The solution that is thrown away

Every field written on this ladder has been the retarded one. The velocity field that extrapolates a charge’s motion, the three terms of an oscillating dipole, the sum over an extended source — all of them are what the source did earlier, arriving now.

That was never derived. It was chosen, at the first line of every one of those calculations, and the choice is invisible because it is made before there is anything to look at.

Maxwell’s equations reduce to a wave equation, and a wave equation is second order in time. Anything second order in time is unchanged by reversing time, so for every solution running outward there is a mirror-image solution running inward, exactly as good, satisfying the same equations with the same sources.

What is actually being assumed #

The general solution of the wave equation with a source is the retarded solution plus any solution of the source-free equation. Writing the field as purely retarded means setting that homogeneous part to zero, and that is a statement about the boundary of the region being solved: no field arrives from outside, ever.

It is a good assumption in the sense that it works. It is not a consequence of anything in the equations, and it is worth seeing how strange it is when stated plainly. The claim is that of all the fields consistent with the sources present, the one nature selects is the one with no incoming component from infinity — a condition imposed at the boundary of the universe, in order to fix what happens on a laboratory bench.

The reason this matters rather than being a technicality is that the equations are otherwise symmetric under time reversal, and the world visibly is not. An equation that only runs forwards is the shape of every irreversible process, and it always turns out that the irreversibility was put in somewhere. In diffusion it is in the initial condition. Here it is in the boundary condition, and the question of what justifies it is old and not closed.

The construction that makes the choice a consequence #

Wheeler and Feynman, in 1945, tried to derive the retarded field rather than assume it.

Their proposal is to take the equations at their word and let every charge emit half a retarded field and half an advanced one. That is time-symmetric, which is what the equations offer, and it seems immediately wrong — it predicts a signal arriving before it was sent.

The rescue is the rest of the universe. Every other charge, driven by that half-advanced and half-retarded field, radiates its own half-and-half in response, and the sum of all those responses turns out to have a particular form near the original source: half retarded minus half advanced.

Adding the two gives a full retarded field and no advanced field at all. The advanced parts cancel exactly; the retarded parts double, which is why the source ends up emitting a whole retarded field rather than half of one.

The result is genuinely attractive: the arrow of time in electromagnetism becomes a property of the universe being a good absorber, rather than a rule about fields. It also explains radiation reaction without a self-force — the recoil on a radiating charge is the absorbers’ advanced fields arriving back at it, which is a force from elsewhere rather than a force from itself, and it dissolves the divergence the self-force ladder runs into.

And the condition it requires is a cosmological one that is probably false. The cancellation needs the future to be a perfect absorber: every outgoing ray must eventually be absorbed by something. In an expanding universe with accelerating expansion that fails, because light emitted now can escape to infinity without meeting anything. Various repairs have been proposed and none is generally accepted; the theory is neither refuted nor established, and it has been in that state for eighty years.

The one place the difference is already used #

The advanced solution appears in a standard calculation that nobody thinks of as unorthodox, and it is where the whole subject first became more than a curiosity.

Dirac, in 1938, asked what force a radiating charge feels from its own field. The retarded field of a point charge is infinite at the charge; so is the advanced one. Their difference is not: the infinities are identical and subtract, and what is left is finite, well defined, and equal to the radiation reaction that energy conservation demands.

So the standard classical expression for radiation reaction — the one containing a third derivative of position, with all the trouble that brings — is derived as $\tfrac{1}{2}(F_{\text{ret}} - F_{\text{adv}})$. An advanced field is not a speculative addition to the theory; it is a term inside the only classical account of what a charge’s own radiation does back to it.

Wheeler and Feynman’s construction is what happens when that combination is taken seriously as a physical field rather than as a regularisation. In their account, $\tfrac{1}{2}(F_{\text{ret}} - F_{\text{adv}})$ is not a subtraction performed to remove an infinity; it is what the absorbers actually send back, and the fact that it is exactly the expression Dirac needed is the argument’s strongest point.

The awkwardness Dirac’s expression carries — pre-acceleration, in which a charge begins to move a fraction of a second before the force arrives — is then not an artefact either. It is a small piece of advanced influence, on the timescale of a classical electron radius over $c$, which is $10^{-24}$ seconds. Whether that is a defect of the classical model or a real remnant of time symmetry is precisely the question this rung cannot settle.

The mirror that returns a wave the way it came #

Optics has a device that emits the advanced solution, and it is sold commercially.

A phase-conjugate mirror reflects a wave by reversing its phase rather than its normal component, so the returning wave retraces the incoming one exactly, undoing whatever distortion the incoming path introduced. Send a beam through a piece of frosted glass onto an ordinary mirror and the returning beam is scrambled twice. Send it onto a phase-conjugate mirror and the return passes back through the same frosted glass and comes out clean.

For a monochromatic wave, reversing the phase is the same operation as reversing time. The device is a steady-state time-reversal mirror, and the wave leaving it is the advanced solution of the wave it received.

The mechanism is a nonlinear one: four-wave mixing in a suitable crystal, where two counter-propagating pump beams and the signal generate a fourth wave whose phase is the conjugate of the signal’s. That the process needs pump beams is the whole of the accounting — the conjugate wave’s energy comes from the pumps, and the arrangement is not a passive object that spontaneously produces incoming waves. It is the same lesson as the spherical cap: an advanced solution is a thing to be built and paid for, and the payment is what makes the world’s ordinary asymmetry survive.

Unlike the rotation a return trip doubles, which is a genuinely non-reciprocal effect, phase conjugation is the reciprocal case pushed to its limit — the return path is not merely available but exactly retraced.

The account that moves the problem rather than solving it #

The other standing explanation for retardation is statistical, and it is worth stating precisely because it is often stated loosely.

A converging spherical wave requires a very particular field over a large surface at an earlier time — every part moving inward, with phases coordinated to arrive together. A diverging wave requires nothing coordinated at all. So the retarded solution is what happens for the same reason a gas fills a box: not because the reverse is forbidden, but because the reverse corresponds to an infinitesimal fraction of the available initial conditions, which is the second law with a probability attached.

The parallel to entropy as a count is exact, and so is the difficulty. Counting says a converging wave is improbable given a random earlier state; it does not explain why the earlier state should be treated as random rather than the later one, since the counting argument is symmetric. Running it backwards says diverging waves are improbable given a random later state, which is false. What breaks the symmetry is an assumption about the past — the same assumption thermodynamics needs and cannot supply from within itself.

So the statistical account and the absorber account fail in opposite directions. One needs a cosmological condition on the future that appears to be false; the other needs a cosmological condition on the past that is assumed rather than derived. Neither is a theorem, both are respectable, and the retarded solution goes on being used in every calculation anybody performs.

The advanced solution, built on purpose #

The strongest argument that advanced solutions are not forbidden is that they can be made.

A time-reversal mirror records a wave on a surface surrounding a region, reverses the recording, and plays it back. What comes out is a wave converging on wherever the original source was — an advanced solution over the enclosed region, built by paying for it on the boundary.

This is not a thought experiment. Mathias Fink’s group built acoustic time-reversal mirrors in the 1990s that refocus an ultrasound pulse onto its source through highly scattering media, and the more scattering the medium, the better the focus — because scattering gives the recording surface access to directions it would not otherwise have seen, which is the same statement the figure makes about the cap.

The lesson is about what the advanced solution costs, and it is not causality. It is information on a surface. A converging wave requires the field to be specified everywhere on a closed surface at an earlier time, which for a spontaneous process nobody arranged is an amount of coordination the world does not supply. Retardation is not selected because convergence is forbidden; it is selected because divergence is what happens when nothing is arranged, in the same sense that a gas fills a box.

What it costs to impose it by hand #

A last piece of evidence that the boundary condition is a real assumption rather than a formality: it has to be paid for, in full, in any simulation.

Integrating Maxwell’s equations on a grid means the grid has an edge, and the edge is a boundary the field will reach. Doing nothing there means the field reflects — which is to say the simulation supplies exactly the incoming wave the retarded condition forbids, and it does so within a few hundred timesteps of anything reaching the wall. Every early computational result in electromagnetics was contaminated by it.

The fix is a boundary that absorbs, and getting one has taken the field forty years. Simple absorbing boundary conditions extrapolate the outgoing wave and work for waves arriving nearly perpendicular; they fail at grazing incidence, and a large simulation has a great deal of grazing incidence. The modern answer is a perfectly matched layer — a shell of fictitious lossy material with the loss written into a complex coordinate stretch, contrived so that its impedance matches the vacuum’s at every angle and every frequency, so waves enter it without reflecting and die inside.

That construction is worth noticing for what it says about the physics. A perfectly matched layer is not a material; no substance has those properties, and it is written in a coordinate transformation rather than in a permittivity. It exists to impose the Sommerfeld radiation condition — no incoming field from infinity — on a domain that has no infinity in it. Nature applies that condition for nothing; a computer has to buy it, at ten or twenty cells of grid on every face and a good deal of care about numerical stability.

The same accounting appears in an anechoic chamber, which is a room built so that its walls do not exist electromagnetically. The wedges of absorber are the physical realisation of the same boundary condition, and they are metres long at low frequencies because the layer has to be thick compared with a wavelength for the same reason the numerical one does.

The laboratory version of the question #

There is one place where the boundary condition stops being philosophy and becomes a measurement, and it is worth ending on because it settles at least the local version of the argument.

Whether an excited atom emits is not decided by the atom.

Purcell noticed in 1946 that an emitter in a resonant cavity should decay faster by $3Q\lambda^3/4\pi^2 V$, and the effect is now the basis of an industry: single-photon sources, low-threshold lasers, and light-emitting diodes whose speed is set by the cavity rather than by the material.

The other direction is the more startling one. Put an atom between two mirrors closer together than half a wavelength and it cannot emit at all, because there is no mode of the correct wavelength for it to emit into. Hulet, Hilfer and Kleppner did it in 1985 with Rydberg atoms between parallel plates and measured the lifetime rising by more than a factor of twenty as the plates were brought together.

So the rate at which a source radiates depends on what is available to receive the radiation — measurably, controllably, by factors of thousands. That is a weaker statement than the absorber theory’s and it is in the same direction, and it is the one that has been demonstrated. Whether the cosmological version is also true remains open; whether the local version is true stopped being open in 1985.

Where this stops being right #

Nothing here decides anything. This rung’s subject is a question, and the honest summary is that the retarded boundary condition is universally used, has never failed, and has no accepted derivation. Wheeler–Feynman is one attempt whose cosmological requirement looks false; the statistical account — that the retarded solution is overwhelmingly more probable given a low-entropy past — is another, and it moves the problem into cosmology rather than solving it.

The absorber construction was drawn in one dimension with a point source. The real argument is an integral over all absorbers in all directions with the right response function, and whether it converges depends on the absorber’s properties as well as on its completeness. The figure shows the cancellation, not the derivation of the term being cancelled.

The time-reversal figures assume a scalar field and no loss. A medium that absorbs cannot be time-reversed, because the reversal would require amplification rather than absorption. Every real time-reversal experiment works in a nearly lossless medium for exactly that reason, and the technique fails in proportion to the loss.

And the Purcell factor is a weak-coupling result. When the coupling is strong enough that the emitter and the cavity exchange the excitation back and forth before it is lost, there is no emission rate to enhance — the system oscillates instead, and the description has to be replaced rather than corrected.

What the drawings leave out #

The pair figure draws two functions and calls one advanced, and nothing in the drawing distinguishes them. That is the point being made and it is also a limitation: the difference between the two panels is which direction time runs, and a static picture of a field profile has no time in it to run. A reader has to supply the direction, and supplying it is exactly the choice the essay is about.

The absorber figure draws a sum of four curves at one instant and hides that three of them are counterfactual — the half-advanced field and the absorbers’ response are not separately observable, only the total is. A picture that draws unobservable components as though they were on the same footing as the observable sum is doing what every decomposition does, and it should not be read as a claim that the pieces are there to be found.

And no figure here shows an absorber. The construction requires every outgoing ray to be intercepted eventually, by matter that is nowhere in any drawing, at times far outside any drawing’s range.

Where the ladder stands #

Four rungs stand on retardation. The first split the field of a charge into what stays and what leaves. The second found the three terms of an oscillating source and the radius where they change places. The third dropped the assumption that a source is heard all at once. This one drops the assumption underneath all three.

The habit worth carrying away is about where an asymmetry lives. When a set of equations is symmetric and the world is not, the asymmetry has been put in as a condition, and finding which one is the whole of the question. In thermodynamics it is the initial state; in electromagnetism it is the boundary at infinity; in both cases the equations are innocent and the answer is somewhere nobody was looking. The test generalises: any time an irreversible result is derived from reversible equations, a step has been taken that was not an equality.

What is left on this ladder is the piece the site has already written from a different direction. The self-force is where the classical theory breaks rather than merely being incomplete, and the absorber account above is one of the two things it was replaced by. The other is quantum electrodynamics, where the boundary condition returns as a choice of vacuum state and the question of this essay reappears wearing different notation.