quickfur wrote:Oooh... This looks like a very promising approach to deriving a workable model of electromagnetism in 4D! Thanks for posting this... I've been trying to find an approach to electromagnetism that isn't tied to a specific dimension of space. If we assume the obvious analogies to special relativity and basic electric charges, we could probably calculate the effect of a current of charged particles to a test charge, and thereby derive 4D-specific equations for magnetism.
quickfur wrote:Very nice stuff. I got scared off bivectors by the wikipedia article, which probably does not do it justice... but seeing them in action has convinced me that they are worth looking into. I suppose 4D rotational mechanics would benefit from using a geometric algebra representation too. I agree with the note on the website that cross products are trouble. They are specific to 3D, and are generally quite fragile and must be handled with care, etc..
quickfur wrote:Now your next challenge is to come up with a workable model of 4D atomic physics. :- The problem is, the last time somebody tried to solve the 4D Schroedinger's equation, it had no local minima except at r=0, which implies that the direct equivalent of a 4D atom would simply collapse and disintegrate.
quickfur wrote:There's also the issue of wave propagation through a 4D medium, which some time ago somebody linked an article about, that indicates that 3D is the only dimensionality of space where a signal transmitted from a point source can be recovered faithfully at a receiver at some distance away. In even dimensions, in particular, the wavefront would induce back-propagation which causes the signal to interfere with itself in a complex way, that makes it difficult to recover the original signal at the receiving end.
These things seem to be suggesting that a workable model of 4D physics may turn out to be far more alien than what one might expect based on a naive generalization from 3D by dimensional analogy.
quickfur wrote:The problem with generalizing precession to 4D is that precession assumes a rotational axis. However, rotational axes don't exist in 4D. Rotation being a 2D phenomenon, if you were to place a spinning object on a 3D hypersurface, the rotation would mainly occupy only 2 out of the 3 horizontal directions, so I wouldn't expect rotational inertia to hold it upright. It would probably just fall sideways along the direction perpendicular to the rotational plane. But then again, I've never managed to work out how to generalize rotational inertia to 4D, so perhaps I'm wrong. But there will definitely be an inhomogenous situation in the horizontal hyperplane, unlike rotation in 3D.
Now, precession in free-floating objects in 4D space is a different story altogether. According to wendy, if such objects have any rotation, they should eventually settle into one of the two chiral Clifford rotations, in which there are an infinite number of stationary (2D) planes. I don't know how you'd even apply the concept of precession to that!
quickfur wrote:Unless, of course, we declare by fiat that gravity continues to obey an inverse square law in 4D. Then the familiar elliptical orbits return, except with a much larger scope of possibilities granted by the extra dimension of space. Sometimes this is a convenient fancy to explore, as it yields many interesting results that are sufficiently similar to our own 3D universe that we understand it more easily, and gain insights into the behaviour of 4D space. However, it's also a rather unrealistic model of how an actual 4D universe might look like.
PatrickPowers wrote:Macdonald leaves adding in the charges as an exercise for the reader. Grrr.
ei|diE = rho/eps0,
but is it
ei|diB = j/eps0 - dtE
or
ei|diB = -j/eps0 - dtE
?
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