Charge and current are one thing

The rung below this one asks what stayed fixed while everything else in the wire argument was transforming, and finds that it is the charge. Energy, momentum, length, duration, density and field strength all change with the observer; the charge on an electron does not, to twenty-one decimal places.

That is an answer and it is also a constraint, because a quantity that does not change is a quantity everything around it has to accommodate.

Take a volume containing a fixed amount of charge. Boost past it, and the charge is unchanged while the volume contracts by $\gamma$. So the charge density is multiplied by $\gamma$ — which is exactly how a time component of a four-vector transforms for an object at rest. Set the charge moving and the current density picks up the space-component behaviour. The pair has no choice about what it is.

What the pair has to be #

Write it out. For a boost of rapidity $w$ along $x$,

$$c\rho' = c\rho\cosh w - J\sinh w, \qquad J' = J\cosh w - c\rho\sinh w,$$

and those are the same two lines as

$$ct' = ct\cosh w - x\sinh w, \qquad x' = x\cosh w - ct\sinh w.$$

Not analogous. The same. So $(c\rho, \mathbf{J})$ is a four-vector, called the four-current, and everything true of four-vectors is true of it — including that it has an invariant length, $c^2\rho^2 - J^2$, which is the same for every observer in the same way the interval is.

That invariant has a name and a meaning. It is $c^2\rho_0^2$, where $\rho_0$ is the charge density in the frame in which the charge is at rest — the proper charge density. Every observer disagrees about $\rho$ and about $J$ and they all compute the same $\rho_0$ from their own pair.

The first rung, done again with no lattices in it #

The opening rung of this ladder derives the magnetic force on a charge beside a wire by transforming to the charge’s frame, where the wire’s two lattices contract by different amounts and the wire is no longer neutral. It is a good argument and it is a story about a particular arrangement — two species of charge, one moving, one not.

The four-current gives the same result with nothing in it about lattices.

A wire neutral in the laboratory has $\rho = 0$ and $J \neq 0$. That is a four-vector with a zero time component in one frame. A four-vector with a zero time component in one frame has a non-zero one in every other, because the transformation mixes them — the figure counts exactly one rapidity out of the whole range at which the density vanishes.

So the wire is charged in the test charge’s frame, and the amount is fixed by the boost. No assumption about what carries the current, no assumption that one species is stationary, no counting of ions. A superconductor, a beam of electrons in vacuum, a plasma with both species drifting — all give the same transformed density, because all that entered was the four-vector.

That generality is the point of the reformulation. The lattice argument is more vivid and it is less general, and a reader who has only met it can reasonably wonder whether the answer depends on the metal.

It does not, and the check is available. A wire made of a superconductor has no lattice of carriers to contract — the current is carried by a condensate with no individual velocities to speak of — and the field outside it is the same field, with the same transformation into a moving frame. A beam of electrons in vacuum has no positive charges at all, so there is nothing for the negative ones to be contracted relative to, and the transformation still gives the right answer. In each case the four-current is what was transformed, and the composition of the source never entered.

The habit that produces is worth naming before the essay is over. An argument that reaches the right answer through a mechanism is more persuasive and less portable than one that reaches it through a transformation property, and the two together are worth more than either: the mechanism says why anybody should believe it and the transformation says how far it goes.

How large the effect is in a real wire #

The transformation is exact and it is worth putting a number on, because the number is what makes the first rung of this ladder so striking.

A copper wire a square millimetre in cross-section carrying ten amps has a drift speed of about seven tenths of a millimetre per second. The rapidity of that is $2.3 \times 10^{-12}$, so $\sinh w$ is the same to twenty-three decimal places, and the charge density appearing in the moving frame is $J\sinh w / c$ — which for a current density of $10^7$ amps per square metre comes to about $8 \times 10^{-14}$ coulombs per cubic metre.

That is a tiny density and it produces a force anybody can feel. The reason is the size of the underlying quantities: the positive and negative charge densities in the wire are each about $10^{10}$ coulombs per cubic metre, so the imbalance is one part in $10^{23}$ — and the electrostatic force between two such distributions is enormous enough that one part in $10^{23}$ of it is an ordinary laboratory force.

Every magnetic effect anybody has ever measured is of that character. Magnetism is not a weak force; it is a colossal force nearly cancelled, with the residue being the part relativity does not cancel. The wire argument says this and the four-current says it in a form where the near-cancellation is visible as an enormous invariant with a small difference between its two parts.

Conservation, as a property of the object #

The four-current has one more thing to say, and it is the part that changes an experimental law into a structural one.

Charge conservation is usually written

$$\frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{J} = 0,$$

and presented as a separate fact. It is not separate. Taking the divergence of the Ampère–Maxwell equation gives it immediately — the divergence of a curl vanishes, and what is left is exactly that expression. Two of Maxwell’s equations are constraints rather than laws of motion and this is the price of consistency between the other two.

In four-vector language the expression collapses to

$$\partial_\mu J^\mu = 0,$$

which is a scalar built from a four-vector and a derivative — one equation, identical in every frame, rather than a time part and a space part that would have to be checked separately.

The distinction between a global conservation law and a local one is worth insisting on because the two are routinely conflated. A global law says a total does not change. A local law says the total in any region changes only by what crosses that region’s boundary. The second implies the first and is much stronger, and it is the second that relativity requires: a global law would have to be checked on a simultaneity slice, and observers disagree about those.

Why it had to be a four-vector #

There is a way of running the argument backwards that is more convincing than the forward version, and it is worth having because it explains why nothing else could have worked.

Suppose $(c\rho, \mathbf{J})$ were not a four-vector — suppose the density transformed some other way. Then $\partial_\mu J^\mu$ would not be a scalar, so an equation setting it to zero would hold in one frame and fail in another. Charge would be conserved for one observer and not for another — a component being mistaken for an invariant, with a conservation law as the casualty.

That is not merely inelegant. Charge conservation is checked to extraordinary precision — the limit on electron decay is a half-life above $10^{26}$ years — and it is checked in laboratories moving at all sorts of velocities relative to one another, including the thirty kilometres a second of the Earth’s orbit. A conservation law that held only in a preferred frame would have shown up.

So the four-vector character of the source is not an aesthetic choice. It is what makes the conservation law frame-independent, and the conservation law is measured.

What continuity forbids that conservation does not #

The steady case makes the strength of the local law visible, and it is worth taking one example all the way.

Imagine a charge disappearing from one end of a laboratory and an equal charge appearing at the other, simultaneously. The total is conserved. Nothing crosses the space between them.

That is forbidden, and the reason is not an extra postulate. If it happened, then in the laboratory frame the total is conserved at every instant — but “at every instant” means on a particular simultaneity slice, and a moving observer slices differently. On the moving observer’s slices the two events are not simultaneous, so there is an interval during which one charge has gone and the other has not arrived, and that observer sees the total change.

So a globally conserved but non-local charge is conserved for one observer and not for another, which is exactly the situation the four-vector structure exists to rule out. Locality is not an aesthetic preference here; it is what makes conservation frame-independent.

The same argument applies to every conserved quantity in a relativistic theory, and it is why they all come with continuity equations rather than with mere totals. Energy, momentum, baryon number and lepton number each have a current whose four-divergence vanishes, and in each case the local statement is the one the theory contains and the global one is a consequence.

The sign of the invariant, and what it classifies #

The invariant $c^2\rho^2 - J^2$ can be positive, negative or zero, and the three cases are physically distinct in the same way the interval’s three cases are.

Positive means there is a frame in which the current vanishes — a distribution of charge that can be brought to rest. A charged capacitor plate is such a case, and the frame is the plate’s.

Negative means there is a frame in which the charge density vanishes but never one in which the current does. A neutral current-carrying wire is such a case, and the figure above shows the single rapidity at which the neutrality occurs.

Zero means neither is possible at any speed. That requires $|J| = c\rho$, which means the charge is moving at $c$ — a beam of massless charged particles, which does not exist, so the case is empty in practice and is exactly the light-like case of the interval with the same emptiness for material sources.

Notice that the classification of sources here has the same three-way structure as the classification of fields on the rung below this one but one, and for the same algebraic reason: an indefinite quadratic form has three sign cases, and which one applies is a property of the object rather than of the observer.

The potentials, which are the same object again #

There is a second four-vector in electromagnetism, and mentioning it here rather than leaving it to the next rung makes the structure visible.

The scalar potential and the vector potential combine into $(\phi/c, \mathbf{A})$, which transforms exactly as $(c\rho, \mathbf{J})$ does. That is not a coincidence: in the Lorenz gauge each component of the four-potential satisfies a wave equation with the corresponding component of the four-current as its source, so the two objects are tied together component by component and must transform alike.

Two consequences follow immediately and both are worth having.

A pure scalar potential in one frame is a vector potential in another. The Coulomb potential of a static charge, boosted, acquires an $\mathbf{A}$ — and the curl of that $\mathbf{A}$ is the magnetic field of the moving charge. The whole of the first rung of this ladder is the statement that a boost of a four-vector with one non-zero component produces one with two.

And gauge freedom is a statement about the four-potential rather than about its pieces. The potentials are not unique, and the transformation that changes them without changing the fields is a four-gradient added to the four-potential — one operation rather than the separate rules for $\phi$ and $\mathbf{A}$ that the three-dimensional treatment gives.

The Lorenz gauge condition is then $\partial_\mu A^\mu = 0$, which is the same expression as charge conservation with the current replaced by the potential. That it is the same expression is why the condition is preserved by a boost and the Coulomb gauge’s is not — a fact that is a nuisance in three dimensions and is obvious in four.

The measurement the whole structure rests on #

Charge invariance is the input, and it is worth saying how well it is known, because everything above is an inference from it.

The direct test is neutrality. If an electron’s charge depended on its speed, an atom whose electrons move at different speeds in different orbitals would not be exactly neutral, and neither would a molecule. Experiments looking for a residual charge on bulk matter — by measuring the deflection of a gas jet in a field, or the acoustic response of a gas to an oscillating field — bound any charge imbalance between a proton and an electron at below $10^{-21}$ of the elementary charge.

The comparison across species is the sharper one. Helium’s electrons move considerably faster than hydrogen’s, and a heavy atom’s innermost electrons move at a substantial fraction of the speed of light — a caesium 1s electron at roughly two fifths of it. If charge varied with speed at all, the neutrality of caesium and of hydrogen could not both hold, and both hold to the same extraordinary precision.

That is the fact the four-vector structure was derived from, and it is the reason to trust the derivation more than the elegance of the result would justify on its own. A structural argument is only as good as its input, and this input is measured better than almost anything else in physics.

Where this stops being right #

The transformation was written for one spatial dimension. The full four-current has three components of $\mathbf{J}$ and the transverse ones are unchanged by a boost along $x$, exactly as transverse coordinates are. Nothing in the argument changes; the arithmetic is longer.

The sources have been treated as continuous. A charge density is an average over a region containing many charges, and the transformation of a single point charge’s contribution requires care — the delta function’s argument transforms too, and getting it right is what makes the four-current of a point particle come out as an integral over its worldline.

Charge invariance was taken from the rung below. It is an experimental result rather than a theorem, and everything here follows from it. If charge depended on speed the whole structure would collapse and the atom, as that rung argues, would not be neutral.

And nothing here is quantum. The four-current of a quantum field is an operator, and its conservation follows from a symmetry of the Lagrangian rather than from the field equations — which is a deeper statement and does not change any of the arithmetic.

What the pictures cannot show #

The boost figure draws two densities against a rapidity, and neither density is something an instrument reads directly. What is measured is a force on a test charge, or a deflection, or a field; the density is inferred from a model of the source. A curve of an inferred quantity against a coordinate choice is two abstractions deep, and the reason it is worth drawing is that the shape — hyperbolic, with a fixed combination — is what the physics constrains.

The spacetime box draws a region with two faces and shows current crossing them, and it has no charges in it. What is conserved is a number attached to a region, and the drawing shows the region and the flux and not the thing being counted.

Where this ladder goes next #

Four rungs stand on field-transformation. The first found magnetism to be electrostatics seen sideways. The second found what a boost can and cannot do to a field. The third asked what stayed fixed. This one asks what kind of object the sources are, given that answer, and finds that they had no choice.

The habit worth carrying away is about the leverage an invariant gives. Once one quantity is known to be observer-independent, the transformation of everything built from it is forced rather than free. Charge invariance plus length contraction determines the transformation of charge density with no further input, and that determines the four-vector character of the source, and that makes charge conservation a scalar equation. Three results from one measured fact.

What is left on this ladder is the fields. Six numbers that mix into each other under a boost, antisymmetrically, are the components of one object as surely as $\rho$ and $\mathbf{J}$ are — and writing them that way turns Maxwell’s four equations into two.