The clock that runs slow because it is warm

The clock that has to slow, and why no clock can refuse derives time dilation for a clock moving at a steady speed and shows that every kind of clock must obey it. The clocks that matter most in physics — atoms and nuclei — are never still. In a gas at room temperature an atom moves at hundreds of metres per second in a direction that changes with every collision; in a crystal a nucleus vibrates about its site, a few thousand billion times a second. Each of those motions dilates the atom’s time.

It is tempting to think the effect averages away. Motion towards an observer blueshifts the light an atom emits and motion away redshifts it, and in a gas with no net flow the shifts cancel on average, leaving a broadened line centred where it would have been. That is true of the first-order Doppler effect and false of time dilation, which slows a moving clock whichever way it moves. A gas of atoms is a crowd of moving clocks, every one of them slow, and the crowd as a whole is slow by an amount set by its temperature.

How much slower a warm atom ticks #

A clock moving at speed $v$ runs slow by the fraction $1 - 1/\gamma$, which for everyday speeds is $v^2/2c^2$. For atoms in thermal equilibrium the average of $v^2$ is fixed by the temperature — half a kT for every way of moving gives $\tfrac12 m\langle v^2\rangle = \tfrac32 kT$ — so the average slowing is

$$\frac{\Delta f}{f} = -\frac{\langle v^2\rangle}{2c^2} = -\frac{3}{2}\frac{kT}{mc^2}.$$

It is the ratio of the thermal energy to the rest energy, and it depends on nothing else.

At room temperature the numbers are tiny in everyday terms and enormous in the terms clocks are now judged by. A hydrogen atom at 300 kelvin runs slow by four parts in a hundred billion; a caesium atom, 132 times heavier, by three parts in ten trillion. The best optical clocks aim at a part in $10^{18}$, which for strontium means holding the atoms below two-thirds of a millikelvin, and for hydrogen below seven microkelvin. Every line on the drawing falls in proportion to the temperature, and every factor of ten of cooling buys a factor of ten in the clock.

The gravitational shift that Pound and Rebka measured across a 22.5-metre tower, marked for scale, is the size of the thermal shift of a caesium atom at a few kelvin. Relativity’s two ways of slowing a clock — motion and height — are of comparable size in the laboratory, and a clock that is to measure one must control the other.

Why the small shift survives and the large one does not #

The first-order Doppler shift of a moving atom is far larger than its time dilation: $v/c$ rather than $v^2/2c^2$. Why, then, is it the second that shifts a clock?

At room temperature the first-order shifts are spread over a few parts in ten million, a million times more than the second-order shift. But they have both signs. The atoms moving towards the observer are balanced by atoms moving away, and in the speeds in a still room — the Maxwell distribution with no net flow — they cancel exactly on average. Sampling twenty thousand atoms leaves an average first-order shift of three parts in a billion, which is the statistical noise of the sample and falls as the sample grows; its expected value is zero. What the first-order shifts do is broaden the line — the Doppler width of every spectral line of a gas — without moving its centre.

The second-order shifts all have one sign, because $v^2$ is never negative. Every atom’s clock is slow, whichever way it moves, and the average of the sample is three parts in ten trillion, the value the formula gives. A symmetric crowd of Doppler shifts cancels; a crowd of time dilations cannot. The two effects also scale differently with temperature — the spread as its square root, the shift in proportion — so cooling helps the second more than the first.

This is the same separation that the shift that survives at right angles describes in the Ives–Stilwell experiment of 1938, which isolated the second-order shift of fast-moving ions by averaging the light emitted forwards and backwards. A thermal gas does the averaging by itself, over every direction at once.

A twin paradox in every collision #

Each atom in a warm gas is, in the language of the twin who comes back younger, a travelling twin. It flies off at a few hundred metres per second, collides, turns round, flies off somewhere else, and after a second of this has made millions of round trips. An atom at rest in the same box — a stay-at-home twin — would have aged more. The twin calculation depends only on the speeds along the path, not on the turnarounds, and for a thermal gas the average over the path is exactly the average over the Maxwell distribution. So the thermal slowing is the twin effect, summed over collisions: a warm gas of caesium atoms, after a year, is younger than a cold one by three parts in ten trillion of a year, about ten microseconds.

That framing removes a possible worry. The atoms accelerate violently in every collision, and it might seem that a clock subjected to such accelerations could run at some other rate. The clock that does not feel the turn states the clock hypothesis — that a clock’s rate depends on its speed and not on its acceleration — and the thermal shift is one of its quieter confirmations: the measured shifts of Mössbauer lines and of trapped ions follow $\langle v^2\rangle$ and nothing else, although the accelerations involved are enormous.

The same shift, read as a mass #

There is a second way to see the thermal shift, and it connects it to $E = mc^2$. A warm gas has more energy than a cold one — $\tfrac32 kT$ more per atom — and mass is a form of energy says that extra energy has mass: each atom, counted with its motion, weighs more by $\tfrac32 kT/c^2$. The fraction by which the gas is heavier is exactly the fraction by which its clocks are slow. The coincidence is not one. A clock’s rate is its energy spacing divided by Planck’s constant, measured in its own frame; seen from the laboratory, the same clock carries extra energy of motion that does not tick, and the ratio of the ticking part to the whole is the time-dilation factor.

For a solid the extra energy is the vibrational energy, zero-point included, and the heavier-by and slower-by fractions again coincide, apart from the factor of a half that comes from counting only the kinetic part of each vibration. The box of light that weighs something makes the same point for trapped radiation: energy in motion inside a body adds to its mass without ticking any of its clocks.

A clock that runs slow at absolute zero #

In 1960, a few months after the discovery of the Mössbauer effect had made it possible to emit and absorb gamma rays with a line width of a part in $10^{12}$, Robert Pound and Glen Rebka, and independently Brian Josephson, pointed out that the nuclei emitting those gamma rays are clocks vibrating in a crystal, and that their vibration must slow them.

A nucleus in a crystal is not a free particle, and its velocity is not given by the gas formula. In a harmonic crystal the average kinetic energy of an atom is exactly half its vibrational energy, and the vibrational energy follows the Debye model’s curve, which includes the motion that remains at absolute zero. The motion that cannot be stopped establishes that a particle in a well cannot be at rest; here that zero-point motion has a relativistic consequence. Iron-57 nuclei in an iron crystal at absolute zero run slow by four parts in ten trillion, because they are still moving.

The shift that can be measured is the difference between two samples, and there the zero-point part cancels if the samples are the same material at the same temperature. What does not cancel is a difference in temperature: near room temperature, heating an iron-57 source by one kelvin lowers its gamma-ray frequency by 2.2 parts in $10^{15}$. Pound and Rebka’s tower was 22.5 metres high and the gravitational shift across it 2.46 parts in $10^{15}$. A temperature difference of 1.1 kelvin between the source at the top and the absorber at the bottom would have produced the whole effect.

They knew it, because they had just measured it. In early 1960 Pound and Rebka heated and cooled an iron-57 source and watched the Mössbauer resonance shift by the predicted amount — a measurement of time dilation for nuclei vibrating at a few hundred metres per second, the first of its kind. In the tower experiment that followed, reported in the clock that runs slow lower down, they monitored the temperatures of source and absorber and corrected for the difference, and they reversed the direction of the gamma rays so that the gravitational shift changed sign while the thermal one did not.

A heat capacity read off a clock #

The temperature coefficient of the shift has a meaning of its own.

Since the shift is half the vibrational energy per atom divided by $Mc^2$, its rate of change with temperature is half the heat capacity per atom divided by $Mc^2$. The two curves in the drawing are therefore one curve, necessarily: the change in a nuclear clock’s rate with temperature is the crystal’s heat capacity, in units of the rest energy. At low temperature the heat capacity rises as the cube of the temperature, and so does the clock’s sensitivity to it; above the Debye temperature it levels off at Dulong and Petit’s three $k$ per atom, and the clock’s coefficient levels at $3k/2Mc^2$. Iron’s Debye temperature is above room temperature, which is why the coefficient Pound and Rebka met was a little under the Dulong–Petit value.

That is a strange instrument when stated plainly. A spectroscopic measurement of a gamma-ray line — a question about frequency — returns a thermodynamic quantity, the heat capacity, because the energy that raises the crystal’s temperature goes partly into the kinetic energy of its nuclei, and the kinetic energy slows their clocks. Why heating a perfect spring changes nothing finds that a harmonic crystal does not expand when heated; its nuclei’s clocks, by contrast, respond to every joule.

How slow each clock’s own motion makes it #

The hydrogen maser, the workhorse of radio astronomy’s time-keeping and of many national time scales, stores its atoms in a Teflon-coated bulb at about forty degrees Celsius, and its atoms move at a few kilometres per second. Its frequency is lowered by four parts in a hundred billion by their motion, a correction that has to be calculated and removed, and that changes by a part in $10^{13}$ for each kelvin the bulb drifts. Keeping the bulb’s temperature stable is part of what makes a maser stable.

Laser cooling changed the scale. Atoms in a caesium fountain clock are cooled to about a microkelvin and tossed upwards, and their thermal time dilation falls to a part in $10^{21}$, far below everything else in the error budget. Strontium cooled only on its strong line would still be slow by a part in $10^{18}$, and optical lattice clocks cool further, on a narrow line, and hold the atoms in a lattice where their motion is quantised. The aluminium ion clock, whose ion is cooled to its ground state of motion in its trap, has had to account for the motion that remains even there — zero-point motion and the driven oscillation of the trap — which is why its builders have measured the time dilation of an ion moving at a few metres per second, and seen the expected slowing. The clock that measures a height describes what such clocks can do; the thermal term in their budget is part of why they can.

Where the thermal picture stops #

Thermal equilibrium. The formulas assume the atoms’ speeds follow the Maxwell distribution at a single temperature. Laser-cooled atoms, trapped ions and atoms in a lattice are not in thermal equilibrium in general, and their mean-square velocity has to be measured or calculated for the actual trap and cooling.

Harmonic crystals. The Debye model treats the crystal as harmonic with a single characteristic temperature. Real crystals have anharmonic vibrations and a spectrum that is not quite Debye’s, and the measured temperature coefficient of a Mössbauer line is used partly to learn about that spectrum; the relation to the heat capacity survives as long as the kinetic energy is half the vibrational energy, which anharmonicity slightly spoils.

Other shifts. In a Mössbauer experiment the chemical environment of the nucleus shifts its line too, by the isomer shift, which depends on the electron density at the nucleus. Comparing a source and an absorber of different composition mixes that shift with the thermal one; Pound and Rebka used the same material on both ends.

Not the only thing temperature does to a clock. A warm room also bathes a clock’s atoms in thermal radiation, whose electric field shifts their levels — the blackbody shift, which for a strontium lattice clock at room temperature is a few parts in $10^{15}$ and is the largest correction such clocks make. It is an effect of the radiation’s field on the atom, not of the atom’s motion, and it has nothing to do with relativity; but it is easily confused with the thermal time dilation because both are “temperature shifts” and both are removed by the same strategy of cooling and measuring. The figures here are the relativistic part alone.

Special relativity only. Everything here is the time dilation of motion. A clock in a gravitational field is slowed as well, and a hot gas in a gravitational field is a system where both act at once — the subject of the column that is hotter at the bottom.

One atom’s clock, fast and slow by turns #

The drawings reduce each clock to one number, an average. What they cannot show is the individual atom: a caesium atom at room temperature is sometimes nearly still and sometimes moving at twice the typical speed, and its clock rate fluctuates accordingly from moment to moment. A clock built from many atoms reports the average, and a clock built from one trapped ion reports that ion’s own history, averaged over the time of the measurement. Nor do the figures show the line itself — the Doppler-broadened profile whose centre the second-order shift moves — or its narrowing when atoms are held in a trap smaller than their wavelength, the Lamb–Dicke regime in which the first-order broadening disappears altogether while the second-order shift remains.

Still open: how far down the motion can be pushed #

Optical clocks now compare to parts in $10^{18}$ and below, and at that level every source of motion counts: the residual thermal motion after cooling, the zero-point motion in the trap, the oscillation the trap’s own fields drive, and the motion of the clock’s reference cavity. For the best ion clocks the time-dilation term is known to a few parts in $10^{19}$, and pushing it further requires cooling that is also measured, since a correction is only as good as the temperature it assumes. Whether time dilation from motion or from gravity will be the limiting systematic effect of the next generation of clocks, and how well a single ion’s zero-point motion can be characterised, are questions being answered experimentally, clock by clock.

The habit worth carrying away is to ask whether a random effect has a sign. Fluctuations that go both ways average to zero and only broaden; fluctuations that go one way average to a shift, however small each one is. Thermal motion does both at once — it broadens every spectral line through the first-order Doppler effect and shifts every clock through time dilation — and a clock that cannot measure its temperature cannot tell its motion from its height.