Six numbers, one object

The rung below this one finds that charge density and current density are the four parts of one object, forced into that shape by charge invariance and length contraction. The fields have no more freedom than the sources did.

Under a boost, the three components of $\mathbf{E}$ and the three of $c\mathbf{B}$ mix into each other. They do not mix into anything else — not into the potentials, not into the sources, not into any seventh quantity. Six numbers that transform among themselves under the Lorentz group are the independent entries of something, and the something is a four-by-four antisymmetric array.

Why six, and why an array #

The counting is the argument, so it is worth doing rather than quoting.

An antisymmetric $n \times n$ array has $n(n-1)/2$ independent entries: the diagonal is forced to zero and each off-diagonal pair holds one number between them. For $n = 4$ that is six.

Six is the number of field components there are. That is not a coincidence to be admired; it is a constraint that fixes what kind of object the field can be. A four-vector has four components, a symmetric rank-two tensor has ten, a general rank-two tensor sixteen. Only the antisymmetric one has six, so if the field components transform among themselves and there are six of them, the field is an antisymmetric rank-two tensor and nothing else is available.

Assign them as

$$F^{\mu\nu} = \begin{pmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0\end{pmatrix},$$

and the transformation of the fields under a boost is the ordinary transformation of a rank-two tensor — one rule, applied to one object, instead of the six separate expressions the three-dimensional treatment carries.

What the boost is doing #

The three-dimensional transformation rules are usually presented as a list to be memorised: components along the boost unchanged, components across it mixed with factors of $\gamma$ and $\beta$. Written on the tensor, they are one operation.

The reason rapidity is the right variable is that a boost is a rotation, through an imaginary angle, in the plane containing the time axis and the boost direction. Ordinary rotations mix the space components of the field among themselves, in the way a boost mixes a time and a space coordinate; boosts mix the time-space components — which is to say $\mathbf{E}$ — with the space-space ones, which is $\mathbf{B}$. Both are rotations of the same array, in different planes of the same four-dimensional space.

That reframing settles a question the earlier rungs left open. Whether a field is electric or magnetic is a question about which components of a tensor happen to be non-zero in a chosen basis, and choosing a basis is choosing an observer. Asking whether a field “really is” electric is like asking whether a vector really points along $x$.

The one component that does not move #

There is a detail in the transformation that the tensor makes obvious and the three-dimensional rules make look arbitrary: the components along the boost are unchanged.

$$E'_x = E_x, \qquad B'_x = B_x.$$

In the three-dimensional presentation that is a fact to be remembered alongside the mixing rules for the other four components. In the tensor it is the statement that a rotation in the $t$–$x$ plane leaves the $y$–$z$ block alone, and does not touch the $F^{tx}$ entry either because both of its indices lie in the plane being rotated. A rotation does not move what lies along its own axis, and it does not move what lies entirely within its own plane.

The consequence is one of the more useful practical facts in the subject. A charge moving parallel to a magnetic field feels no force, at any speed, in any frame — because the parallel component is what it was and the perpendicular ones can be made to vanish. The pitch angle of a particle spiralling along a field line is therefore a meaningful quantity, and the whole of guiding-centre motion in a magnetised plasma rests on the parallel and perpendicular parts being genuinely separate.

The same statement explains why a boost along a wire changes nothing about its magnetic field’s magnitude at a fixed distance while a boost across it changes everything. Which components are along the motion is the only question, and it is answered before any arithmetic.

Two invariants, and no others #

The rung below this one but one found two quantities that no boost can change: $\mathbf{E}\cdot\mathbf{B}$ and $E^2 - c^2B^2$. The tensor form explains why there are two and why there are not three.

A tensor’s invariants are the scalars that can be built from it by contracting all its indices. For an antisymmetric rank-two tensor there are exactly two available constructions:

$$F_{\mu\nu}F^{\mu\nu} \propto E^2 - c^2B^2, \qquad F_{\mu\nu}\tilde{F}^{\mu\nu} \propto \mathbf{E}\cdot\mathbf{B},$$

where $\tilde{F}$ is the dual — the array with the electric and magnetic entries interchanged. Any other scalar anybody constructs from $F$ turns out to be a function of those two, because there is nothing else to contract with.

So the three-way classification of fields is complete rather than provisional. A field is characterised, up to a Lorentz transformation, by two numbers, and everything else about it is a description of the observer. That is a much stronger statement than “these two happen not to change”, and it is not available without the tensor.

The dual is worth a sentence of its own because it makes an asymmetry visible. Swapping $\mathbf{E} \to c\mathbf{B}$ and $c\mathbf{B} \to -\mathbf{E}$ leaves the homogeneous Maxwell equations unchanged and would leave the inhomogeneous ones unchanged too — if there were magnetic charges. The one missing charge is the whole of the difference, and in tensor language it is the statement that $\partial_\mu \tilde{F}^{\mu\nu} = 0$ while $\partial_\mu F^{\mu\nu} = \mu_0 J^\nu$.

Four equations becoming two #

The most consequential thing the tensor does is not to the fields but to the equations, and it changes what kind of statements two of them are.

The inhomogeneous pair — Gauss’s law and the Ampère–Maxwell law — collapse into

$$\partial_\mu F^{\mu\nu} = \mu_0 J^\nu.$$

One equation, with a free index running over four values, whose time component is Gauss’s law and whose space components are Ampère’s. Taking $\partial_\nu$ of both sides gives zero on the left, because $F$ is antisymmetric and $\partial_\mu\partial_\nu$ is symmetric, so $\partial_\nu J^\nu = 0$ — charge conservation, as the rung below derives it, falling out of the antisymmetry in one line.

The homogeneous pair — no magnetic charge, and Faraday’s law — collapse into a cyclic identity on $F$, and the identity is satisfied automatically the moment $F$ is written as a four-curl of the four-potential.

That second collapse is the interesting one. It means two of Maxwell’s equations are not independent physical laws at all: they are the statement that the field comes from a potential, and any $F$ built that way satisfies them whatever the potential is. Two of the four were already recognised as constraints rather than laws of motion in three-dimensional language; the tensor form says why, and reduces them from a condition to be maintained to an identity that cannot fail.

The Lorentz force, in the same object #

The transformation is only half of what the tensor buys. The other half is the force law, which in three dimensions is a sum of two unrelated-looking terms and in four is one contraction.

$$\frac{\mathrm{d}p^\mu}{\mathrm{d}\tau} = q\,F^{\mu\nu}u_\nu.$$

The four-velocity contracted with the field tensor gives the four-force, and its space components are $q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$ while its time component is the rate at which the field does work. So the two halves of the Lorentz force and the power delivered by it are three faces of one expression, and the fact that the magnetic part does no work becomes the statement that the time component picks up only the electric entries — which is a consequence of where the components sit in the array rather than a separate observation.

There is a further consequence that the three-dimensional form conceals entirely. Contracting the four-force with the four-velocity gives $q F^{\mu\nu}u_\mu u_\nu$, which vanishes identically because $F$ is antisymmetric and $u_\mu u_\nu$ is symmetric. A four-force orthogonal to the four-velocity is one that changes a particle’s momentum without changing its rest mass — so the antisymmetry of the field tensor is what guarantees that electromagnetic forces accelerate particles rather than altering what they are.

That is a considerable amount of physics from one algebraic property. It is also the reason the antisymmetry is not a convenient accident: a symmetric field tensor would produce forces that changed rest masses, and there would be no stable particles to have a theory about.

What the same counting does elsewhere #

The move that produced the tensor — count the components, ask what object has that many — is worth applying twice more, because the answers are the two other objects the subject needs and they arrive the same way.

Ten components: the stress-energy tensor. An energy density, three components of momentum density and six of stress — the pressures and shears — make ten quantities that mix into one another under a boost. Ten is what a symmetric rank-two tensor has, and that is what the stress-energy is. Its symmetry is not decorative either: the equality of the off-diagonal entries is the statement that angular momentum is conserved.

Four components: the potential. A scalar potential and three components of a vector one, mixing under a boost, is the four-potential of the rung below. Four is what a vector has, and the field tensor is its four-curl — which is why the homogeneous Maxwell equations are identities rather than laws.

In each case the number came first and the identification followed, and in each case the identification carried a consequence nobody had put in: antisymmetry gave charge conservation and the constancy of rest mass, symmetry gave angular momentum conservation, and the vector character gave gauge freedom.

That is the strongest argument for the whole apparatus. A formalism that only rewrote known results in fewer symbols would be a convenience. One in which the shape of the object implies conservation laws that were separate statements beforehand is doing work, and the work is what carries over into every field theory written since.

What it costs to say it this way #

The tensor formulation is not free, and its costs are worth naming because they are why the three-dimensional version is still taught.

The array hides the geometry. A picture of field lines shows what a field does to a charge; a four-by-four array of numbers does not. Everything about the direction of a force, the shape of a dipole field, or the sense of an induced current has to be recovered by contracting with something, and the recovery is not visual.

It makes some easy things harder. Computing the field of a wire from the tensor is worse than computing it from Ampère’s law, and nobody does. The formulation earns its place on questions about frames and about structure, not on questions about a particular configuration.

And the index conventions are a genuine hazard. The sign of the metric, the placement of the indices and the factors of $c$ differ between texts, and a sign error in $F^{\mu\nu}$ produces a magnetic field pointing the wrong way with nothing to indicate it. That is the standard reason the components are written out explicitly, as they are in the figures here, rather than left as a definition.

Where the notation came from, and what it replaced #

The formulation is older than special relativity’s acceptance and younger than Maxwell’s equations, and the sequence is worth knowing because it shows what the tensor added.

Maxwell wrote twenty equations in twenty variables in 1865, in scalar components. Heaviside and Gibbs reduced them to the four vector equations everybody now learns, in the 1880s, and that reduction was a large simplification made entirely within three dimensions.

Minkowski wrote them in four in 1908, three years after Einstein’s paper, and the point he was making was not economy. It was that the Lorentz transformation is a rotation in a four-dimensional space with an indefinite metric, and that electromagnetic quantities are geometric objects in that space. The equations became shorter as a side effect of their becoming statements about geometry.

Einstein was initially unimpressed — he is supposed to have called it superfluous learnedness — and changed his mind entirely. Ten years later general relativity was written in the language Minkowski had supplied, and could not have been written without it: a theory in which the metric itself is dynamical needs objects defined independently of any coordinate system, and the tensor is what such an object is.

So the answer to “is this just notation” is historical as well as structural. The notation was invented to say something, the something turned out to be the foundation of the next theory, and the brevity was never the argument.

Where this stops being right #

The tensor is a field, not a particle theory. Everything here treats $F$ as a classical object with a definite value at each point. The quantum theory replaces it with an operator, and the two invariants become operators whose measurement is subject to the usual constraints.

Antisymmetry is a statement about the vacuum equations. In a medium the useful objects are two tensors rather than one — $F$ built from $\mathbf{E}$ and $\mathbf{B}$, and $G$ built from $\mathbf{D}$ and $\mathbf{H}$ — and the constitutive relation between them is what carries the material’s properties. It is not a Lorentz-invariant relation for a moving medium, which is where the subject becomes genuinely difficult.

Gravity is not included. In curved spacetime the partial derivatives become covariant ones and the equations keep their form, which is a strong statement and is not the whole story: the stress-energy of the field then sources the curvature, and the system is no longer linear.

And the counting argument assumed four dimensions. In $n$ dimensions the field would have $n(n-1)/2$ components, and the dual would be a tensor of rank $n-2$ rather than rank two — so the second invariant exists as it does only in four. That is one of several places where four dimensions is doing more work than it appears to.

What the pictures cannot show #

The array figures draw sixteen numbers, and the object being drawn is a set of six numbers plus a rule about how they sit. Every zero on the diagonal and every minus sign across it is a consequence of the antisymmetry rather than a fact about a field, so most of what the picture contains is structure and not content.

Nor can any figure show that the transformation is a rotation. A rotation in a plane containing the time axis is a hyperbolic one, and its orbits are hyperbolas rather than circles — which the component plot shows as unbounded growth and which no drawing can make look like a rotation, because the thing that is rotating includes a direction the paper does not have.

What the ladder has arrived at #

Five rungs stand on field-transformation. Magnetism as electrostatics seen sideways; the two invariants and what a boost cannot do; charge as the one quantity that does not transform; the sources as a four-vector forced by that; and now the fields as a tensor forced by how they mix.

The habit worth carrying away is about counting components. When a set of quantities transforms among itself and nothing else, count them — the count usually determines what the object is, because very few objects have any given number of components. Six under the Lorentz group means an antisymmetric rank-two tensor and nothing else. Four means a vector. Ten means a symmetric tensor, which is what the stress-energy is. The count is a stronger constraint than it looks, and it is available before any of the physics is worked out.

What is left on this ladder is what the field carries rather than what it is. The tensor has a stress-energy of its own, built from $F$ quadratically, and it is the object that makes the field’s momentum a mechanical quantity — the reason light has a pressure, and the reason a field can be said to be somewhere at all.