Illustrated Physics
physics, drawn
Relativity

The one quantity a boost leaves alone

Energy, momentum, length, duration, density and field strength all change when the observer moves. Electric charge does not, and the whole of the field-transformation argument rests on it — so it is worth asking what the evidence is.
14 min read 5 figures What stays the sameMade into an instrument

Assumes: Magnetism is electricity seen sideways · The field nobody can transform away

Magnetism is electricity seen sideways rests on a step that is usually taken without comment. A neutral wire carrying a current is neutral because the positive and negative charge densities cancel; boost into the frame of the drifting electrons and the two lattices contract by different factors, so the densities no longer cancel and the wire acquires a charge.

That argument uses the charge itself as fixed and the volume as the thing that changes. If the charges had changed too — if a moving electron carried more charge than a stationary one, the way it carries more energy — the whole derivation would give something else, and probably nothing.

So charge invariance is load-bearing, and it deserves an argument of its own.

Two things that change and one that does not. How a boost treats a piece of charged matter: the charge density rises by the Lorentz factor because the same charges occupy a contracted length, and the length falls by the same factor. At β = 0.6 the density is 1.250 times what it was and the length is 0.800 times, and their product is one to fourteen decimals across the whole range drawn. So the total charge is the same number in every frame, and it is the only quantity in the transformation that is. Charge density is the time component of a four-vector and transforms like an energy; charge itself is a scalar, and nothing about the observer changes it.
Fig. 1 What a boost does to a charged object: the charge per unit length rises by the Lorentz factor and the length falls by the same factor, and the product is one to fourteen decimals across the whole range. Charge density is not invariant. Charge is.

Two things that change, and a product that does not

The first thing to separate is charge from charge density, because almost every confusion here is the two being run together.

Charge density is the time component of a four-vector — the four-current, whose spatial part is the current density — and it transforms exactly as an energy does, gaining a factor of γ\gamma. That is not a subtlety; it is what makes a neutral wire charged in the electrons’ frame, and it is why the field nobody can transform away has to be careful about which quantities are invariants.

Charge is a scalar. It is the density integrated over a volume, and the volume contracts by exactly the factor the density grows by, so the product does not move. The figure verifies the cancellation rather than asserting it, and the verification is trivial — which is the point. Nothing is being balanced. The two factors are reciprocal by construction, because contraction and density are the same fact stated twice.

Why it is a count

The same charges, counted on a different now. 5 charges on worldlines of assorted speeds, with two observers' notions of one moment drawn across them: the horizontal slice and the slice tilted at β = 0.6. Each worldline crosses each slice exactly once, and it must, because no charge travels at light speed and no permitted slice is steeper than the light cone. So the two observers count the same charges — not the same number by accident, but the same worldlines. That is the whole of why charge is invariant: it counts objects, and the objects are there whichever way the counting surface is tilted. Energy is an amount carried on each worldline and can differ; a count cannot.
Fig. 2 Charges on worldlines of assorted speeds, with two observers’ notions of one moment drawn across them. Each worldline crosses each slice exactly once, and must, since no charge travels at light speed and no permitted slice is steeper than the light cone. The two observers count the same charges — the same worldlines, not merely the same number.

The deeper reason for the invariance is not about factors cancelling at all, and the spacetime picture makes it obvious.

Charge counts things. A total charge is a sum over the charged objects present, weighted by ±1\pm 1 or by whatever fractions the objects carry, and a sum over objects is not the kind of quantity a change of viewpoint can alter. Energy is different because it is an amount carried by each object, and how much each carries depends on the frame.

The formal version is that the total charge is the integral of the four-current over a spacelike slice, and the divergence of the four-current vanishes — which is conservation of charge. A vanishing divergence means the integral over any two slices that can be deformed into each other is the same, and every pair of inertial slices can be.

The picture is the same statement without the calculus. Tilt the slice and it crosses each worldline at a different event, but it crosses each worldline. It cannot miss one, because a worldline is timelike and a simultaneity slice is spacelike, and a timelike line meets every spacelike plane exactly once. There is nowhere for a charge to hide from one observer’s now that it does not hide from the other’s.

That also says exactly which assumption charge invariance depends on: that charge is conserved, and that charge is carried by things that go slower than light. Neither is optional, and neither is separable from the rest of electromagnetism.

The measurement that settles it

How neutral an atom actually is. For each species, two numbers on a logarithmic axis: the charge imbalance an atom would carry if a particle's charge grew with its speed the way its energy does, and the experimental bound on the imbalance. hydrogen: predicted 2.66e-5 of an elementary charge against a bound of 1e-21, a factor of 2.7e+16; helium: predicted 1.07e-4 of an elementary charge against a bound of 1e-21, a factor of 1.1e+17; argon: predicted 8.74e-3 of an elementary charge against a bound of 1e-19, a factor of 8.7e+16; caesium: predicted 9.18e-2 of an elementary charge against a bound of 1e-17, a factor of 9.2e+15. The inner electrons of an atom of atomic number Z move at about Zα, so the effect would grow as Z² and heavy atoms would be conspicuously charged. Every one of these is neutral to a precision that excludes it by ten or more orders of magnitude. The neutrality of bulk matter is the most precise null measurement in physics, and it is a measurement of exactly this.
Fig. 3 For each species, two numbers on a logarithmic axis: the charge imbalance an atom would carry if a particle’s charge grew with its speed, and the measured bound on the imbalance. The inner electrons of an atom of atomic number Z move at about Zα, so the effect would grow as Z², and every one of these is neutral to a precision that excludes it by ten or more orders of magnitude.

An argument from the structure of the theory is not evidence, and the evidence for charge invariance is extraordinarily good.

The test is the neutrality of matter. An atom contains as many electrons as protons, and the electrons move at speeds the protons do not — the innermost electron of an atom of atomic number ZZ moves at about ZαZ\alpha, which is a hundredth of light speed for helium and two-fifths of it for uranium. If charge grew with speed, the electron’s would exceed the proton’s, and the atom would carry a net negative charge growing as Z2Z^2.

The size of that imbalance is easy to compute and the figure computes it: about 10−410^{-4} of an elementary charge for helium and much more for heavy atoms. The measured bounds are around 10−2110^{-21} elementary charges per atom.

Those bounds come from a beautiful class of experiments. A gas is allowed to flow out of a vessel, and any charge it carried away is measured on the vessel by an electrometer; or a jet of gas is passed between charged plates and its deflection measured; or an acoustic resonance is driven in a gas by an oscillating field, which would couple only if the molecules were charged. The neutrality of matter is the most precise null measurement in physics, and the reason it is done at all is precisely to test statements like this one.

The Z² scaling makes it a sharp test rather than a general reassurance. A theory in which the charge depended on speed would predict a pattern across the periodic table, and the pattern is absent along with the effect.

The obvious experiment, and why nobody uses it

The obvious experiment, and why it is the wrong one. The electric field a current-carrying copper wire would produce at 10 mm if the charge of a moving electron grew with its speed the way its energy does. The conduction electrons drift at about 6.242 mm per second at 10 amperes, so the effect goes as the square of ten to the minus eleven: 6.24e-7 volts per metre at 10 amperes and 9.99e-6 at 40. That is at the edge of what a good electrometer can reach and no better, which is the honest verdict on the obvious experiment. An atom's inner electron moves 1.8e+8 times faster than the drift, so an atom is 3.1e+16 times the instrument a wire is — and that is why the neutrality of matter, rather than the neutrality of a live cable, is where the bound comes from.
Fig. 4 The field a current-carrying copper wire would produce at a centimetre if charge grew with speed. Conduction electrons drift at a fraction of a millimetre per second, the effect goes as the square of that, and the answer is a hundredth of a microvolt per metre at forty amperes — at the edge of what an electrometer reaches and no better.

There is an experiment that looks much more direct and turns out to be much worse, and it is instructive because it is the one most people propose.

A current-carrying wire is neutral. If the moving electrons carried more charge than the stationary lattice ions, it would not be, and the wire would produce a field. That sounds like an easy measurement — take a mains cable and an electrometer.

The trouble is the speed. Conduction electrons drift at something like six millimetres per second at ten amperes, which is 2×10−112\times10^{-11} of light speed, and the effect goes as the square of that. The predicted field is around 10−610^{-6} volts per metre, which is right at the boundary of measurable and comes with every stray-charge problem an electrostatic measurement has.

Compare with the atomic test: helium’s inner electron moves at 1.5×10−21.5\times10^{-2} of light speed, seven hundred million times the drift speed, so its β2\beta^2 is 5×10175\times10^{17} times larger. An atom is a vastly better relativistic instrument than a wire, and it is available in bulk.

That comparison is worth carrying beyond this question. The slowness of the drift is what makes magnetism is electricity seen sideways startling — an effect at 10−1110^{-11} of light speed producing a force everyone has felt — and it is the same slowness that makes the wire useless here. The difference is that magnetism survives because the charges are so many that the cancellation is between two enormous numbers, and the charge-invariance violation would not be amplified the same way.

The same count at another tilt

The same charges, counted on a different now. 4 charges on worldlines of assorted speeds, with two observers' notions of one moment drawn across them: the horizontal slice and the slice tilted at β = 0.5. Each worldline crosses each slice exactly once, and it must, because no charge travels at light speed and no permitted slice is steeper than the light cone. So the two observers count the same charges — not the same number by accident, but the same worldlines. That is the whole of why charge is invariant: it counts objects, and the objects are there whichever way the counting surface is tilted. Energy is an amount carried on each worldline and can differ; a count cannot.
Fig. 5 Four charges rather than five, at different speeds, with the second observer at half light speed instead of six-tenths. Both slices again cut every worldline once. Nothing about the count depends on how many charges there are, how fast they move, or how far the slice is tilted — which is why the invariance is exact rather than approximate.

Redrawing the count at other numbers is worth doing because the argument’s strength is that it has no numbers in it.

The energy of the system in the two frames is different, and by a different factor for each charge. The momentum is different. The separations between the charges are different, and the order in which two of them pass a given point can be different. What is the same is how many worldlines there are, and no rearrangement of the picture threatens that.

This is a good example of a quantity being protected by topology rather than by dynamics. Nothing about the forces between the charges was used, nothing about what they are made of, and nothing about how they move beyond the fact that they move slower than light. A conclusion that survives that much ignorance is usually one worth building on.

It also explains a fact that would otherwise look like a coincidence. Charge is quantised and charge is invariant, and the two go together: a quantity that comes in indivisible units is a count, and a count is exactly the kind of thing a boost cannot change. A continuously divisible charge could in principle have transformed; the integers cannot.

What would break

It is worth listing what a speed-dependent charge would cost, because the list is longer than one derivation.

Gauss’s law would stop being a law about the enclosed charge. The flux through a surface would depend on how the charges inside were moving, and the whole apparatus of counting field lines would fail.

Conservation of charge would become frame-dependent. A process conserving charge in one frame would create it in another, and there is no consistent electrodynamics with that property.

Atoms would be charged and matter would not hold together. A gram of helium contains 1.5×10231.5\times10^{23} atoms; at 10−410^{-4} of an elementary charge each, the mutual repulsion would exceed anything chemistry could resist by an astronomical margin. Ordinary matter existing at all is a bound on this, and a crude version of it is stronger than most laboratory measurements.

Magnetism would stop following from electrostatics. The field nobody can transform away classifies what a boost can do to a field, and every step of that classification takes the sources as fixed.

And electromagnetism would not be a gauge theory. Charge invariance is the conservation law that the theory’s gauge symmetry produces, by the argument a symmetry hands over, and a charge that depended on the observer would leave the symmetry with nothing to conserve.

The last one is why the invariance is not treated as a separate empirical fact by anyone building a theory. It comes with the structure, and the experiments are testing the structure.

The other quantities, for comparison

Setting charge beside the rest of the transformation’s cast is the quickest way to see how unusual it is.

Length and duration change, and change in opposite senses, so that the interval between two events survives — which is the length that depends on when and its partner in time. Neither is invariant on its own.

Energy and momentum change together and preserve the mass, which is the same shape of statement one dimension up. A system’s mass is invariant and its energy is not.

The electric and magnetic fields mix into each other and preserve two combinations, which is what makes the classification of fields possible at all.

Proper time is invariant, and it is the closest analogue to charge in the list: both are quantities that a single object carries and that every observer computes the same. The difference is that proper time is invariant because it is defined by an object’s own clock, while charge is invariant although it is defined by an integral over all space.

And the speed of light is invariant by postulate, which is where the whole structure comes from.

The pattern is that invariance is rare and always has a reason. For proper time the reason is that it is measured along a worldline; for mass, that it is the energy in a particular frame; for charge, that it counts. Anything that looks invariant without a reason of that kind is usually a quantity nobody has boosted yet.

Where the model stops

The atomic estimate uses a hydrogenic inner electron. Real inner electrons are screened, correlated, and in heavy atoms relativistic already, so the ZαZ\alpha speed is a first approximation. The conclusion survives by so many orders of magnitude that the approximation is irrelevant, which is the useful kind of margin.

The bounds quoted are on the difference between the proton’s and the electron’s charge, together with the neutron’s charge, and the published limits separate the two in ways this figure does not. A violation of charge invariance is one way to produce a difference and not the only one, so the experiments constrain a class of theories rather than this one alone.

The wire calculation assumes one carrier species and a free-electron density. Real conduction is more complicated, the effective carrier density is not exactly the atomic one, and the sign can differ in a semiconductor. None of it moves the answer by the ten orders of magnitude that would matter.

And this is all flat spacetime. Charge remains invariant in general relativity, and the argument becomes a statement about the flux through a closed surface rather than an integral over a global slice, since there are no global slices. The conclusion is more robust than the derivation given here.

How the bound is actually reached

The experiments deserve a paragraph of their own, because a limit of 10−2110^{-21} elementary charges per atom is not obtained by weighing anything.

The best of them use gas efflux. A sealed metal vessel holds a gas and is mounted inside a shield with its potential monitored. The gas is let out through a valve, carrying with it any net charge its molecules had, and the vessel’s potential is watched. Roughly 102210^{22} molecules leave, so a per-molecule imbalance of 10−2110^{-21} elementary charges is a total of about ten electrons’ worth of charge — small, and within reach of a good electrometer with a long integration and a patient experimenter.

The amplification is the whole trick, and it is the same one that makes the neutrality of matter so well constrained: a tiny per-particle quantity multiplied by Avogadro’s number becomes an ordinary laboratory quantity. The difficulty is entirely in the systematics — contact potentials, triboelectric charging in the valve, ions in the gas — and the published bounds are limited by those rather than by sensitivity.

A second family uses acoustics. Drive a gas with an oscillating electric field and, if the molecules were charged, a sound wave would be launched; listen for it at the driving frequency with a microphone. That version has the advantage of a narrow band and a phase-sensitive detection, and it reaches comparable limits by an entirely different route.

Two methods with different systematics agreeing on a null result is what makes the bound credible, and neither has ever seen anything.

What the pictures cannot show

The counting figure draws worldlines as though a charge were a particle with a definite trajectory, which is a classical picture the argument does not need. What it needs is a conserved current, and a current is available in quantum field theory with no trajectories anywhere. The drawing makes an argument about topology look like an argument about particles.

The neutrality figure plots a predicted number against a bound, which are different kinds of quantity. The bound is a limit, not a measurement, and drawing it as a point invites the reading that somebody measured a charge of 10−2110^{-21}. Nobody did; they failed to measure one, very precisely.

A third omission is the sign. Every figure treats charge as a signed count and none of them shows why the signs matter: the invariance would be equally true of the total number of charged particles, which is not conserved, and it is the signed sum that the continuity equation protects. Pair creation makes two more charged particles and changes the total charge by nothing, which is exactly the distinction the drawings cannot carry — their worldlines all begin at the bottom edge and end at the top, and none of them begins in the middle.

Where the ladder goes next

The field-transformation ladder began with magnetism is electricity seen sideways, where a relativistic effect at 10−1110^{-11} of light speed produces a force anyone can feel, and continued with the field nobody can transform away, where the invariants of the field decide what a boost can and cannot do. This rung asks what stayed fixed while the fields were being transformed. The rungs after it: the four-current as an object rather than a pair of quantities; the transformation of the potentials, which carries the same information in a form gauge symmetry acts on directly; and the field tensor, in which the two invariants and this one become the whole of the classification.

The habit worth carrying away is that a derivation’s fixed points are worth as much scrutiny as its moving ones. Everything in the wire argument transforms except the charge, and the one thing held fixed is the one nobody checks — which is generally where an argument’s real content is.

Part 3 of 5

This essay is one argument about Field transformation. The others:

What a boost can and cannot do to a field. The electric and magnetic magnitudes of three fields, plotted against each other as the observer is boosted from -0.98c to 0.98c across them. Each field slides along a hyperbola, because E² − c²B² does not change: the recomputed value drifts by at most 2.6e-15 over every point drawn. The diagonal is E = cB, and which side of it a field starts on is permanent. Below it there is a speed at which the electric field vanishes; above it, one at which the magnetic field does; on it, a wave that no observer can slow, dim or unbalance. The field nobody can transform away Part 2 — A wire's magnetic field is an electric field seen from the wrong frame, and a charged plate's electric field is a magnetic one seen the same way. Neither trick works on a light wave. Two combinations of E and B are the same for every observer, and which side of one line a field sits on is a fact nothing about the observer can alter. Charge density and current, mixing like time and space. The charge density and the current density of a wire, against the rapidity of the frame they are measured in, starting from cρ = 0 and J = 2. They mix by exactly the transformation that mixes a time and a space coordinate — a hyperbolic rotation — and the combination c²ρ² − J² is unchanged at every rapidity, checked here to nine decimal places. A wire that is neutral in the laboratory is charged in every other frame, at exactly one rapidity out of all of them, and that single fact is the mechanism the first rung of this ladder tells as a story about two contracted lattices. Here it is a coordinate change. Charge and current are one thing Part 4 — The rung below asks what a boost leaves alone and answers charge. That answer forces the next one: a fixed charge in a contracting volume gives a density that transforms like a time component, and a current that transforms like a space one. So charge density and current density are the four parts of one object — and conservation of charge stops being an extra law and becomes the condition that makes the object exist.

Where this goes next

The same wire, seen twice at 0.6c. Above: the wire in the laboratory. The lattice is at rest and the electrons drift, so the electrons are the contracted ones — and the wire is neutral, which means their contracted spacing is what the manufacture of a neutral wire produced. Below: the same wire seen by something moving with the electrons at 0.6c. Now the electrons are at rest and the spacing between them stretches by γ = 1.250, while the lattice moves and its spacing contracts by the same factor. The two densities no longer cancel and the wire is charged. Nothing was done to the wire; the only thing that changed is who is looking, and the magnetic force in the first frame is the electric force in the second. Magnetism is electricity seen sideways The force on a charge moving beside a current-carrying wire is magnetic in the laboratory and purely electrostatic in the charge's own frame. Both calculations give the same answer, and the drift speed that makes them agree corresponds to a Lorentz factor differing from one in the twenty-sixth decimal place. What a boost can and cannot do to a field. The electric and magnetic magnitudes of three fields, plotted against each other as the observer is boosted from -0.98c to 0.98c across them. Each field slides along a hyperbola, because E² − c²B² does not change: the recomputed value drifts by at most 2.6e-15 over every point drawn. The diagonal is E = cB, and which side of it a field starts on is permanent. Below it there is a speed at which the electric field vanishes; above it, one at which the magnetic field does; on it, a wave that no observer can slow, dim or unbalance. The field nobody can transform away A wire's magnetic field is an electric field seen from the wrong frame, and a charged plate's electric field is a magnetic one seen the same way. Neither trick works on a light wave. Two combinations of E and B are the same for every observer, and which side of one line a field sits on is a fact nothing about the observer can alter. A spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light. The length that depends on when, and is not really about length A moving object is measured shorter. The contraction is real, it is not an illusion of light travel time, and it turns out to be a disagreement about simultaneity wearing a different costume.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Charge densityCharge invarianceConservation lawsElectric fieldFour-vectorLength contractionThe Lorentz transformationNeutralityReference framesSimultaneity