The bill that arrives when the pushing stops

The argument about a charge sitting on a table ends with a loose thread. Whether a charge on a table glows settles who sees radiation from a uniformly accelerated charge, and in passing it names a tension it does not resolve: the radiation reaction force on such a charge is zero, and the power it radiates is not. The force a charge exerts on itself derived that force from the requirement that radiated energy be paid for, and the payment it found depends on how fast the acceleration is changing. A steady acceleration changes at no rate at all.

So for as long as a push holds steady, a charge sends energy away at the rate a charge that turns must glow computes, and not a single watt of it is taken from the charge’s motion. The energy has to come from somewhere. The answer is that it is lent — by the field that travels with the charge — and repaid later, by whatever is doing the pushing, at the moment the push changes. The loan has a name, the Schott term, after George Adolphus Schott, who wrote it down in 1912, and it has been quietly balancing every calculation of radiation reaction since.

One identity, three terms #

The whole of it fits on a line. For a charge of mass $m$ the reaction force is $m\tau\dot{\mathbf a}$, where $\tau$ is $6.27\times10^{-24}$ seconds for an electron — the time light takes to cross two-thirds of the classical electron radius. In the same units Larmor’s formula says the power radiated is $m\tau a^2$. The rate at which the reaction force takes energy from the motion is minus its dot product with the velocity. And the product rule supplies the rest:

$$m\tau a^2 = -\,m\tau\,\dot{\mathbf a}\cdot\mathbf v + \frac{d}{dt}\big(m\tau\,\mathbf a\cdot\mathbf v\big).$$

That is nothing more than $\frac{d}{dt}(\mathbf a\cdot\mathbf v) = \dot{\mathbf a}\cdot\mathbf v + a^2$, multiplied through by $m\tau$.

Read as accounting, the left side is what leaves for infinity. The first term on the right is what the reaction force takes out of the charge’s kinetic energy. The second is the rate of change of a quantity that depends on where the charge is in its motion rather than on anything it is losing: the product of its acceleration and its velocity, scaled by $m\tau$. That quantity is the Schott energy, up to a sign convention that varies from author to author. Radiated equals taken plus lent, at every instant.

Nothing in the identity is approximate, and none of it is new physics. It is calculus applied to Larmor’s formula and the Abraham–Lorentz force together. What it does is make “who pays for the radiation?” answerable moment by moment, and the answer turns out to depend entirely on the shape of the motion.

It also explains where the reaction force came from in the first place. The force a charge exerts on itself obtained it by requiring the work done over a whole cycle to equal the energy radiated over that cycle, and an integration by parts threw away a boundary term — this one — because over a whole cycle $\mathbf a\cdot\mathbf v$ returns to where it started. The derivation answered the question it asked. The Schott term is what it was entitled to discard, and it is the whole of the story whenever the motion is not a cycle.

A push that rises, holds and stops #

The cleanest case is a charge that starts at rest and is pushed with an acceleration that rises smoothly to a steady value, holds there, and falls smoothly back to zero. Measure time as a fraction of the push and power in units of $m\tau a_0^2$, where $a_0$ is the steady acceleration, and the drawing no longer depends on which charge it is or on the value of $\tau$.

The push has three stretches and each tells a different story.

While the acceleration rises, the charge has barely begun to move. The reaction force points along the growing acceleration — forward — and so it does a little work on the charge rather than taking any: 0.025 units, handed to the motion. The Schott balance grows slightly faster than the radiation, because the field is being loaded as well as drawn on.

While the push holds, $\dot{\mathbf a}$ is zero and the reaction force is zero — not small, but zero. The radiated power sits at one unit and the Schott rate sits exactly on top of it. The charge radiates at the full rate while its motion is untouched by the radiation, which is the situation whether a charge on a table glows left unexplained.

While the push stops, $\dot{\mathbf a}$ is large and negative and the charge is moving at its fastest. The reaction force now points backwards, against the motion, and takes energy quickly: its peak is several times the radiated power, and over the stopping stretch it takes 0.775 units. That is more than everything radiated over the whole push, 0.750, and the excess is precisely the 0.025 it gave away at the start.

The radiation of the steady stretch is paid for afterwards, when the push is withdrawn. And “paid” has a literal meaning. Whatever supplies the push — a field between two plates, a laser, a magnet — has to do more work while it reduces its force than the charge’s kinetic energy alone would require, because in that interval the charge is being braked by its own field. The bill is presented to the agent, and it is presented late.

The shape of the ramps changes how the bill is spread out and not what it comes to. Over the stopping stretch the Schott balance falls from $m\tau a_0 v$ to zero, whatever the ramp, so the reaction force must take the energy radiated during the stop plus that whole balance. A faster stop means a taller spike for a shorter time and the same total.

The running total, and the moment it closes #

Accumulated, the loan becomes an area. Radiated energy climbs from the moment the push begins. The reaction force’s total dips very slightly while the push starts and then stays flat through the whole steady stretch. The gap between the two curves is, at each instant, the value of $m\tau\,\mathbf a\cdot\mathbf v$ — which is the check the running totals are held to at every step. When the push begins to stop, the gap is at its largest, 0.70 units, and that number is nothing but the steady acceleration, one, times the velocity the charge has reached by then, 0.70.

It then closes in the time the push takes to stop, and it closes completely, because a charge whose acceleration has returned to zero has $\mathbf a\cdot\mathbf v = 0$ whatever its velocity. The account balances between any two moments at which a·v has the same value, and in general between no others. A push still on at the moment the books are inspected has an open balance, and there is no inconsistency in that; the balance is a statement that the field has not yet been settled with.

For a real electron the amounts are absurd, as every number in this subject is at laboratory accelerations. Held in a field of 100 kilovolts per metre for a nanosecond, an electron accelerates at $1.8\times10^{16}$ m/s², gains $9.0\times10^{-17}$ joules — about 560 electronvolts — and radiates $1.3\times10^{-30}$ joules, fourteen orders of magnitude less. The accounting is not interesting because the energy is large. It is interesting because conservation has to hold to every decimal place, and without the third term it would fail, for as long as the push held steady, by exactly the amount radiated.

Where the balance is held #

The Schott term belongs to the charge’s field rather than to its motion, and it is worth being clear about what that claim says and what it does not.

The field of an accelerating charge splits into two parts. One falls off as $1/r^2$, travels with the charge, and carries nothing away; the other falls off as $1/r$, is proportional to the acceleration, and carries energy to infinity. The energy density of the field is quadratic in the total, so it contains a cross term between the two parts, proportional near the charge to $\mathbf a\cdot\mathbf v$ — falling off too fast to reach infinity and too slowly to be negligible nearby. When a charge accelerates along its direction of motion that cross term is being drawn down, and the energy it gives up is what leaves as radiation during a steady push. When the acceleration returns to zero the reaction force restores it. That is the usual reading: the Schott energy is energy in the near field, lent to the radiation and replaced from the motion.

Where the energy of a field actually is establishes the principle this reading rests on — that the energy of a field is somewhere, with a density, and not merely a number attached to a configuration. The Schott term extends that from static fields to the neighbourhood of a moving charge, and it inherits a difficulty on the way. The Coulomb self-energy of a point charge is infinite, and the finite Schott term has to be separated from an infinite background by a subtraction that is itself a choice. That is why careful people disagree about whether the Schott energy is located in the field or is an entry that happens to have the right value, and the disagreement goes back to the renormalisation of the electron’s mass on which the force a charge exerts on itself is built.

What is not in dispute is the identity. Everyone writes it down; the argument is only about which of its terms has earned the word energy.

An oscillator that pays in the middle of its swing #

Everything so far has been a single push. A charge oscillating along a line is the case that matters for every antenna ever built, and it is where the Schott term shows what it is for.

For $x = x_0\cos\omega t$ the radiated power goes as $\cos^2\omega t$ and peaks at the turning points, where the charge is momentarily at rest and its acceleration is largest. The reaction force’s take goes as $\sin^2\omega t$ and peaks at the centre of the swing, where the charge is moving fastest and radiating nothing. The two are a quarter of a cycle apart. The Schott rate, $\cos 2\omega t$, is the difference between them: energy is taken from the motion at the centre, lent to the field, and sent away at the ends.

Averaged over a whole swing, the radiated power and the reaction force’s take are equal and the Schott term contributes nothing. That is why the reaction force could be derived by averaging, as it historically was, and why no calculation of an antenna’s radiation resistance ever mentions Schott: a steady oscillation settles its account twice in every period.

The shape of that argument is familiar from alternating-current circuits, and the connection is exact rather than decorative. A capacitor takes energy in during one quarter of a cycle and returns it in the next; its power averages to zero and its instantaneous power does not, and engineers call that reactive power and keep it separate from the real power a resistor dissipates. The distance where a field changes its mind shows the same division around a small antenna, where the near field stores energy that sloshes back and forth every cycle and shows up in the antenna’s impedance as reactance, while the radiation field carries power away and shows up as resistance. The Schott term is the reactive power of a single charge. A lone electron has an impedance, and this is its imaginary part.

The quarter cycle between taking and sending is also the quarter cycle the frequency that gets an answer finds between a driving force and the displacement it produces at resonance. In both, the energy that averages away is the energy stored, and the energy that survives the average is the energy that leaves.

A circle has nothing to borrow #

Between a charge oscillating on a line and a charge going round a circle lies every ellipse, and following the Schott term through them shows which of those two cases is the special one.

A charge held by a spring-like force towards a centre moves on an ellipse. On a circle its velocity and acceleration are at right angles at every moment, so $\mathbf a\cdot\mathbf v$ is zero and the Schott term does not merely average away — it never appears. Every watt the charge radiates is taken from its motion at the moment it is radiated. As the orbit flattens, velocity and acceleration acquire components along one another and the Schott term starts to swing, twice per orbit, with an amplitude that grows until the ellipse degenerates into a line and the swing reaches twice the mean radiated power. That limit is the oscillator above.

The circle is the case that the largest radiating machines on Earth are built around. An electron in a storage ring moves on a circle at nearly the speed of light and radiates so much that the machine exists for the radiation, and every joule of it is honest drag, replaced turn by turn by radio-frequency cavities pushing along the orbit. The bending magnets, as the force that does no work explains, cannot supply any of that energy, and the Schott term cannot lend any either. The classical collapse of a hydrogen atom in sixteen picoseconds, computed in a charge that turns must glow, is the same case: a circular orbit, spiralling in because every emitted joule is subtracted from the motion as it goes.

A linear accelerator is the opposite extreme. There the velocity and acceleration are parallel, the Schott term is as large as it can be, and during the steady part of each accelerating section the reaction force is zero. For a single electron at any gradient anybody can build, neither effect is measurable; the distinction between the ring and the line is nevertheless exact, and it is what the classical picture of where a storage ring’s energy goes rests on.

At any speed, for as long as it lasts #

The push drawn at the start was slow enough for Newtonian bookkeeping. The steady stretch is the part worth testing at high speed, because a steady acceleration held long enough makes any speed at all.

The relativistic reaction force, completed by Dirac in 1938, has two terms. One is proportional to the rate of change of the four-acceleration — the Schott term — and the other to the square of the proper acceleration, directed along the four-velocity, which is the four-momentum carried away as radiation. For motion at constant proper acceleration, the hyperbolic worldline of the ship that never arrives at c, the two cancel identically. The figure evaluates both along the worldline and finds their difference zero to the precision of the differencing, at every speed up to 0.995 of light’s.

So the result for a steady push is not a low-speed accident. A charge held at constant proper acceleration feels no radiation reaction at any speed. The external force supplies its kinetic energy and nothing else, and the energy it radiates — rising as the hyperbolic sine of its rapidity, measured in the frame it started from — is matched exactly by a Schott energy falling as the negative of the same function.

That falling curve has no floor, and that is informative. An acceleration held for ever drives the Schott energy negative without limit, which is another sign that the eternally accelerated charge is an idealisation with pathological features, reached in whether a charge on a table glows by a different road. Every real charge began accelerating at some moment and will stop, and the account for a real charge closes when it does. The connection to the observer-dependence argued there is direct. The radiation of a uniformly accelerated charge goes into the region beyond the horizon of the wall of silence behind a rocket, and an observer riding with the charge sees neither the radiation nor the debt; both lie on the far side of a surface that observer never sees past.

The numbers stay what they always are. For an electron held at one gravity, the radiated energy is $4.4\times10^{-31}$ of its kinetic energy by the time it reaches three-quarters of the speed of light. The ledger is exact to all orders in the speed and unmeasurable at every one of them.

Where the ledger can be trusted, and where it cannot #

The charge is a point, and its mass has been renormalised. The reaction force comes from subtracting an infinite self-energy, and the Abraham–Lorentz equation that results has runaway solutions and preacceleration. The pushes drawn here sidestep both by prescribing the motion and asking what the forces must be, rather than solving for a motion; the identity holds for any prescribed trajectory. The reduced-order form due to Landau and Lifshitz, which removes the runaways, gives the same ledger to the order in $\tau$ at which the classical theory means anything.

Changes faster than τ mean nothing. The ramps drawn here each last a fifth of the push. Nothing in the argument survives a ramp lasting a few times $10^{-24}$ seconds, by which point a classical point electron has stopped being a description of anything and quantum electrodynamics is required.

Only the last figure is relativistic. The worked electron ends its push at 0.047 of the speed of light, where Newtonian bookkeeping is good to a fraction of a per cent. The result for a steady push survives relativity exactly, as the last figure shows; the ramp and the oscillator acquire corrections that change their numbers and not their shape.

The meaning of the Schott term is contested. That it is near-field energy is the majority reading, supported by explicit calculations of the field energy for particular motions. That it is an accounting entry with no location is held by others. The figures show the identity and cannot choose between the readings.

Radiation is defined in one frame. Every energy here is measured in the inertial frame the charge started from, far from the charge — the observer for whom Larmor’s formula is written. For an observer accelerating with the charge, as the argument about the table showed, the question of how much is radiated does not have the same answer, and neither does the question of how much is owed.

What a graph of powers leaves out #

Every figure here plots energy against time, and the unsettled part of the subject is about space: where the balance is held, how far from the charge, and in which part of the field. A map of the cross term between the bound and radiated fields around a charge in the middle of its steady stretch would show whether the balance sits within a few multiples of $c\tau$ of the charge or spreads through the region the radiation has not yet reached. That is a map, not a curve, and none of the figures is one.

Nor do the figures show the agent doing the pushing. The extra work done while a push is withdrawn is real work by a real field — a pair of charged plates discharging slightly differently from the way they would with no radiating charge between them — and the ledger places the work correctly without showing where it goes. And every amount drawn in physical units is so small that no picture of it could suggest a measurement. The accounting is exact, and there is nothing to weigh.

Still open: whether the balance has an address #

The identity is more than a century old and its terms are not in doubt. What remains open is whether the Schott energy is localised — energy with a density at particular places in the field — or an entry that balances the account without being anywhere. Calculations for hyperbolic motion place it in the field on the far side of the co-moving horizon, which is suggestive and specific to that motion. For a general trajectory there is no agreed division of the field energy into a bound part, a radiated part and a Schott part, because the split between bound and radiated fields depends on the observer and the point charge’s infinite self-energy has to be removed before anything can be divided.

The question sharpens in two directions. One is the detector: an apparatus that absorbs energy from the field near an accelerating charge either registers the Schott balance or it does not, and the answer is a fact about what that apparatus couples to rather than about the field. The other is gravity. In a curved spacetime “radiated” means “reaching infinity”, and a spacetime without the right kind of infinity has no clean way to separate what leaves from what is lent.

The habit worth carrying away is a way of reading apparent failures of conservation. When energy seems to appear from nowhere during a steady process, look for a quantity that changes only when the process changes. The Schott term is invisible to every average and every steady state, which is exactly why a derivation made over a cycle could discard it — and why a charge pushed steadily can seem to radiate for free.