The gas law relativity leaves exact

The body that has no temperature when it moves and the count that no observer can disagree about asked what thermodynamics looks like to an observer moving past a hot body, and found that temperature is a rest-frame quantity while entropy is agreed by everyone. Later essays heated matter until it made particle pairs, let heat diffuse no faster than light, and followed the sound speed a neutron star needs, which in an ideal gas creeps up towards $c/\sqrt{3}$ as the gas gets hotter and never reaches it.

This essay stays in the gas’s own frame and heats the gas itself, until its particles move at nearly the speed of light, and asks which of the familiar laws of the ideal gas survive. Ferencz Jüttner worked out the distribution of speeds in such a gas in 1911, six years after special relativity, and the answer is uneven in an instructive way. Almost every property of the gas changes — the distribution of speeds, the energy per particle, the heat capacity, the inertia. One law does not change at all, and it is the one first written down for slow gases in the seventeenth century: the pressure is exactly nkT.

Speeds that cannot pass c #

In a gas in equilibrium the probability of a particle’s state is weighted by Boltzmann’s factor $e^{-E/kT}$, as the speeds in a still room found for air, and the only change relativity makes is to the energy: $E = \gamma mc^2$ rather than $\tfrac12 mv^2$. Counting states by momentum, which is what the phase space of a particle is made of, gives Jüttner’s distribution,

$$f(\gamma)\,d\gamma \propto \gamma^2\beta\, e^{-\gamma mc^2/kT}\,d\gamma,$$

normalised by a Bessel function. When kT is small compared with the rest energy the exponential confines the particles to γ close to one and the distribution is Maxwell’s. When kT is comparable with mc² the high-energy tail is populated, and since every particle’s speed is below c, the distribution of speeds is squeezed against that ceiling: at kT equal to the rest energy the most probable speed is 0.98c, at three times it 0.996c.

For electrons, whose rest energy is 0.511 MeV, kT equals mc² at about six thousand million kelvin. Such temperatures occur in the plasma falling into black holes and in the universe’s first seconds, and the distribution matters wherever hot electrons scatter light, because their energies decide how far up in frequency they push it.

The pressure is a count of p·v #

Pressure is a rate of arrival found the pressure of a gas by counting the momentum its particles carry across a surface each second. A particle with velocity v carrying momentum p contributes in proportion to the product of its momentum along one direction and its velocity along the same direction, and averaging over directions gives

$$P = \tfrac13\, n\,\langle \mathbf{p}\cdot\mathbf{v}\rangle.$$

This holds at any speed; nothing in it assumed the particles were slow. For slow particles p·v is mv², twice the kinetic energy, and the pressure is two-thirds of the kinetic energy density. That is the step the usual derivation takes, and it hides what is actually computed. The velocity is the derivative of the energy with respect to momentum, $v = dE/dp$, so p·v is $p\,dE/dp$, and its average under Boltzmann’s weighting can be done for any relation between E and p by integrating by parts:

$$\left\langle p_x \frac{\partial E}{\partial p_x}\right\rangle = kT.$$

The factor $\partial E/\partial p_x\, e^{-E/kT}$ is $-kT$ times the derivative of the exponential, and moving the derivative onto $p_x$ leaves kT. Three directions give $\langle p\cdot v\rangle = 3kT$, and the pressure is $nkT$ — whatever the particles’ mass, speed or energy. The figure checks it by averaging over Jüttner’s distribution: three, to four decimal places, from a thousandth of the rest energy to three hundred times it.

A distribution that had to be checked #

Jüttner’s result looks inevitable now, but it was argued over for most of a century. The difficulty is that a distribution of speeds depends on the frame in which it is written, and a relativistic gas has a preferred frame, the one in which it is at rest as a whole, only by virtue of its own motion. Several alternatives were proposed, differing in how the phase space of a relativistic particle should be counted, and they predict different tails at high energy. Because no laboratory gas of free particles reaches temperatures where the difference shows, the question was settled in 2007 not by experiment but by simulation: a gas of relativistic particles colliding elastically, started far from equilibrium and left to settle, reached Jüttner’s distribution and not its rivals.

That is the right kind of evidence for a statement of statistical mechanics, and it also shows where the content lies. The distribution follows from two ingredients, Boltzmann’s weighting of each state by $e^{-E/kT}$ and the counting of states as equal volumes of momentum, and the second is the one that was in doubt. Count states uniformly in momentum, and Jüttner’s distribution follows; count them any other way, and the collisions in a simulation correct it.

A pile-up in one variable only #

The squeezing of the speed distribution against c is real, but it is a feature of the variable as much as of the gas. Plotted against momentum instead of speed, the same distribution is smooth and unbounded: the number of particles with momentum between p and p + dp is proportional to $p^2 e^{-E(p)/kT}$, which for a hot gas falls off as a plain exponential in p with no ceiling at all. A particle’s momentum can grow without limit while its speed approaches a fixed value, so an interval of momentum at high energy maps onto a tiny interval of speed, and the probability that was spread over the first is crowded into the second.

Which picture is more useful depends on the question. For the pressure, the momentum carried across a surface, what matters is p·v, and both factors enter. For the light a hot electron scatters, the energy it can give a photon depends on its γ, which grows with momentum, and the long tail in momentum is what makes a hot plasma’s scattered spectrum extend so far. The speed distribution’s pile-up, striking as it is, matters mainly for effects that depend on speed alone, such as how quickly particles cross a region.

What equipartition actually shares #

The integration by parts is the equipartition theorem in its general form, and it says something more careful than the version usually taught. Half a kT for every way of moving gave each quadratic term in the energy kT/2. The general statement is that each momentum component gets kT of $p\,\partial E/\partial p$. For a quadratic energy, $p\,\partial E/\partial p$ is twice the energy, so the energy per component is kT/2. For any other dependence, the share of energy is something else, as the share that is not half a kT found for other shapes of potential.

For an ultra-relativistic particle, $E = pc$, and $p\,\partial E/\partial p$ is the energy itself. Each direction then gets kT of energy, three in all — twice the slow gas’s three halves. In between, the figure shows the kinetic energy per particle rising smoothly from the first value to the second as the particles become relativistic. The pressure, meanwhile, was never tied to the energy at all; it was tied to p·v, which is what heat distributes, and so it stays at nkT while the energy doubles. The photon gas of the gas that nobody counted is the extreme case, with pressure one-third of its energy density, and an ultra-relativistic gas of massive particles approaches the same ratio from the other side.

A heat capacity that doubles #

The heat capacity follows directly. A cold gas of point particles takes three halves of k per particle per degree; an ultra-relativistic one takes three. The rise is gradual, reaching most of the way by the time kT equals the rest energy. Because the pressure is still nkT, the work done by a heated gas expanding at constant pressure is still k per particle per degree, so the heat capacity at constant pressure is always one more than at constant volume, and their ratio, the adiabatic index, falls from five-thirds to four-thirds.

That fall matters to stars. A self-gravitating ball of gas is stable only if its adiabatic index stays above four-thirds, the condition the heat that turns into matter found being violated when heat goes into making particle pairs. An ideal gas of relativistic particles approaches four-thirds from above without making anything, which is why a star supported mainly by the pressure of relativistic particles — its radiation, or its relativistic electrons — is close to the edge of stability. The same index fixes the sound speed whose approach to $c/\sqrt{3}$ the neutron-star essay drew.

The virial that a star cannot escape #

The same average appears in the most important balance in astrophysics. Weighing what cannot be put on a scale used the virial theorem to relate a bound system’s motions to its gravity, and the kinetic term in that theorem is, exactly, the sum of p·v over the particles — the same quantity that gives the pressure. For slow particles it is twice the kinetic energy, and a star in equilibrium has a total energy of minus its kinetic energy: it is firmly bound, and must lose energy to contract.

For relativistic particles p·v becomes the energy itself, not twice it, and the bookkeeping changes. A star whose pressure comes from relativistic particles has a total energy approaching zero, barely bound, which is the virial-theorem version of the adiabatic index approaching four-thirds. It is also why the mass no cold matter can hold up exists: in a white dwarf heavy enough for its degenerate electrons to be relativistic, their pressure can no longer grow faster than gravity under compression, and above the Chandrasekhar mass there is no equilibrium at all. The ideal gas law and the instability of the heaviest stars are the same average read two ways.

Heavier when hotter #

The energy of a hot gas has inertia, as the box of light that weighs something found for radiation, and in a moving fluid the inertia is not the energy alone but the energy plus the pressure times the volume, the enthalpy. Pushing a hot gas means pushing its pressure’s work along with it. Per particle the enthalpy is $mc^2K_3/K_2$, which for a cold gas is the rest mass plus a small correction, $5kT/2$, and for a hot one approaches $4kT$, four times the temperature in energy units and with the rest mass negligible.

That matters for the jets of plasma that leave the neighbourhoods of black holes and neutron stars, which are hot enough that their inertia is mostly heat. A jet’s speed, how it decelerates as it pushes into the surrounding gas, and how its shocks heat it further, all depend on its enthalpy rather than its rest mass, and the relativistic equations of fluid dynamics are written with exactly that quantity where Newtonian ones have the density.

Where gases are relativistic #

The electron is the particle for which the corrections matter in practice, because its rest energy is small. The gas between the galaxies of a large cluster is at about a hundred million kelvin; its electrons have kT about a fiftieth of their rest energy, mildly relativistic, and when they scatter the photons of the cosmic microwave background, the distortion they leave in its spectrum — the Sunyaev–Zel’dovich effect — carries relativistic corrections that observers must include to infer the gas’s temperature correctly. In the flows of gas spiralling into black holes the electrons can reach ten times their rest energy, and the X-rays they make by scattering lower-energy light are shaped by the Jüttner distribution’s tail.

The protons in the same places are two thousand times heavier and far from relativistic, so a hot plasma can be a relativistic gas of electrons and a classical gas of ions at once, and even at different temperatures, since the two exchange energy slowly when collisions are rare. Only in the first microsecond of the universe were the protons and their constituents relativistic too.

Reading a temperature of thousands of millions of degrees #

No thermometer can be put into such a gas, so its temperature is read from its light, and the reading depends on the distribution. A hot electron that scatters a low-energy photon boosts it, on average, by a factor of about $\tfrac43\langle\gamma^2\beta^2\rangle$ in energy, and repeated scatterings in a cloud of hot electrons turn a beam of soft photons into a spectrum stretching up to energies of order kT, with a cutoff whose shape reflects the electrons’ energy distribution. The X-ray spectra of black holes accreting gas from a companion star show exactly such cutoffs, at a hundred kiloelectronvolts or so, and fitting them with scattering by a Jüttner gas gives electron temperatures near a thousand million kelvin, where kT is a fifth of the rest energy and the relativistic corrections are already tens of per cent.

The microwave background seen through a cluster of galaxies gives the same kind of reading at a gentler temperature. The cluster’s electrons scatter a small fraction of the background’s photons to higher frequencies, leaving a deficit at low frequencies and an excess at high ones, and the precise frequency at which the distortion crosses from one to the other shifts with the electrons’ temperature through the relativistic part of their distribution. Measuring that shift is one of the few ways to take the temperature of the gas between galaxies independently of its X-rays.

In both cases the pressure, nkT, is what holds the gas up against gravity, and it is the quantity the arguments about equilibrium use; the distribution’s shape is what the light reveals. The two were computed above from the same weighting, and the exactness of one is what makes the other worth fitting.

Where the ideal gas stops #

The figures describe a gas of non-interacting point particles in equilibrium, obeying classical Boltzmann statistics. At the temperatures where electrons are relativistic, other effects intervene: pairs of electrons and positrons appear when kT approaches the electron’s rest energy, adding particles whose number is not fixed, and that is the regime of the pair-instability essay rather than of this one. At high densities quantum statistics replaces Boltzmann’s, and the degenerate electrons of a white dwarf are relativistic because of their density, not their temperature, with a different equation of state. And real hot plasmas are rarely in equilibrium: their electron distributions have tails built by acceleration rather than by collisions.

The pressure’s exactness has a domain too. It rests on the gas being ideal, with no forces between the particles beyond brief collisions, and on Boltzmann’s weighting. Interactions add a correction to the pressure, the virial correction, at any speed; quantum statistics changes the integration by parts. Within the ideal classical gas, nkT is exact at every temperature.

Still open: what temperature a relativistic plasma settles at #

In the most extreme astrophysical plasmas — the gas flowing into the black hole at the centre of the Galaxy, imaged by the Event Horizon Telescope — collisions between particles are so rare that electrons and ions do not share their energy in the time the gas takes to fall in. Their temperatures can differ by large factors, and the ratio between them, which sets how bright the gas looks in radio waves, is not fixed by any equilibrium argument. Simulations of such flows have to assume a prescription for how the heat is divided, and which one is right is being argued with the help of the images. Whether these electrons even have a Jüttner distribution, or one with a power-law tail, is part of the question.

The habit worth carrying away is to ask what a conservation or equipartition law actually conserves or shares. Heat shares out $p\,\partial E/\partial p$, kT per direction, whatever the particles’ energy–momentum relation, so the pressure n⟨p·v⟩/3 is exactly nkT at every temperature — while the kinetic energy per particle runs from 3kT/2 to 3kT, the heat capacity doubles and the speeds crowd against c, 0.996c at kT = 3mc². The ideal gas law was never about energy, which is why relativity leaves it alone.