The column that is hotter at the bottom

The zeroth law of thermodynamics is the one that makes temperature a quantity at all. Two systems each in equilibrium with a third are in equilibrium with each other, and the thing they share is called their temperature. A thermometer works because of it. A whole subject rests on the idea that when heat has finished flowing, the temperatures are equal.

In 1930 Richard Tolman, and then Tolman and Paul Ehrenfest together, showed that this is not true in a gravitational field. A column of gas, a tank of water or a star, left alone until nothing in it changes any more, is not at one temperature. It is warmer at the bottom. What is the same everywhere is not the temperature $T$ but the temperature multiplied by the local rate of clocks:

$$T\,\sqrt{g_{00}} = \text{constant}.$$

Near the Earth that factor is $1 + gh/c^2$ to excellent accuracy, and the gradient is about a part in $10^{16}$ per metre. No thermometer will ever see it on a laboratory bench. But the relation is not a correction to thermodynamics that can be left out at low precision. It is what equilibrium is, and it follows from three different directions — from light, from counting states, and from the weight of heat — each of which is drawn below.

Light that climbs arrives as a cooler blackbody #

The first route is the one in which the relation is almost forced. Put a cavity of thermal radiation at the bottom of a field and another at the top, and connect them with a tube so that light can pass. A clock lower down runs slow, so light that climbs arrives at a lower frequency than it left with. The question is what that does to a thermal spectrum.

The spectrum that arrives is computed without any appeal to temperature: every frequency is multiplied by the ratio of clock rates, and the number of photons occupying each mode is carried along unchanged, which is what propagation along a ray does to thermal light. The result is not the old spectrum dimmed. It is a Planck spectrum at a temperature lower by exactly the ratio of clock rates, and it matches Planck’s law at 2400 K to one part in $10^{14}$. The dashed curve shows that this is not a matter of taste: scaling the original spectrum down to the same height gives a curve that peaks at the wrong frequency.

That settles the question for the top cavity. A thermometer at the top, bathed in light from below, reads 2400 K. If the top cavity were at 3000 K it would radiate more down the tube than it received, and energy would flow downward. Equilibrium between the two requires the top to be at 2400 K, cooler by the same factor as its clocks are faster. The Planck form surviving the shift is the property that a moving body’s radiation does not have, and the difference between the two cases is worth a section of its own.

Why a shift in every direction keeps the spectrum thermal #

Planck’s law says that the average number of photons in a mode of frequency $\nu$ is $1/(e^{h\nu/kT} - 1)$. Frequency and temperature enter only through their ratio. So any process that multiplies every photon’s frequency by the same factor, while leaving the number of photons in each mode alone, turns a Planck distribution at $T$ into a Planck distribution at that factor times $T$ — not approximately, but identically, which is why the carried spectrum in the figure matches to the fourteenth decimal place rather than to the third.

Both conditions hold for light climbing a static field. The number of photons per mode is conserved along a ray in empty space; it is the quantity that brightness theorems conserve, and no lens can increase it. And the redshift between two heights is the same for light arriving from every direction, because it depends only on the ratio of clock rates at the two places and not on the path. A thermometer at the top is therefore surrounded by light that is thermal in every direction at one temperature, which is exactly the condition for it to settle at that temperature.

A moving body fails the second condition. Its light is blue-shifted ahead and red-shifted behind, so each direction is a Planck spectrum at a different temperature, and a thermometer bathed in the mixture has no single temperature to agree with. That is the whole of the difference between motion, which has no transformation law for temperature, and height, which does. The photon gas’s own peculiarity — that its number is set by the temperature rather than fixed — plays no part: the argument would go through for any radiation whose spectrum depends on frequency over temperature.

Two cavities and one conserved energy #

The second route does not use light at all, and it shows why the result is not a quirk of radiation. Equilibrium is the state of greatest entropy consistent with whatever is conserved, and the only subtle step is saying what is conserved.

Energy measured locally is not. A joule of heat carried down into a field arrives as more than a joule, measured by local clocks and rulers, because it has fallen through the potential. What the field conserves is the energy as a distant observer would book it, the local energy multiplied by the local clock rate. Heat has weight, and the bookkeeping has to include what the weight is worth.

The entropy of each cavity is a function of its own local energy, and its local temperature is fixed by the rate at which that entropy grows. The figure moves conserved energy from one cavity to the other and asks where the total is greatest. The search finds the peak at a share of 0.339 held at the top, and there the top cavity’s temperature is 0.800 of the bottom’s — the ratio of clock rates to six figures. The place where the two thermometers would agree, at a share of 0.556, is a place where the total entropy is still 1.82 per cent below its maximum, and moving energy downward would increase it.

The derivation needs nothing about photons. Maximising a sum of entropies at fixed total energy sets the derivatives of each entropy with respect to the conserved energy equal, and each such derivative is one over the local temperature divided by the local clock rate. So $T\sqrt{g_{00}}$ is equal throughout any system in equilibrium in a static field, whatever it is made of. The entropy is a count, and a count is the same number to every observer; what differs from place to place is how much energy it costs, and the temperature is that cost.

Why a uniform column would be an engine #

The third route is the oldest in spirit, and it is the one that shows the relation is not optional.

Suppose a tall column were at one uniform temperature. Take a small quantity of heat from the top, lower it in an insulated box to the bottom, and extract the work its weight does on the way down — it is small, the heat’s energy times $gh/c^2$, and it is not zero. At the bottom, deliver the heat to the column. Heat at the same temperature as the column goes in without any cost in entropy. Now let the column restore itself: if a uniform temperature were equilibrium, conduction would carry the heat back to the top, and nothing would resist it. The cycle ends where it began, having converted heat drawn from a single temperature into work, which the second law forbids.

The only way out is for the bottom to be warmer. Then the heat carried down, delivered at a temperature below the column’s, has to be pumped in, and the work of pumping is exactly what lowering it provided. The gradient that closes the loophole is $gh/c^2$ per unit height, the same number as before.

A gradient of that kind was argued about once already. In 1876 Josef Loschmidt claimed that a gas column in gravity must be colder at the top in ordinary kinetic theory, because each molecule slows as it climbs. Maxwell and Boltzmann showed that it is not: the slow molecules fail to reach the top, the fast ones arrive slowed, and the distribution of speeds that survives is exactly the same distribution at every height — which is why the air thins with height in the way the speed distribution predicts. Classical equilibrium is isothermal. Relativity reinstates a gradient of the sign Loschmidt wanted, at a size smaller than his by the ratio of a molecule’s thermal energy to its rest energy.

How big the effect is, body by body #

The fractional temperature gradient at a surface is the surface gravity divided by the square of the speed of light. That single number decides whether the effect is a curiosity or a structural fact.

On the Earth it is $1.1\times10^{-16}$ per metre, and across ten kilometres of atmosphere a part in $10^{12}$. The real atmosphere cools by about seven kelvin per kilometre for reasons that have nothing to do with equilibrium, because it is heated from below and stirred, and the Tolman gradient is beneath that by a factor of more than a hundred billion. At the surface of the Sun it is thirty times larger and still negligible. A white dwarf, with a surface gravity about a hundred thousand times the Earth’s, reaches a part in $10^{9}$ across its thin atmosphere.

A neutron star is different in kind. Its surface gravity is about $10^{12}$ metres per second squared, the gradient is $1.4\times10^{-5}$ per metre, and across the outer kilometre of the star the equilibrium temperature changes by more than a per cent. The surface figures in the chart are Newtonian and underestimate the true value there, and inside the star the field is strong enough that no expansion in $gh/c^2$ is adequate.

Inside a star that has reached one temperature #

Models of how neutron stars cool, which are compared with the surface temperatures of observed stars decades and centuries after the supernovae that made them, describe the interior after its early thermal relaxation as isothermal. What they mean is that $T\sqrt{g_{00}}$ is uniform — the redshifted temperature — and the local temperature is not.

The figure uses the exact interior metric of a star of uniform density — the simplest relativistic star there is, and one whose central clock rate falls to zero at a compactness of 8/9, which is the largest any static star can have. At compactness 0.34, close to that of a neutron star of 1.4 solar masses and 12 kilometres, the centre of an isothermal star reads 1.31 times the temperature three radii out and 1.13 times the surface. At compactness 0.6 the centre reads 1.41 times the surface. The outside of each curve was checked against the local gravitational acceleration: at a radius half as far out again as the surface, the gradient in the logarithm of temperature per unit of proper distance equals the local acceleration over $c^2$.

A real neutron star is far from uniform in density, and its interior temperature profile comes from a detailed equation of state. The statement that survives every model is the one drawn: a star that has stopped exchanging heat internally is hotter at its centre than at its surface, and the difference is a matter of geometry rather than of heating.

At a horizon the temperature has no ceiling #

The relation takes its most extreme form outside a black hole. A black hole radiates as a body at the Hawking temperature, measured by an observer far away — the horizon has a temperature and an entropy — and a thermometer lowered towards it on a rope, and held still, sits in that radiation in equilibrium at every height. By the Tolman relation its reading is the Hawking temperature divided by the local clock rate, which goes to zero at the horizon.

The solid curve rises without limit as the horizon is approached: 1.41 times the Hawking temperature at twice the horizon radius, 3.3 times at 1.1. The dashed curve is a different quantity with a different origin. Holding a thermometer still near a black hole requires an enormous upward acceleration, and an accelerated observer finds empty space to be a thermal bath at a temperature proportional to that acceleration. Near the horizon the two curves merge. The equilibrium temperature there is the temperature of hovering, and the Hawking radiation seen from far away is what that bath looks like after climbing out.

That identification is one of the places where the relation earns its keep. The thermal atmosphere that a held observer sees just outside the horizon is where calculations of the hole’s entropy locate the entropy, and the fact that its temperature diverges at the horizon nothing marks is why those calculations need a cutoff a small proper distance above the horizon — an unresolved part of the story rather than a finished one.

What the relation assumes #

The field is static. The relation holds where clock rates at each place do not change with time, so that the redshift between two places is a fixed number. In a rotating system there is a version with the rotation included; in a field that changes, such as a collapsing star, there is no equilibrium to have a temperature in.

The system has reached equilibrium, and most never do. An atmosphere heated from below, an ocean driven by winds, and a star still cooling all carry temperature gradients vastly larger than this one, maintained by flows of heat. The relation says where those gradients would end if every flow stopped, not what they are.

Heat flow is redefined, not just the endpoint. If equilibrium has a gradient, then the rule that heat flows down a temperature gradient must be amended, and Carl Eckart did so in 1940: heat flows down the gradient of the redshifted temperature. A column at uniform local temperature in a field carries a steady flow of heat downward, driven by gravity alone, until the bottom is warm enough to stop it.

Temperature is measured locally. Every temperature in the figures is what a thermometer at rest at that place reads. A distant observer looking at the bottom cavity through a telescope sees its light redshifted and infers the top’s temperature for it — which is the statement that the redshifted temperature is uniform, arrived at from outside.

What the figures cannot show #

The spectrum and entropy figures use a ratio of clock rates of 0.8, which exists only within a few kilometres of a neutron star’s surface or near a black hole. At any height on Earth the arrived spectrum would lie on top of the original to the width of the line, and the entropy peak would sit on top of the equal-temperature line. The figures show what the relation says, at a scale where it can be seen, and the gradient chart is there to say how far that scale is from any experiment.

Nor can a static figure show how equilibrium is approached. The relation is a statement about the end state. How long a column takes to reach it — and for a neutron star’s core that is a question of years to centuries, set by the conductivity of degenerate matter — belongs to transport rather than to thermodynamics, and nothing drawn here computes it.

Still open: a direct measurement #

The Tolman–Ehrenfest relation has never been measured directly. Every piece of it rests on measurements — the gravitational redshift is measured from a lift shaft to a centimetre of height, the Planck form of thermal radiation to extraordinary precision, and the second law everywhere — and the derivations above show it cannot fail without one of those failing. But no experiment has placed two thermometers at different heights in one system in equilibrium and read a difference, because on Earth the difference is a part in $10^{16}$ per metre, and no thermometer approaches that resolution.

Whether it can be tested directly is a live question for exactly the reason that makes it interesting. Proposals look for systems in which the effective gravity is large or the effective speed of light small — laboratory analogues in which sound or a condensate’s excitations play the part of light, and a flowing fluid plays the part of the field — where a Tolman-like relation should hold with the analogue’s own “speed of light”. What such an analogue could establish, and what it could only illustrate, is argued over.

The habit worth carrying away is to ask what is conserved before asking what is equal. Equilibrium equalises the derivative of entropy with respect to the conserved quantity, and a temperature is only equal across a system when energy measured locally is the conserved quantity. In a gravitational field it is not, and the zeroth law survives by being about $T\sqrt{g_{00}}$ — which a clock thrown upward that comes back older than one in orbit is already measuring, one tick at a time.