The speed that stops while the energy keeps coming

The push that does not point where the body goes found that Newton’s second law survives relativity only in the form $F = dp/dt$: a steady force keeps increasing a body’s momentum without limit, while its speed approaches but never reaches the speed of light. Speeds that refuse to add found the same limit from another direction, in the arithmetic of combining velocities. Both are statements of a theory. This essay is about the experiment that put the two halves of the statement — speed limited, energy not — into one measurement, and why the second half was the part that needed doing.

Speed against energy #

An electron accelerated through a potential difference of $V$ volts gains a kinetic energy of $V$ electronvolts, whatever theory is used to describe it: that is just energy conservation, and the work done by the field. Newton’s mechanics says that kinetic energy is $\tfrac12 mv^2$, so the speed should grow as the square root of the energy without limit. Relativity says the kinetic energy is $(\gamma - 1)mc^2$, with $\gamma = 1/\sqrt{1-v^2/c^2}$, so the speed approaches $c$ and stops.

For an electron, whose rest energy is half a megaelectronvolt, the two predictions part company almost at once. Newton’s speed reaches $c$ at a quarter of a megaelectronvolt and goes on to seven and a half times $c$ at fifteen. Relativity’s speed is already 86 per cent of $c$ at half a megaelectronvolt and changes hardly at all after that: between 1.5 and 15 megaelectronvolts the energy grows tenfold and the speed by three and a half per cent.

By 1964 nobody seriously doubted which was right. Walter Kaufmann had measured the deflection of fast electrons from radium in electric and magnetic fields from 1901, and Alfred Bucherer had done it more carefully in 1908, both finding that the ratio of charge to mass seemed to fall with speed in the way relativity’s momentum required. Every particle accelerator built since the 1930s had been designed on relativistic dynamics and worked. What Bertozzi set out to do, at the Massachusetts Institute of Technology, was a direct demonstration for teaching — filmed as The Ultimate Speed — in which the speed and the energy were both measured, independently, on the same electrons.

Where Newton’s formula comes from #

Newton’s kinetic energy is not wrong so much as incomplete. Expand relativity’s $(\gamma - 1)mc^2$ for small speeds and the first term is $\tfrac12 mv^2$; the next is $\tfrac38 mv^4/c^2$, which is a hundredth of the first when the speed is about a sixth of $c$. For anything a person can throw, drive or launch into orbit, the correction is a part in a thousand million or less, and Newton’s formula is exact for all practical purposes. For an electron it fails long before the electron is fast by any everyday standard: a television tube’s electrons, accelerated through twenty-five thousand volts, already move at three-tenths of the speed of light and carry eight per cent more energy than $\tfrac12 mv^2$ says.

The failure is not gradual in any interesting sense. The two formulas agree to within two per cent up to about $0.15c$ and disagree by a factor of two and a half at $0.86c$; past that, Newton’s speed for a given energy runs off past $c$ while relativity’s barely moves. The electron is the natural test particle precisely because its rest energy is so small that ordinary laboratory voltages carry it across the whole range, from where the formulas agree to where they disagree by factors of ten.

A clock inside every electron #

The $\gamma$ that relates an electron’s energy to its rest energy is the same $\gamma$ by which a moving clock runs slow. An electron of 15 megaelectronvolts has $\gamma = 30$: its energy is thirty times its rest energy, and any clock it carried would run thirty times slower than the laboratory’s. Energy, in relativity, is rest energy multiplied by the rate at which the body’s own time is being stretched. That is a strange sentence and an exact one, and it means that Bertozzi’s calorimeter was, in effect, measuring how slowly the electrons’ clocks were ticking, while his oscilloscope measured how fast they were going.

Unstable particles make the connection visible. A pion lives 26 nanoseconds at rest; in a beam of pions at a few gigaelectronvolts, with $\gamma$ of a few tens, it travels hundreds of metres before it decays, and the distance is the energy, not the speed, since the speed is $c$ to within a part in a thousand in both cases. A beam line designer choosing how far a pion beam will survive reads the answer off the energy alone.

A race over eight and a half metres #

Bertozzi’s electrons came in short pulses from a Van de Graaff generator, at up to 1.5 megaelectronvolts, and for the higher energies from a linear accelerator that could take them on to 15. A pulse passed an electrode at the start of an 8.4-metre flight path, starting the trace on an oscilloscope, and arrived at an aluminium disc at the end, marking its arrival. Light takes 28.0 nanoseconds to cover that distance.

At half a megaelectronvolt the electrons took about 32 nanoseconds, slower than light by a seventh. At 1.5 megaelectronvolts they took 29, and at 15 they took 28 — indistinguishable from light’s time with the oscilloscope available, as relativity predicts, since the difference is three hundredths of a nanosecond. Newton’s prediction at 15 megaelectronvolts was 3.7 nanoseconds. A factor of ten increase in energy had bought a nanosecond, and the times had stopped falling.

The speeds alone, though, leave an escape route that a determined sceptic could take. Perhaps the accelerator simply failed to give the electrons the energy it was supposed to. Perhaps, above some speed, the field stops being able to push on a charge — Newton’s mechanics would then survive, with a cap on what any field can do. Nothing in a time of flight distinguishes “the electron has fifteen megaelectronvolts and travels at $0.9995c$” from “the electron could only be given a quarter of a megaelectronvolt and travels at $c$”.

The heat in the target #

That second possibility is not as silly as it sounds. Every accelerator works by pushing on a charge with an electric field, and the force a field exerts on a charge, $qE$, does not mention the charge’s speed. If the push simply stopped transferring energy beyond some speed, a Newtonian electron would also be stuck below $c$, for a different reason. The two pictures make opposite predictions only about where the energy goes, and the time of flight cannot see energy.

So Bertozzi measured the energy too. The aluminium disc at the end of the flight path was a calorimeter: the electrons stopped in it, and their energy went into heating it. Knowing the number of electrons from the charge they delivered, and the heat from the disc’s temperature rise, gave the energy per electron. At 1.5 and 4.5 megaelectronvolts it matched the energy the accelerators had supplied, within the precision of a teaching experiment of about ten per cent.

The energy arrived. And the speed at which it arrived, put into Newton’s $\tfrac12 mv^2$, accounted for a quarter of a megaelectronvolt at most — at 4.5 megaelectronvolts, a nineteenth of what the disc measured. The discrepancy is not a correction to Newton; it is a different relation. Energy is not, at high speed, tied to speed at all in any useful way. It is tied to $\gamma$, which goes on growing when the speed has nowhere left to go.

That is the half of the experiment the slogan leaves out. “Nothing can go faster than light” sounds like a limit on how much can be done to a particle. The heat says the opposite: there is no limit on how much energy, or momentum, a particle can be given. There is only a limit on how fast it goes while carrying it, and the relation between the two quantities, which seemed the most obvious thing in mechanics, turns out to be a low-speed approximation that fails before a quarter of a megaelectronvolt.

Kaufmann’s and Bucherer’s experiments had pointed at the same thing in the language of their day, a “mass” that grew with speed. Kaufmann’s measurements of 1906 seemed at first to favour a rival theory, Max Abraham’s rigid spherical electron, over the Lorentz–Einstein formula, and for a few years the question was argued from data of disputed accuracy; Bucherer’s measurements of 1908, and better ones in the following decade, settled it for the Lorentz–Einstein form. The language of a speed-dependent mass has since been largely abandoned, because the only mass that does not depend on the observer is the rest mass, and what grows is the energy and the momentum. Bertozzi’s experiment measures those directly, which is one reason it has outlived the arguments that preceded it.

What each nine costs #

Close to the speed of light the shortfall $1 - v/c$ is about $1/2\gamma^2$, and $\gamma$ is proportional to the energy, so the shortfall falls as the inverse square of the energy. Each additional nine in $v/c$ — each tenfold reduction in the shortfall — therefore costs a factor of $\sqrt{10}$, about 3.16, in energy. An electron reaches $0.9c$ at 0.66 megaelectronvolts, $0.99c$ at 3.1, $0.999c$ at 10.9 and six nines at 361. A proton, 1,836 times heavier, needs 1,836 times as much energy for each: 20 gigaelectronvolts for three nines.

The figure shows why talking about speed becomes pointless at accelerator energies. The axis of speed is crushed against $c$; the axis of energy runs on for decades. Mass is a form of energy found that a body’s rest energy $mc^2$ sets its scale; the figure is the same statement in dynamic form, since $\gamma$, the energy in units of the rest energy, is what the nines measure.

The end of the line is light #

Follow the curve far enough and the electron behaves almost exactly like light. At fifteen megaelectronvolts its energy is thirty times its rest energy, its momentum is its energy over $c$ to within a part in two thousand, and its speed is $c$ to within five parts in ten thousand. Its rest mass has become a small correction to everything about it. A photon with a momentum is the limit of the same relation, a particle with energy and momentum and no rest mass at all, whose speed is $c$ exactly whatever energy it carries; there the decoupling of speed from energy is complete, and the energy of a photon is told only by its frequency.

The relation $E^2 = (pc)^2 + (mc^2)^2$ holds the whole family together. A slow body’s energy is nearly all rest energy and its momentum is $mv$; a fast one’s is nearly all momentum times $c$; in between, the electron of Bertozzi’s experiment travels the whole way in a few megaelectronvolts. No sharp transition separates “particles” from “light” in this description. There is one curve, and the rest mass says where on it a given energy puts the particle.

The race that is no longer a race #

At the energies of the largest colliders the speeds are so close to $c$ that a race against light is lost by millimetres over the distance light travels in a second. The electrons of CERN’s Large Electron–Positron collider, at 104.5 gigaelectronvolts in its final year, fell short of $c$ by $1.2 \times 10^{-11}$ — 3.6 millimetres in three hundred thousand kilometres. The Large Hadron Collider’s protons, at 6.8 teraelectronvolts and a Lorentz factor of 7,250, fall 2.85 metres behind in a second.

What the accelerator designers worry about at those energies is not speed at all. The time a particle takes to go round a 27-kilometre ring is the circumference over $c$ to a part in a hundred million, the same at injection as at full energy, so the radio-frequency system that pushes the particles runs at a nearly constant frequency — something impossible in the early cyclotrons, where the particles’ rising speed mattered. What grows is the momentum, which sets how strong the bending magnets must be, and for electrons the radiation they emit going round, which the flash a circling charge sends once a turn found grows as the fourth power of the energy and which capped LEP’s energy at a little over a hundred gigaelectronvolts.

The same shortfall, measured over cosmic distances, becomes a test of physics at energies no accelerator reaches. The photons that raced for seven billion years arrived together from a gamma-ray burst and showed that light of different energies travels at the same speed to a precision no laboratory could match. For massive particles the shortfall is a measurement of mass: neutrinos from the supernova of 1987, travelling for 168,000 years, arrived within a few seconds of one another, which caps the neutrino’s mass, since a heavier neutrino of a given energy would have fallen measurably behind.

Where the electron’s energy goes, and where the measurement stops #

Bertozzi’s calorimeter was crude by modern standards: a thermocouple on an aluminium disc, a beam current measured to a few per cent, an energy balance good to about ten. Electrons stopping in aluminium also make X-rays, which can escape the disc and carry energy away, and the backscattered electrons that bounce out of the surface take some too; both were estimated and small at the energies used. The flight times were limited by the oscilloscope’s speed and the length of the pulses, which is why above a few megaelectronvolts the experiment could only say “about 28 nanoseconds”.

None of that matters to the argument, because the argument rests on the gap between two curves that differ by factors of ten. The calorimeter needed only to show that the energy was arriving in proportion to the voltage; the clock needed only to show that the speed was not. Modern measurements of the same relation are made every day in every accelerator and particle detector, which measure momentum by bending in a magnetic field, velocity by time of flight or by the angle of Cherenkov light, and energy by total absorption in calorimeters, and identify particles from the mismatch between them; the light a charge makes by changing medium found such methods telling electrons from pions at a few gigaelectronvolts.

The relation is also checked every year in teaching laboratories with apparatus far simpler than Bertozzi’s. Electrons from a strontium-90 source, emitted with a spread of energies up to a couple of megaelectronvolts, are sorted by momentum in a semicircular magnetic spectrometer — the radius of their path fixes the momentum — and their kinetic energy is measured in a scintillator at the exit. Plotted against each other, the points fall on the relativistic curve $E^2 = (pc)^2 + (mc^2)^2$ and well away from Newton’s parabola, and the momenta at which they part company are within reach of a bench and an afternoon.

The domain of the experiment is a charged particle accelerated by a static or radio-frequency field and stopped in matter, with energies from a fraction of its rest energy to many times it. Within that domain, kinetic energy grows without limit while speed approaches $c$, and the two are related through $\gamma$, not through $\tfrac12 mv^2$.

Still open: whether the limit is exact at the highest energies #

The approach to $c$ is tested to astonishing precision at accelerator energies and, through cosmic rays and gamma rays, at energies millions of times higher, and no departure has been found. Some proposals for physics at the Planck scale predict small modifications to the relation between energy and momentum at very high energies, which would change how fast the most energetic particles travel or allow processes that strict relativity forbids, such as a high-energy photon decaying. Searches through the arrival times of gamma-ray bursts’ photons, the energies of the most energetic cosmic rays and the behaviour of astrophysical neutrinos at PeV energies have pushed the scale of any such effect beyond the Planck energy for some kinds of modification, and not for others. The most energetic cosmic-ray protons recorded, near $3 \times 10^{20}$ electronvolts, have Lorentz factors of about $3 \times 10^{11}$ and fall short of $c$ by a few parts in $10^{24}$, a regime where nothing but the relation itself has ever been tested; whether relativity’s relation holds exactly, or is the low-energy face of something else, is open.

What is not open is what Bertozzi’s two measurements showed together. Given 0.5 to 15 MeV, electrons crossed 8.4 m in 32.5 down to 28.03 ns — never faster than light’s 28.0 — while the heat they delivered rose in proportion to the energy supplied, far beyond the 0.256 MeV that ½mv² could ever account for at those speeds; energy keeps arriving, at a cost of $\sqrt{10}$ times more for each further nine in v/c, and only the speed stops responding. The ultimate speed is real, and it is not a limit on what can be given to a particle. It is a limit on what the giving can do to its speed.