered at this point, it can be concluded that v moves with constant speed in a circle in this sphere. And this implies that the average value of the spatial components of the 4-dimensional velocity is 0, and the average t is 1.

The first part is clear: our planet on average does not fly away from the Sun, so its average speed is zero. The second part is more complicated: the usual time t moves forward with an average speed of 1 relative to the new time s, but its rate of change oscillates sinusoidally.

Integrating both parts,



we obtain



for a certain constant vector a . The equation says that the position of r harmoniously oscillates around the point a . Since adoes not change with time, it is a conserved quantity. This is called the Laplace — Runge — Lenz vector.

Often people start with the law of inverse squares, show that the angular momentum and the Laplace – Runge – Lenz vector are conserved, and use these conserved quantities and the Noether theorem to show the presence of a 6-dimensional symmetry group. For solutions with negative energy, this turns into a group of turns in 4 dimensions, SO (4). After working a little more, you can see how the Kepler problem is coupled to a harmonic oscillator in 4 dimensions. This is done through reparametrization of time.

I liked Horasnon's approach more, because it starts with the reparameterization of time. This allows you to effectively show that the elliptical orbit of the planet is the projection of a circular orbit in four-dimensional space onto three-dimensional. Thus, 4-dimensional rotational symmetry becomes apparent.

Goranson transfers this approach to the law of inverse squares in n-dimensional space. It turns out that elliptical orbits in n dimensions are projections of circular orbits from n + 1 dimensions.

He also applies this approach for positive-energy orbits, which are hyperbolas, and for zero-energy orbits (parabolas). For hyperbolas, the symmetry of Lorentz groups is obtained, and for parabolas, the symmetry of Euclidean groups. This is a known fact, but it is noteworthy how simple it is derived using the new approach.

Mathematical Details

Due to the abundance of equations, I will put a frame around the important equations. The basic equations are energy conservation, strength and the change of variables that give:



Start with energy conservation:



then use



to get



A little algebra - and we get



This shows that the 4-dimensional velocity



remains on the sphere of unit radius centered at (1,0,0 , 0).

The next step is to take the equation of motion



and rewrite it using the primes (derivatives with respect to s) and not the points (derivatives with respect to t). We start with



and differentiate to get



Now we use a different equation for



and we get



or



therefore



Now it would be nice to get the formula for r ''. First, we will calculate



and then differentiate. We will



connect the formula for r ", something will be reduced, and we will get.



Remember that the conservation law says



and we know that t '= r. Therefore,



and



We



get Since t' = r, then we get



both need.

Now we get a similar formula for r '' ' . Let's start with



and differentiate



Connect the formulas for r' 'and r' ' '. Something is reduced and it remains



Integrate both parts and get



for some constant vector a . This means. It is interesting that both the vector r and its norm r oscillate harmoniously.

The quantum version of the planetary orbit is a hydrogen atom. Everything that we calculated can be used in the quantum version. See Greg Egan, The ellipse and the atom for details .

See John Baez, Mysteries of the gravitational 2-body problem for details on the history of this task .

And all this also has to do with quantum physics, supersymmetry and Jordan algebra!