Why do you assume light is electromagnetic?
Why not? Unfortunately, we can't measure the speed of 4D light, but if we find a contradiction in the equations somewhere down the track, then we'll know that light isn't electromagnetic. Anyway, lets define 4D light as a 4D electromagnetic wave, if only because it's a shorter word.
I'm still trying to find a website that explains electromagnetic waves in 3D. There's a perfect explanation on my uni's website, but it requires a password, so for now I'll try to explain it myself.
I'll start with Gauss's laws, because it's easiest. Gauss's law for any field says that the flux through a closed surface is equal to the charge enclosed. There's no such thing as a magnetic charge, so this means the magnetic flux is 0.
In 4D, the only difference is that the "surface" is actually a 3D surcell. So we integrate over a volume, not an area.
We don't yet know whether tetric monopoles exist, but that doesn't matter, because we're looking at an electromagnetic wave on it's own, far away from any electric or tetric charges. So for any gaussian surface we pick, there are no charges of any kind enclosed, and Gauss's law simply becomes:
Int
E.d
V = 0
Int
B.d
V = 0
Int
T.d
V = 0
(
E,
B,
T and
V are vectors)
Faraday and Ampere's law are more complicated. With no current enclosed, they are:
Int
E.d
l = - d(flux B) / dt
Int
B.d
l = mu*eps d(flux E) / dt
This is how my textbook (Young and Freedman 11th ed.) shows that electromagnetic waves can exist (in 3D).
Draw a 3D graph, with an E pointing in the y direction and B pointing in the z direction, and the wave propagating in the x direction. We assume the wave is travelling at a constant speed c, without saying what value c might be. During a time dt, the wave front moves ahead c.dt
Then you draw a box in the xy plane, that extends behind and in front of the wave, and integrate the electric field anticlockwise around the box,
Here's an example. The first vertical line is the y axis, the second is a wave front, and the third is the same wave front after a time t. The dots represent the B field coming out of the page, behind the wave front. There is also an electric field pointing up, also behind the wave front.
- Code: Select all
y----->cdt<----
|........|...|
|...g------------f ^
|...|....|...| | |
|...|....|...| | a
|...|....|...| | |
|...h------------e v
|........|...|
O______________x
fg and he are perpendicular to the field, ef is outside the field, so the integral is simply
Int
E.d
l = |
E.gh | = - |E|*a
Now by Faraday's law, this means -Ea = - d (flux B)/dt
the change in magnetic flux through the rectangle, during the time dt. This is the area that the wave front sweeps out, a*c dt.
So we get -Ea = -Bac, or E=Bc
Now we do something similar on the xz plane, and from Ampere's law we get B= mu*eps*cE.
Together, these give B = mu * eps *c *Bc
or c^2 = 1/(mu * eps)
Now, there are many possibilities for a 4D analogue of Maxwells equations, and I've only tried a few of them. But, using the same reasoning as above, I've found values for all the important integrals etc.
I assumed that the three fields are all perpendicular, looked at the planes xy,xz and xw each in turn, and drew a box.
for the xy plane,
Int E dl = -Ea,
d (flux B) / dt = Bac
d (flux T) / dt = Tac
for xz,
Int B dl = Ba,
d (flux E) / dt = Eac
d (flux T) / dt = Tac
and xw,
Int T dl = -Ea,
d (flux E) / dt = Eac
d (flux B) / dt = Bac
The set of equations I decided on was this. Note that a changing E field generates a B field, a changing B field generates a T field, and a changing T field generates an E field.
Int
E.d
l = - d(flux B) / dt
Int
B.d
l = d(flux T) / dt
Int
T.d
l = mu*eps*tau * d(flux E) / dt
This results in
Ea = Bac
Ba = Tac
Ta = mu*eps*tau * Eac
The first two give E = Tc^2
And the third gives
T = mu*eps*tau * Tc^3
so c^3 = 1/(mu*eps*tau)
mu is in Webers per Amp meter, and epsilon is in
C^2/Nm^2. If we don't change these, then tau should be simply m/s, but I don't think that's right. For instance, Wb is the unit of flux, but flux is now an integral over volume, not area.
