Exploring Magnetism in 4+1D: What Does a Magnetic Field Look Like?

[granpa]

what would a magnetic field look like in 4+1D?

In 3+1D the field lines simply rotate around the axis which points along the direction of motion of the electron. But in 4+1D this is not defined. But if magnetism is simply a result of relativity then there should be a way to make the equations work in 4+1D.

Surely someone somewhere must have worked this out by now.

maybe instead of 'lines' of force one would instead use 'planes' of force??

 

[Thaakisfox]

The answer can be best seen from the covariant formulation of electrodynamics. The electromagnetic field is represented as a two index antisymmetric tensor, called the field-strength tensor. Electrodynamics can be readily generalized to arbitrary dimensions in this formulation.
The point is that 3+1 D is so special that the magnetic field can be represented as a vector. The point is that if we look at how the magnetic field is situated in the field strength tensor (in 3+1 for now) we see that it is an antisymmetric tensor itself.
Now there is a theorem in linear algebra relating antisymmetric tensors, called the Hodge-dualism (of course there is more to this). And it so happens that there in 3D there is a one to one correspondence between a two index antisymmetric tensor and a axial vector (this has to do with why there is such a thing as a vector product in 3D, a vector product can also be defined in 7D, the reasons of these are even deeper and related to Clifford algebras.). Hence the magnetic field which is actually not a vector can be *represented* as a vector in 3+1. So in extra dimensions we cannot imagine it as a vector.

 

[granpa]

so its a psuedovector? (in 3+1D)

 

[Thaakisfox]

Yep, It's a pseudo vector.

 

[granpa]

Pardon my stupidity, but the 3+1D psuedovector for magnetism,
does it point in the direction of the magnetic field lines or
does it point along the axis that the magnetic field lines curl around
(i.e. the direction of motion of the electron)

edit:This is seeming more and more like a really dumb quesion. I am sure it must be pointing along the magnetic field lines. But my thinking is that a bivector is also a pseudovector and the magnetic field lines 'curl' or rotate around the axis. In 4D rotation is also a bivector. I'm probably just confusing myself here.

no wait, a bivector is a tensor but rotation in 4D reduces to a psuedovector in 3D.
I hate tensors.

 

[Thaakisfox]

So by "field lines" you mean you mean the lines of the electric field?

 

[granpa]

I was thinking magnetic field lines.

see my edited post below

 

[Thaakisfox]

The field lines are by definition the trajectories of the vector. i.e. the magnetic vector is tangent to the magnetic field line by definition, similarly for the electric part.

 

[granpa]

Ok. Thank you. You've answered by question.
In higher dimensions its a tensor.

 

[granpa]

This whole line of reasoning has made me think about tensors in a new light.
Tensors have always been mysterious to me but I think I understand them better now.
If a vector is a one dimensional line in a higher dimensional space then a tensor (like a bivector) would be a plane or manifold in a higher dimensional space.
Maybe I should look into clifford algebra again.