The spring that becomes a light-clock

The harmonic oscillator is the most useful idealisation in physics, and its most useful property is that its period does not depend on how far it swings. A mass on a spring with a force exactly proportional to displacement returns to its starting point in the same time whether it is pulled back a millimetre or a metre, which is why every minimum looks like a parabola is such a powerful statement: anything near equilibrium keeps time. The property rests on Newton’s second law in its simplest form, force equals mass times acceleration, with the mass fixed. Relativity replaces that law with force equals the rate of change of momentum, with momentum growing without limit as the speed approaches $c$. Keep the spring perfect — exactly Hooke’s law, at any extension — and let only the mass be relativistic, and the most reliable clock in physics stops keeping time.

The equation, and the natural unit of amplitude #

The oscillator’s energy is the relativistic energy of the mass plus the energy stored in the spring:

$$E = \sqrt{p^2c^2 + m^2c^4} + \tfrac12 k x^2.$$

It is conserved, and Hamilton’s equations give the motion: the rate of change of momentum is $-kx$, exactly as for a Newtonian spring, and the velocity is $pc^2/E_{\text{mass}}$, which approaches $c$ as the momentum grows and never passes it. The slow oscillator has angular frequency $\omega_0 = \sqrt{k/m}$, and the natural unit in which to measure its amplitude is $c/\omega_0$ — the distance light travels while the slow oscillator’s phase advances by one radian. An oscillator whose amplitude is small compared with $c/\omega_0$ never moves at an appreciable fraction of $c$, because its peak speed is $\omega_0$ times the amplitude. One whose amplitude is comparable with $c/\omega_0$ reaches relativistic speeds, and one whose amplitude is many times $c/\omega_0$ would, in Newtonian mechanics, move faster than light.

For a mass on a laboratory spring, $c/\omega_0$ is hundreds of thousands of kilometres, and the effect is unobservable. For an electron in the field of an intense laser, or an electron oscillating in a plasma, $\omega_0$ is so large that $c/\omega_0$ is micrometres, and relativistic amplitudes are routine. The idealisation of a perfect spring is then the idealisation of a restoring force proportional to displacement, which is what a uniform plasma supplies to its electrons.

A speed with a ceiling, a momentum without one #

The first figure’s triangle wave is the most visible consequence, and the reason for it is plainest in the velocity.

Released from its greatest extension, the mass is pulled back by the full spring force, and its momentum grows at the rate the force sets, just as a Newtonian mass’s would. At first its speed grows in proportion. Within a short distance the speed is close enough to $c$ that further momentum adds almost nothing to it: the spring keeps doing work, and the work goes into the mass’s inertia, its Lorentz factor rising to 5.5 at the centre of this swing. Through the middle of the swing the mass crosses at nearly constant speed, just under $c$, and on the far side the spring decelerates it — which, starting from a large momentum, takes time and distance before the speed falls appreciably — and then reverses it sharply near the far end. The position therefore changes nearly linearly in time for most of the cycle, and the turnarounds are brief. That is a triangle wave.

The same behaviour is what a ship accelerating at constant thrust shows: its speed creeps towards $c$ while the quantity that keeps growing without limit is its rapidity, its momentum, its energy. The oscillator is that journey made twice a cycle in opposite directions.

A period that grows until it is a light-crossing time #

A triangle wave of amplitude $a$ traversed at speed $c$ takes $4a/c$ per cycle — the time light takes to cross the swing four times. So at large amplitude the period grows in proportion to the amplitude, and the oscillator becomes a clock whose rate is set not by the spring but by the size of the swing.

The period is computed from the energy integral, $T = 4\int_0^a dx/v(x)$, with the speed at each position fixed by energy conservation, and it is checked against the time between zero crossings of the integrated motion. At small amplitude the first correction is a lengthening by three-sixteenths of the square of the peak speed over $c$ — a part in ten thousand when the mass reaches 2 per cent of the speed of light — and at large amplitude the period merges into the light-crossing line. Between the two the curve rounds the corner at an amplitude of about $c/\omega_0$, where the Newtonian peak speed would have been exactly $c$.

The loss of isochronism looks at first like the pendulum’s familiar failing, whose period also grows with amplitude. The resemblance is misleading about the cause. A pendulum’s period grows because its restoring force falls short of proportional at large angles — the force law is anharmonic. Here the force law is exactly harmonic at every extension, and the period grows because the inertia is not constant. A relativistic spring is anharmonic in its mass rather than its force, and there is nothing a better spring could do about it. Isochronism is not a property of Hooke’s law; it is a property of Hooke’s law combined with Newton’s second law in its slow form.

Where the three-sixteenths comes from #

The first correction to the period can be found without integrating anything, and the reasoning shows why it is positive. For a slow oscillator the energy of the mass is $mc^2 + p^2/2m - p^4/8m^3c^2 + \cdots$, the expansion of the square root, and the first relativistic term is negative: at a given momentum, a relativistic mass has slightly less kinetic energy than a Newtonian one, which is the same statement as saying it moves slightly slower for the momentum it carries. Moving slower, it takes longer to cover the swing. Averaging the extra term over one cycle of the slow motion, where the momentum varies as a sine, gives a fractional lengthening of the period of $\tfrac{3}{16}(\omega_0 a/c)^2$. At a peak speed of a tenth of $c$ the period is two parts in a thousand longer than the slow one, and the figure’s integrated curve and the dotted correction agree until the peak speed approaches half of $c$.

The sign is the useful part. Relativity always lengthens the period of an oscillator whose force law is fixed, because it always makes a mass harder to accelerate than Newton says. A clock built on a mechanical oscillator therefore always runs slow in proportion to the square of its peak speed over $c$, independently of any motion of the clock as a whole — a correction that is utterly negligible for any mechanical clock and is the whole of the story for an electron oscillating at relativistic amplitude.

Where a relativistic spring is found #

A perfect spring at relativistic amplitudes sounds like a thought experiment, and the nearest real case is common. In a plasma, the free electrons can be displaced as a whole relative to the heavy ions, and the displaced charge sets up an electric field that pulls them back in proportion to the displacement — a restoring force exactly linear in the displacement, for a uniform slab, at any amplitude. Released, the electrons oscillate at the plasma frequency, the frequency below which the plasma reflects radio waves, and for a dense laboratory plasma that frequency is so high that $c/\omega_0$ is a few micrometres.

A laser pulse of the intensities now routinely available drives those electrons to relativistic speeds in a single oscillation. What happens then is this essay’s oscillator: the electrons’ motion lengthens its period, their oscillation acquires harmonics, and the wave they form behind the laser slows and steepens. The relativistic lengthening of the period is visible directly as a drop in the plasma’s effective frequency, which lets a laser propagate through a plasma that would reflect it at low intensity — relativistic transparency — and the steepening is what ends in wave breaking. Electrons driven straight by a strong laser, without the plasma’s restoring force, trace figure-eight paths and radiate harmonics of the laser’s frequency; the plasma’s linear restoring force is what makes the motion the symmetric triangle drawn here.

Ellipses that become lenses #

The state of an oscillator is a point in the plane of position and momentum, and a cycle traces a closed curve there.

A slow harmonic oscillator traces an ellipse, because its energy is a sum of squares of position and momentum. The relativistic energy is not quadratic in momentum — at large momentum it grows linearly — and the curves of constant energy change shape accordingly. Near the turning points, where the momentum is small, the energy is still nearly quadratic and the curve still rounds; but the momentum at the centre grows much faster with amplitude than a Newtonian oscillator’s, since it must carry the spring’s energy as $pc$ rather than $p^2/2m$, and at a given position the curve bulges out to momenta far larger than the ellipse would reach. The lens shape is a picture of the same fact as the triangle wave: momentum is what the spring adds, and momentum, unlike velocity, has no ceiling.

The area enclosed by the curve is still conserved and still fixes the cycle’s action, which is the quantity quantum mechanics counts. The levels of a relativistic oscillator are therefore not evenly spaced, as a slow oscillator’s are, but crowd as the action grows — the same loss of isochronism, seen in the spectrum instead of in the timing, and part of why the state that swings like a pendulum is special to a harmonic potential with a fixed mass.

The overtones a perfect spring puts into its own motion #

A sine wave contains one frequency. A triangle wave contains many, and the relativistic oscillator’s motion acquires them as it changes shape.

A mass oscillating on a perfect spring at relativistic amplitude radiates, if it is charged, not at one frequency but at the fundamental and all its odd multiples — which is how the oscillator’s shape becomes observable in the light it gives off. The harmonics are odd because the motion is symmetric under reflection through the centre: the swing to the left is the mirror image of the swing to the right, half a period later, and even harmonics cannot survive that symmetry. An oscillator pushed past its parabola does the same thing for a different reason — its force law is cubic — and produces the third harmonic first. Here the third harmonic is produced by an exactly linear force acting on a mass whose inertia varies through the cycle, and the two effects are indistinguishable in the spectrum they generate.

How much a rider ages #

A clock riding on the oscillating mass is moving, so it runs slow, and the amount depends on how fast it moves through the cycle.

The rider’s time per cycle is the integral of $dt/\gamma$ around the cycle, and it is checked against the same quantity accumulated step by step along an integrated trajectory. At large amplitude the rider spends nearly the whole cycle at a Lorentz factor of order the amplitude squared, so it records a small fraction of the period, and the twin who travels ages less with every swing. The rider is accelerated violently at each turnaround, and the calculation assumes — as the clock hypothesis does for every such case — that its rate depends only on its instantaneous speed and not on its acceleration. For muons circulating in a storage ring under accelerations of $10^{18}$ times the Earth’s gravity, that assumption has been tested and holds.

The numbers make the asymmetry concrete. At an amplitude of ten times $c/\omega_0$ one cycle lasts 6.43 slow periods by the bystander’s clock and about 0.38 of a slow period by the rider’s. Over a thousand cycles the bystander ages more than six thousand slow periods and the rider fewer than four hundred, and nothing about the arrangement is symmetric: the rider is the one being accelerated back and forth, and the bystander never moves. It is the twin arrangement run as a steady state rather than as a single round trip, and it is how the lifetimes of fast unstable particles held on a closed path are extended in practice.

Where the ideal spring is ideal #

No real spring is Hooke’s law at relativistic amplitudes. A material spring breaks long before its end moves at an appreciable fraction of $c$, and a material restoring force cannot even propagate faster than sound in the material. The model is realised by fields, not springs: the restoring force on an electron in a uniform plasma, displaced from its ions, is linear in the displacement, and that is where the relativistic oscillator is found.

The mass does not radiate. A charged mass oscillating at relativistic speed radiates strongly, losing energy and changing its motion; at large amplitudes in real systems the radiation reaction is not small, and the motion drawn here is the motion of a neutral mass or of a charge whose radiation is ignored.

The spring’s own inertia is ignored. A real restoring medium carries energy and momentum of its own, and a plasma’s electrons all oscillate together, so the relevant motion is collective. In a plasma this appears as a drop in the oscillation frequency with amplitude — the plasma frequency divided by the square root of the electrons’ Lorentz factor — and the collective wave eventually breaks.

The scale the shapes were divided by #

The first figure divides each curve by its own amplitude and period so that only the shape is compared, and that choice hides the scale. The largest oscillation is fifty times the amplitude of the smallest and takes more than six times as long per cycle; drawn to a common scale, the slow sine would be a flat line at the bottom of the triangle wave’s first rise.

Nor can a plot of position against time show where the energy is. At the turning points it is all in the spring; at the centre it is all in the mass; and at large amplitude the mass’s share is almost entirely inertia rather than anything that shows up as speed. The triangle wave looks like a mass moving at constant speed, which in Newtonian terms would carry constant kinetic energy, and the real energy flow through the cycle is invisible in it.

Still open: how far relativistic plasma waves can be pushed #

Laser-driven plasma accelerators use exactly this physics: an intense laser pulse drives the plasma’s electrons into oscillation, the oscillation forms a wave travelling behind the pulse at nearly the speed of light, and electrons riding the wave are accelerated to gigaelectronvolts in centimetres. The accelerating field grows with the oscillation’s amplitude until the wave breaks — until electrons in the oscillation overtake the wave that carries them — and the relativistic lengthening of the period sets where that happens. How to drive the wave close to breaking without losing its regularity, and how to keep the accelerated electrons’ energies narrowly spread when the wave’s shape depends so strongly on its amplitude, are the central engineering questions of the field, and the answers so far come largely from simulation and experiment rather than from any closed theory.

The habit worth carrying from here is to ask which ingredient of a familiar result is doing the work. A spring keeps time because force is proportional to displacement and acceleration is proportional to force — and relativity removes the second proportionality while leaving the first untouched, which is enough to turn the best clock in physics into one set by the speed of light.