Feb2-13, 07:49 PM
I'm trying to model the electro-magnetic vector potential, does the following come close for 1 + 1 dimension spacetime? See sketches below.
Consider an elastic string, under tension, between two fixed points A and B that lie on the z axis. Let the distance between points A and B be large. Let the motion of the string in the x and y directions be given by ψ_x(z,t) and ψ_y(z,t). The string is such that both ψ_x(z,t) and ψ_y(z,t) satisfy the wave equation ψ(z,t),tt = c^2ψ(z,t),zz.
Now confine the motion of the string so that it moves freely on the surface of a cylinder of radius r, see below. Now let the position of the string be given by θ(z,t), with θ(A,t) and θ(B,t) = 0, and let θ(z,t) = 0 when the string is at rest. Define an orientation on the cylinder so we know which direction positive θ is.
Now let charge density ρ(z,t) interact with the constrained string above in the following way, where ever we have positive (negative) charge density there is a perpendicular force on the string in the positive (negative) θ direction. The magnitude of the force in the θ direction on a segment of the string from z to z + dz is proportional to the charge density at z times dz. See below.
The 1 + 1 dimensional vector potential A_0 will go like θ. the electric field will go like θ,z and A_z will go like θ times the velocity of the charge?
Let there be two separated and localized charge densities, positive and negative, that act on the string in the manner above, see below.
I think the 3 + 1 dimensional version of the above follows in a straight forward way.
Thanks for any help or suggestions!
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