Whether a charge on a table glows

A charge that turns must glow establishes Larmor’s formula: a charge with a proper acceleration radiates power proportional to the square of that acceleration. The floor that cannot be told from gravity establishes the equivalence principle: standing still in a gravitational field is locally indistinguishable from accelerating in empty space.

Put the two together and something is wrong. A charge sitting on a table has a proper acceleration of one gravity, so it should radiate — for ever, with nothing supplying the energy. And a charge dropped from the table is locally inertial, so it should not radiate, although it is accelerating relative to everything that stayed behind.

Both halves are uncomfortable, and the argument about them has run since the 1950s.

The size of the thing being argued about #

Before the physics, the arithmetic, because it decides what kind of question this is.

An electron held motionless at the Earth’s surface radiates about $5\times10^{-52}$ watts. That number is not small in the way laboratory numbers are small; it is small in a way that removes the question from the laboratory entirely.

Held for the age of the universe, the charge would have radiated $2\times10^{-34}$ joules. A single photon of green light carries $4\times10^{-19}$. So the total is not merely undetectable; it is nine orders of magnitude below the smallest amount of energy the electromagnetic field can carry away in one piece.

This is not an experimental question and never has been. It is a question about whether two well-tested pieces of theory are consistent with each other, which makes it a question about definitions — and, as it turns out, about one definition in particular.

What radiation is a property of #

The resolution begins by noticing that “does it radiate” has been treated as though it had an observer-independent answer, and it does not.

Radiation is defined by energy flowing away and not coming back: a flux through a distant surface, integrated over a long time. Both of those — “distant” and “long” — are statements about a region of spacetime rather than about a point, and different observers slice spacetime differently.

For a uniformly accelerated charge the calculation has been done, and the answer is not ambiguous. The full Larmor power flows out to an inertial observer far away. A detector riding with the charge, accelerating alongside it, registers nothing.

Where the radiation goes #

The reason the co-accelerating detector finds nothing is geometric, and it is the same geometry as the wall of silence behind a rocket.

An observer with a steady acceleration has a horizon: a surface behind them beyond which no signal ever reaches them, however long they wait. Its distance is $c^2/a$, which at one gravity is nearly a light-year.

The radiation from a uniformly accelerated charge goes into that region. It crosses the horizon and is gone, from the accelerated observer’s point of view, in the same sense that the interior of a black hole is gone. The observer riding with the charge therefore finds a static field and no radiation, and is not wrong — the radiation exists and is not in their universe.

That is the whole resolution of the first half. A charge held stationary in a gravitational field radiates as measured by a freely falling observer, and does not as measured by another observer standing on the ground beside it, and both statements are correct because the two observers are asking about different regions.

The energy question dissolves with it. The observer who sees no radiation has no energy to account for. The observer who sees radiation is falling freely, sees the charge accelerating away, and can attribute the energy to whatever is holding the charge up — the table, ultimately the ground, ultimately the source of the gravitational field.

The second half, and its detector #

The freely falling charge is the mirror image and its resolution is the same sentence read the other way.

A charge in free fall is locally inertial, so a detector falling with it finds nothing. A detector held stationary nearby is accelerating relative to the charge, and does register radiation.

That is not a contradiction unless one insists that radiation is a property of the source. It is not: it is a property of the pair — source and detector — or more carefully, of the source and the region over which the flux is evaluated.

Where an acceleration is large enough to matter #

It is worth marking the accelerations at which the same formula is not a philosophical matter, because they exist and they are large.

An electron in a storage ring is bent by a magnet — a force that does no work, as the force that does no work requires, and that accelerates all the same — on a circle of tens of metres at nearly light speed. Its proper acceleration is $3\times10^{21}$ metres per second squared — twenty orders of magnitude beyond a centrifuge — and it radiates so much that the machines are built around the radiation rather than around the beam. Every synchrotron light source in the world is Larmor’s formula used as an instrument.

That comparison settles a question the paradox tends to obscure: the formula itself is not in doubt. It is tested to high precision wherever the acceleration is large enough to give a measurable answer, which is every accelerator ever built and every antenna. What is in doubt is nothing about the formula and everything about how to apply the word “radiation” when the acceleration is supplied by gravity and the observer is embedded in the same field.

A law can be exactly right and its interpretation still be unsettled, and the gap between the two is usually widest where the effect is smallest — because that is where nothing forces the issue.

The same structure, one field over #

The quantum version of the same statement is the temperature of an acceleration, and it is worth putting beside this because it makes the classical result look less like a special pleading.

An accelerating detector in empty flat space registers a thermal bath. An inertial detector at the same event registers nothing. Nobody regards that as a paradox any more, because the calculation is unambiguous and its interpretation is agreed: what counts as a particle depends on how the field’s modes are split into positive and negative frequency, and that split depends on the observer’s notion of time.

The classical radiation problem has exactly that structure with the quantum removed. What counts as radiation depends on how the field is split into a bound part that travels with the charge and a free part that escapes, and that split depends on the observer.

The two effects share their geometry. The horizon at $c^2/a$ appears in both, the Unruh temperature is proportional to the same acceleration, and the observer who finds no radiation is the observer who finds a warm vacuum. That is not a coincidence: the thermal bath and the missing radiation are the same fact seen through different formalisms.

The bound field and the free field #

The technical statement behind “which part to call radiation” is worth setting out, because it is a distinction that recurs everywhere in electromagnetism.

The field of a moving charge splits into two pieces. One falls as the inverse square of the distance, carries the charge along with it, and represents no energy leaving — the velocity field, or bound field. The other falls as the inverse first power, is proportional to the acceleration, and carries energy away at a rate that does not fall off with distance because the flux through a sphere is the field squared times the area. That is the radiation field.

The split is unambiguous for a charge whose acceleration begins and ends. It is not unambiguous for one that has been accelerating for ever, because there is no era in which the field is purely bound to compare against, and because the two pieces are defined relative to a frame — a decomposition into $1/r$ and $1/r^2$ parts depends on which $r$, measured by whom.

For an observer in the charge’s own accelerated frame, the field of a uniformly accelerated charge is static. Not approximately static: exactly, and it has been known since Born that it looks in those coordinates like a Coulomb field with a distortion. A static field has no radiation in it by any definition.

The disagreement is therefore not about the field but about which coordinates the split is performed in, and the reason it took so long to say that clearly is that in every other application of the formula the choice is obvious and nobody had to make it explicit.

What is still argued about #

The account above is the majority position and is not universal, which is worth saying plainly.

The disagreements are about definitions rather than about calculations. Everybody agrees on the field of a uniformly accelerated charge — it was written down by Born in 1909. What is disputed is which part of it to call radiation, whether the radiation reaction force on a uniformly accelerated charge is zero (it is, since the Abraham–Lorentz force depends on the rate of change of acceleration), and how to reconcile a zero self-force with a non-zero radiated power.

That last one is a real tension and its usual resolution is a Schott term: an energy stored in the near field that is being exchanged with the radiation, and that vanishes over a complete cycle but not over a stretch of steady acceleration. It is bookkeeping rather than mystery, and it is bookkeeping several careful people have disagreed about.

A question that has stayed open for seventy years without a single experiment being proposed is a question about language. That is not a dismissal — the language has to be got right before the next problem can be stated — but it explains why the literature is philosophical in tone and why the numbers in the first figures are the ones that matter most.

What the argument was originally for #

The puzzle was not raised as a curiosity. It was raised because it appeared to threaten the equivalence principle itself, and following that thread shows what the resolution actually protects.

The threat runs like this. Suppose a supported charge radiates and a freely falling one does not. Then a laboratory could distinguish gravity from acceleration by watching a charge: put it on the bench and look for radiation. The equivalence principle would be false, and general relativity would be built on a foundation that electromagnetism contradicts.

The resolution removes the threat by denying its premise. Nobody in the laboratory can look for the radiation, because a detector in the laboratory is accelerating with the charge and sits inside the same horizon; the radiation is outside it. To see the radiation one must be freely falling, and a freely falling observer is one who describes the charge as accelerating — which is the case where radiation was expected all along.

No single observer ever finds the combination that would be a contradiction. That is the same shape as every other resolution in relativity: two accounts that would conflict if they could be compared, and a geometry that prevents the comparison being made. The pole and the barn, the twins, the mutual contraction — all of them are settled by noticing that the two observers are not talking about the same events.

The equivalence principle survives, and what has been given up is the idea that radiation is something a source does.

Where the model stops #

The acceleration is taken as uniform and eternal. A charge that has been accelerating for ever is an idealisation with pathological features, including a field that does not fall off properly. Every real charge started somewhere, and the transient from that beginning is part of what a distant observer eventually sees.

Gravity is treated through the equivalence principle alone. A real gravitational field is not a uniform acceleration: it has curvature, the equivalence holds only over a small region, and over the light-year in the horizon figure the Earth’s field is nothing like uniform. So the strict statement about a supported charge on Earth requires a calculation in the Schwarzschild geometry, which has been done and agrees.

The charge is a point. The self-energy of a point charge diverges, the Abraham–Lorentz force it produces has runaway solutions, and everything about radiation reaction inherits those problems — which the force a charge exerts on itself is about.

And the treatment is classical. In quantum electrodynamics the question becomes one about the response of a detector rather than about a field configuration, which is cleaner and which is why the Unruh formulation is the one most people now argue in.

And the equivalence principle is being used beyond its warrant. It is a statement about a region small enough that tidal effects are negligible, and the horizon in the second figure is a light-year away. So the sentence “a supported charge is equivalent to an accelerated one” is being applied over a region where the equivalence does not hold, which is why the honest version of the argument is done in a specific spacetime rather than by appeal to the principle. That the answers agree is a result rather than an assumption.

Nothing here concerns a charge in a real orbit. A charge orbiting a mass is not in uniform acceleration and does radiate, unambiguously, to everybody — the orbit is not a geodesic in the presence of its own field, and the radiation drains it. That is a different and much easier question, and it is the one that matters for any astrophysical situation with charges in it.

What the pictures cannot show #

Every figure here plots a scalar against an acceleration, and the content of the argument is about which region of spacetime a quantity is evaluated over. A diagram of the accelerated charge’s field with the horizon drawn on it would carry that, and it is a spacetime diagram rather than a graph — the drawing this essay most wants and does not have.

The budget figure compares energies and cannot show that the comparison is with a quantum rather than with a sensitivity. A detector a thousand times better would not help: the radiated total is below one photon, and there is no such thing as a fraction of one.

A third omission is time. All five figures are steady-state: a charge that has always been accelerating at a fixed rate, a horizon that has always been where it is. The interesting cases are the transients — a charge switched from inertial motion into acceleration, or released from a support into free fall — and those have a rich structure that the steady picture erases. A charge dropped from a table emits a burst as it changes state, and that burst is unambiguous and would be seen by everybody. It is also, for one electron under gravity, of order $10^{-52}$ joules.

Where the ladder goes next #

The radiating-charge ladder began with a charge that turns must glow, which is Larmor’s formula and the pattern it radiates into, and continued with the force a charge exerts on itself, where conservation demands a reaction force with unpleasant properties. This rung asks whether the formula survives contact with gravity, and finds that it does at the cost of radiation stopping being a property of the source. The rungs after it: the Schott energy and the complete accounting for a uniformly accelerated charge; the detector-based formulation, in which the question becomes what a particular apparatus registers; and radiation in a curved spacetime, where “energy flowing to infinity” needs an infinity to flow to and does not always have one.

The habit worth carrying away is that a quantity everybody uses may not be observer-independent. “Does it radiate” has no answer until somebody says who is asking, and the eighty years of argument are the time it took to notice that the question was missing a word.