The count that no observer can disagree about

The argument that a moving body has no temperature ends with a quantity that has no transformation law because it has no definition. That is an unusual place for a subject to stop, and the way out is to ask what does survive.

The four at the top of that figure are not a miscellaneous list. They are the same kind of thing, and naming the kind is the whole of this essay.

Why a count cannot transform #

Entropy is the logarithm of the number of microscopic arrangements consistent with what is known about a system. It is a count, and that is the whole of the argument for its invariance.

Two observers in relative motion disagree about lengths, durations, energies and simultaneity. They do not disagree about how many of something there is. That is the same reason a loop’s circulation counts a whole number of turns whatever path it took. If a box contains a particular number of photons, both count that number; if a configuration space contains a particular number of states consistent with the constraints, both count that number.

It is worth noticing which other invariants are on the list for the same reason. Electric charge is invariant — and that invariance is a measured fact rather than a definition, tested to extraordinary precision by the neutrality of atoms whose electrons move far faster than their nuclei. The number of particles is invariant. A baryon number, a lepton number, a winding number: all invariant, all counts — and a winding number’s invariance under deformation is the same statement one level down.

Against that, the quantities that do transform all have a length, a duration or an energy in them, and those are the things a boost mixes.

So the rule underneath the figure is simple to state. A quantity transforms when it is built from the geometry and does not when it is built from counting, and the invariants of relativity divide almost perfectly along that line.

What follows, computed #

With entropy invariant and energy-momentum a four-vector, everything about a boosted box of radiation follows.

Volume contracts. One dimension shortens by the Lorentz factor, so $V = V_0/\gamma$.

Energy rises. For radiation isotropic in its own frame the total momentum is zero there, so the energy is the time component of a four-vector whose invariant length is the rest energy with no spatial part, and $E = \gamma E_0$.

Energy density rises as the square. Energy up by $\gamma$, volume down by $\gamma$: $u = \gamma^2 u_0$.

Entropy density rises as the first power. Entropy unchanged, volume down by $\gamma$: $s = \gamma s_0$.

Those last two are the pair that matters, and putting them side by side answers the question the previous essay left open.

The relation that having a temperature consists of #

For blackbody radiation at rest, $u = aT^4$ and $s = \tfrac43 aT^3$. Eliminating the temperature gives

$$u = \frac{3}{4}\left(\frac{3s}{4a^{1/3}}\right)^{4/3}\Big/ a^{1/3} \propto s^{4/3},$$

with a coefficient built only from the radiation constant. That relation is what having a temperature amounts to, written without mentioning one. A system whose energy density and entropy density satisfy it has a temperature, and the temperature is what the constant of proportionality can be solved for.

Under a boost the energy density goes as $\gamma^2$ and the entropy density as $\gamma$, so the combination $u/s^{4/3}$ goes as $\gamma^{2/3}$ — which is not one. The relation fails.

That is the previous essay’s conclusion arrived at from the other side, and it is sharper. There is no temperature for the boosted radiation, and the reason is not that a thermometer reads different things in different directions; it is that the two densities no longer stand in the relation a temperature would be defined by. Both are perfectly well-defined quantities, both are computed here from invariants, and their ratio is simply not what it would have to be.

A worked case: the radiation in a moving cavity #

The abstract statement is easier to trust once run on a specific object, and a cavity is the specific object thermodynamics was built on.

Take a cubic metre of cavity radiation at 1,000 kelvin. In its own frame the energy density is $aT^4 = 7.6\times10^{-16}\times10^{12}$, which is $7.6\times10^{-4}$ joules a cubic metre, so the box holds 0.76 millijoules. Its entropy is $\tfrac43 aT^3 V$, which is $1.0\times10^{-6}$ joules per kelvin. It holds about $2\times10^{16}$ photons.

Now boost to a frame in which the box moves at four fifths of light speed, a Lorentz factor of $5/3$.

The photon count is unchanged: $2\times10^{16}$. Both observers count photons and get the same answer, which is what makes the rest of it work.

The entropy is unchanged: $1.0\times10^{-6}$ joules per kelvin. It is 3.6 times Boltzmann’s constant per photon in either frame, because both of those numbers are counts.

The energy is 1.27 millijoules, up by the Lorentz factor.

The volume is 0.6 cubic metres, down by it.

So the energy density is 2.1 millijoules a cubic metre — up by $25/9$ — and the entropy density is $1.7\times10^{-6}$ joules per kelvin per cubic metre, up by $5/3$.

Ask the second frame what temperature a blackbody with that energy density would have: $(u/a)^{1/4}$ gives 1,290 kelvin. Ask what temperature a blackbody with that entropy density would have: $(3s/4a)^{1/3}$ gives 1,186 kelvin. The two answers differ by nine per cent, and there is no third calculation that would reconcile them. That is the failure of the relation, in numbers, for one box.

What the second law becomes #

Entropy being invariant is what makes the second law a relativistic statement at all, and it is worth seeing why it is not automatic.

The second law says entropy does not decrease with time. Time is frame-dependent, so the law as stated is about a quantity that transforms and an ordering that may not be agreed on. Two things rescue it.

The quantity does not transform. Every observer computing the entropy of a system at a given moment gets the same number, so the two observers are comparing the same quantity even though they disagree about which moment.

And the events being compared are timelike-separated. A system’s entropy now and the same system’s entropy later are events on the system’s own worldline, and the order of two timelike-separated events is agreed on by every observer. So every observer sees the same numbers in the same order.

What is not covered is a pair of spacelike-separated entropy changes — one here and one far away — whose order is genuinely negotiable. The second law makes no claim about their sequence, and it does not need to: it is a statement about each system’s own history, and each system’s own history is ordered.

That is a cleaner resolution than the second law usually gets, and it comes entirely from the invariant being a count.

The one thing that must be handled carefully #

There is a step in the argument above that is easy to make wrongly, and it caught careful people.

The entropy of a system is defined on a simultaneity slice — it is the count of arrangements of the system at one moment. Two observers use different slices. For a system in a steady state that does not matter, because every slice gives the same answer; for a system whose entropy is changing, two observers slice a different set of events and are, strictly, counting different things.

The repair is to define the entropy on the system’s own rest-frame slice and treat it as a scalar attached to the worldline, which is what relativistic thermodynamics does. The entropy of a fluid element is defined where the element is at rest, it is a scalar, and what appears in the equations is an entropy four-current whose time component in any frame is the entropy density there.

That construction gives back everything above — the density rising as $\gamma$, the total being invariant — and it is what makes the argument respectable rather than merely plausible. The count is invariant; which count is being taken requires a convention; and the convention that works is the one attached to the system rather than to the observer.

What the invariance decides about the old argument #

The previous essay leaves three transformation laws standing and says none is refutable. Entropy’s invariance narrows that, and it is worth seeing exactly how far.

Every derivation of a temperature transformation starts from $dS = \delta Q/T$ and requires it in both frames. With $S$ invariant, $dS$ is invariant, so

$$\frac{\delta Q}{T} = \frac{\delta Q_0}{T_0}$$

exactly. The product of the two unknown transformation factors is fixed. Whatever heat transforms as, temperature transforms as the same thing.

That is a real constraint and it explains the structure of the old dispute. Planck took $\delta Q = \delta Q_0/\gamma$ and got $T = T_0/\gamma$; Ott took $\delta Q = \gamma\,\delta Q_0$ and got $T = \gamma T_0$; Landsberg took both invariant. All three satisfy the constraint, which is why all three are self-consistent, and the constraint is all that entropy’s invariance supplies.

What would settle it is a definition of heat that is not a convention, and there is not one. Splitting an energy change into heat and work requires saying what work is being done, a body moving at constant velocity has a rate of doing work that depends on the frame, and there is no frame-independent place to draw the line.

The modern resolution is to stop drawing it. Relativistic thermodynamics works with the energy-momentum tensor and an entropy four-current, from which rest-frame quantities are extracted when they are wanted and no frame-dependent split is ever made. That is a refusal to answer rather than an answer, and it is the right one: the question was about a boundary somebody had drawn.

Why this was not obvious in 1907 #

Entropy’s invariance is stated in the modern literature as though it were self-evident, and it was not treated as settled when the argument about temperature began.

The reason is the order things were understood in. In 1907 entropy was a thermodynamic quantity defined by $dS = \delta Q/T$ — by a process, not by a count. Boltzmann’s identification of it with the logarithm of a number of arrangements was twenty-five years old, it was not universally accepted, and Boltzmann had died the year before with the atomic hypothesis still contested.

From the process definition, entropy’s transformation is exactly as ambiguous as heat’s and temperature’s, because it is built out of them. Planck’s derivation therefore had to assume entropy invariant — which he did, on the grounds that the number of states must be the same for everyone, so he had the modern argument and used it as a premise rather than as a conclusion.

What changed is that the counting definition became the primary one. Once entropy is a number of arrangements, its invariance is not an assumption to be justified but a triviality, and what is left to argue about is only the split of an energy transfer into heat and work.

That is a reversal worth noticing. The quantity that seemed most obscure turned out to be the only one with an unambiguous transformation, and the two that seemed concrete turned out to be conventions. Heat and temperature are what a nineteenth-century physicist could measure; entropy was the abstraction. Under a boost, the abstraction is the survivor.

Rigid walls do not exist, and the transformation of heat is still open #

The box has been assumed to have rigid walls, and there are no rigid walls in relativity. A real container’s front and back are at different phases of any process, its length is what a particular observer measures, and the radiation inside it is in equilibrium with a boundary that is not simultaneous with itself. For a box small compared with a light-crossing time of anything happening, the idealisation is fine.

And the radiation has been assumed to remain in equilibrium in its own frame. That is what makes the rest-frame entropy well defined. A radiation field that is not in equilibrium anywhere has an entropy that depends on the coarse-graining chosen, and the invariance argument still works — a count is a count — but the count being taken is a matter of choice.

Nor does any of this fix the transformation of heat. Entropy invariant plus $dS = \delta Q/T$ ties the transformation of $Q$ to that of $T$, and leaves both undetermined. What the modern formulation does is refuse to split the energy transfer into heat and work at all in a frame-dependent way, and work instead with a covariant energy-momentum flux — from which the rest-frame quantities can be extracted when they are wanted.

The same argument outside thermodynamics #

The pattern — that counts are invariant and measurements are not — is worth following out of this subject, because it is where several results that look unrelated come from.

The number of nodes in a standing wave is a count, so two observers watching the same vibrating string in relative motion agree about it, though they disagree about the wavelength and the frequency. That is not a triviality: it is what makes a mode label meaningful, and it is why quantum numbers are the same in every frame.

The number of times a field winds round a point is a count, and it is invariant under any continuous change of the field as well as under a boost. Every topological charge in physics — a vortex’s circulation quantum, a domain wall’s charge, a flux quantum — is protected for this reason.

And the number of quanta in a mode is invariant, which is what made the argument about the boosted cavity work above and is worth stating separately, because it does the same job in the Unruh effect run the other way: an accelerating observer counts a different number from an inertial one, and that is a statement that the two are not related by a boost. Acceleration is where the invariance of a count stops, and that is one of the sharper ways to say why an accelerated frame is genuinely different from a moving one.

The exception is instructive. Under a boost, “the state of a mode” is the same state and the counting is the same counting. Under acceleration, the mode decomposition itself changes — what one observer calls a positive-frequency mode the other calls a mixture — so the two are not counting the same things, and the count is not invariant because the objects being counted are not the same objects.

A theorem about counting, drawn as a row of bars #

They cannot show that the invariance is a theorem about counting rather than about thermodynamics. The figures draw entropy as one bar among several, with a factor of one beside it, and nothing in that picture distinguishes it from a quantity that happens to have a transformation factor of one. What distinguishes it is that no factor could be otherwise.

Nor can they show the slicing. Every bar is a quantity attached to a system at a moment, and “at a moment” is exactly the thing two observers disagree about. The figure is drawn for a steady state, where the disagreement does not bite, and the careful version of the argument is a statement about four-currents rather than about bars.

And they cannot show that the energy density is not the whole story. A boosted photon gas has a momentum density and a shear stress as well, all parts of one tensor, and the energy density is one component of it. Two boxes with the same energy density and different momentum densities are different physical situations, and a bar chart of densities has no place to put the difference.

Still open: what the entropy of a gravitational field is, and whether it counts #

The argument here rests on entropy being a count, and every case where that is unproblematic is a case where the microstates are known. For a gravitational field they are not.

A black hole has an entropy, proportional to the area of its horizon, which is not allowed to shrink, and it behaves exactly like an entropy in every thermodynamic argument it appears in. Whether it is a count, and of what, is the central question of the subject: string theory supplies a count for certain idealised black holes that comes out right, the loop-quantum-gravity programme supplies a different one, and neither covers the astrophysical case.

The stakes here are specific. If black-hole entropy is a count, it is invariant for the reason above, and the second law extended to include horizons is a relativistic statement of the same kind. If it is not a count, the argument has to be made some other way. The generalised second law — that the sum of ordinary entropy and horizon area never decreases — has survived every attempt to violate it, which is evidence that it is a count of something, and is not a proof.

The habit worth carrying away is about where invariance comes from. A quantity is invariant when it is a number of things rather than a measurement of things, and asking which of the two a quantity is settles its transformation law faster than deriving one. Entropy, charge, particle number and every topological index are counts; energy, length, duration and every density are measurements; and the dividing line runs through the middle of thermodynamics.