Four-current and charge conservation

Keep signature $(+,-,-,-)$ and $x^0=ct$. Define $J^\mu=(c\rho,\mathbf j)$, with charge density rho in coulombs per cubic metre and current density j in amperes per square metre. Multiplying rho by c makes all four components share the latter units. Under a passive boost along x, J transforms exactly like any other four-vector. The density equation contains $Vj_x/c^2$; the current equation contains $V\rho$. Both combinations have the correct units before the common gamma factor is applied.

For one species moving with ordinary velocity v, its convection current is $\mathbf j=\rho\mathbf v$. If its proper density is $\rho_0$, the four-current is $J^\mu=\rho_0u^\mu$. A mixture need not have any frame where total current vanishes. Oppositely charged populations can carry a net current while their total charge density cancels. Treat each population consistently before summing; using one species' rest density as if it were the net laboratory density is a frequent source of incorrect wire and beam calculations.

The four-divergence $\partial_\mu J^\mu$ expands to $\partial_t\rho+\nabla\cdot\mathbf j$ because the c factors cancel in the temporal derivative. Setting it to zero says that charge can accumulate only when more charge enters a region than leaves. Integration over a fixed laboratory volume and the divergence theorem yield the boundary-flux version. The surface normal points outward, so positive outward flux decreases the enclosed charge. This sign is more reliable than a memorized verbal rule about incoming or outgoing current.

A total-charge statement alone would allow charge to vanish at one place and appear instantly elsewhere. Local continuity is stronger: intermediate transport must account for the change. For an isolated system, sufficiently rapid decay of current at spatial infinity gives a time-independent total charge in each inertial frame. Boundary assumptions matter. A finite open region can lose enclosed charge while obeying exact continuity; that does not violate charge conservation. The lost charge has simply crossed the chosen boundary.

Relativistically, total charge is the flux of the four-current through a spacelike hypersurface. With the present normalization it is $Q=c^{-1}\int J^\mu d\Sigma_\mu$. On a constant-time slice this reduces to the familiar integral of rho over volume. Continuity makes the result independent of deformations of the slice when no current escapes through the side boundary. This is how different observers can agree on total isolated charge while disagreeing on the density and on which separated events are simultaneous.

A stationary uniform charge cloud illustrates the density effect simply. An observer moving along positive x sees the charges move along negative x, measures a density enhanced by gamma, and measures a negative x current. The cloud's contracted volume compensates for the density increase, leaving its total charge unchanged. For an infinite cloud, an infinite total charge is not a useful invariant number; the local transformation remains valid. Always distinguish local density calculations from finite-system global charge claims.