The body that has no temperature when it moves

Relativity has an account of nearly every quantity that changes when the observer moves. Lengths contract, durations dilate, energy and momentum mix, densities transform, electric and magnetic fields turn into one another, and every one of those has a definite law with a Lorentz factor in it.

Temperature does not, and the reason is not that nobody worked hard enough.

The list is worth having in front of one, because it makes the gap conspicuous. A moving rod is shorter by a definite factor; a moving clock runs slow by a definite factor; a moving charge’s field is flattened by a definite factor; the energy and momentum of a moving body are the parts of one object with a definite rule for mixing them. Every one of those is a theorem with a proof and a number.

In 1907 Planck and Einstein, independently, derived that a moving body’s temperature is $T_0/\gamma$ — that a moving body is colder. The derivation was accepted for fifty-six years. In 1963 Ott derived $\gamma T_0$, that a moving body is hotter. In 1966 Landsberg argued that it is unchanged. None of the three contains an error.

What is actually there, computed #

Set the transformation laws aside and ask what an observer would see.

A blackbody at $T_0$ in its own frame radiates isotropically with the Planck spectrum, which is a statement about spectra and not about heat. Let it move past at speed $\beta$. Radiation arriving from the direction $\theta$ in the observer’s frame is Doppler-shifted by a factor $D(\theta) = 1/\gamma(1-\beta\cos\theta)$, which is greater than one ahead and less than one behind.

The Planck form has a property that decides everything. It is exactly invariant under a Doppler shift: the spectrum arriving from any direction is again a Planck spectrum, at temperature $T_0 D(\theta)$.

So the moving body is a perfect blackbody, in every direction, at a different temperature in each. Nothing has been lost, nothing has been smeared, and there is no single number to report.

A thermometer placed in that radiation comes to some equilibrium, and what it comes to depends on how much of the sky it sees and in which directions. A small absorber facing forward reads 173 kelvin; one facing backward reads 58; and the forward direction is where nearly all the light is for anything moving fast, one seeing the whole sky reads something in between that depends on how it weights the directions. All three are correct readings of a real temperature of the thermometer, and none of them is a property of the body.

Why the sum is not a blackbody either #

The natural repair is to average over the sky and call the result the temperature. It does not work, and the reason is arithmetic rather than conceptual.

A sum of Planck spectra at different temperatures is never a Planck spectrum. That is the same statement as a sum of exponentials at different rates never being an exponential, and it has the same short proof: the family is not closed under addition.

The departure is largest where it is easiest to see. At short wavelengths a Planck spectrum falls off exponentially, so a hotter contribution — even a small one from the forward hemisphere — dominates the tail by orders of magnitude. The averaged spectrum has a blue excess no single temperature can reproduce.

And the departure is not a technicality. It is the measurement by which a moving blackbody could be distinguished from a stationary one of any temperature, and it is what makes the question well posed observationally: there is no temperature, and the evidence that there is none is a spectrum that no temperature fits.

The one place this is measured #

The universe supplies a blackbody, and something moving relative to it.

That dipole is measured to four significant figures and is removed before anything else in the microwave sky is looked at — and what is left after it is removed is the pattern the sky is written in. What it measures is the Earth’s velocity — the Sun’s, plus the Galaxy’s, plus the Local Group’s — relative to the frame in which the background is isotropic.

Two things about it bear on the argument.

The background is a blackbody in every direction, and each direction has a different temperature. That is exactly the situation drawn above, at a very small $\beta$, and it is the observational fact rather than an idealisation. Every measurement of the background’s spectrum reports a Planck spectrum and a temperature, and the temperature depends on which patch of sky was measured.

And the second-order term has been seen. The exact shift is not a cosine: expanding it gives a dipole at first order in $\beta$ and a term at second order, which at that speed is 2.07 microkelvin. That term is a quadrupole in the sky, it is a prediction with no free parameters, and separating it from the cosmological quadrupole is one of the standard checks a microwave experiment makes.

How the dipole is used, and what it is not #

The microwave dipole is the only measurement in the subject, and it is worth being precise about what it measures, because it is routinely over-read.

It measures a velocity relative to a state, not relative to space. The background is a physical system filling the universe, it has a frame in which it is isotropic, and the Earth moves at 369 kilometres a second relative to it. That no more contradicts relativity than measuring a ship’s speed through water contradicts it. The laws are identical in every frame; the background’s state is not, and a state may perfectly well pick out a frame.

The frame it picks out is local. The background is isotropic at each place for an observer at rest with respect to the matter there, and the matter is not all moving the same way. The measured dipole is the sum of the Sun’s motion round the Galaxy, the Galaxy’s motion within the Local Group, and the Local Group’s motion toward a large-scale overdensity — three contributions of comparable size, known separately, which is why the total can be decomposed.

And it is the best-measured velocity in astronomy. Four significant figures, from an amplitude of three millikelvin on a background of two point seven, which is an extraordinary measurement to have made twice with different instruments and got the same answer.

What the dipole is not is evidence for an aether. An aether would have been a medium for light, whose rest frame the speed of light was supposed to be measured relative to. The microwave background is not a medium for anything: light travels through it at exactly $c$ in every frame, and what its rest frame does is the ordinary thing any material’s rest frame does.

What the three laws were each computing #

The three proposals are not nonsense and it is worth saying what each is.

All three start from the first law and require that $dS = \delta Q/T$ hold in every frame. Entropy is invariant, for reasons the invariance of a count is about, so whatever $\delta Q$ transforms as fixes what $T$ transforms as. And $\delta Q$ has no unambiguous transformation either, because splitting a change in energy into heat and work requires deciding what counts as work when the body is moving — and a body being pushed along has a rate of doing work that depends on the frame.

Planck and Einstein took heat to transform as $\delta Q_0/\gamma$, which follows if the energy transferred is treated as the time component of a four-vector whose spatial part is fixed by the body’s momentum. Then $T = T_0/\gamma$.

Ott took the transfer of heat to a moving body to include the work done in keeping it moving at constant velocity while its mass rises, giving $\delta Q = \gamma\,\delta Q_0$ and $T = \gamma T_0$.

Landsberg argued that a quantity defined by counting — as entropy is, and as the ratio $\delta Q/T$ must be if entropy is — should have both of its parts invariant, giving $T = T_0$.

Each is a self-consistent bookkeeping and the three differ in where a boundary is drawn. That is the diagnosis: the disagreement is about a definition, and it survived sixty years because no measurement distinguishes definitions.

Why the Planck form survives a shift at all #

The invariance the whole argument rests on is worth a paragraph, because it is not obvious and it is the reason the situation is as clean as it is.

Specific intensity per unit frequency divided by the frequency cubed is a Lorentz invariant. That is a theorem about photon phase-space density and it holds for any radiation whatever — it is why surface brightness is conserved along a ray in flat space, and it is the same statement that makes a beam’s brightness impossible to increase with lenses.

Apply it to a Planck spectrum. The observed intensity at frequency $\nu$ is $D^3$ times the emitted intensity at $\nu/D$, so

$$I_\nu^{\text{obs}}(\nu) = D^3 \cdot \frac{2h(\nu/D)^3}{c^2}\cdot\frac{1}{e^{h\nu/DkT_0}-1} = \frac{2h\nu^3}{c^2}\cdot\frac{1}{e^{h\nu/k(DT_0)}-1},$$

which is a Planck spectrum at $DT_0$. The cube from the invariance cancels the cube in the Planck prefactor exactly, and the shift moves into the exponential as a change of temperature.

That cancellation is specific to the Planck form. A spectrum of any other shape is Doppler-shifted into a spectrum of a different shape, and only a blackbody is shifted into a blackbody. It is the same fact that makes the microwave background’s spectrum a measurement rather than a fit: whatever redshift it has suffered since it was emitted, it remains a Planck spectrum, and its temperature is the only thing that changed.

Equilibrium in the rest frame, free streaming, and heat left open #

The body is assumed to be in equilibrium in its own frame, which is what makes each direction’s spectrum Planckian. A body that is not in equilibrium to begin with has no temperature in any frame, and the argument here is about the case where there is one to start with.

The radiation is treated as free-streaming. A real moving object is surrounded by its own emission and by whatever is around it, and the spectrum an observer sees depends on the geometry of the whole arrangement. The clean statement is about the radiation emitted by a moving blackbody surface, measured far away.

And nothing here settles the transformation of heat. The argument is that temperature has no frame-independent definition for a moving body, which leaves the bookkeeping question open rather than answering it. A choice still has to be made to do relativistic thermodynamics at all, the usual modern one is to work with covariant quantities — a four-vector for heat flow and a temperature four-vector whose magnitude is the rest temperature — and that choice reduces to Planck’s in the cases where the old argument was conducted.

Which other quantities this happens to #

Temperature is not the only thermodynamic quantity to lose its definition under a boost, and the pattern of which do and which do not is worth having in one place, because it is not the pattern one would guess.

Pressure survives. A fluid’s pressure in its own rest frame is a scalar, and it appears in the stress-energy tensor as a diagonal element in that frame. It transforms as part of a tensor, with no ambiguity, and it is the quantity relativistic hydrodynamics is written in terms of.

Chemical potential survives, for the same reason as entropy: it is an energy per particle, and both the energy and the particle count are defined in the rest frame, so their ratio is.

Heat capacity does not. It is a derivative of heat with respect to temperature, and neither of those two quantities has an unambiguous transformation, so their ratio has two ambiguities rather than none.

And the free energies do not, being combinations of an energy that transforms and a temperature that does not have a law. The Gibbs free energy of a moving body is as ill-defined as the temperature in it.

What the survivors have in common is that they are defined in one frame — the rest frame — and then carried around as components of a tensor or as invariants. What the casualties have in common is that they are defined by a process whose split into heat and work is frame-dependent.

That is the general rule and it is more useful than any of the individual results. A thermodynamic quantity has a relativistic meaning when it can be defined by pointing at a state, and does not when its definition requires pointing at a transfer.

What a moving thermometer actually does #

The argument so far is about what is there. A measurement is a separate question and it has a definite answer, which is worth working out because it shows how much of the ambiguity is real.

Put a small perfectly absorbing sphere in the radiation of a body moving past. It absorbs from every direction, weighted by the Doppler-shifted intensity in each, and it re-emits isotropically in its own frame. It comes to a steady state when the two balance, and the temperature it settles at is the fourth root of the absorbed flux divided by the Stefan–Boltzmann constant.

That is a definite number, it is computable, and it is a property of the sphere and its situation rather than of the moving body. Change the sphere’s own velocity and the number changes; make it a flat plate facing forward and it changes; put it behind a shield and it changes again.

So there is no ambiguity about what a thermometer reads and there is no thermometer reading that is a property of the moving body. Those two statements together are the content of this essay. A measurement’s result being well defined does not make the quantity it was supposed to measure well defined, and here the measurement is of the instrument’s equilibrium with a field rather than of any temperature the field has.

The same is true of the microwave background, at a level that matters: a detector pointed at one patch of sky measures that patch’s temperature, and the sky’s temperature is a map rather than a number. It is only because the variation is a part in eight hundred that the map is usually quoted as one figure with an error bar — and everything cosmology does with the background is in what is left after the number is subtracted.

The thermometer, which is where a temperature actually comes from #

They cannot show the thermometer. Every figure here draws the radiation, and what a temperature measurement returns is the equilibrium state of an instrument placed in it — which depends on the instrument’s absorptivity, its shape, and which directions it faces. Two thermometers at the same point in the same radiation field can read different temperatures if they look different ways, and that is not an instrumental defect.

Nor can they show that the rest frame is special without being preferred. The body has a frame in which its radiation is isotropic, and that frame is picked out by the body’s own state rather than by any law. Nothing about relativity is violated: the laws are identical in every frame and the state is not, which is the same situation as a body of water having a rest frame.

And they cannot show the second-order term at its true size. The dipole figure draws the exact shift and a pure cosine, and the difference between them is two microkelvin on a three-millikelvin dipole — a part in sixteen hundred, which is a fraction of a line width at the scale drawn. The number is printed because it cannot be seen.

Still open: what temperature means for a gas that is flowing #

The problem here has a working counterpart in relativistic hydrodynamics, where a fluid element’s temperature has to be defined while the fluid moves relative to the frame the equations are written in, and the modern treatment defines it in the element’s own rest frame and carries a four-velocity alongside.

That works and it leaves a residual question about how heat conduction is described. The naive relativistic generalisation of Fourier’s law allows a temperature disturbance to propagate faster than light and is unstable, and the repairs — adding a relaxation time, so that a heat flux takes a finite time to respond — introduce a parameter that is not fixed by the equilibrium thermodynamics. Which of the several proposed formulations is right, and how their parameters relate to microscopic physics, is an active question in the modelling of heavy-ion collisions and of neutron-star mergers, where the fluid is genuinely relativistic and the dissipation genuinely matters.

The habit worth carrying away is about quantities that turn out not to exist. When several careful derivations of one quantity disagree and no measurement separates them, the first thing to check is whether the quantity is defined at all. Temperature is a property of equilibrium, a boost destroys the isotropy that equilibrium requires, and sixty years of argument was about the transformation law for something that a boost had already taken away.