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

[mr_e_man]

Well, spin-stabilization (of a 4D magnetic planet's orientation) remains a possibility. But I have no idea how long it would last, before energy losses cause the spinning to slow down and allow the planet to flip over, or before the magnetic moment changes in some other way....

[PatrickPowers]

I think if the star and the planet are both magnetized and not charged, then the force is attractive, and thus not helpful in stabilizing the orbit. If they're oriented such that the force is repulsive, then the planet will twist around until it's attractive. And if the magnetism is too weak to rotate the planet, then it's too weak to be relevant at all.

Now I'm suspecting that my 2D simulations would be unstable when a dimension is added. The orbit itself might flip over, and align with the field.

You are definitely correct that nothing like this will work in odd dimensional spaces. Magnetic fields in odd dimensional spaces are not orientable so they will move about until they attract one another maximally and reduce their potential energy. Even dimensional spaces are a different matter. Magnetic fields are orientable (they surely are in 4D. I think this is true in all even dimensional spaces but I'm not sure.) That means that two magnets can be fundamentally incompatible. The attraction between then can't be particularly strong. I'm not optimistic but think this is worth looking into. Unlike charge the magnetic fields of heavenly bodies can be extremely strong and stable.

You got me thinking.
And calculating. I derived some complicated formulas for the interaction between two magnets. Should I start a new topic?
But the details don't seem to matter here.

Any bivector in 4D, such as a magnet's moment (I hesitate to call it a "dipole moment"), can be written in the form

M = A e₁e₂ + B e₃e₄

where A and B are scalars and the e's are orthonormal vectors. Its wedge-square is

M∧M = 2AB e₁e₂e₃e₄,

and this quadvector doesn't change when M is rotated (though it does change when M is reflected). The sign of AB tells whether M is right-handed or left-handed. So, yes, two magnets can be "fundamentally incompatible" in some sense. However, attraction between them is just as possible as repulsion (of the same strength). That's because M can be rotated and end up as -M. Rotate by an angle θ in the e₁e₃ plane:

M(θ) = A (e₁cosθ + e₃sinθ)e₂ + B (-e₁sinθ + e₃cosθ)e₄
M(180°) = - M(0°)

My geometric algebra was never much and I've forgotten what little I knew but I'm going to engage with this anyway.

Let's set aside magnetic fields for a while and start with the simpler case of planetary rotations in 4D. Let's say that you have two planets. Each has a faster plane of rotation a slower plane of rotation. You somehow have the power to move these planets any way you like. You move the planets so that the faster planes are in the same plane and rotating with the same sense. You can always do this. Move the planets so that the two slow planes are also coplanar. There are two cases. Either the two slow planes are rotating in the same sense or they are rotating in the opposite sense. Call them sync or antisync.

Start all over but this time align the two slow planes first with the same sense of rotation. You can always do this. Move the planets so that the two fast planes are also coplanar. If the planes were sync before they still will be. If the planes were antisync before they still will be. This sync/antisync property is an invariant.

Taking it a step further, you move the planets so that the faster planes are in the same plane and rotating with the same sense BUT that sense is the opposite of what it was before. You can always do this. Maybe it is more clear if you leave the two planets alone and move yourself so that you see the faster planes from an opposite viewpoint. Same thing. Then since the sync/antisync thing is invariant, the slow plane must also be seen in the opposite sense of what you saw before.

Now let's go to magnetism. The magnetic field of Earth is generated by a geodynamo. As the planet very slowly cools the hot liquid iron of the inner core rises toward the surface, carrying heat. The Coriolus effect causes this rising plume to rotate. This generates a magnetic field. Rotation is 2D planar so the fields are also 2D planar. Our Earth has three main plumes under Canada, Siberia, and between Australia and Antarctica. They all generate magnetic fields but magnetic fields sum together so at any point we observe only one plane. The planes of rotation of the plumes are not aligned with Earth's rotational plane. This is why the Earth's magnetic plane is not aligned with it's rotational plane. The Earth's magnetic plane also moves around as the relative strengths of the plumes vary.

On 3D Earth only one plane is possible so there is a direction/dimension in which there is no magnetic force, we call that the pole. On 4D Earth there is no such thing. At all points there are two planes of magnetic force. I seem to recall you told me that the plane of maximum force will always be perpendicular to the plane of minimal force. That is, it's a mathematical thing that has nothing to do with the plumes and so forth. While on a rigid planet the rotational planes will be the same everywhere, the magnetic planes are "flexible". This happens on our real 3D Earth. The relation of the magnetic plane to the rotational plane changes both with time and with where one is on Earth. There are complicated systems to compensate for this.

Magnetism works like this. Let's say you have your two planets lined up with the fast planes coplanar and spinning with the same sense. The magnetic fields will repel one another. You can easily confirm this with refrigerator magnets, something I have done. If you reverse one then they will be spinning with opposite senses and will attract one another. I like to think of spinning wheels. If they are spinning in the same plane with the same sense then if they touch there will be much friction, wailing and gnashing of teeth. If they are spinning in the same plane with the opposite sense and they touch then friction will be much less. If such should happen to be spinning the same speed there will be no friction at all.

When we consider both planes of rotation there are two cases. Suppose the planets are in "sync." Then either both planes repel or both planes attract. If the planets are "antisync", then one plane repels and the other attracts. Their interaction will always be weakened by this. You can't change the sense of one rotation without also changing the sense of the other.

It seems to me that in some theoretical case where the two planes of magnetism are exactly the same strength everywhere then there should be no net magnetic force between the two planets no matter what you do. This theoretical case would never happen in real life with its messy geodynomos. Instead what will happen is that the planets will "try" to minimize potential energy by "seeking" the state with maximal attraction. That might take billions of years but it will happen.

[mr_e_man]

It seems that you're not taking account of the fact, that the force between two magnets depends not only on their relative orientation, but also on their relative position. If you place two 3D magnets next to each other, they'll repel; but if you place one on top of the other (without changing its orientation), they'll attract.

[PatrickPowers]

It seems that you're not taking account of the fact, that the force between two magnets depends not only on their relative orientation, but also on their relative position. If you place two 3D magnets next to each other, they'll repel; but if you place one on top of the other (without changing its orientation), they'll attract.

"If they are spinning in the same plane...coplanar."

I keep going back and forth as to whether or not what I wrote about 4D "orientability" is nonsense or truth. I suppose I'll get it someday.

[PatrickPowers]

Why even-dimensional spaces have two kinds of spin:

Start with a 4D sphere with dimensions w,x,y, and z. We have an object rotating in the wx and yz planes. Is it possible to rotate the coordinates so that one rotation reverses and not the other? No. Rotating the coordinates in either the wx or yz planes doesn't change the sense. To reverse the sense in one plane it is necessary to choose one dimension from {w,x} and the other from {y,z} to get wy, wz, xy, or xz. Rotating the coordinates in that plane pi radians will reverse the sense of both rotations so the parity of the rotations doesn't change. It's invariant. This is true no matter how many dimensions one has as long as their number is even.

Sorry I've expressed this so awkwardly, informally, and badly.

[mr_e_man]

Let's set aside magnetic fields for a while and start with the simpler case of planetary rotations in 4D. Let's say that you have two planets. Each has a faster plane of rotation a slower plane of rotation. You somehow have the power to move these planets any way you like. You move the planets so that the faster planes are in the same plane and rotating with the same sense. You can always do this. Move the planets so that the two slow planes are also coplanar. There are two cases. Either the two slow planes are rotating in the same sense or they are rotating in the opposite sense. Call them sync or antisync.

Start all over but this time align the two slow planes first with the same sense of rotation. You can always do this. Move the planets so that the two fast planes are also coplanar. If the planes were sync before they still will be. If the planes were antisync before they still will be. This sync/antisync property is an invariant.

Taking it a step further, you move the planets so that the faster planes are in the same plane and rotating with the same sense BUT that sense is the opposite of what it was before. You can always do this. Maybe it is more clear if you leave the two planets alone and move yourself so that you see the faster planes from an opposite viewpoint. Same thing. Then since the sync/antisync thing is invariant, the slow plane must also be seen in the opposite sense of what you saw before.

This all looks correct. Though, if the two fast planes are coplanar, then the two slow planes are also coplanar automatically. It's just the orthogonal complement.

Denote the two planets' rotational velocity bivectors Ω and Ψ. "Sync" means that the two quadvectors (or "pseudoscalars") Ω∧Ω and Ψ∧Ψ have the same sign, and "antisync" means they have opposite signs.

Why even-dimensional spaces have two kinds of spin:

In 6D, consider the pseudoscalar Ω∧Ω∧Ω.
In 8D, consider Ω∧Ω∧Ω∧Ω.
Etc.

Magnetism works like this. Let's say you have your two planets lined up with the fast planes coplanar and spinning with the same sense. The magnetic fields will repel one another. You can easily confirm this with refrigerator magnets, something I have done. If you reverse one then they will be spinning with opposite senses and will attract one another. I like to think of spinning wheels. If they are spinning in the same plane with the same sense then if they touch there will be much friction, wailing and gnashing of teeth. If they are spinning in the same plane with the opposite sense and they touch then friction will be much less. If such should happen to be spinning the same speed there will be no friction at all.

(My objection is not yet.)

First, we need to keep a distinction between the magnetic moment (which may vary in time, but not in space, as it's just a bivector summarizing the electric currents flowing in the object), and the magnetic field (which varies in space). The field close to the object may be complicated, but far away it's fairly simple to describe in terms of the magnetic moment. It's an idealization or approximation.
I think a planet's magnetic moment is likely to be at least roughly aligned with its rotational velocity.

If the two magnetic moments are (Ae₁e₂ + Be₃e₄) and (Ce₁e₂ + De₃e₄), and the displacement between the magnets is re₁, then the force is proportional to (AC - BD)e₁/r⁵. Ignoring the slow plane e₃e₄, we get AC/r⁵, which is positive (repulsive) or negative (attractive) according to the signs of A and C.

When we consider both planes of rotation there are two cases. Suppose the planets are in "sync." Then either both planes repel or both planes attract. If the planets are "antisync", then one plane repels and the other attracts.

No. If they're "sync", with rotations in the same sense in both planes, then the plane containing the displacement vector produces repulsion (the AC term above), but the other plane produces attraction (the -BD term above). If they're "antisync", then both planes produce repulsion, or both attraction.

We're not even considering cases where the displacement vector isn't aligned with the two planes, or where the magnetic moments aren't aligned (so there are four planes with different orientations). In such cases e.g. the force could be sideways, neither attractive nor repulsive.

It seems to me that in some theoretical case where the two planes of magnetism are exactly the same strength everywhere then there should be no net magnetic force between the two planets no matter what you do.

Well, at least if A = B and C = D and everything is aligned, then indeed the force is 0.

[PatrickPowers]

When we consider both planes of rotation there are two cases. Suppose the planets are in "sync." Then either both planes repel or both planes attract. If the planets are "antisync", then one plane repels and the other attracts.

No. If they're "sync", with rotations in the same sense in both planes, then the plane containing the displacement vector produces repulsion (the AC term above), but the other plane produces attraction (the -BD term above). If they're "antisync", then both planes produce repulsion, or both attraction.

OK, I just made an arbitrary assignment like the "right hand rule." In geometric algebra you don't get a choice. That's fine with me.

We're not even considering cases where the displacement vector isn't aligned with the two planes, or where the magnetic moments aren't aligned (so there are four planes with different orientations). In such cases e.g. the force could be sideways, neither attractive nor repulsive.

In 3D there's no sideways force between two magnets, but AFAIK it could happen in 4D. I'm going to ruminate on that. There is however usually a sideways force on charged particles.

[PatrickPowers]

In 3D there's no sideways force between two magnets, but AFAIK it could happen in 4D. I'm going to ruminate on that. There is however usually a sideways force on charged particles.

I'm wrong again. What magnets seek to do is if possible align their magnetic planes and unless in a local minimum get closer together. So the path taken depends on the initial conditions and the shape of the magnetic fields, which can be all sorts of shapes. So there can be all manner of paths taken.

I used to have a set of magnets where one was suspended horizontally in the air above another magnet. It was shaped like a stretched out horizontal top. There was a neutral vertical baseplate that the point of the top pressed against. It was possible to spin it in the air, as the top was in a magnetic local minimum.

[PatrickPowers]

While munching a rice cake it occurred to me that spin CAN be orientable in some odd-dimensional spaces. It goes like this...

In our last episode we say that in an even-dimensional space reversing one spin always reverses exactly one other spin as well. So we have invariant parity of the signs of rotations. In odd dimensional spaces it is also possible to move one's point of view along the pole axis. Passing through the origin apparently reverses all the spins changing the sign of all of the spins. If the number of spins is even then parity is still invariant. Parity changes only if the number of spins is odd. So spin is orientable unless the number of dimensions N is equal to 4x-1.

Spin isn't orientable in dimensions 3, 7, 11, 15... but is everywhere else.

Now it occurs to me that if you have a plane that isn't rotating then it is possible to change the sign of one and only one other rotation. In such a condition spin isn't orientable no matter how many dimensions you've got. Since it is possible to not magnetize one or more planes, this means it is always possible to make a non-orientable magnet.