The mass a charge's field gets wrong by a third

Where the energy of a field actually is puts the energy of a charged capacitor in the space between its plates, at $\varepsilon_0E^2/2$ per cubic metre. Applied to a single charge, the same density gives the charge’s own field an energy. For a charge $q$ spread over a thin spherical shell of radius $a$, the field is zero inside and Coulombic outside, and the energy outside is

$$U = \frac{q^2}{8\pi\varepsilon_0 a}.$$

Energy has inertia, so this field should resist being accelerated as though it had a mass $U/c^2$. J. J. Thomson noticed in 1881 that a moving charged sphere is harder to accelerate than an uncharged one, and by 1900 the hope that all of the electron’s mass might be of this kind — that inertia itself was electromagnetic — was a serious research programme. The programme ran into a number it could not explain. Computing the electromagnetic mass from the field’s momentum instead of its energy gave four-thirds of the answer.

The momentum is in the broadside field #

A charge moving at velocity $\mathbf v$ has a magnetic field $\mathbf B = \mathbf v\times\mathbf E/c^2$, and wherever there are crossed electric and magnetic fields there is momentum, at a density $\varepsilon_0\mathbf E\times\mathbf B$ per unit volume. The momentum of something that is not moving meets field momentum where it is least expected; here it is where it would be expected, around a moving charge, and the only question is how much.

The left panel is the entire discrepancy, and it is a matter of direction. The energy density of the field is the same in every direction, because the electric field of a slowly moving charge is still almost spherically symmetric. The momentum density is not. The magnetic field of a moving charge is zero straight ahead and straight behind, where $\mathbf v$ is parallel to $\mathbf E$, and largest broadside, so the momentum density goes as $\sin^2\theta$ times a factor of two relative to the energy density. Averaged over every direction, $\sin^2\theta$ is two-thirds; twice two-thirds is four-thirds.

The right panel says where the mass lives by distance, and here the two agree. Both densities fall as the fourth power of distance, so both have the same radial profile, $1 - a/r$ of the total within radius $r$. Half the electromagnetic mass of a charged shell is within twice its radius, and nine-tenths within ten times. The mass is not in the charge; it is in a region around the charge a few times its size.

Two masses for one body #

For any body whatever, the mass that its momentum implies and the mass that its energy implies must be the same: $p/v = E/c^2$, at every speed. That is what makes $E^2 - p^2c^2$ an invariant and the mass a property of the body rather than of the observer, the relation the invariant that survives a boost builds on.

The field of a charged shell fails that test at every speed below light’s. At rest the momentum says four-thirds and the energy says one. As the shell moves faster the field’s energy rises faster than a simple $\gamma$ would give, as $\gamma(1 + \beta^2/3)$, because the field flattens into the plane perpendicular to the motion — the flattening the field that points where the charge is now draws — and the magnetic energy grows. The momentum rises as $\frac{4}{3}\gamma\beta$. The ratio closes towards one only in the limit of light speed, where it is irrelevant.

These numbers are not a formula quoted and plotted. The dots are the energy and momentum of the boosted Coulomb field, integrated over all of space outside the Lorentz-contracted shell, at each speed, and they sit on the closed-form curve to a few parts in a hundred thousand. The field’s energy and momentum really are what the curve says.

A rest mass that depends on who is watching #

The same failure has a sharper form. If the field’s energy and momentum were a four-vector, $E^2 - p^2c^2$ would be the same in every frame. For the field alone it comes out as $U^2(1 - \beta^2/9)$: the rest mass of the field, computed by an observer moving at 0.9c, is 4.6 per cent smaller than computed by an observer at rest with it. A rest mass cannot depend on the frame it is computed in. Whatever the field’s energy and momentum are, together they are not the energy and momentum of anything that could exist on its own.

That is the clue. The field of a charged shell cannot exist on its own, because a charged shell cannot. Every part of the charge repels every other part, and the force read off a surface that touches nothing puts a number on it: the field presses outward on the shell with a pressure $\varepsilon_0E^2/2$. Something must hold the charge in. Henri Poincaré said so in 1905 and added a cohesive stress to the electron — a negative pressure inside it, of whatever origin, just strong enough to balance the electrostatic repulsion.

The stress carries energy and no momentum #

A stress is part of the energy–momentum of a body, in the same way as a field is, and it transforms in the same way when the body moves. Max von Laue proved in 1911 the general version of what Poincaré’s stress achieves: a body in static equilibrium has energy and momentum that form a four-vector only if the stresses inside it integrate to zero. The field of a charged shell has an integrated stress of $U/3$ in each direction; the cohesive stress must supply $-U/3$ to make the body stable, and the moment it does, the sum becomes a four-vector.

The ledger shows how the repair is made, and it is not what might be guessed. The stress does not carry the missing momentum. Its momentum is zero at every speed, because in a moving body a stress’s energy and its tension enter the momentum with opposite signs, and for this stress they cancel exactly. What the stress carries is energy: $U/3$ at rest, falling as the body moves. The field’s momentum was right all along, at four-thirds. It was the field’s energy that was short, by exactly the amount the stress supplies, and with the stress included the energy says four-thirds too.

The rule behind that cancellation is general and worth having. For any body at rest with internal energy $E_0$ and an integrated stress $S$ along the direction of motion, the momentum when it moves at $\beta c$ is $\gamma\beta(E_0 + S)/c$ and the energy is $\gamma(E_0 + \beta^2 S)$. Tension counts negatively and pressure positively. The field of the shell has $E_0 = U$ and $S = +U/3$, so its momentum carries the factor $\frac43 U$ while its energy starts at $U$ — the whole discrepancy in one line. The stress has $E_0 = U/3$ and $S = -U/3$, so its momentum is zero and its energy is $\gamma U(1-\beta^2)/3$, the falling curve in the ledger. Add them and $S$ totals zero, which is von Laue’s condition, and the body’s energy and momentum become $\gamma\cdot\frac43 U$ and $\gamma\beta\cdot\frac43 U/c$ — a four-vector.

So the electromagnetic mass of a charged shell, done properly, is $\frac{4}{3}U/c^2$ — provided the cohesive stress has the particular equation of state that makes its rest energy $U/3$, like a vacuum energy with its pressure equal to minus its density. A different model of what holds the charge together gives a different rest energy for the stress and a different total. There is also a different fix altogether, developed by Fermi in 1922 and by Rohrlich in 1960, which redefines the field’s energy and momentum so that they transform properly on their own and gives the mass $U/c^2$. Both are consistent. Which one is “the” electromagnetic mass depends on what holds the charge together, and the field alone cannot answer that.

A quarter-century argument, settled by a theorem about stresses #

The 4/3 was not a curiosity in 1905. Max Abraham’s model of a rigid spherical electron, Hendrik Lorentz’s model of an electron that contracts with its motion, and several others made different predictions for how the electron’s mass rises with speed, and Walter Kaufmann was deflecting fast electrons in electric and magnetic fields to tell them apart — the experiments the push that does not point where the body goes follows through to their resolution. Lorentz’s electron, the one relativity turned out to require, had the 4/3 problem; Abraham’s did not, and for a few years that looked like an argument for Abraham.

The argument was misplaced. Relativity does not care what an electron is made of; it requires that whatever it is made of, the total energy and momentum form a four-vector. The field is not the whole electron, so the field alone has no obligation to be one. Von Laue’s theorem turned the discrepancy from a failure of electrodynamics into a statement about stability: any charged body that holds together contains non-electromagnetic stresses, and those stresses are part of its mass.

The same theorem appears wherever energy sits inside a body under tension. The box of light that weighs something has light pressing on the walls of a box, and the walls’ tension is what makes the box’s energy and momentum transform as a four-vector; the light alone would not. In both cases the mass belongs to the equilibrium, not to the thing inside it.

How much of an electron its field would be #

The last question is whether any of this describes a real electron.

Classically, the field energy grows as $1/a$ without limit as the charge is squeezed. It equals the electron’s whole mass for a shell 1.41 femtometres in radius, half the classical electron radius, which is where that length comes from. But scattering experiments see no structure in the electron down to about $10^{-18}$ metres, a thousand times smaller, and a shell that size would have field energy 1,409 times the electron’s measured mass. The only way to keep the classical picture is to give the electron a negative “bare” mass of nearly the same size, cancelling to leave the observed mass — a subtraction of two enormous numbers that the force a charge exerts on itself meets as mass renormalisation, and whose remnant is the runaway solution of the radiation-reaction equation.

Quantum electrodynamics changes the shape of the curve rather than its existence. Victor Weisskopf showed in 1939 that the electron’s self-energy in the quantum theory grows only as the logarithm of the radius at which it is cut off, not as its inverse: the virtual electron–positron pairs the field creates around the charge screen it, and the energy of the field inside the reduced Compton wavelength, 386 femtometres, is spread rather than concentrated. Cut off at $10^{-18}$ metres, the self-energy is 4.5 per cent of the electron’s mass. Cut off at the Planck length, where every model runs out at once, it is 18 per cent. The classical divergence is a power law and cannot be tamed; the quantum one is a logarithm and can be absorbed. Renormalisation takes the observed mass as an input and never needs to know the cut-off, and the theory’s predictions — the electron’s magnetic moment to twelve digits among them — come out right.

Where field energy really is a mass #

The programme of 1900 failed for the electron and succeeded, in a form nobody then imagined, for the particles made of quarks. The mass that is missing shows a helium nucleus weighing less than its parts by its binding energy — a mass deficit of field energy, negative because the fields there bind rather than repel. The proton is the case where the field energy is not a correction but nearly the whole. Its three quarks have masses adding to about one per cent of its own; the other ninety-nine per cent is the energy of the strong field between them and of their confined motion, computed from the theory of that field on large computers and found to agree with the measured mass to a few per cent. Almost all the mass of ordinary matter is the kind of mass this essay is about — field energy with an inertia — only the field is not electromagnetic.

The electromagnetic part still leaves a mark, and it is a small one with large consequences. The proton is charged and the neutron is not, so the proton’s own electric field adds to its mass, by something between 0.6 and 1 megaelectronvolt depending on how the calculation is done. If nothing else differed, the proton would be the heavier of the two, and a free proton would decay into a neutron. What saves chemistry is that the neutron’s quark content is heavier by more than the electromagnetic difference, and the neutron ends up heavier by 1.29 megaelectronvolts. The electromagnetic self-energy of a nucleon — the same field energy that gives a charged shell its $U$ — is one of the two nearly cancelling terms that decide whether hydrogen exists.

That calculation meets the four-thirds problem in a modern form. A proton’s electromagnetic mass cannot be computed from its field alone, because the field is held in by the strong interaction, and the strong interaction’s stresses are part of the answer — exactly von Laue’s lesson, applied to a real particle. The quoted numbers come from treating both together, and the spread between them is largely a spread in how that is done.

Where the charged shell stops #

The shell is a model. The 4/3, its angular origin and the repair by stresses belong to a sphere of charge of definite radius. A uniformly charged ball gives the same 4/3 with a different $U$; a charge distribution that is not spherically symmetric gives a mass that depends on the direction of motion. The electron, as far as anyone can measure, has no size at all.

Slow acceleration. Every figure is for uniform motion. A charge whose acceleration changes fast enough that light crosses it in a time comparable to the change sees its own field lag behind, and that lag is the radiation reaction of the force a charge exerts on itself, whose inertial term is exactly this four-thirds electromagnetic mass.

The stress is unspecified. Poincaré gave the cohesive stress no origin, and the figures give it only the equation of state that makes the arithmetic of the ledger work. What actually holds a real particle together is a question for the theory of that particle; for the electron, in quantum electrodynamics, the question dissolves into renormalisation rather than being answered.

A stress that is not a material #

The figures account for the field’s energy and momentum and a stress’s energy and momentum, and never show the stress as a thing. A negative pressure inside a sphere of radius a femtometre, strong enough to balance an outward electrostatic pressure of order $10^{30}$ pascals, is not a material, and no figure could honestly depict it as one. The angular figure draws densities at one radius and cannot show that the magnetic field, and with it the momentum, is being carried along with the charge rather than recreated; and the radius figure plots a classical and a quantum answer on one axis as though they answered the same question, when the quantum one is a statement about a quantity — the self-energy — that is not itself observable.

Still open: whether a particle’s mass can be computed #

The programme of 1900 was to compute the electron’s mass from its charge. It failed classically because the answer depended on an unmeasured radius, and quantum electrodynamics made the failure respectable by showing that the answer depends on it only logarithmically and that no measurement is affected. Whether the masses of the fundamental particles are computable at all — from a deeper theory, as the proton’s mass is computed from the strong interaction’s field energy to a few per cent — is open. For the proton, nearly all the mass is field energy of the kind this essay is about, and computing it took lattice calculations on supercomputers. For the electron, nothing now known says what would set the number.

The habit worth carrying away is to check that a thing is closed before asking for its invariants. A part of a system can have energy and momentum that do not transform together, and when that happens the missing piece is the part holding it in place. The field of a charge fails the four-vector test by a third, and the third is not an error in the field. It is the energy of whatever the field is pushing against.