Contents

Definition

The electromagnetic field on a spacetime $X$ is mathematically modeled by a circle bundle with connection $\nabla$ on $X$. If the underlying bundle is trivial, or else on local coordinate patches ${ℝ}^{n}↪X$ over which it is so, this connection is equivalently a differential 1-form $A\in {\Omega }^{1}\left({ℝ}^{n}\right)$.

This is then called the electromagnetic potential of the electromagnetic field (sometimes: “vector potential” or “gauge potential of the electromagnetic field”).

$F≔dA$F \coloneqq \mathbf{d}A

is the actual field strength of the electromagnetic field.

On a 4-dimensiona Minkowski spacetime with its canonical coordinates $\left\{t,{x}^{1},{x}^{2},{x}^{3}\right\}$, the electromagnetic potential $A$ is naturally expanded into corredinate components, traditionally written as

$A=\varphi dt+{A}_{1}d{x}^{1}+{A}_{2}d{x}^{2}+{A}_{3}d{x}^{3}\phantom{\rule{thinmathspace}{0ex}}.$A = \phi \mathbf{d}t + A_1 \mathbf{d}x^1 + A_2 \mathbf{d}x^2 + A_3 \mathbf{d}x^3 \,.

Here

• $\varphi$ is the electric potential

• $\stackrel{⇀}{A}=\left[{A}_{1},{A}_{2},{A}_{3}\right]$ is the magnetic potential

(for this choice of coordinates).

References

Section 5 of

Revised on April 26, 2013 19:14:20 by Urs Schreiber (82.169.65.155)