The force a charge exerts on itself

The rung below this one takes Larmor’s formula and applies it to an electron in orbit: the charge accelerates, so it radiates, so it loses energy, so the orbit shrinks, and the atom is gone in sixteen picoseconds. The arithmetic is impeccable and the conclusion is false, and the usual moral is that classical physics needed replacing.

There is a second problem in that calculation, and it is entirely classical. Larmor’s formula is derived by counting the energy flowing out through a distant sphere, which is where the field’s energy is taken to be. Nowhere in the derivation is there a force on the charge. The energy loss was put in by hand — the orbit was made to shrink at the rate the field carries energy away — and the classical theory ought to be able to say what pushes the charge to make that happen.

That is a satisfying answer to how big the effect is, and a disastrous one to what the effect does. This essay is about the disaster, because it is the clearest case in the subject of a theory diagnosing its own domain of validity — and the boundary it draws is at a distance no classical picture of an electron could have survived to anyway.

Where the force has to come from #

Conservation supplies the missing statement, as it does wherever a ledger has to balance. Over an interval in which the motion is periodic, or in which the acceleration starts and finishes at zero, the work done by whatever force acts on the charge must equal minus the energy radiated:

$$\int F_{\text{rad}}\,v\,\mathrm{d}t = -\int \frac{\mu_0 q^2 a^2}{6\pi c}\,\mathrm{d}t.$$

Integrating the right-hand side by parts turns $a^2 = \dot v \cdot \dot v$ into $-\dot a\cdot v$ plus a boundary term that vanishes under the stated conditions. Comparing the integrands then gives

$$F_{\text{rad}} = \frac{\mu_0 q^2}{6\pi c}\,\dot a = m\tau\dot a, \qquad \tau = \frac{\mu_0 q^2}{6\pi m c} = \frac{2r_e}{3c}.$$

Two things about that derivation are worth naming immediately. It fixes the force only on average over an interval, so the instant-by-instant form is a guess — a very natural one, and not forced. And the result depends on the third derivative of position, which no equation of motion in mechanics — not one of them — has ever contained.

The τ that comes out is not a free parameter. It is built from the charge, the mass, the speed of light and the permeability of free space, with no room for adjustment, and it can be written three ways that are the same number: μ₀q²/6πmc, two thirds of the classical electron radius divided by c, and two thirds of the time light takes to cross the region in which a sphere of charge would store its whole rest energy. The generator checks the first two against each other before drawing anything, because four constants written from memory is exactly the kind of arithmetic that fails silently — the first version of it had a factor of c² too many and produced a radius of 3 × 10⁻³² metres.

One time, and only one #

Before integrating anything it is worth asking how many free numbers the theory could possibly have had. A classical point charge is described by exactly three constants: its charge, through the combination $q^2/4\pi\varepsilon_0$, which carries units of energy times length; its mass; and the speed of light. There is one way to build a length out of those three and one way to build a time, and no way at all to build a second of either. So a classical theory of a point charge does not merely happen to have a characteristic time in it — it cannot avoid having one, and the only thing left to compute is the numerical factor in front. That factor is two thirds, and everything else in τ was fixed before the calculation began.

What is missing from that list is as informative as what is on it. Planck’s constant is not there. The τ above is built with no quantum mechanics anywhere, which is why it is legitimate to speak of the classical theory failing on its own terms rather than being superseded on somebody else’s — the inconsistency is internal, and it would be there in a world where ħ was zero.

The size of the number is what makes the failure so hard to reach. Six times $10^{-24}$ seconds is not a short time by the standards of laboratory physics; it is not on the same axis. The shortest light pulses anybody has produced are a few tens of attoseconds, which is around ten million time constants long, and a pulse that short is already the product of a decade of technique. An electron in the strongest focused laser field ever built sees a force changing over roughly $10^{-15}$ seconds, still a hundred million τ. The ratio in the first figure is that comparison and nothing else: τ over the time the force takes to change, which is the only dimensionless quantity the problem has.

The equation, and the free particle that will not stay still #

Put the self-force into Newton’s second law and the result is

$$m(a - \tau\dot a) = F_{\text{ext}}.$$

Set the external force to zero and this does not say the acceleration is zero. It says $\dot a = a/\tau$, whose general solution is an acceleration growing exponentially with time constant τ.

This is not a subtlety. A free particle that spontaneously accelerates violates conservation of energy, conservation of momentum and every reasonable expectation, and it is not a marginal solution: it is the general one. The physical answer — zero acceleration for a free particle — is one exact initial condition out of a continuum, and any deviation from it in the last decimal place grows by a factor of e every 6 × 10⁻²⁴ seconds.

Why a third derivative was always going to appear #

A third-order equation of motion looks like an error, and there is a reason for it that has nothing to do with electromagnetism in particular.

The full system here is not a particle. It is a particle and a field, and the field has infinitely many degrees of freedom of its own, each with its own history. Writing an equation for the particle alone means eliminating all of them — solving the field’s equations, feeding the answer back, and asking what is left. Eliminating a dynamical variable from a coupled system does not generally leave a system of the same kind behind: what it leaves is an equation with memory, in which the force at one instant depends on an integral over the past of the motion. A term proportional to $\dot a$ is the first thing such an integral gives up when it is expanded for a motion that changes slowly compared with the memory time.

Seen that way the awkward features stop being surprising in the same order they appeared. The memory time is τ, and a slow expansion of a memory kernel is exactly an expansion in τ, which is what the reduced-order form below turns out to be. The runaway is the pathology an expansion acquires when it is treated as an exact equation rather than as the leading term of one — the same disease as any perturbative series solved in closed form outside its radius. And a memory integral over the past cannot produce pre-acceleration, which is the strongest hint available that the third-order equation is not merely difficult but wrong in kind.

The cure, and what it costs #

The runaway can be removed. Integrating the equation forward requires an initial acceleration; integrating it backward from the future requires instead that the acceleration go to zero long after the force stops, which is a perfectly reasonable demand and picks out exactly one solution:

$$a(t) = \frac{1}{m\tau}\int_t^{\infty} e^{-(t'-t)/\tau}\,F(t')\,\mathrm{d}t'.$$

The acceleration now depends on the force at times later than t.

The condition that removed the runaway deserves a second look, because of what it is. Ordinary mechanics takes an initial position and an initial velocity and predicts the future. This equation is third order, so it needs a third piece of information, and the piece supplied was a statement about the end — that the acceleration must die away eventually. Trading an initial condition for a final one is precisely how a problem stops being an initial-value problem, and pre-acceleration is what that trade looks like from inside.

So the choice is between a free charge that accelerates for ever and a charge that starts moving before it is pushed. Neither is acceptable and one of them has to be taken, which is the clearest possible signal that the equation is not a fundamental one.

There is a third option that nobody takes, and it is worth naming because it explains why the first two are the only candidates. The equation could be read as an integro-differential relation with a finite charge distribution in it, in which case the self-force becomes a sum over the forces one part of the charge exerts on another, retarded by the time light takes to cross the object. That calculation is finite, causal and free of runaways — and it introduces a size for the electron, together with a set of internal stresses holding it together that no purely electromagnetic theory can supply. Poincaré’s stresses were exactly that admission: the classical electron cannot be made of electromagnetism alone.

And the interval is the diagnosis. The acausality lasts for τ, and light crosses 1.88 femtometres in that time. That length is not a coincidence: it is two thirds of $r_e = \mu_0 q^2/4\pi m$, the radius at which a sphere of charge would have an electrostatic self-energy equal to $mc^2$. The classical theory has one length in it built out of the charge and the mass, and that length is where it stops making sense — a self-consistency that is much more satisfying than the failure it announces.

Three lengths, each 137 times the next #

The classical electron radius is one of three lengths that can be built for an electron, and the pattern they make is the shape of the whole difficulty.

The largest is the Bohr radius, 52.9 picometres, which is the size of an atom and the scale on which the rung below this one watched an orbit collapse. Next, smaller by a factor of 137, is the reduced Compton wavelength, 386 femtometres — the length below which asking where an electron is stops being a well-posed question, and the wavelength any particle carries evaluated at the speed of light. Smaller again by another 137 is the classical electron radius, 2.82 femtometres, and two thirds of it is what light crosses in τ.

The factor is the same both times because it is the fine-structure constant, and the three lengths are $a_0$, $\alpha a_0$ and $\alpha^2 a_0$. That is not a numerical accident; it is the statement that one dimensionless number governs how strongly a charge couples to a field, and that each step down the ladder is one more power of that coupling.

The ordering is what matters for this essay. The scale at which the classical self-force becomes incoherent is the smallest of the three, so anything trying to reach it has already passed the scale at which the electron stopped being a point in a much more thorough way. The classical theory is not being repaired at the boundary it announces — it is being replaced well before it, by a theory that never had a self-force problem because it never had a trajectory to put one on.

Where the effect is real, and what it is not #

Small is not zero, and the reaction is measured all the time — as an energy loss rather than as a force.

The place it is most visible is a storage ring, where the loss per turn is an unmissable fraction of the beam energy and has to be replaced every lap by radio-frequency cavities. That is radiation reaction as a bill rather than as a differential equation, and it is paid without anybody solving the Abraham–Lorentz equation, because the loss over one turn is enormously larger than the correction to the motion within one turn.

The reaction is not a drag. Its direction is set by $\dot a$, not by $v$ or by $a$. For steady circular motion $\dot a$ points opposite to $v$ and the reaction is a retarding force, which is what a charge on a circular path feels. For a charge in uniform acceleration — the case the equivalence principle is built on — $\dot a$ is exactly zero and the self-force vanishes — while the radiation does not, because $a^2$ is not zero. That case is one of the oldest arguments in the subject, and the resolution is that energy is being borrowed from and returned to the near field rather than steadily supplied.

The field separates into two pieces distinguished by how they fall away. The velocity field falls as $1/r^2$ and carries energy that stays with the charge; the acceleration field falls as $1/r$ and carries energy that leaves. Only the second is radiation, and only the second has to be paid for — so the self-force this essay is about is the reaction to the $1/r$ piece alone, and the far larger $1/r^2$ piece contributes nothing to it whatever.

What replaced it, and what did not #

The modern account does not repair the Abraham–Lorentz equation so much as decline to need it. Quantum electrodynamics computes the same energy loss without ever writing an equation of motion for a point charge, and the divergent self-energy that produced τ is absorbed into the measured mass — the procedure called renormalisation, which is what “the electron’s mass” has meant since.

What does survive intact is the radiation itself. The wave and its speed, its transverse structure and the energy it carries are all untouched by any of this — the Larmor formula gives the right answer and is confirmed everywhere from a radio antenna to a synchrotron. What fails is only the attempt to write down the back-reaction on the charge as an ordinary force in an ordinary equation of motion, and that failure is a failure of the point-particle idealisation rather than of electrodynamics.

There is a useful test of whether a correction of this kind is being used inside its domain, and it costs nothing to apply. Compute the ratio in the first figure for the case in hand; if it is small, the reduced-order equation is the right tool and the exact third-order one is a trap; if it is not small, no classical equation of motion is going to help, because a force varying that fast is a force varying over a distance at which the charge is not a point. There is no middle band in which the full Abraham–Lorentz equation is both necessary and trustworthy. That absence is the practical content of everything above — a term can be simultaneously real, measured, and never worth writing exactly.

And the reduced-order equation is what is actually used. Substituting $F_{\text{ext}}/m$ for $a$ inside the small correction term gives $m a = F_{\text{ext}} + \tau\dot F_{\text{ext}}$, which has no runaways, no pre-acceleration and no third derivative. It is not exact and it is not derived; it is the first term of an expansion in τ, and it is legitimate precisely because the ratio in the first figure is never large. The Landau–Lifshitz form of that equation is what every accelerator and plasma calculation uses.

Every classical model has a length at which it stops, and the classical electron radius is this one’s — arrived at not by asking how big an electron is but by asking where the self-energy of a point charge becomes comparable with its rest energy. At $2.8\times10^{-15}$ m it is a hundred times larger than anything an experiment has resolved, which is the honest statement of the situation: the theory announces its own breakdown at a scale it is already known to be wrong about.

And nothing above has any quantum in it, which is the honest limit. The classical picture fails at 1.9 femtometres; the quantum description of an electron fails to be a point much earlier, at the Compton wavelength of 386 femtometres, which is 137 times larger. So the regime where classical radiation reaction misbehaves is one that quantum mechanics has already taken over by two orders of magnitude — and the classical inconsistency is therefore a symptom that was never going to be treated on its own terms.

That ordering is worth stating carefully because it is easy to get backwards. It is not that the classical theory breaks down and quantum mechanics repairs it at the same scale. It is that the classical theory would break down if it were pushed to a scale it never gets to, because something else has intervened long before.

The rung after this one #

Two rungs of this ladder have now been about a charge that accelerates because something makes it. What neither has asked is what happens when the acceleration is not the charge’s own doing — a charge sitting still in a field that is itself changing, or a charge at rest in a gravitational field, where a clock lower down runs slow, which by the equivalence principle is accelerating and by every static argument is not. Whether such a charge radiates, and who would see it if it did, is a question that has been answered several times in incompatible ways, and the reason is that “radiates” turns out to depend on the observer.