The Four-Current and Four-Potential

Electromagnetism is the theory that forced relativity. Maxwell's equations single out one speed, $c=1/\sqrt{{\mu }_0{ϵ}_0}$, with no reference to any frame, and reconciling that with mechanics is what produced the Lorentz transformation.¹ The reward for building the tensor machinery of the earlier modules is that the whole theory collapses into a few index equations in which Lorentz invariance is manifest. This module rebuilds it. The starting point is the source of the fields: charge and current.

Throughout, the metric signature is $(-,+,+,+)$, coordinates are $x^{\mu }=(ct,x,y,z)$ so that $x^0=ct$, and the derivative operator is ${\partial }_{\mu }=\partial /\partial x^{\mu }=\left(\frac{1}{c}{\partial }_t,\nabla \right)$. Indices are raised and lowered with ${\eta }_{\mu \nu }=\mathrm{diag}(-1,+1,+1,+1)$, following the four-vector conventions of Module 2.

Charge and current as one object

Charge density $\rho$ and current density $\vec{J}$ are not separate quantities in different frames. A static line of charge, seen by a moving observer, is a current; a stationary charge cloud, boosted, carries both a compressed density and a flow. The transformation that ties them together is the same one that ties $ct$ to $\vec{x}$, so they must be the components of a four-vector.

Take a cloud of charge with proper charge density ${\rho }_0$, the density measured in the local rest frame of the charge. In that frame there is no current. Give the cloud four-velocity $U^{\mu }=\gamma (c,\vec{u})$, where $\vec{u}$ is its ordinary velocity and $\gamma =(1-u^2/c^2{)}^{-1/2}$. The four-current is

Its components in a general frame are the ordinary charge and current densities:

The time component $J^0=c\rho$ carries the density; the space components carry the flow. The factor $\gamma$ in $\rho =\gamma {\rho }_0$ is length contraction of the charge distribution: the same charge occupies a volume shrunk by $1/\gamma$ along the motion, so the density measured by a moving observer is larger by $\gamma$. Charge itself is a Lorentz invariant — every observer counts the same number of elementary charges — but charge density is not, because volume is not.

The cleanest illustration is the one-dimensional case. A wire at rest carries charge per unit length ${\lambda }_0$ and no current. View it from a frame moving at speed $v$ antiparallel to the wire, so the charges stream past at $+v$. The spacing between charges contracts by $1/\gamma$, so the linear density becomes $\lambda =\gamma {\lambda }_0$, and a current $I=\lambda v=\gamma {\lambda }_{0}v$ appears where there was none. Density and current are the same physical charge seen through two frames.

Charge conservation as a divergence

The statement that charge is neither created nor destroyed is the continuity equation,

Charge that leaves a region must flow across its boundary. In four-vector form this is the vanishing four-divergence of the current:

The four-divergence ${\partial }_{\mu }{J}^{\mu }$ is a Lorentz scalar: its value is the same in every frame. Charge conservation is therefore not a separate law in each frame but a single invariant statement. If charge is conserved for one observer, the tensor equation ${\partial }_{\mu }{J}^{\mu }=0$ guarantees it for all.

The geometric content is a flux balance. Integrate ${\partial }_{\mu }{J}^{\mu }=0$ over a four-dimensional region of spacetime and apply the divergence theorem: the net flux of $J^{\mu }$ through the boundary vanishes. For a box that is a spatial volume $V$ swept from time $t_1$ to $t_2$, the flux through the two spacelike faces is the total charge in $V$ at each time, and the flux through the timelike sides is the current leaving through the walls. Setting the two equal gives

the integral form of charge conservation.

The four-potential

The electric and magnetic fields derive from potentials. The scalar potential $\varphi$ and the vector potential $\vec{A}$ produce the fields through

These two potentials assemble into the four-potential

That $A^{\mu }$ is a genuine four-vector is not obvious from the definitions of $\varphi$ and $\vec{A}$ alone; the justification is that the field equations written in terms of $A^{\mu }$ take the form of a four-vector equation, which the next section establishes. The four-potential is the fundamental variable of the covariant theory. The fields $\vec{E}$ and $\vec{B}$ are extracted from its derivatives.

Gauge freedom

The potentials are not unique. The fields $\vec{E}$ and $\vec{B}$ are unchanged by the gauge transformation

for any scalar function $\chi (t,\vec{x})$. In four-vector form this is a single shift,

with ${\partial }^{\mu }=(-\frac{1}{c}{\partial }_t,\nabla )$. The magnetic field is unchanged because $\nabla \times \nabla \chi =0$; the electric field is unchanged because the added pieces $-{\partial }_t\nabla \chi$ and $\nabla {\partial }_t\chi$ cancel. Different four-potentials related by ${\partial }^{\mu }\chi$ describe identical physics.

Gauge freedom is a redundancy, and it can be used to simplify the equations. The Lorenz gauge is the condition

a manifestly Lorentz-invariant statement (it is the vanishing four-divergence of $A^{\mu }$). Any potential can be brought to Lorenz gauge: if ${\partial }_{\mu }{A}^{\mu }=f\ne 0$, choose $\chi$ solving $\square \chi =-f$, and the shifted potential satisfies the condition. Even within Lorenz gauge a residual freedom remains — any $\chi$ with $\square \chi =0$ preserves it — but the condition is enough to decouple the field equations.

The sourced wave equation

Writing Maxwell's equations for the potentials, in Lorenz gauge, gives a single equation. The two inhomogeneous Maxwell equations, Gauss's law $\nabla \cdot \vec{E}=\rho /{ϵ}_0$ and the Ampère–Maxwell law $\nabla \times \vec{B}={\mu }_0\vec{J}+\frac{1}{c^2}{\partial }_t\vec{E}$, when expressed through the potentials and simplified with ${\partial }_{\mu }{A}^{\mu }=0$, both reduce to the same form. Define the d'Alembertian, the four-dimensional wave operator,

Then the potentials satisfy

Component by component, this is one wave equation for the scalar potential and one for each component of the vector potential:

The check on the time component uses $A^0=\varphi /c$ and $J^0=c\rho$: $\square (\varphi /c)=-{\mu }_{0}c\rho$ gives $\square \varphi =-{\mu }_0{c}^2\rho =-\rho /{ϵ}_0$, which is the wave equation above, and reduces to Poisson's equation ${\nabla }^2\varphi =-\rho /{ϵ}_0$ in the static limit.

The solution is the retarded potential: the field at a point is set by the source at the retarded time $t-|\vec{x}-\vec{x}{}^{\prime }|/c$, the earlier moment from which a signal at speed $c$ reaches the point. Disturbances in charge and current propagate outward as waves at $c$, the light speed built into the wave operator. That electromagnetism predicts waves at $c$ in every frame is the puzzle Module 1 opened with, now written as a manifestly invariant equation.

Summary

• Charge density and current density are the components of one four-vector, the four-current $J^{\mu }=(c\rho ,\vec{J})={\rho }_0{U}^{\mu }$. Density and current are frame-dependent shadows of the same charge; a boost turns a static density into a current.
• Charge conservation is the single invariant statement ${\partial }_{\mu }{J}^{\mu }=0$, the vanishing four-divergence, equivalent to the continuity equation and to a flux balance on any spacetime region.
• The scalar and vector potentials combine into the four-potential $A^{\mu }=(\varphi /c,\vec{A})$, defined only up to a gauge shift $A^{\mu }\to A^{\mu }+{\partial }^{\mu }\chi$ that leaves the fields untouched.
• The Lorenz gauge ${\partial }_{\mu }{A}^{\mu }=0$ is Lorentz invariant and reduces the field equations for the potentials to the wave equation $\square A^{\mu }=-{\mu }_0{J}^{\mu }$, whose solutions propagate at $c$.

The next lesson builds the gauge-invariant field strength $F^{\mu \nu }$ from the derivatives of $A^{\mu }$ and reads $\vec{E}$ and $\vec{B}$ off its components.