mr_e_man wrote:PatrickPowers wrote:mr_e_man wrote:
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°)
mr_e_man wrote: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 wrote: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.
Why even-dimensional spaces have two kinds of spin:
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.
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.
mr_e_man wrote: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.
mr_e_man wrote: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.
PatrickPowers wrote: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.
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