Vector potential A_mu from scalar function theta(x_mu)?

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Suppose we have a scalar function θ(x,y,z,t) of space and time where theta is some angle (0≤θ≤2π) that represents the compact coordinate of a 3 dimensional space (x,y,z) filling membrane at the space time point (x,y,z,t) in a compact space dimension w. Suppose that charge density "pushes" on the membrane in the compact dimension w with a force that is proportional to the magnitude of the charge density. The force only has a component in the w direction, the direction of the force in w is given by the sign of the charge density. This function can be graphed as a single clock hand at each point of spacetime (x,y,z,t). Define a positive charge such that if we move away from the charge θ increases, it would be opposite for a negative point charge. Hope I have been clear enough.

Can a function θ(x,y,z,t) yield the electromagnetic vector potential A_μ(x,y,z,t) by suitable mathematical operations on θ? I think not but I am stuck trying to show this can or cannot work.

Thanks for any help!
 

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Can a function θ(x,y,z,t) yield the electromagnetic vector potential A_μ(x,y,z,t) by suitable mathematical operations on θ?
You can get a vector function from a scalar function by taking a gradient...
 
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You can get a vector function from a scalar function by taking a gradient...
But in this case we would have ##F_{\mu\nu}=0##.
 
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