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Mathematics for Physics

165 Four-Current

A four-vector formed from charge density and current density that is used to package the sources of the electromagnetic field into one Lorentz-covariant object.

Unified charge-current representation. Charge density and current density 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 keep charge and current transforming together between inertial frames.

The four-current is

\[ J^{\mu} = (c\rho,\, \mathbf{J}) \]

where

  • \(J^{\mu}\) is the four-current.
  • \(\rho\) is the charge density.
  • \(\mathbf{J}\) is the current density.
  • \(c\) is the speed of light.

Covariant conservation of charge. Local conservation of charge is the vanishing four-divergence of the four-current. The continuity equation is that conservation law written as a spacetime derivative. This principle is used to keep charge conserved in every inertial frame.

The covariant continuity equation is

\[ \partial_{\mu} J^{\mu} = 0 \]

where

  • \(\partial_{\mu}\) is the spacetime derivative.

Proper-velocity scaling. The four-current of a moving charge distribution is proper charge density times four-velocity. Proper charge density is the charge density in the rest frame of the charges. Four-velocity is the rate of change of spacetime position with proper time. This principle is used to show that ordinary \(\rho\) rises because moving volumes contract.

The proper-density form is

\[ J^{\mu} = \rho_{0} U^{\mu} \]

where

  • \(\rho_{0}\) is the proper charge density.
  • \(U^{\mu}\) is the four-velocity.

The four-current as the wave source. The four-current is the source of the four-potential wave. The d’Alembertian is the Lorentz-invariant wave operator. 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.
  • \(A^{\mu}\) is the four-potential.
  • \(\mu_{0}\) is the permeability of free space.

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

165.1 References

  1. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — current \(4\)-vector \(J^{\alpha}=(c\rho,\mathbf{J})\); \(\partial_{\mu}J^{\mu}=0\); \(\square A^{\mu}=-\mu_{0}J^{\mu}\).
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(J^{\mu}=(c\rho,J_{x},J_{y},J_{z})\); continuity equation; \(J^{\mu}=\rho_{0}U^{\mu}\).
  3. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — four-current; charge conservation; wave equation.
  4. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — four-current; proper charge density.