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.
quickfur wrote:why is it that a point mass in 3D produces a quadratic well, whereas a point mass in 4D produces a cubic well? If curvature depends only on the magnitude of a point mass, why can't it result in a quadratic gravity well instead of a cubic one?
quickfur wrote:But the whole point of this exercise was to postulate a geometrical source of gravity rather than the field model, which we already know results in a 1/r^(n-1) law of gravity.
quickfur wrote:Another way of thinking about this, is this thought experiment: suppose space was 4D yet stable orbits exist. If Einstein were to live in such a universe, what kind of theory would he have come up with to explain it? How different would General Relativity look if Einstein had invented it in a 4D universe with stable orbits? Which parts of the theory would be different? Can these differences be logically consistent, without leading to bizarre consequences that negate the other parts of a 4D physics derived in analogy from 3D?
mr_e_man wrote:Yesterday I did some numerical simulations.
A constant interstellar magnetic field, acting on an electrically charged planet orbiting a star with 1/r^3 gravity, seems to stabilize the orbit!
The orbit is in the opposite direction to the magnetic field (bivector). So the force of magnetism points outward, against gravity.
If the planet drifts too far inward, then it speeds up, and since the magnetic force is proportional to velocity, magnetism pushes the planet outward.
If the planet drifts too far outward, then it slows down, and since the magnetic force is proportional to velocity, magnetism relaxes and allows gravity to pull the planet inward.
If the magnetic field is not constant but is produced by the star and decreases with distance, then it should be even more helpful in stabilizing the orbit. However, given that our 3D Sun negates its magnetic field every 11 years, perhaps this shouldn't be relied on to sustain a 4D solar system.
The simulations were 2D. I haven't considered what would happen if the orbital and magnetic planes weren't aligned, or if the magnetic field wasn't planar, or if there was also an electric field.
mr_e_man wrote:
And of course we must ask how the planet retains its charge, and how that would affect life on its surface.
Ions in space, of the same charge as the planet, would be repelled. Ions of the opposite charge would be attracted, and become part of the planet and neutralize it.
PatrickPowers wrote:But magnetism doesn't depend on charge. Consider kitchen magnets. Planets are more or less like that.
Let me mention that I'm no expert on magnetism.
mr_e_man wrote:Yesterday I did some numerical simulations.
A constant interstellar magnetic field, acting on an electrically charged planet orbiting a star with 1/r^3 gravity, seems to stabilize the orbit!
The orbit is in the opposite direction to the magnetic field (bivector). So the force of magnetism points outward, against gravity.
If the planet drifts too far inward, then it speeds up, and since the magnetic force is proportional to velocity, magnetism pushes the planet outward.
If the planet drifts too far outward, then it slows down, and since the magnetic force is proportional to velocity, magnetism relaxes and allows gravity to pull the planet inward.
[...]
mr_e_man wrote:PatrickPowers wrote:But magnetism doesn't depend on charge. Consider kitchen magnets. Planets are more or less like that.
Let me mention that I'm no expert on magnetism.
https://en.wikipedia.org/wiki/Lorentz_force
The magnetic force on an object with charge q and velocity v is
f = q B•v,
where B is the magnetic field, which can be considered either as a bivector or as an antisymmetric matrix.
The magnetic field has many effects. I'm considering its effect on an electrically charged object. You're considering its effect on a magnetized object. (Note that some planets are not magnetic, particularly Venus.)
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.
PatrickPowers wrote:I've never heard tell of a heavenly body with a static charge. Even here on the surface of the Earth how often does one come across any object with a static charge. It is theoretically possible but static charges tend not to persist.
mr_e_man wrote:A constant interstellar magnetic field, acting on an electrically charged planet orbiting a star with 1/r^3 gravity, seems to stabilize the orbit!
[...]
The simulations were 2D. I haven't considered what would happen if the orbital and magnetic planes weren't aligned, or if the magnetic field wasn't planar, or if there was also an electric field.
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.
PatrickPowers wrote:I've never heard tell of a heavenly body with a static charge. Even here on the surface of the Earth how often does one come across any object with a static charge. It is theoretically possible but static charges tend not to persist. A positively charged heavenly body would attract negative ions until it was neutral again. [...]
I've always seen those laws about charged particles applied to elementary particles and ions. Molecules might be as large as that goes.
If you could have such a static charge then all you would have to do would be to have two heavenly bodies with the same polarity and they would always repel one another. That would definitely stabilize the orbit.
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.
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