line-shaped planets in 4d

[DonSoreno]

If a 4d planets where made of (very long) 1d-strings, and every point on the string had mass, then gravity would decrease by 1/r^2 and I think we would get stable orbits in the orthogonal 3-space.
Then we would have strings orbiting parallel strings.
The strings would need to resist bending, so that they don't collapse under their own weight.
If all strings are parallel, this would degenerate to the 3d case.
Could this work or am I missing something?

[quickfur]

It could work, but (1) as you pointed out yourself, what would stop these linear planets from collapsing into spheres? and (2) what would maintain their relative orientation to each other so that orbits would still work? because if two such planets / bodies were to lie in orthogonal planes, for example, the orbital paths may not work out as they do in 3D.

[mr_e_man]

(2) is not a problem, because we only need one line-shaped star. A spherical planet would orbit the star with a 1/r^2 force.
But as it orbits it would drift along the line in a helical motion, and would eventually reach the end of the line. Then I suppose it would get pulled back toward the line, and start moving along the line in the opposite direction.

Perhaps the star should be a closed loop instead of a line.
Far from the star, since the loop appears as a point, the force would be approximately 1/r^3. Near the star, since the loop appears as a line, the force would be approximately 1/r^2. An orbit would probably be like a torus knot.


What about a 2D plane-shaped star? The gravitational force would be 1/r^1, which also allows stable orbits.

[quickfur]

One could also pose the question in 3D: what if 3D planets are linear instead of spherical? We could probably come up with all sorts of interesting consequences for them... but the thing is, we don't observe them in the 3D universe. Meaning that they are not a stable construct, at least as far as natural occurrence is concerned.

And if we're going to go with arbitrary shapes that confer stability, why not just cut to the chase and declare planetary orbits in 4D stable by fiat? A large part of 4D speculations on this forum is based on the assumption of stable planets; the unlikeliness/impossibility of such in a 4D universe obtained by dimensional analogy from 3D didn't stop us then, why should it stop us now?

As I mentioned in another thread, it could be possible to work around Gauss's Law (the 1/r^(d-1) law of force attenuation) by postulating a suitable space-time in which gravity is caused by the curvature of space such that a 1/r^2 gravity well is produced. It wouldn't be the exact analogue of general relativity anymore (because AIUI GR generalized to 4D will result in a 1/r^3 law for gravity), but it would be consistent.

[mr_e_man]

But the 1/r^3 gravitational force is not necessarily fatal, if other forces are involved.

Either the shape of the star (not a sphere), or a planet's orbit (not an inward spiral), could be held against gravity by some force like magnetism.

[quickfur]

The problem with non-spherical stars or planets is that it's hard to find a justification for them, unless you're willing to throw out conventional physics.

But if you're going to throw out conventional physics, might as well go all the way and reinvent physics from scratch. Forget about planets and orbits, create a whole new universe from first principles. The result will probably look nothing like anything we know or understand, but it would work. (In fact, I've been working on and off on an alternative universe in n dimensions that's not based on physics as we know it, but based on symmetry.)

Or just invent a fantasy world where we declare things that we want to exist by fiat, and call it a day. I mean, after all, if we're already not sticking with the consequences of physics generalized to 4D, why tie our hands behind our backs?

The whole point behind generalizing 3D physics to 4D is to use things familiar to us to explore 4D geometry. For that purpose, it doesn't really matter if 4D orbits aren't actually stable; we just declare it so by fiat and use that to explore 4D geometry. But if we're going to create a 4D universe that's native and self-consistent, then we can't get away from grappling with fundamental issues like these, at which point the question becomes, do we want to stick with the dimensional analogue of 3D physics and be bound to the consequences which don't always go our way, or should we just throw out the whole thing and invent a 4D universe from first principles?

Alternatively, if we're going to stick with the dimensional analogue of 3D physics, another approach could be to ask the question, if orbits and atoms are not stable, then what is stable? This will probably lead to a completely new, alien world that's totally unfamiliar to us, but it will also be an interesting exploration of the consequences of physics as we know it applied to a 4D world.

[mr_e_man]

I'm saying we should stick with the consequences of the straightforward generalization of 3D physics.

Even in 3D, it's conceivable that a loop-shaped star can exist. https://physics.stackexchange.com/quest ... oidal-star

When electromagnetism is described without the vector cross product (e.g. with geometric algebra or tensor algebra), dimensional generalization is easy. For example, the magnetic force, on a particle with electric charge q and velocity v, is F=qBv, where the magnetic field B can be considered as an antisymmetric matrix, or as a bivector (in which case we should take the vector part of Bv, discarding the trivector part).

I'm not sure about the stability of atoms, or how quantum spin generalizes to other dimensions....

[mr_e_man]

I have been playing with TestTubeGames Gravity Simulator. It is only 2D, but it allows you to modify the force law.

[quickfur]

Atoms as we know them are inherently unstable in 4D. This isn't just something we calculated here on this forum; I've seen other research papers and peer-reviewed sources that show that Schrödinger's equation for the hydrogen atom does not have non-zero minima in 4D. Meaning that electron orbitals cannot exist; they collapse onto the nucleus. So chemistry as we know it is impossible in 4D.

If a 4D universe is going to have atoms and matter and chemistry and stuff, it's going to have to be based on something completely different from what we know in the 3D universe.

This is why I said, if we're going to assume the existence of stable orbits and stable matter in 4D, we're already declaring the existence of these things by fiat. So we might as well just drop the pretense and build a world where these basic objects are simply declared to exist, and not waste energy trying to rationalize them in terms of a generalization of 3D physics that clearly does not work the way we expect in 4D.

[quickfur]

P.S. even waves (including electromagnetic radiation) in 4D work in unexpected ways in 4D. If we were to study the behaviour of wave propagation in 4D, we will discover that signals cannot be transmitted faithfully: back-echoes and other complex phenomena will cloud the signal at the receiving end, such that what the receiver sees will be a highly modified signal from what the transmitter sends. This has far-reaching consequences, because it means that sight, which works by light bouncing off objects reaching the eye's retinal cells, works differently in 4D: a single pulse bouncing off the object shows up as a series of echoing pulses in the eye. So clear vision is impossible. Similarly, clear audio signals like sound are impossible; they will arrive with echo distortions. Radio transmissions will also not work the way they do in 3D; they may not even be possible unless there's a way of cancelling out the echoes.

Basically, a lot of things do not work the same way in 4D if you derive them from first principles. To get anything close to what we observe in the 3D universe we pretty much have to throw out most of physics and declare a lot of things by fiat.

[mr_e_man]

I see your point, and I mostly agree.
Still I see value in going as far as possible with the given laws of physics.

Dimensional generalization of the given 3D laws is not unique. If I recall correctly, General Relativity in 2D implies that there is no gravity, only curvature of space; a massive particle doesn't attract other particles but is just a "cone point", where the total angle is less than 360°. This conflicts with the 1/r^(n-1) force law. And this supports your point; 1/r^(n-1) (or perhaps 1/r^2 for all dimensions) must be declared and not derived. We have to choose which laws are "correct". Yet some choices, such as 1/r^3 gravity in 4D, seem more natural than others.

[mr_e_man]

Atoms as we know them are inherently unstable in 4D. This isn't just something we calculated here on this forum; I've seen other research papers and peer-reviewed sources that show that Schrödinger's equation for the hydrogen atom does not have non-zero minima in 4D. Meaning that electron orbitals cannot exist; they collapse onto the nucleus. So chemistry as we know it is impossible in 4D.

Schrödinger's equation itself may not be the correct generalization.

Complex numbers are 2D. The imaginary unit is a bivector, related to the electron's spin.
(I wanted to point to David Hestenes' site, but it doesn't seem to be working. See if any of these links work. I'll also add this and this.)
So we need to know what spin is in 4D, before Schrödinger's equation can be applied.

[quickfur]

Well, if you're prepared to throw out Schrödinger's equation, then what difference is that from just declaring the existence of things by fiat?

It goes back to what I said: a 4D universe that looks like our own via dimensional analogy cannot be arrived at via a generalization of our physics to 4D. The only way you can get that is by making essentially arbitrary modifications to physics. At which point we might as well just drop the pretense and just say that we're declaring X, Y, Z by fiat, and we're working backwards to find how X, Y, Z can be justified. This is no different from constructing a fantasy world in which we postulate a priori things that we'd like to have in it. The justification / internal consistency is retroactively constructed.

And there's nothing wrong this approach either. This is 4D speculation, not real physics with the scientific approach. We don't have an actual 4D universe that we can measure and investigate using scientific methods. We're not obligated to follows the "rules" of science (whatever that even means).

[mr_e_man]

As I said, I mostly agree.

[PatrickPowers]

I'm going to start a new thread about what could happen in a Universe with elementary particles but without atoms.

[PatrickPowers]

So we need to know what spin is in 4D, before Schrödinger's equation can be applied.

As far as I know no one knows what quantum spin is here in 3D. I told some physicists that quantum spin had to be taking place in an even dimensional space (because it is orientable). They didn't know what I was talking about and demanded a reference. It seems to me that no one knows what that space might be. String theory hypothesizes seven more dimensions but no one knows how such might relate to the real world.

[quickfur]

Not only quantum spin, but people don't even know what is "waving" in Schrödinger's equation. The most they can say is that the square of the wave's magnitude represents the probability the particle will be found in that location. But what is that quantity that's varying, nobody can explain.

This is why I'm skeptical that we actually understand what's going on; all we have is a model that's sufficiently sophisticated to predict the behaviour of the world with great accuracy, but we don't actually understand what is being described. Just like a high-resolution monitor has enough pixels to reproduce the image of a face, but the pixels themselves are not the face, they are merely a digital approximation of the face. Contrary to what the materialists would have us believe.