Light in 4D: Electromagnetism is not enough

[daemonflower]

The "4D countryside" thread prompted me to think about the properties of light in four dimensions. Sad to say, I was stopped right at the beginning.

The bad news: The Maxwell equations, from which all properties of electromagnetism are derived (in particular the existence of electromagnetic waves), don't make sense in four dimensions.

For those who missed Electromagnetism 101, and those who remember it only dimly (like me), here are the Maxwell equations:

• div E = 0
• curl E = -d_tB
• div B = 0
• c² curl B = d_tE

where E is the electric field, B the magnetic field, d_t partial derivation after the time t, div the divergence operator (div E = d_xE_x+d_yE_y+d_zE_z) and curl, well the curl operator which is defined as the cross product of the Nabla operator with its argument. [1]

The trouble here is that the cross product of two vectors is not defined in four dimensions. Think about how the cross product is defined in 3D:

Code: Select all

            | ex ey ez |
a x b = det(| ax ay az |)
            | bx by bz |


where a and b are two vectors, e_x, e_y and e_z are the unit vectors in x, y and z direction, and a_i and b_i are the respective components of a and b. det() means we take the determinant of a matrix. In more intuitive terms, the cross product of two vectors is perpendicular to both of them (unless they're parallel, in which case the cross product is zero).

Now a determinant can only be done on a square matrix. So in four dimensions, we can only perform this operation:

Code: Select all

    | ex ey ez ew |
det(| ax ay az aw |)
    | bx by bz bw |
    |  ?  ?  ?  ? |


which means that you can't do a cross product of two vectors (only three - that makes sense if you demand that the cross product should be perpendicular to all vectors), which means an operation like curl E is not defined. That means that the Maxwell equations are not valid in 4D, and we have nothing from which we can derive a wave equation for electromagnetic waves.



I figure there are three ways to go from here:

• Accept that there are no light waves in a four-dimensional universe. This seems a rather unsatisfying thought to me. After all we're talking about how four dimensional beings would see things quite a lot on this forum, and we'd have to scrap all that talk about "seeing" without light.
  On the other hand, this might be appealing to a certain breed of Science Fiction authors who use hyperspace as a way to circumvent the universal speed limit. After all, where there's no light, there's no speed of light either. You send your space ships away at the speed of imagination - if you can get the atoms they're made of to stick together without EM waves...
• Redefine the cross product so that it can work on two vectors:
  

  Code: Select all

              | ex ey ez ew |
            |  1  1  1  1 |
a x b = det(| ax ay az aw |)
            | bx by bz bw |


  
  I haven't thought that through yet. Maybe it would lead to light rays in three dimensions becoming light "disks" in four.
  The reason I haven't thought about it much is that I believe that to redefine the cross product in that way you'd need a good reason, and I can't think of one. Just as you rotate about a plane and not an axis in 4D, you'd take a cross product of a vector not with a line (vector), but with a plane spanned by two vectors. At least that is my intuitive understanding of the cross product.
• We can introduce a third component of the force analogous to electromagnetism. Let's call that field X (for "extradimensional mystery field"[2]). Then we can write four-dimensional Maxwell equations as
  
  • div E = 0
  • div B = 0, as before
  • cross(nabla, E, X) = -d_tB
  • c² cross(nabla, B, X) = d_tE
  Again, I haven't done the math yet, but I believe this would lead to a wave equation analogous to the electromagnetic wave equation in many aspects, including that the direction of propagation is perpendicular to all fields making up the wave.

I leave these thoughts to you for discussion. What can be done to save the eyesight of those poor 4D beings? Maybe I've overlooked solutions, too. I'd be interested in them. And I hope I get around to doing some math on my proposed solutions soon, to see where they lead to...

[1] Sorry if I don't write everything down using the correct symbols here, writing vector analysis terms in BBCode is a bit tedious.
[2] For all you smartypantses out there: I know it's not extradimensional. I just wanted something to scan with "X" and that is what I came up with
[3] I see the idea of a third force has come up before. But neither could I find a deduction of why it is needed, nor did I see a reference to either of the alternatives I mentioned.

[wendy]

Leo Young write "Systems of Units in Electricity and Magnetism" in 1969. Its a very interesting book, since it deals with EM units in six dimensions (base units). He also does not regard maxwell's equations as the basis of electrics, but deals with these from the source pattern. I will follow this order too. Consider this, and see what can be modified for higher dimensions.

1. Charge and Field.

Electrics and Magnetics were supposed to derive from a charge/field model, akin to gravity. One has charge Q, P, (in verbers) and fields E, H (in galvins/ft). field is the gradient of a potential V, U (Galvins). Charge differs from mass, in that it can be either sign, and not dependent on the inertial mass of the thing.

2. Flux, Flux density, and 'rationalisation'

The flux model is similar to light, supposes that the field is the result of a radiant flow of carriers from the source. The decay in intensity of the source is due to larger spheres having larger surface, the total flux emitted Psi, Phi [Byots] in an instant is the flux density eg D, B [Byots per sq ft], mutiplied by the surface area [eg sq. ft.]. One can measure either flux at unit distance from unit source, (eg foot-candle = intensity at one foot from a candle) [gives unrationalised units], or by total flux emitted [eg byot of flux emitted by a verber of charge: rationalised]. The constant of rationalisation is the surface of a sphere of n dimensions and diameter 2, divided by the unit of area [eg S3 = 4pi P2 = 8pi T2 = 16 C2].

3. The electric constant, magnetic constant, etc.

Electric and magnetic fields arise from the conversion of flux into field: ie D = ec.E, B = mc.H. Space is said to 'permit' (e) the conversion of electric flux to field, and permeable (m) to convert magnetic flux to field, There is a difference in units between permitivity ; permeability (seconds/foot), and various capacitivities (Edison-second/foot). Coulomb's constant is radial capacitivity. It is the capacititivity of a sphere of radius r, eg one coulomb-foot is the capacitivity of a sphere of radius one foot, is the unit in the ESU.

4. Dipoles.

Because charge comes with sign, it is possible to have a moment of charge that is independent of displacement. This happens because the gross charge is zero, and only the charge distribution provides the moment. One has then Electric Dipole and Magnetic Dipole (p, j, in verber-feet), and the corresponding electric, magnetic moments (based on flux), as k, m. The volume-densities of these gives polarisation (P, J, in verber / sq ft) and electrication, magnetisation K, M, (in oerstedts per sq ft).

Dipoles can arise naturally from the presence of external fields. "Displacement" originally stood for the vector P, the density of charge displaced by the field. It was only after rationalisation that displacement came to represent the density of electric flux.

5. Linkage and the EM velocity constant

In an electromagnet, magnetism arises from an electric current going around a vector area. That is, one can realise oerstedt-sq ft as the product of oerstedt and the vector area in sq ft. It is possible to wrangle all of this around to get an EM velocity constant.

The EM velocity constant is such that two currents in Vb/l ft, give the same force electrostatically, as Vb/t s. The measure l/t is the EM velocity constant. You can easily consider two currents in the definition of the ampere (which gives 'C/s), and then find the number of C/m that give the same force electrostatically. The ratio of C/s ÷ C/m gives m/s, which is converted to ft/s. This was first measured by Weber and Kausreich at 3.107 E10 mm.

One does not draw from this that charges in a current travels at the em velocity. A current of x C/s can be implemented by a very tiny flow of a large number of electrons, while C/ft is implemented by a very faint excess of charges in the wire.

6. Light and Relativity

Maxwell showed that waves from a vibrating EM source propogate at the EM Velocity, and noting the closeness of the WK value above to the speed of light, supposed that light moved in the same medium. Maxwell's equations are not transitive by Newtonian Relativity. Yet in the limit (when c=inf), it is. That is, Newtonian relativity is the limit of Einstein relativity when c -> oo. Einstein did not disprove Newton, since Newtonian physics is a limit with c=inf, which is pretty much the case in the real world.

Notes for Dimensionalists

Note that the Maxwells equations, and the definitions usually found in books of electronics are later adjustments to simplify calculations. Follow the examples above to get an idea,

a. Charge could become complex in higher dimensions, eg E+iH. Q+iJ

b. The em linkage could be entirely different.

c. Note that units are sq, cu. You may need different units, eg T2, T3 or P3, P4. area and volume here relate to bounding surface and content, not a hedrix and chorix respectively.

d. Vector surface exists in all dimensions. However, one can not have "current" flowing around a sphere.