The field nobody can transform away

A current-carrying wire is electrically neutral and exerts a magnetic force on a moving charge beside it. Ride along with that charge and the magnetism is gone, because the charge is now at rest, and yet the force is still there — it has become electric, produced by a net charge density the moving frame sees on a wire the laboratory calls neutral.

That is the standard demonstration that the two fields are one object seen from different angles. It invites a conclusion which is false: that any magnetic field can be boosted away, and that the labels are therefore arbitrary. They are not. There are three kinds of electromagnetic field, the classification is the same for every observer, and no boost moves a field from one kind to another.

Below the diagonal 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 two numbers #

Under a boost, the field components split into those along the motion and those across it. The parallel components are unchanged; the perpendicular ones mix:

$$\mathbf{E}'_\perp = \gamma(\mathbf{E} + \mathbf{v}\times\mathbf{B})_\perp, \qquad \mathbf{B}'_\perp = \gamma\left(\mathbf{B} - \frac{\mathbf{v}\times\mathbf{E}}{c^2}\right)_\perp .$$

Both fields change, both can be made larger or smaller, and either can be made zero in the right circumstances. Two combinations survive all of it:

$$E^2 - c^2B^2 \quad\text{and}\quad \mathbf{E}\cdot\mathbf{B}.$$

These are the two Lorentz invariants of the field, and they are the complete set: any quantity built from $\mathbf{E}$ and $\mathbf{B}$ that every observer agrees about is a function of these two.

The first invariant does the classifying. If $E^2 - c^2B^2 < 0$ the field is magnetic-dominated: there is a frame in which $\mathbf{E}$ vanishes entirely, and the wire is the standard case. If it is positive the field is electric-dominated and there is a frame with no magnetic field, which is a charged plate seen by a moving observer. If it is zero the field is null, and there is no frame of either kind.

The second invariant decides whether either frame exists at all. If $\mathbf{E}\cdot\mathbf{B} \ne 0$ the two fields have a component along each other, and since a boost can only mix perpendicular components, that alignment cannot be removed. Such a field is neither electric nor magnetic in any frame, and every field with both a parallel electric and a parallel magnetic component is of this kind.

The wire, and what it says about magnitudes #

The classification has a quantitative face that is worth seeing on the wire itself, because the numbers there are absurd.

For a wire carrying ten amps, $cB$ is of order $10^{-4}\times 3\times10^8$, about $6\times10^4$ volts per metre, while $E$ is essentially zero. So $E^2 - c^2B^2$ is large and negative, the field is magnetic-dominated by an enormous margin, and the boost that removes the electric part is the drift velocity — a fraction of a millimetre a second. That is why the trick works so cleanly on a wire and why the required speed is so absurdly small.

The velocity that removes the electric field #

The classification says that a frame with no electric field exists when the first invariant is negative. It is worth writing down what that frame is, because the answer turns out to be a quantity plasma physics uses every day for what looks like a different reason.

Take $\mathbf{E}$ perpendicular to $\mathbf{B}$ and ask for the boost that cancels the electric field. The transformation gives $\mathbf{E}' = \gamma(\mathbf{E} + \mathbf{v}\times\mathbf{B})$, so the requirement is $\mathbf{E} = -\mathbf{v}\times\mathbf{B}$, whose solution is

$$\mathbf{v} = \frac{\mathbf{E}\times\mathbf{B}}{B^2}.$$

Its magnitude is $E/B$, which is $c$ times $E/cB$ — and that is less than $c$ exactly when the field is magnetic-dominated. So the condition for the frame to exist and the condition for that velocity to be attainable are the same condition, stated twice. When the field is electric-dominated the formula returns a speed above light, which is the arithmetic refusing rather than failing.

The expression is the $\mathbf{E}\times\mathbf{B}$ drift, and it is normally derived quite differently: a charged particle in crossed fields does not go where either field points but sideways, at exactly this velocity, independently of its charge, its mass and its energy. That independence is startling in the usual derivation and inevitable in this one. The drift velocity is not a property of the particle because it is not about the particle at all — it is the velocity of the frame in which there is no electric field to push anything, and in that frame every particle simply circles.

Which makes the classification operational. In a magnetic-dominated region a plasma drifts, coherently, at a velocity that is the same for ions and electrons and therefore carries no current. In an electric-dominated region there is no such frame, the two species are accelerated apart, and a current flows. The boundary between the two behaviours is the line in the opening figure, and crossing it is what a reconnection region does.

The field that cannot be simplified #

A plane electromagnetic wave has $E = cB$ exactly and $\mathbf{E}\cdot\mathbf{B} = 0$ exactly. Both invariants are zero, which puts it on the diagonal in the hero figure and on the boundary between the two classes.

Maxwell’s added term is what makes a wave’s two fields self-sustaining, and it also fixes their relationship: in a plane wave $E = cB$ exactly, and the two are perpendicular. That places a wave precisely on the diagonal of the invariant plane — neither electric-dominated nor magnetic-dominated — which is the one case no boost can move off, and the reason a light wave looks like a light wave in every frame.

The consequences are strange and are worth stating one at a time. A light wave cannot be brought to rest, which is familiar and is the observation the whole subject was built on. It also cannot be made purely electric or purely magnetic, cannot be made to have its two fields at any angle other than a right angle, and cannot have their ratio changed from $c$. A boost toward the wave increases both amplitudes by the Doppler factor and a boost away decreases both; the relation between them is untouched.

That invariance is the field-theoretic form of the statement that the speed of light is the same in every frame. The wave’s speed being $c$ and its two fields being in the ratio $c$ are the same fact expressed twice, and neither can be altered without altering the other.

The intensity of a wave is very much frame-dependent, even though its classification is not. A synchrotron’s beam is concentrated forward by four powers of the Doppler factor, so the same radiation is feeble in one frame and blinding in another — and none of that changes the invariant, which stays zero. What a boost can change and what it cannot are different questions, and this essay is about the second.

Reading the hyperbola #

The trajectory each field traces in the hero figure is worth reading in detail, because its shape carries the whole argument.

A dipole’s field is magnetic-dominated everywhere outside its source, so every point of it lies on one side of the diagonal — and no boost carries any of those points across. That is how the classification is read off a picture: not by looking at which field is larger in whichever frame one happen to be in, but by asking which side of the diagonal the point sits on, which no observer can dispute.

Start with a magnetic-dominated field and boost. Both magnitudes rise or fall, the point slides along a hyperbola opening toward the horizontal axis, and it approaches the diagonal asymptotically without ever touching it. The electric part can be made zero — that is where the hyperbola crosses the vertical axis — and beyond that crossing it grows again with the opposite sign. So there is exactly one frame in which the field is purely magnetic, up to the freedom of moving along the field, and every other frame sees both.

Now start on the diagonal. There is no crossing to reach: the hyperbola has degenerated into the line itself, and boosting merely slides the point up and down it. Both magnitudes go to zero as the observer chases the wave and to infinity as the observer meets it, and their ratio never budges. A wave can be made arbitrarily faint and arbitrarily fierce and never made into anything else.

Where the classification is used #

The invariants are not bookkeeping. They answer questions about what is possible.

A charge in a magnetic-dominated field spirals forever without gaining energy, because there is a frame in which the electric field vanishes entirely and a purely magnetic field does no work. That is where the classification earns its keep: it says, without solving anything, whether a configuration can accelerate a particle indefinitely or only turn it.

The same sign also settles whether a field configuration can be described by a potential drop at all, and whether an observer can be found for whom nothing accelerates — questions that read as though they are about apparatus and are answered by arithmetic on two numbers. That is the practical content. A plasma physicist asking whether a configuration can accelerate particles, an accelerator designer asking whether a given field arrangement can impart energy, and an astrophysicist asking whether a pulsar’s magnetosphere can pull charges off a surface are all asking the sign of $E^2 - c^2B^2$, and none of them has to specify a frame to ask it.

In an ideal conductor the electric field is zero in the frame of the moving fluid, which is exactly the statement that the field there is magnetic-dominated. So a collapsing conducting cloud carries a field that stays on the same side of the diagonal throughout — the classification is preserved by the dynamics as well as by boosts, and that is why flux freezing and this invariant are two statements about one situation.

The second invariant has its own uses, and one of them is a genuine test of quantum electrodynamics. A field with both invariants non-zero — crossed electric and magnetic fields that are not perpendicular — can, at sufficient strength, create electron–positron pairs from the vacuum, and the threshold is written entirely in terms of the invariants because it must be the same for every observer. A null field cannot do it at any intensity, which is why no laser however powerful pair-produces on its own and why the experiments collide two of them.

The field’s invariant traces hyperbolae in the plane of $E$ against $cB$ for the same reason and by the same algebra that intervals trace hyperbolae in spacetime: a boost is a hyperbolic rotation, and what it preserves is a difference of squares. The two diagrams are the same diagram with different labels, which is worth knowing because everything true of one is true of the other.

Why there are exactly two #

That the list stops at two is not an observation but a count, and the count is worth doing because it explains why no third invariant has ever been found lurking.

The field has six components — three electric and three magnetic — and the Lorentz group that mixes them has six parameters of its own: three rotations and three boosts. A rotation can be used to point the electric field along a chosen axis, which uses up two of them; another rotation about that axis places the magnetic field in a chosen plane, using a third. The three boosts can then be spent removing three more components. Six components, six parameters, and what is left over is nothing — except that the group does not act freely, and the shortfall is exactly two functions that cannot be moved.

Those two are the invariants. There is no room for a third, because there is nothing left to be invariant: any further quantity is a function of the two, and the classification into magnetic-dominated, electric-dominated and null is complete because it exhausts the possible signs. It is the same style of argument that says a system of two particles has one relative velocity and no more, and the same style of argument that fixes the number of independent elastic constants a crystal class permits.

One of the two is not quite a scalar #

A small precision about the second invariant is worth having, because it is the reason it behaves differently from the first in several places.

$E^2 - c^2B^2$ is a scalar in the full sense: it is unchanged by rotations, by boosts, and by a reflection of the coordinates. $\mathbf{E}\cdot\mathbf{B}$ is not. Under a reflection the electric field, being a vector, reverses along the reflected axis while the magnetic field, being an axial vector built from a cross product, does not — so their dot product changes sign.

A quantity that is invariant under proper transformations and changes sign under reflection is a pseudoscalar, and the distinction has consequences. A physical situation and its mirror image have the same first invariant and opposite second ones, so any effect proportional to $\mathbf{E}\cdot\mathbf{B}$ is one that distinguishes left from right. That is why it appears in the description of processes that are not mirror-symmetric and is absent from those that are, and why a term in a theory built on it is a term that violates parity.

It also explains a small asymmetry in how the two invariants are used. The sign of the first is a classification everybody quotes; the sign of the second is rarely quoted on its own, because it depends on a handedness convention as much as on the field. What is quoted is whether it is zero, which is a convention-free statement, and that is precisely the question the classification needs answered.

Where the two come from #

The count above says there are exactly two invariants and does not say where they come from, and the route is short enough to sketch.

The six field components are not six unrelated quantities: they are the components of a single antisymmetric four-by-four array, with the electric field in the row and column that involve time and the magnetic field in the purely spatial part. A change of frame acts on it the way a rotation acts on any array with two indices, so the question of what survives becomes the standard one of what combinations of an array’s components are unchanged by such a transformation.

Contracting the array with itself gives one number, and it is the first invariant. Contracting it with its dual — the array obtained by swapping the roles of the electric and magnetic parts — gives the second. There is no third, because an antisymmetric array in four dimensions has only those two independent contractions, which is the algebraic version of the counting argument above.

The reformulation is worth more than the tidiness. It makes the transformation law a single line rather than two, it makes the invariants obvious rather than found, and it makes the near-symmetry between the two fields into a statement about an array and its dual. What it does not do is make the asymmetry disappear: one of the four field equations has a source and the other does not, and no rearrangement of the array supplies the magnetic charge that would make it symmetric.

Where the model stops #

The invariants classify the field at a point. A field that is magnetic-dominated here and electric-dominated a metre away is perfectly possible, and there is then no single frame simplifying both places at once. The neat picture of “boost into the frame where the field is purely magnetic” is available only where the field is uniform over the region of interest.

Two invariants are the complete set for the classical field and not for what it is made of. The pair above is complete for $\mathbf{E}$ and $\mathbf{B}$; a full description of an electromagnetic configuration includes its sources, and those bring their own invariants.

Nothing here is quantum. The classification is a statement about a classical field configuration, and at field strengths near 10¹⁸ volts per metre the vacuum itself stops being linear: light scatters off light, the invariants acquire corrections, and a strong enough null field is no longer exactly null. That scale is called the Schwinger field and it has not been reached, though focused lasers are now within four orders of magnitude of it.

And a null field is a knife edge. Real light is never exactly a plane wave: a focused beam, a pulse of finite length, and any superposition of waves travelling in different directions all have non-zero invariants. That two counter-propagating beams give a field which is not null is what makes standing light waves able to do things a travelling one cannot, including trapping a neutral atom.

What the pictures cannot show #

The hero figure plots two magnitudes against each other and discards all the directional information. Two fields at the same point in that plane can be quite different objects — one with $\mathbf{E}$ perpendicular to $\mathbf{B}$ and one with them at a small angle — and only the second invariant tells them apart. A complete picture needs both invariants and would be two-dimensional in a different way.

Nor can any of these figures show a field being transformed. The boost is a change of description, not a process, and drawing it as a trajectory in a plane — which is what the hero figure does — invites the reading that something moves along the curve. Nothing does; the curve is the set of answers a family of observers would report, all at once.

Where this ladder goes next #

The rung below showed that magnetism is what electrostatics looks like from a moving frame, which raised an obvious question the essay did not answer: whether the reverse always works. It does not, and this rung is the boundary — a pair of numbers that no observer can change, and a classification into three kinds that no observer can dispute.

The rungs above are the ones that treat the field as a single object rather than as two. The field tensor, in which $\mathbf{E}$ and $\mathbf{B}$ are six components of one antisymmetric array and the transformation is a rotation of it. The dual, which is what produces the second invariant and makes the near-symmetry between electricity and magnetism explicit — along with the one asymmetry, that there are no magnetic charges. And the stress-energy of the field, which carries momentum and is what makes radiation pressure a mechanical quantity rather than an analogy.

The habit worth carrying away is a question to ask of any quantity. Is it a component or an invariant? Components can be transformed away, made larger, made zero; invariants cannot, and a physical statement that depends only on invariants is a statement about the world rather than about a description of it. The energy of a particle is a component and its mass is an invariant. The electric field is a component and $E^2 - c^2B^2$ is an invariant. Almost every apparent paradox in relativity is a component being mistaken for an invariant.