The four-current, charge-current density, and covariant continuity

Intuition Beginner

Special relativity unifies space and time into a single 4-dimensional framework. The charge density $\rho$ and current density $\mathbf{J}$ are similarly unified into a single four-component object called the four-current:

$$j^{\mu }=(c\rho ,J_x,J_y,J_z)$$

The time component is the charge density (times $c$), and the spatial components are the current density. Just as the position four-vector $(ct,x,y,z)$ combines space and time, the four-current combines charge and current.

The continuity equation (charge conservation) becomes beautifully simple in four-vector notation:

$${\text{div}}_4(j)=0$$

where ${\text{div}}_4$ denotes the four-dimensional divergence.

This single equation says: charge is conserved. In any reference frame, at any point in spacetime, the four-divergence of the current vanishes. The fact that it can be written as a single four-vector equation means it is automatically Lorentz-invariant: if charge is conserved in one frame, it is conserved in all frames.

Visual Beginner

3D quantity	4D quantity	Component
Charge density $\rho$	$j^0=c\rho$	Time component
Current density $J_x$	$j^1=J_x$	Space component
Current density $J_y$	$j^2=J_y$	Space component
Current density $J_z$	$j^3=J_z$	Space component
Continuity equation	four-divergence of $j$ equals zero	Lorentz scalar

Worked example Beginner

A uniform line charge with density $\lambda$ lies along the $x$-axis in frame S (at rest). In this frame, $\rho =\lambda \delta (y)\delta (z)$ and $\mathbf{J}=0$. The four-current is $j^{\mu }=(c\lambda \delta (y)\delta (z),0,0,0)$.

Now boost to frame S' moving with velocity $v$ in the $x$-direction. The Lorentz transformation mixes the time and $x$ components:

$$j^{\prime 0}=\gamma (j^0-\beta j^1)=\gamma c\lambda \delta (y^{\prime })\delta (z^{\prime })$$ $$j^{\prime 1}=\gamma (j^1-\beta j^0)=-\gamma \beta c\lambda \delta (y^{\prime })\delta (z^{\prime })=-\gamma v\lambda \delta (y^{\prime })\delta (z^{\prime })$$

So in the boosted frame, there is both charge density ${\rho }^{\prime }=\gamma \lambda \delta (y^{\prime })\delta (z^{\prime })$ (enhanced by length contraction) and current density $J_x^{\prime }=-\gamma v\lambda \delta (y^{\prime })\delta (z^{\prime })$ (the line charge is now a current). The four-current correctly captures both effects.

Check your understanding Beginner

Formal definition Intermediate+

The four-current density is defined as:

$$j^{\mu }=(c\rho ,\mathbf{J})$$

where $\rho (\mathbf{r},t)$ is the charge density and $\mathbf{J}(\mathbf{r},t)$ is the current density. Under a Lorentz boost with velocity $\mathbf{v}$:

$$j^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{j}^{\nu }$$

Explicitly, for a boost along $x$: $j^{\prime 0}=\gamma (j^0-\beta j^1)$, $j^{\prime 1}=\gamma (j^1-\beta j^0)$, $j^{\prime 2}=j^2$, $j^{\prime 3}=j^3$.

The continuity equation in covariant form.

$${\partial }_{\mu }{j}^{\mu }=\frac{1}{c}\frac{\partial j^0}{\partial t}+\nabla \cdot \mathbf{J}=\frac{\partial \rho }{\partial t}+\nabla \cdot \mathbf{J}=0$$

This is the local statement of charge conservation. The integral form is $\frac{dQ}{dt}=-\oint \mathbf{J}\cdot d\mathbf{a}$.

The inhomogeneous Maxwell equations in covariant form. Using the Faraday tensor $F^{\mu \nu }$ 10.06.01:

$${\partial }_{\mu }{F}^{\mu \nu }={\mu }_0{j}^{\nu }$$

This single tensor equation encodes both Gauss's law ($\nu =0$) and the Ampere-Maxwell law ($\nu =1,2,3$). The continuity equation follows by taking the divergence: ${\partial }_{\nu }{\partial }_{\mu }{F}^{\mu \nu }={\mu }_0{\partial }_{\nu }{j}^{\nu }$. Since $F^{\mu \nu }$ is antisymmetric, ${\partial }_{\nu }{\partial }_{\mu }{F}^{\mu \nu }=0$, giving ${\partial }_{\nu }{j}^{\nu }=0$.

Transformation of charge density. For a boost along $x$:

$${\rho }^{\prime }=\gamma \left(\rho -\frac{v}{c^2}{J}_x\right),J_x^{\prime }=\gamma (J_x-v\rho )$$

A pure charge density ($\mathbf{J}=0$) in one frame becomes both charge and current in a boosted frame. The total charge $Q=\int \rho d^{3}r$ is a Lorentz scalar: $Q^{\prime }=Q$.

Key derivation Intermediate+

Derivation (Covariant continuity equation from Maxwell's equations).

Theorem. The continuity equation ${\partial }_{\mu }{j}^{\mu }=0$ follows from the inhomogeneous Maxwell equations ${\partial }_{\mu }{F}^{\mu \nu }={\mu }_0{j}^{\nu }$.

Proof. Take the four-divergence of both sides:

$${\partial }_{\nu }{\partial }_{\mu }{F}^{\mu \nu }={\mu }_0{\partial }_{\nu }{j}^{\nu }$$

The left side involves the contraction of the symmetric tensor ${\partial }_{\nu }{\partial }_{\mu }$ with the antisymmetric tensor $F^{\mu \nu }$. For any symmetric $S_{\mu \nu }$ and antisymmetric $A^{\mu \nu }$: $S_{\mu \nu }{A}^{\mu \nu }=0$ (each term cancels with its transposed partner: $S_{12}{A}^{12}+S_{21}{A}^{21}=S_{12}{A}^{12}+S_{12}(-A^{12})=0$). Therefore ${\partial }_{\nu }{\partial }_{\mu }{F}^{\mu \nu }=0$, and:

$${\partial }_{\nu }{j}^{\nu }=0$$

This is the continuity equation, derived as a consequence of Maxwell's equations, not an additional postulate. $\square$

Bridge. The four-current is the source term in the covariant Maxwell equations, playing the same role as $\rho$ and $\mathbf{J}$ in the 3D formulation but with manifest Lorentz invariance. The foundational insight is that charge conservation (${\partial }_{\mu }{j}^{\mu }=0$) is not an independent law but a consequence of the inhomogeneous Maxwell equations and the antisymmetry of $F^{\mu \nu }$. The central message is that the four-current transforms as a four-vector, ensuring that charge conservation holds in all reference frames automatically. Putting these together, the four-current provides the source for the Faraday tensor 10.06.01, and the stress-energy tensor 10.06.03 is built from both $F^{\mu \nu }$ and $j^{\mu }$. The Lorentz transformation of E and B fields 10.06.04 follows from the transformation of $F^{\mu \nu }$, which is sourced by $j^{\mu }$.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib has Lorentz transformations and Minkowski space but not the four-current as a physical quantity, the covariant continuity equation, or the transformation laws for charge/current densities. lean_status: none.

Advanced results Master

The gauge-covariant derivative. In the presence of an electromagnetic field, the ordinary derivative is replaced by the gauge-covariant derivative $D_{\mu }={\partial }_{\mu }+iq{A}_{\mu }/\hslash$ (where $q$ is the charge). The four-current for a charged field $\psi$ is $j^{\mu }=iq({\psi }^{\ast }{D}^{\mu }\psi -\psi (D^{\mu }\psi {)}^{\ast })$. This expression is automatically conserved (${\partial }_{\mu }{j}^{\mu }=0$) by the equations of motion for $\psi$.

Charge conservation as a Noether current. The continuity equation ${\partial }_{\mu }{j}^{\mu }=0$ is the Noether current associated with the global $U(1)$ symmetry of the Lagrangian: $\psi \to e^{i\alpha }\psi$ (global phase rotation). The conserved charge is $Q=\int j^0{d}^{3}x$, which is the total electric charge. This connects electromagnetism to the gauge principle: local gauge invariance ($\alpha \to \alpha (x)$) requires the existence of the electromagnetic field $A_{\mu }$.

The charge-current tensor. For a system of charged particles, the total angular momentum is conserved only if the current satisfies additional conditions. The full description involves the energy-momentum tensor $T^{\mu \nu }$ 10.06.03 and the angular momentum tensor $J^{\mu \nu \lambda }=x^{\mu }{T}^{\nu \lambda }-x^{\nu }{T}^{\mu \lambda }$.

Synthesis. The four-current is the covariant source for the electromagnetic field, unifying charge density and current density into a single geometric object. The foundational insight is that charge conservation (${\partial }_{\mu }{j}^{\mu }=0$) is not an additional law but a consequence of the antisymmetry of $F^{\mu \nu }$, and being a four-vector equation, it is automatically Lorentz-invariant. The central message is that the four-current transforms as a four-vector under Lorentz boosts, mixing charge and current in exactly the way that space and time mix. Putting these together, the four-current provides the source for the inhomogeneous Maxwell equations 10.06.01, drives the stress-energy tensor 10.06.03, and determines the transformation of fields between frames 10.06.04.

Full proof set Master

Proposition (Lorentz invariance of the continuity equation). If ${\partial }_{\mu }{j}^{\mu }=0$ holds in one frame, it holds in all Lorentz-transformed frames.

Proof. The continuity equation is ${\partial }_{\mu }{j}^{\mu }=0$, a Lorentz scalar. Under a Lorentz transformation $x^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{x}^{\nu }$: ${\partial }_{\mu }^{\prime }={\Lambda }_{\mu }{}^{\nu }{\partial }_{\nu }$ and $j^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{j}^{\nu }$. So ${\partial }_{\mu }^{\prime }{j}^{\prime \mu }={\Lambda }_{\mu }{}^{\nu }{\Lambda }^{\mu }{}_{\rho }{\partial }_{\nu }{j}^{\rho }={\delta }_{\rho }^{\nu }{\partial }_{\nu }{j}^{\rho }={\partial }_{\nu }{j}^{\nu }=0$. The contraction ${\Lambda }_{\mu }{}^{\nu }{\Lambda }^{\mu }{}_{\rho }={\delta }_{\rho }^{\nu }$ follows from the defining property of Lorentz transformations: ${\Lambda }^{\mu }{}_{\nu }{\eta }_{\mu \rho }{\Lambda }^{\rho }{}_{\sigma }={\eta }_{\nu \sigma }$. $\square$

Connections Master

• Faraday tensor 10.06.01 is the field quantity; the four-current is the source: ${\partial }_{\mu }{F}^{\mu \nu }={\mu }_0{j}^{\nu }$.
• Four-vectors 10.05.03 provide the framework; the four-current is a specific four-vector.
• Stress-energy tensor 10.06.03 involves both $F^{\mu \nu }$ and $j^{\mu }$ in the Lorentz force equation.
• Displacement current 10.03.04 is part of the inhomogeneous equation ${\partial }_{\mu }{F}^{\mu \nu }={\mu }_0{j}^{\nu }$.
• Field transformations 10.06.04 follow from the Lorentz transformation of $F^{\mu \nu }$, which is sourced by $j^{\mu }$.

Historical & philosophical context Master

The four-current was introduced by Minkowski in 1908 as part of his geometric formulation of special relativity. The identification of $j^{\mu }=(c\rho ,\mathbf{J})$ as a four-vector was a key insight: it showed that charge and current are the same physical quantity viewed from different reference frames, just as space and time are aspects of a single spacetime.

The proof that charge conservation follows from Maxwell's equations (rather than being an independent principle) is sometimes attributed to Minkowski, though the continuity equation itself was known to Maxwell. The covariant formulation made the proof transparent: the antisymmetry of $F^{\mu \nu }$ forces ${\partial }_{\nu }{j}^{\nu }=0$.

The deep connection between charge conservation and gauge invariance was discovered by Weyl (1918) and formalised by Noether (1918): global phase symmetry $\psi \to e^{i\alpha }\psi$ implies a conserved current, and local gauge symmetry $\alpha \to \alpha (x)$ requires the existence of the electromagnetic field. This gauge principle is the foundation of the Standard Model of particle physics.

Bibliography Master

• Minkowski, H., "Die Grundgleichungen fur die elektromagnetischen Vorgange in bewegten Korpern," Nachr. Ges. Wiss. Gottingen (1908), 53-111.
• Noether, E., "Invariante Variationsprobleme," Nachr. d. Konig. Gesellsch. d. Wiss. zu Gottingen (1918), 235-257.
• Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
• Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge, 2017).
• Landau, L. D. and Lifshitz, E. M., The Classical Theory of Fields, 4th ed. (Butterworth-Heinemann, 1975).