Magnetic Fields of Planets in even dimensions

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Magnetic Fields of Planets in even dimensions

Postby anderscolingustafson » Wed Feb 26, 2014 6:34 pm

In an odd number of dimensions such as 3 dimensions or 5 dimensions when a planet rotates there are two points at opposite ends of the planets surface that are stationary. In 3 dimensions a planet will tend to orientate its magnetic field so that the North and the South magnetic field are close to the two points that don't. In an even number of dimensions there would not be two points on the surface of a planet that remain stationary but it would be possible for every point on the surface of a planet to move. Would a planet in an even number of dimensions still have a magnetic field and if so how would the magnetic field of a planet with an even number of dimensions orientate itself?
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Re: Magnetic Fields of Planets in even dimensions

Postby quickfur » Wed Feb 26, 2014 7:08 pm

That depends on how electromagnetism even works above 3D. I don't think we figured that out yet. For one thing, magnetic fields in 3D involve a cross product between two 3D vectors derived from the circulation of the electric field, but this is very specific to 3D. Only in 3D do you have a binary cross product. In 2D, the cross product has only a single argument -- and it's not clear how electromagnetism would even work like that (which of the two vectors should be chosen for the cross product?). In 4D, the cross product requires 3 arguments, but there are only 2 vectors available. What then? Where does the 3rd vector come from? And the further up the dimensions you go, the more complicated this problem becomes. In 5D, for example, the cross product requires 4 arguments but only two vectors are available! So where would you get the other 2 vectors from?
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Re: Magnetic Fields of Planets in even dimensions

Postby ICN5D » Wed Feb 26, 2014 8:17 pm

That's actually a pretty neat concept. The more I think about it, the more my mind places three magnetic poles around a 4D planet. It's interesting to see you also point out that, quickfur. This means there would have to be a third charge, in addition to + and - .Strangely enough, this sort of resembles the three charges of quarks in an atomic nucleus. When this happens, there is am increase of possible charge potentials.

In 3D Magnetics, + and - , 2 unique magnetic poles on 3D planet, used with 2D vector

+ : positive charge
- : negative charge

+- : neutral charge


In 4D Magnetics, + and - and ^ , 3 unique magnetic poles on 4D planet, used with 3D vector

+- : plus-minus charge
+^ : plus-carat charge
-^ : minus-carat charge

+-^ : neutral


In 5D Magnetics, + and - and ^ and *, 4 unique magnetic poles on 5D planet, used with 4D vector

+-^ : plus-minus-carat charge
+-* : plus-minus-asterisk charge
+^* : plus carat-asterisk charge
-^* : minus-carat-asterisk charge

+-^* : neutral
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Re: Magnetic Fields of Planets in even dimensions

Postby anderscolingustafson » Thu Feb 27, 2014 5:45 pm

quickfur,

Is an argument produced by the cross product the same as a magnetic charge?
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Re: Magnetic Fields of Planets in even dimensions

Postby quickfur » Thu Feb 27, 2014 6:13 pm

anderscolingustafson wrote:quickfur,

Is an argument produced by the cross product the same as a magnetic charge?

I'm not sure I understood your question.

Maxwell's equations describe (classical) electromagnetism, and two of them connect (the strength and direction of) the electric field with (the strength and direction of) the magnetic field. These two equations basically state that the strength of the electric field is proportional to the change in the magnetic field, and vice versa, and that the direction of one is perpendicular to the other. Or more precisely, a magnetic field in a certain direction induces a rotation of the electric field in the perpendicular plane, and vice versa.

There is a fundamental dependence on space being 3D implicit in these equations, in that 3D is the only dimension where a vector uniquely defines a rotation (i.e., the vector points in the direction of the rotational axis). In 2D, rotations can only happen around a point, so when you have a vector, there is no rotation possible around it. In 4D, on the other hand, fixing a single vector does not uniquely define a 2D rotational plane; it only determines a 3D hyperplane, and there are 3 principal rotations (along with all their oblique combinations thereof) possible. So, a single vector is insufficient to determine the plane of rotation. So basically, in 2D Maxwell's equations will most of the time have no solution (or contradictory solutions), whereas in 4D, it does not have unique solutions (it is under-determined), so it cannot fully describe what happens to one field when the other changes.

Basically, electromagnetism as we know it is strongly bound to space being 3D; to make it work in 4D would require fundamental changes that will probably make it unrecognizable as electromagnetism to us.
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Re: Magnetic Fields of Planets in even dimensions

Postby granpa » Sat Mar 01, 2014 5:47 am

In 4d the magnetic field is no longer a vector field but rather becomes a bivector field
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Re: Magnetic Fields of Planets in even dimensions

Postby ICN5D » Sat Mar 01, 2014 6:16 am

That would make sense, because of the cross-combined charge potentials. Between the three poles, there are three charges with two parts each. The bivector has two parts as well. Nice connection. Then, I suppose, a tetrapolar 5D magnetic field has a trivector due to the three parts of each charge. Hmm, interesting.
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Re: Magnetic Fields of Planets in even dimensions

Postby wendy » Sat Mar 01, 2014 7:41 am

lecky and magnety in higher dimensions are interesting, but a lot of the 3d theory relies on the rot(x) = curl(x), which has no reflex in 4d. So one has to look at something different.
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Re: Magnetic Fields of Planets in even dimensions

Postby PatrickPowers » Mon Mar 25, 2019 6:27 am

The first step is to see familiar 3D magnetism as a bivector field. That is, as an ordered pair of vectors. This is actually a more natural way of looking at it. One thinks of planes of magnetism instead of poles. This was all worked out by Clifford but the more succinct cross product idea caught on instead, where the dual of the plane is used. It is convenient but works only in 3D. The bivector representation works in any number of dimensions. In 4D two bivectors are needed. It is quite analogous to rotations. In 4D rigid rotations are described by two bivectors. The main difference is that magnetic fields aren't rigid. I don't know geometric algebra well enough to know exactly how I works out.

At any rate, the magnetic field would be generated by geodynamos powered by the Coriolus force. The Coriolus is also a double bivector. On a Clifford planet the two would be of equal magnitude. After that it grows complicated. Nobody really understands the details of even our own 3D Earth's magnetic field. There seem to be three competing geodynamos, or perhaps three groups of geodynamos, all of which are centered at latitude 60. On 4D Earth there is no particular reason for the two magnetic bivectors to be perpendicular or of equal magnitude, and there would surely be local variations.

I find perplexing the absence of unique planes of rotation on Clifford planets. How would it affect all this?
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Re: Magnetic Fields of Planets in even dimensions

Postby PatrickPowers » Fri Aug 09, 2019 4:27 am

PatrickPowers wrote:I find perplexing the absence of unique planes of rotation on Clifford planets. How would it affect all this?


On a planet with an isoclinic rotation the Coriolis force is of the same magnitude everywhere. Its plane is parallel to the surface of the planet.

The same is then true of true of the magnetic field that is generated by the Coriolis force. Everywhere on the surface of 4D Earth it is like 3D Earth's magnetic field at its poles. It would be of limited use for navigation, reducing the number of unknown directions from 3 to 2. Better than nothing. It could be used to find the direction in which one is rotating through space, but that's all. This direction depends on ones location on the planet. It's not like our east-west. Everything is relative, not absolute. An absolute background could be imposed, but it would be artificial.

This magnetic field would protect the planet from charged particles, I think. They come in at an angle, not at the poles.
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