Would solar systems be "flat" in any manner in 4d?

[Vector_Graphics]

Occupying one 2d plane, two distinct 2d planes, or what?
This is assuming:
- Planets and stars can form in 4d
- The inverse square law is followed (and thus, that orbits behave identically to in 3d, as two-body motion is planar).

[quickfur]

Those are pretty big assumptions

But let's say we assume they hold, for the sake of argument. Then solar systems, or more precisely, planetary systems, assuming they coalesced from the gravitational condensation of gas that initially formed the central star and that the orbiting planets result from leftover angular momentum in the system, then probably it would be planar? I'm not 100% sure. Perhaps it could exist in two distinct planes ala the clifford double rotation. But it's hard to know without more exact parameters that one could plug into a Monte Carlo simulation to see what kinds of results are obtained.

One interesting thing about a planetary system in two 2D planes is that the planets in one plane would, as far as the other plane is concerned, occupy the stationary plane intersecting at the origin (the star), so the collective mass of those planets would effectively act as additional gravity to the planets in the other plane. Furthermore, since they would oscillate above and below the other plane within their own orthogonal plane, planets in the other plane would experience an oscillating gravitational force perpendicular to their orbital plane. The net force is addtional gravity in the direction of the star. But I'm not sure if this perpendicular oscillating force would destabilize the orbit within the plane or not. Depending on the relative orbital periods, maybe it could actually act as a stabilizer? Again, hard to say without a Monte Carlo simulation to observe the possibilities.

[Vector_Graphics]

Yeah. Usually, though, I'd imagine planets being far enough apart that it doesn't really matter that much.

[PatrickPowers]

The mundane 3D case was the great problem of the 19th century of Laplace and Lagrange so even that is well beyond me. Today the SS has been found to be (mildly) chaotic. So the 4D case is out of reach. I guess the thing to do is make a simulation and see if you can get a stable solution. My guess is that a solution with two planes could be done and could arise naturally.

[PatrickPowers]

The mundane 3D case was the great problem of the 19th century of Laplace and Lagrange so even that is well beyond me. Today the SS has been found to be (mildly) chaotic. So the 4D case is out of reach. I suppose the thing to do is make a simulation and see if you can get a stable solution. My guess is that a solution with two planes could be done and could arise naturally.

[steelpillow]

The strict 2D case (flatland) is a simple inverse law and not an inverse square law. The square arises in 3D because of the spherical surface at the orbital distance.
Nevermind. A 2D orbit in 4D is unstable. In 3D, a wobble out of the plane merely tilts the plane slightly. But in 4D it will begin to couple orbital energy into other planes in a complicated way, and the orbit soon breaks up. So you'd need some mechanism (an extra law of physics, perhaps) which prevents such wobbles from happening at all.
I guess you could then come up with something fancy, like an orbit in the wx plane and a closer-in harmonic orbit, i.e. say half or a third the period, in the yz plane. A bit like Lissajous figures in 4D.
This is possible because the distance from the star is the usual function of wxyz so being close in the y,z orbit does not mean you are close in your w,x orbit. But you do also have to take account of the wy, wz, xy and xz orbital planes as well, which is where harmonics become essential to keeping order.

[quickfur]

It's well known that if gravity obeys an inverse cube law, as it would in 4D if we assume gravity is the result of the exchange of (possibly virtual) gravitons, then orbits are inherently unstable. The consequences have been worked out here on this forum before; there are 5 cases:

1) The planet's path diverges, i.e., there is no orbit, it flies off into space.
2) The planet's path spirals outwards, each iteration being a constant distance farther than the previous. Eventually it will also fly off into space, but in a more controlled way.
3) The planet's path is a perfect circle. This is the only case where there is a stable orbit, but it's a local maximum, meaning that the slightest perturbation will destroy its stability and send it into the other 4 unstable cases. I.e., a speck of dust landing on the planet from space will knock it off its perfect circular orbit into one of the other cases. Not to mention the extreme unlikelihood that a perfectly circular orbit would arise spontaneously in a hypothetical star formation scenario.
4) The planet's path spirals inwards, each iteration being a constant distance closer than the previous. Eventually, it will collide with the star it orbits.
5) The planet's path converges to the star, i.e., there is no orbit, it just crashes into the star.

The cause of the lack of stable orbits (besides the impractical perfect circular case) is that the 1/r^3 gravity well can never be perfectly balanced by the mv^2 component of the orbiting body's momentum. There is always a leftover term that will cause the orbit to be unstable. As long as gravity obeys a 1/r^3 law, stable orbits are inherently impossible.

The only way around this is to somehow force gravity to obey a 1/r^2 law instead. It would likely require an unnatural by-fiat imposition of some arbitrary law that violates the flux law or otherwise postulates a completely different mechanism for gravity. Assuming this is done (and this is a huge, huge, huge assumption), then we could have stable orbits in 4D in the analogous way to 3D orbital systems (e.g., orbital paths are conic sections, the stable ones among which would be the circular and elliptical cases).

One possibility that I've come up with in seeking a solution to this conundrum is Einstein's idea behind general relativity: in our 3D universe, it seems awfully convenient that acceleration follows a square law, and gravity also follows a square law. (And furthermore, inertial mass equals gravitational mass, even though there's no a priori reason for such an equivalence.) Einstein thus made the leap of considering what if acceleration is gravity, and gravity is acceleration. Thus, he arrived at general relativity, where gravity isn't a conventional force per se, but is caused by a curvature of space, and the path of an object under gravitational acceleration is actually its inertial path in curved spacetime. If we take this idea in its most radical form, we could postulate that if in our hypothetical 4D universe a similar thing holds, where 4D gravity is the consequence of a quadratically-varying curvature of space, then we could imagine a scenario where gravity actually obeys an inverse square law rather than the inverse cube law implied by the flux theory of force. I.e., our 4D gravity would be exactly the same as motion in an inertial frame, and it's the curvature of the space itself that causes the 1/r^2 shape of the gravity well. This would give us stable 4D orbits, and possibly also give rise to a bunch of unusual consequences that could be rather interesting to explore.

[steelpillow]

Didn't Kaluza and Klein independently study just that? They added an extra dimension to General Relativity and found that, in order to stabilise things, you had to somehow "compactify" it to tiny size. The remarkable outcome of that was that, alongside conventional General Relativity, the equations of electromagnetism just appeared. Unfortunately, so did something known as the Kaluza-Klein scalar, and nobody has ever been able to detect anything matching its possible properties.
So yeah, any extensive 4D space would have to explain gravity - or its equivalent - some other way that would allow stable "de-compactification". Things might be a bit dark in there though, unless you added some form of light back in as well.