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I came across a YouTube Video by minutephysics, a YouTuber with a Ph.D. degree in physics. In this video, he explains that the solar system is approximately flat because of two reasons:

1) conservation of angular momentum,

2) our universe has three spatial dimensions and the same does not hold in a universe with four spatial dimensions.

I completely agree with the first point, and I think it would be nice to add that under the constraints of conservation of momentum and angular momentum, the total energy is minimized if all particles lie in the same plane. I have doubts about the second point, which the video barely explains anything.

In summary, I have two questions that I need help with.

1) If we can visualize angular momentum in 3D as a vector, how should we think about its counterpart in 4D, a tensor? And what about the law of conservation of angular momentum.

2) What is the shape of the solar system in a universe with four spatial dimensions?

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1) If we can visualize angular momentum in 3D as a vector, how should we think about its counterpart in 4D, a tensor? And what about the law of conservation of angular momentum.

In general, angular momentum is a geometric object called a rank $2$ differential form. To oversimplify a bit, you can think of it as a plane, along with a signed magnitude. In three dimensions, and only in three dimensions, every plane corresponds to a vector (i.e. the normal vector), so we can also think of angular momentum as a vector.

2) What is the shape of the solar system in a universe with four spatial dimensions?

Since angular momentum is a plane in any number of dimensions, the conservation law argument actually works the exact same way: if the initial total angular momentum lies in a plane, then the solar system ends up orbiting in that plane.

At least, it would, but gravitational orbits in four spatial dimensions are not stable, or more precisely they are neutrally stable, with particles easily drifting into drastically larger or smaller orbits. For this reason, it's unclear if anything like a solar system could form at all.

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  • $\begingroup$ I checked the link about nonstable orbits. Is it true that in four spatial dimensions, the solar system would have no planets but the Sun alone because particles either fall into the center of attraction or escape the gravity well for good? $\endgroup$ – hb12ah Mar 30 '20 at 3:40
  • $\begingroup$ @hb12ah Honestly, I'm not totally sure, because the orbit isn't technically unstable, it's neutrally stable, exactly on the dividing line. I've heard people claim solar systems can't form, but I imagine it would take a simulation to really know for sure. $\endgroup$ – knzhou Mar 30 '20 at 3:41
  • $\begingroup$ only four spatial dimensional is neutrally stable, five spatial dimensional is not, is that correct? $\endgroup$ – hb12ah Mar 30 '20 at 3:45
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    $\begingroup$ @hb12ah That's right! You definitely can't have a solar system in 5 and up. $\endgroup$ – knzhou Mar 30 '20 at 3:45
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    $\begingroup$ @hb12ah Nah, as the particle comes in, it just means that the radius goes to zero while the speed goes to infinity, keeping the angular momentum constant. This kind of situation is known as a "singularity". It really just means that the math breaks down, so we don't know what would happen. $\endgroup$ – knzhou Apr 4 '20 at 3:54

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