4D electromagnetism

[quickfur]

I've been thinking about how one might go about deriving a consistent electromagnetism for 4D space. Maxwell's equations aren't much help, because they rely on the cross product which is a 3D-specific operation.

What I want to know is, what is the underlying cause of Maxwell's equations? I.e., why does the electric field affect the magnetic field the way it does? What is it about the electric field that causes magnetism, and vice versa? I'm asking because if the underlying cause of electromagnetism is dimension-independent, then we can simply solve the underlying system for 4-space and we should in theory obtain a workable version of 4D electromagnetism.

[wendy]

I've fiddled about with 4D physics, etc.

Generally, the best approach is to not use L,M,T,Q etc as base units, but rather use a coordinate system where size is carried entirely on one dimension. You do this by choosing the bulk of the base units that remain constant over a cosmological scale. I use density (as a square), velocity, and time. The bulk of magnitude is carried by time.

One can further simplify the scale by putting D = 200, V=10, T=1. L becomes 11, M becomes 233, Time becomes 1, Q becomes 122 (emu), or 132 (esu). It is the ratio of 132-122 = 10, which defines the EM velocity constant (c).

We have, then things like density (200), pressure (220), velocity (10), remaining constant over the dimensions.

The measures like surface (22), volume (33), mass (233), force (242), energy (253), power (252), increase by a measure 11 for each dimension: so in 4d, they are 244, 252, 264 and 263 respectively.

The constant of gravity G, is found to be -202 in all dimensions, that GM = eg 233-202 = 31 in N3, or 244-202 = 42 in 4D. 42, and 21 play interesting roles in 4D.

Charge is held to be a surface feature, 122. In 4D, this becomes 133 = L.sqrt(M)= (l^3).sqrt(D). This gives things like current as 132, voltage as E/Q = 131, capacitance as eg V/Q = -2. and capacitivity (f/m), as -13. One can see resistance is 1, and that F, H are then as per usual, freq/resist, and freq.resist. respectively. Of course, all this happens regardless of the value for charge, since this is from LMT and a single variable Q.

[quickfur]

[...]Generally, the best approach is to not use L,M,T,Q etc as base units, but rather use a coordinate system where size is carried entirely on one dimension.

What do L, M, T, Q stand for?

Also, what I was looking for was more of a qualitative model from which the relationship between electricity and magnetism can be derived in a dimension-independent way. I forget the details, but I remember reading somewhere a while ago something about a sphere with a vector pointing from its center to its north pole, and how many different orientations result when you consider all possible combinations of quarter turns of the sphere such that the direction of the vector at the end is preserved. This was somehow related to why there is such a thing as an electric charge in the first place. Unfortunately, I've forgotten what the line of reasoning was. But the point is, I'd like to start from something like this, which is much easier to generalize to higher dimensions than Maxwell's equations, and derive a 4D-specific model of electromagnetism.

[wendy]

L M T Q are basic dimensions: length, mass, time, charge.

It's pretty hard to get some sort of quantive relation, because you have to guess what stays constant over the dimensions. The 3D equations rely on 2 (ie Biot.seconds) = N-1 (Franklins), for example, which means that it does not port very well to 4D. At least this has been my experience in calculations.

Some equations still work well. For example, the dot-product of flux [dot] area-norm is still proportional to the enclosed charge. You still have something like vector-surface of a ring [ampere's law for a loop], but a ring is something that bounds a surface-fragment, is N-2, not 1D. When E and H are the same measures, so that one could write, eg "E+iH" then the relation between "electrical current" and "magnetic line-density" disappears, i think.

Curl in 4D does not work all that well: the cross of two vectors is N-2. You keep adding vectors until this becomes 1.

Of course, this all supposes that 4-space is homogeneous: adding something like pointwise swirlbobs (so that space behaves like CE2 = complex-euclidean 2D) changes everything. I shudder to think what this chances to bring.

This is basically how i try to do it.

1. Charge and gravity are radiant-flux force-fields. That is, from a distance x from a point charge/mass, the force is proportional to the the density of flux radiating from the source. The total field is the sum of such forces. You can use this in all dimensions.

2. Gravity attracts but is positive only (even "antimatter", say, is positive to gravity), charge repells but comes in either sign.

3. Charge arises from minor shifts in bulk volume, not massive shifts in small volumes. A C-m is really a GC × nm, for example.

4. Electricity and Magnetism are permitive, that is, the ratio of flux density to force field is set by the medium where the conversion takes place. eg D = eE and B = µH apply. The exact nature of e and µ is not clear, but we can assume at least some case of isotropic constant (eg vacuum).

5. Dipoles exist, since it is possible to displace the +/- dipoles at an atomic scale (ie GC . nm = C,m).

6. A ring (closed N-2 manifold), or region bounding a surface (N-1 space) has a vector-surface that is constant in size and direction, and independent of the surfaces that span said ring. [Mathematical proof exists for this]. (the 3D case has the current on the loop × vector-area = magnetic dipole thus produced).

Much of this is used to frame physics in every space, not just euclidean space.

[quickfur]

After reading this article on Wikipedia, I got an idea for how to derive 4D electromagnetism. The basic idea is to accept special relativity, and then given electrostatic charges, derive magnetism. The underlying method is to analyse different scenarios, such as moving charges in a wire, taking into account Lorentz contraction in the various reference frames, which gives rise to an uneven distribution of charges that cause a magnetic field to arise orthogonal to the direction of motion. While this may or may not be the "real" origin of magnetism, it's a useful way of deriving it from the other two assumptions.

Now, given that the derivation is actually dimension-independent (except that the number of dimensions should be at least 2), this should be easy to generalize to the 4D case. I don't have time to actually work it out right now, but it's already clear that 4D electromagnetism will have some interesting properties. For one thing, it seems that the 4D magnetic field will no longer be a vector field, but a bivector field, due to the extra degree of freedom. You can no longer produce a bipolar magnet using coils, because the magnetic field generated by a coil of wires carrying moving electric charges will be circular (cf. duocylinder) rather than parallel. In fact, it may not be possible to produce bipolar magnets at all (but I've yet to verify this).