The charge that passes as a flash of light

The field nobody can transform away establishes two numbers that every observer agrees on for any electromagnetic field: $E^2 - c^2B^2$ and $\mathbf E\cdot\mathbf B$. For light both are zero. For the field of a charge at rest, the first is positive and the second zero, and because they are invariant, no change of frame can turn a Coulomb field into light. That is a theorem, and it holds at every speed.

It is also, in a sense, beside the point. A charge going past an atom at nearly the speed of light delivers a field that the atom cannot distinguish from a pulse of light: thin, perpendicular to the line of motion, with its electric and magnetic parts equal in strength and at right angles. Enrico Fermi saw this in 1924 and used it to treat the collisions of fast charged particles with atoms as though the atoms were being illuminated. Carl Friedrich von Weizsäcker and Evan James Williams made the idea exact ten years later, and it has since become the method by which particle colliders are used as photon colliders. This essay follows the field of a passing charge until it becomes light in everything but name.

What a passing charge delivers #

Sit at a point a distance $b$ from the straight path of a charge moving at speed $v$, and record the field as it goes by. The field of a uniformly moving charge is the Coulomb field of its rest frame, transformed — which magnetism is electricity seen sideways and the field that points where the charge is now derive — and at the observer’s point it has three parts: a component across the path, a component along it, and a magnetic field circling the path.

At low speed the transverse field rises and falls over a time of order $b/v$, a broad bump barely distinguishable from the static field the charge would have at its closest approach. As the speed rises the bump becomes a spike. At closest approach the transverse field is $\gamma$ times the Coulomb field at rest, because the moving charge’s field is compressed towards the plane perpendicular to its motion; and it lasts about $1.53/\gamma$ times $b/c$, because the observer is inside that compressed region for only that long. At $\gamma = 30$ the spike is thirty times the Coulomb field and a fiftieth of the light-crossing time $b/c$ long.

The two changes cancel in the one quantity a slow observer most cares about: the impulse delivered across the path, which is the area under the curve. It is $2q^2/4\pi\varepsilon_0 bv$ at every speed, to the precision of the integration — the same result collisions are easier than forces would give from the momentum transferred, with no reference to how the field is shaped in time. What the speed changes is not how much the passing charge delivers but how quickly: a slow charge gives a gentle push, a fast one a hammer blow of the same total.

Three components and one object #

The three parts of the passing field are not independent. Six numbers, one object shows that the electric and magnetic fields are the six components of a single tensor, and a boost along the direction of motion leaves the component along the motion unchanged while it multiplies the transverse electric field by $\gamma$ and creates a magnetic field of $\beta$ times it. In the charge’s own frame there is only the Coulomb field, spherically symmetric; everything in the drawing above is that one field seen from a frame in which the charge moves.

That is why the pulse can be predicted so completely. Nothing about how the charge got its speed, or what it is, enters: a proton, a lead nucleus and an electron at the same Lorentz factor deliver pulses of the same shape, differing only in their charge. The magnetic part of the pulse exerts no force on a stationary charge in the target, but it is always there, and it is what makes the pulse look like radiation to anything sensitive to both fields at once — a magnetic moment, a moving electron, or a detector that measures the flow of energy rather than the field itself.

How close the field comes to being light #

Light has two defining properties as a field: its electric and magnetic parts are equal and perpendicular, so $E^2 - c^2B^2$ is zero; and it has no field component along its direction of travel. The passing field approaches both. Its magnetic field is $\beta$ times its transverse electric field, so the invariant, integrated over the passage, falls as $1/\gamma^2$ — to a fifteenth of a per cent at $\gamma = 10$ and less than a part in a million at $\gamma = 1000$. The longitudinal field, the one component light does not have, carries a share of the energy that falls the same way.

The theorem that a Coulomb field can never become light survives, and becomes irrelevant. The invariant is positive in every frame, but by $\gamma = 1000$ it is a part in a million of what any measurement of the field would register. An atom, a nucleus or a detector struck by the pulse responds to it as it would to a pulse of electromagnetic radiation with the same shape — a thin sheet of crossed, equal fields travelling past at the speed of light — and cannot tell the difference.

The field squeezed into a pancake #

The same compression can be seen at a single instant rather than at a single place.

At rest a charge’s field is the same in every direction. In motion it is γ times stronger across the direction of motion and $\gamma^2$ times weaker along it, and the energy crowds into a thin disc: half of it within 0.61 degrees of the transverse plane at $\gamma = 30$. The disc travels with the charge, at nearly the speed of light, and a sheet of field travelling at the speed of light with its electric and magnetic parts crossed and equal is what a plane wave is. The only difference from a pulse of radiation is that this one is tied to the charge that carries it, and will never leave it.

The same compression, in the charge’s radiation rather than its bound field, is what the sky that crowds into a cone finds for the light a fast source emits, and what the flash a circling charge sends once a turn turns into synchrotron radiation. A fast charge compresses everything it carries into the forward direction or the transverse plane, and the factor is always a power of $\gamma$.

The colours in a passing flash #

A pulse has a spectrum, and the spectrum of this one is where Fermi’s idea begins.

The spectrum of the transverse pulse is flat — the same energy at every frequency — up to a frequency of about $\gamma c/b$, and falls away exponentially above it. That is the familiar trade of sharpness has to be paid for: a pulse $\gamma$ times shorter contains frequencies $\gamma$ times higher. At $\gamma = 30$ the flat part reaches past fifty times $c/b$; for a charge passing an atom at a distance of an ångström, that is well into the X-ray band.

A flat spectrum of radiation is what a beam of photons with an energy distribution going as one over the energy looks like. Divide the energy at each frequency by the energy of one photon of that frequency, $\hbar\omega$, and the passing field becomes a count of photons, many at low energy and fewer at high, cut off near $\hbar\gamma c/b$. That is the method of equivalent quanta: the effect of a fast charge on anything it passes equals the effect of that swarm of photons. An electron passing an atom ionises it as a spectrum of photons would; a nucleus passing another nucleus excites it as a gamma-ray beam would.

The photons a passing nucleus carries #

The number of equivalent photons goes as the square of the passing charge, and that makes heavy nuclei extraordinary sources.

A proton at the Large Hadron Collider carries a thin cloud of equivalent photons reaching up to about two teraelectronvolts, a few hundredths of a photon per logarithmic interval of energy. A lead nucleus, with 82 times the charge, carries $82^2 = 6{,}724$ times as many below its cut-off — about a hundred per logarithmic interval at a gigaelectronvolt — though its cut-off is lower, because its larger size keeps any other nucleus at least fourteen femtometres away and its Lorentz factor per nucleon is smaller.

When two lead nuclei pass within a few nuclear radii of each other without touching — an ultraperipheral collision — their photon clouds pass through each other, and occasionally two photons collide. The one medium that was supposed to add exactly describes the measurement this made possible: the scattering of light by light, predicted in 1936 and observed by the ATLAS and CMS experiments in 2017, using exactly these photons. Photon–photon collisions have also produced pairs of W bosons and pairs of muons, with rates the equivalent-photon counting predicts.

Nuclei broken by a passing flash #

The photons carried by a lead nucleus are energetic enough to do nuclear physics. A photon of ten to twenty megaelectronvolts, absorbed by another lead nucleus, sets its protons oscillating against its neutrons in the giant dipole resonance, and the excited nucleus shakes off one or two neutrons. When two lead beams cross at the LHC, this happens far more often than the nuclei actually collide: the cross-section for one nucleus to be broken up electromagnetically by the other’s photon cloud is around two hundred barns, against about eight barns for the nuclei to touch. Most of the lead nuclei the collider loses from its beams are lost to photons, not to collisions.

The effect is so predictable that it is used as a measuring stick. Detectors placed at zero degrees, far along the beam line, count the neutrons emitted when both nuclei are broken up at once, and the rate, computed from the equivalent-photon spectrum and measured photonuclear cross-sections, calibrates how many collisions the machine is delivering. At lower energies the same physics is called Coulomb excitation: a heavy ion passing a target nucleus too far away to touch it lifts it into excited states with the photons in its field, and the gamma rays that follow are used to measure the shapes of nuclei.

The history runs the other way. Fermi’s paper of 1924 was about something humbler: the energy an alpha particle loses when it passes atoms, which he computed by asking how strongly each atom absorbs light at the frequencies in the passing field. The method was an approximation for a slow alpha particle, and it becomes more accurate the faster the particle, because the faster the particle the more nearly its field is light.

The rise a fast particle’s trail cannot keep #

The same counting explains a detail of how fast charged particles lose energy in matter, and it carries the idea to its limit.

A fast charged particle passing through a gas ionises atoms near its path, and the energy it loses per unit length first falls as its speed rises and then, past a minimum near $\gamma \approx 3$, rises again slowly, as the logarithm of $\gamma$. The rise is the pancake. A faster particle’s field reaches, at the frequencies an atom can absorb, to a distance of about $\gamma c/\omega$, so atoms farther and farther from its path are struck by a pulse short enough to ionise them. The number of atoms within reach grows, and the energy lost grows with it.

In a dense medium the rise stops. The atoms near the path are polarised by the pulse, and their polarisation screens its field at large distances — the effect Fermi identified in 1940 — so the reach no longer grows with $\gamma$ and the energy loss levels off at a plateau. The relativistic rise in a gas and its absence in a solid are used in particle detectors to tell particles of the same momentum but different mass apart: a pion and a kaon of the same momentum have different $\gamma$, and in a gas they leave trails of measurably different density.

Where the passing field stops being light #

Straight-line motion. The charge is taken to move uniformly, undeflected by what it passes. A charge that is deflected radiates real photons of its own — bremsstrahlung — and the equivalent-photon picture then describes only the part of the interaction in which the charge is not disturbed.

Classical fields. The photon counting treats the field’s energy at each frequency as a number of quanta, which is valid when many photons are involved or when only the probability of absorbing one is wanted. The spectrum is exact; the interpretation as a photon beam is a convenient equivalence, not a claim that the photons are there before anything absorbs them.

A minimum distance. The photon numbers depend on the distance inside which the calculation is cut off — the nuclear radius, or the distance at which the target’s own structure matters — and they grow only logarithmically as it shrinks. The figures state the cut-offs used.

Photons small compared with the charge’s energy. The counting assumes each equivalent photon takes a small share of the passing charge’s energy, so the charge is not slowed by the photons it gives up. That holds far below the cut-off for heavy nuclei; for an electron, which can give most of its energy to one photon, the spectrum near the top needs the corrections of a full quantum calculation, and the equivalent-photon method is used there only as a first approximation.

Vacuum. In a medium, the passing field polarises the matter it crosses, and the pulse it delivers is screened at large distances; that is the density effect above, and it changes the spectrum at low frequencies.

The flattened sheet Fermi pictured #

Every figure is a field at a point or a spectrum at a distance, and none draws the thing Fermi pictured: a sheet of crossed electric and magnetic field lines travelling with the charge, flattened into a disc, sweeping past a stationary atom. Nor do the figures show what the atom does when the sheet arrives — the absorption of one equivalent photon, the ejection of an electron, the excitation of a nucleus — which is the process the counting is for. And the photon figure plots numbers of photons per logarithmic interval without showing the other half of an ultraperipheral collision: two such clouds, crossing, and the rare pair of photons that meet inside them.

Still open: what the photon clouds can reach #

Photon–photon collisions at the LHC reach energies of hundreds of gigaelectronvolts and are clean in a way proton collisions are not: two photons carry no debris of quarks and gluons, so what they make can be seen against an almost empty background. How far they can be pushed to look for particles that couple to light but hardly to anything else — axion-like particles, for which the light-by-light scattering measurements already set limits — depends on luminosities and detector coverage for very forward particles that are still being extended. Whether the photon clouds of heavy nuclei will find anything the proton collisions have missed is open.

The habit worth carrying away is to ask what an observer actually receives, not what the source is. A theorem about what a field can be transformed into says nothing about how closely it can resemble something else, and a field that differs from light by a part in a million behaves, to anything that meets it, as light. Fermi’s move was to stop asking what a passing charge is and to count what it delivers — and what it delivers is photons.