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Mathematics for Physics

167 Four-Potential

A four-vector whose time and space parts are the scalar and vector potentials that is used to package the electromagnetic potentials into one Lorentz-covariant object.

Relativistic unification of the potentials. The electric scalar potential and the magnetic vector potential form one four-vector. A four-vector is a four-component object that transforms as spacetime coordinates do under a Lorentz transformation. This principle is used to treat the potentials as one spacetime quantity.

The four-potential is

\[ A^{\mu} = \left(\dfrac{V}{c},\, \mathbf{A}\right) \]

where

  • \(A^{\mu}\) is the four-potential.
  • \(V\) is the electric scalar potential.
  • \(\mathbf{A}\) is the magnetic vector potential.
  • \(c\) is the speed of light.

Local gauge invariance. Adding the spacetime derivative of a scalar function to \(A^{\mu}\) leaves \(\mathbf{E}\) and \(\mathbf{B}\) unchanged. Gauge invariance is that redundancy of the potential. This principle is used to choose the Lorenz gauge \(\partial_{\mu}A^{\mu}=0\) and simplify the equations.

Generation of the field tensor. The field tensor is the four-dimensional curl of the four-potential. The electromagnetic field tensor is the antisymmetric \(4\times 4\) array of \(\mathbf{E}\) and \(\mathbf{B}\). This principle is used to obtain Faraday’s law and the absence of magnetic monopoles as identities.

The field tensor from the four-potential is

\[ F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} \]

where

  • \(F_{\mu\nu}\) is the covariant field tensor.
  • \(\partial_{\mu}\) is the spacetime derivative.

The sourced wave equation. In the Lorenz gauge, the four-potential obeys a wave equation sourced by the four-current. The four-current is the four-vector of charge density and current density. This principle is used to compute radiation from accelerating charges.

The potential wave equation is

\[ \square A^{\mu} = -\mu_{0} J^{\mu} \]

where

  • \(\square\) is the d’Alembertian.
  • \(\mu_{0}\) is the permeability of free space.
  • \(J^{\mu}\) is the four-current.

Note: Also called the potential \(4\)-vector. Also called the \(4\)-vector potential.

167.1 References

  1. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — potential \(4\)-vector \(A^{\alpha}=\left(\dfrac{V}{c},\mathbf{A}\right)\); gauge invariance; \(F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\).
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(A^{\mu}=\left(\dfrac{V}{c},A_{x},A_{y},A_{z}\right)\); Lorenz gauge; wave equation.
  3. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — four-potential; \(F=dA\); gauge transformations.
  4. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — four-potential; field tensor from \(A^{\mu}\).