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Relativistic thermodynamics — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A blackbody in every direction, at a different temperature in each. The spectrum of a blackbody at 100 kelvin in its own frame, seen by an observer it is moving past at 0.5 of the speed of light, in 5 directions. Each curve is a Planck spectrum exactly — the Planck form survives a Doppler shift, with the temperature multiplied by the shift — and the temperatures run from 57.74 kelvin looking one way to 173.21 looking the other. So the body is a perfect blackbody in each direction and has no single temperature. A thermometer placed in the radiation reads something between, and what it reads depends on where it is put and on how much of the sky it sees — which is the reason a transformation law for temperature was argued about for sixty years without being found.

    The body that has no temperature when it moves

    Energy, momentum, length, duration and field strength all change when the observer moves. Temperature was argued about for sixty years, with three transformation laws proposed and each defended by people making no mistake. The resolution is that a moving blackbody is a perfect blackbody in every direction at a different temperature in each — so a thermometer's reading depends on where it is put, and the quantity the law was for is not there.

    part 1 · relativity
  2. What a boost leaves alone, and what it does not. How each quantity of a box of blackbody radiation changes when the observer moves at 0.8 of the speed of light, a Lorentz factor of 1.667, on a logarithmic axis with one at the centre. The top four do not change at all, and the reason is the same in each case: they are counts, or logarithms of counts, or invariants built from four-vectors. A number of photons is a number, and every observer arrives at the same number. The rest change, by powers of the Lorentz factor that follow from the first four. And the entry that matters is the last two together: the energy density rises as the square of the factor while the entropy density rises as the factor itself, so the ratio between them that would define a temperature does not stay fixed — which is why the boosted radiation cannot be a blackbody at any temperature at all.

    The count that no observer can disagree about

    A moving body, it turns out, has no temperature. What it does have is an entropy, and every observer agrees about it — because entropy is the logarithm of a count of arrangements, and a count is a number. That one invariant, with energy and momentum being parts of one object, is enough to compute everything a temperature could not: what happens to the energy density, the entropy density, and the relation between them that having a temperature consists of.

    part 2 · relativity
  3. The bath does push back, by an unmeasurable amount. The retarding force on a perfectly absorbing body moving through isotropic radiation at 2.725 kelvin, against its speed, for three areas. Moving through a bath of radiation is not free: the radiation arriving from ahead is blue-shifted and more intense and the radiation from behind is red-shifted and weaker, so the body absorbs more momentum from the front than from the back and decelerates. A square metre at half the speed of light feels 3.7e-14 newtons. That is the reason a preferred frame exists without relativity being violated: the laws are the same in every frame and the radiation is not — it is a physical system with a state, and its state picks out the frame in which it is isotropic, exactly as a body of water does.

    The bath that pushes back

    Moving through a bath of radiation is not free. The light arriving from ahead is blue-shifted and more intense and the light from behind is weaker, so a body absorbs more momentum from the front than from the back and slows down. That drag picks out the frame in which the radiation is isotropic — without violating relativity, because the laws are the same in every frame and the radiation is not.

    part 3 · relativity
  4. Equilibrium is where the entropy peaks, and there the temperatures differ. Two cavities of radiation, one high in a gravitational field and one low, free to exchange energy, with the clock at the bottom running at 0.8 of the rate of the one at the top. What is conserved is the energy either would deliver to a distant observer, so energy held at the bottom counts for 0.8 of its local value. Across: the share of that conserved energy held at the top. Above: the total entropy of the two gases. Below: the temperature a thermometer in the top cavity reads, as a fraction of one in the bottom cavity. The entropy peaks at a share of 0.339, found by search, and there the top cavity is at 0.800 of the bottom's temperature — the clock-rate ratio exactly. Where the two local temperatures are equal, at a share of 0.556, the entropy is 1.82 per cent below its peak and energy still flows downward, into the deeper cavity.

    The column that is hotter at the bottom

    Two bodies in equilibrium have the same temperature — that is what equilibrium was supposed to mean. In a gravitational field it is false. A column left alone until nothing in it changes is warmer at the bottom by exactly the factor by which clocks there run slow, a part in ten million billion per metre on the Earth and more than a per cent across the outer kilometre of a neutron star, and near a black hole's horizon the equilibrium temperature grows without limit.

    part 4 · relativity
  5. The stiffness that falls below four-thirds. The first adiabatic exponent Γ₁ — how steeply pressure rises when a gas is squeezed without heat escaping — of hot oxygen with its radiation and its equilibrium electron–positron pairs, against temperature, at densities of 10³, 10⁴, 10⁵, 10⁶ g/cm³. At 10³ g/cm³ its least value is 1.222, near 1.2 billion kelvin; at 10⁴ g/cm³ its least value is 1.235, near 1.3 billion kelvin; at 10⁵ g/cm³ its least value is 1.285, near 1.8 billion kelvin; at 10⁶ g/cm³ its least value is 1.339, near 3.1 billion kelvin. A star can hold itself up only if its core's Γ₁ averages above 4/3, the value for a gas of light alone; a core that dips below it has no stable size, and the pair production that pulls it below is strongest where the gas is hot and thin — the cores of the most massive stars.

    The heat that turns into matter

    Thermodynamics counts particles and relativity lets them be made. Heat a gas to a billion kelvin and its light carries enough energy to create electrons and positrons, which the gas then holds in equilibrium like any other ingredient. The heat spent making them raises neither the temperature nor the pressure, so the gas grows soft — softer, in the cores of the most massive stars, than a star can survive. Those stars collapse, ignite their oxygen all at once and blow themselves entirely apart; and the same pairs, vanishing a few seconds after the Big Bang, left the universe's light hotter than its neutrinos by a factor that is the cube root of eleven over four.

    part 5 · relativity
  6. Heat that arrives everywhere at once, and heat that waits. The temperature spread from a point of heat released at time zero, in one dimension, at times 0.5, 2, 8 relaxation times, from the diffusion equation (dashed) and from Cattaneo's telegraph equation (solid) with the same diffusivity, distances in units of the signal speed times the relaxation time. At 0.5, the diffusion equation has put 62 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 78 per cent is still in the two fronts; at 2, the diffusion equation has put 32 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 37 per cent is still in the two fronts; at 8, the diffusion equation has put 5 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 1.8 per cent is still in the two fronts. The fronts, drawn as the vertical marks, carry heat outward at the signal speed and decay; behind them the telegraph solution fills in and, after many relaxation times, becomes the diffusion equation's Gaussian.

    The heat that arrives before it could

    Release a pulse of heat at one point, and the diffusion equation — the law that has described heat flow since Fourier — says the temperature rises everywhere at once, however far away. A thermometer a kilometre off registers a change the instant the heat is released. That is an infinite signal speed, and relativity forbids it. The flaw is harmless for a kettle and fatal for a relativistic fluid. The repair is a relaxation time: heat flow takes a moment to respond to a temperature gradient. With it the equation carries heat at a finite speed, sends a sharp front ahead of the spreading warmth, and turns short-wavelength heat into a damped wave.

    part 6 · relativity
  7. The stiffness a two-solar-mass star demands. The heaviest neutron star the model matter can support, in solar masses, against the constant square of the sound speed assumed above 300 MeV/fm³, as a fraction of c². At the conformal value, one-third (dashed), it is 1.92 solar masses; the heaviest star the model can make rises to 2.97 at the causal limit. To hold up the 2.08 solar masses of J0740+6620 (shaded) the square of the sound speed must exceed 0.394 of c² somewhere above that density — above the conformal one-third, by this model's reckoning. The conclusion depends on the model's low-density part, which the next figure varies.

    The sound speed a neutron star needs

    Heat any gas without limit and the speed of sound in it climbs towards a ceiling: not the speed of light, but light's speed divided by the square root of three, the sound speed of a gas of massless particles. For decades it was a natural guess that no matter could beat that ceiling, since the densest matter ought to behave like a gas of free quarks. Neutron stars say otherwise. A star of twice the Sun's mass, packed into a ball twenty-five kilometres across, can only be held up if its core is stiffer than the ceiling allows — unless nuclear physics fails at densities where it is believed to hold. Somewhere inside the heaviest neutron stars, sound almost certainly travels faster than it can in any free gas.

    part 7 · relativity
  8. What the heat shares out, and what it does not. Two averages per particle in the Maxwell–Jüttner gas, in units of kT, against kT/mc² on a logarithmic scale: the momentum times the velocity, p·v, whose average over the three directions gives the pressure, and the kinetic energy. The first is 3 at every temperature, computed here by averaging over the distribution and equal to 3 to four decimal places across the whole figure — so the pressure is exactly nkT, the ideal gas law, from a gas colder than a star to one hotter than the early universe. The kinetic energy is not shared out the same way. It is 1.506 kT when the gas is cold, Boltzmann's 3/2, and rises to 3.00 kT when it is ultra-relativistic, because for a fast particle the energy is p·c and p·v, which is what equipartition hands out, is the whole of it rather than twice it.

    The gas law relativity leaves exact

    Heat a gas until its particles move at nearly the speed of light and almost everything Maxwell and Boltzmann said about it changes. The speeds pile up just below c instead of spreading out. The energy per particle climbs from three halves of kT to three. The heat capacity doubles, and the ratio of heat capacities falls from five-thirds to four-thirds. One law survives untouched: the pressure is still exactly nkT, at a thousand degrees or a hundred thousand million. The reason it survives is the reason the others change. What heat hands out equally to every way of moving is not energy but momentum times velocity, and for slow particles those happen to differ only by a factor of two.

    part 8 · relativity
  9. What one scattering does to a photon's energy. The average fractional change in a photon's energy when it scatters off an electron in a gas at temperature T, against the photon's energy: (4kT − ε)/mc², for gases at 10⁶, 10⁷ and 10⁸ K. A photon below 4kT gains energy, taking it from the electrons' motion; one above loses it, as recoil, to the electrons. The crossing sits at ε = 4kT: 0.34 keV at 10⁶ K, 3.45 keV at 10⁷ K, 34.47 keV at 10⁸ K. The changes per scattering are small, a few per cent at most here, but a gas whose electrons scatter each photon many times, or that is bathed in light for long enough, is heated or cooled until the gains and losses balance over the whole spectrum.

    The temperature a gas takes from the light around it

    A thin, ionised gas bathed in light settles at a temperature that has nothing to do with how much light there is and everything to do with its colour. Every time a photon scatters off an electron, the two exchange a little energy: a photon softer than four times the electrons' thermal energy gains, a harder one loses by recoil. Balanced over a spectrum, the rule sets the gas at a quarter of the light's mean photon energy. That puts gas near a quasar at tens of millions of kelvin, and it caps the heating once photons approach the electron's own rest energy. For blackbody light the rule falls four per cent short, and the shortfall is the photon being a boson.

    part 9 · relativity

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