Fiber bundles and the fiber for Electromagnetism.

Spinnor

Gold Member
The circle is the fiber over Minkowski spacetime for electromagnetism?

I want to make connection to the classical vector potential via some " picture" involving this circle. Does the following come close?

Can I consider a 3 dimensional surface in C_1XMinkowski space that at a given slice of time has position in the circle dimension of θ(X,t)? Obviously a different observer would see a Lorentz transformed θ(X,t) --> θ'(X',t')

Given the right properties could this surface encode the physics of the classical vector potential, A_μ?

I am thinking we could define kinetic energy at X and time t as being proportional to θ(X,t),t? The potential energy would have parts from the divergence of θ(X,t), ∇°θ(X,t) and the curl of θ(X,t), ∇Xθ(X,t)?

So The electric field goes as something like ∇°θ(X,t) + θ(X,t),t

Edit, we need a vector above and θ(X,t),t is not a vector. Damn!

2nd edit, ∇θ(X,t),t is a vector, does that work?

3rd edit, we can't take the curl of a scaler, ∇Xθ(X,t)?

And the magnetic field goes like ∇Xθ(X,t)?

How do you simply get electromagnetism from the circle?

Mathematics is a very precise language and I apologize for not being more precise.

Thanks for any help!

Last edited:
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Spinnor

Gold Member
So to recap, can we take θ(X,t) as fundamental and by various differential operations to θ(X,t) get the four components A(X,t)_μ?

Thanks for any help!

Gold Member

Spinnor

Gold Member
So to recap, can we take θ(X,t) as fundamental and by various differential operations to θ(X,t) get the four components A(X,t)_μ?

Thanks for any help!
This thread should be killed, and the above is false?

Spinnor

Gold Member
So to recap, can we take θ(X,t) as fundamental and by various differential operations to θ(X,t) get the four components A(X,t)_μ?

Thanks for any help!
If θ(X,t) were the position of an element of the surface along the circle we might also need a vector to map the possible movement of the surface in space? If the surface were "pushed" at two points in opposite directions, in the circle direction then, with the surface having the right physics, we would get a flow of the surface which could be represented by a vector, F(X,t)?

Then F(X,t) and θ(X,t) might after proper manipulation give us something like A_μ?

Spinnor

Gold Member
If θ(X,t) were the position of an element of the surface along the circle we might also need a vector to map the possible movement of the surface in space? If the surface were "pushed" at two points in opposite directions, in the circle direction then, with the surface having the right physics, we would get a flow of the surface which could be represented by a vector, F(X,t)?

Then F(X,t) and θ(X,t) might after proper manipulation give us something like A_μ?

Do we get the electric field goes like ∇°θ(X,t) + F(X,t),t
Magnetic field goes like ∇XF(X,t) ?

Spinnor

Gold Member
Do we get the electric field goes like ∇°θ(X,t) + F(X,t),t
Magnetic field goes like ∇XF(X,t) ?
I see flow going to zero in the plane whose points are equidistant from the opposite forces, even as the forces in the circle direction change with time. We need some other F, %^(

Spinnor

Gold Member
If we use forces to distort the surface in the top post we store energy in the surface. If we now move past this distorted surface, the stored energy in the distorted surface now transforms as a four-vector and we see both energy and momentum? So it now seems to me that there should be a function θ(X,t) that describes the "shape" of the surface as a function of space and time that encodes all the physics of something like A_μ? If we apply a force at one point of the surface that is like charge? If the force moves that is like current? The surface above with the right physics will act like the electromagnetic field?

Thanks for any help!

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