The bath that pushes back

The two essays before this one establish what a moving observer sees when they look at a bath of radiation: a blackbody in every direction, hotter ahead and cooler behind, with no single temperature and with an entropy every observer agrees about.

This essay is about what the bath does back.

Where the force comes from #

A body at rest in isotropic radiation feels no net force. Light arrives equally from every direction, each photon delivers momentum, and the deliveries cancel.

Set the body moving. In its own frame the radiation is no longer isotropic — it is blue-shifted and intensified ahead and red-shifted and weakened behind, which is exactly the situation the first of these essays draws — and for anything much faster the whole sky crowds into a forward cone. So the momentum arriving from the front exceeds the momentum arriving from the back, and there is a net retarding force.

For a perfectly absorbing body of cross-section $A$ moving at $\beta$ through radiation of energy density $u$, the force is

$$F = \frac{4}{3}\,u A \beta\gamma^2,$$

which for small $\beta$ is the classical result and which is what the figure draws. Three features of it are worth separating.

It is proportional to the speed, not to the intensity alone. A body at rest feels nothing however bright the bath, because the anisotropy is created by the motion. A pressure is there either way; a net force is not.

It contains no property of the body except its area. A perfect absorber is a perfect absorber, and the drag on it is the same whatever it is made of.

And it is enormously weak in the case that exists. The microwave background has an energy density of $4.2\times10^{-14}$ joules per cubic metre — about a quarter of an electronvolt per cubic centimetre — so a square metre moving at half the speed of light through it feels $3.7\times10^{-14}$ newtons. A spacecraft would take longer than the age of the universe to notice.

The structure of the argument is the same one optical molasses uses on a single atom: two beams that cancel at rest, a Doppler shift that makes the one ahead louder, and a friction proportional to the speed. The difference is the bath — a whole sky at one temperature rather than two lasers — and the consequence is the same form of force with none of the resonance.

Why a preferred frame is not a contradiction #

The force above is the sharpest form of a fact that is often misstated: the microwave background picks out a frame, and relativity is untouched.

The distinction is between the laws and the state. Relativity says the laws are the same in every inertial frame. It does not say that everything in the universe has to look the same in every frame, and nothing in it could — the universe contains matter, and matter has a rest frame.

A body of water picks out a frame. A swimmer can tell whether they are moving relative to it, by the drag, and nobody has ever thought that a problem for relativity. The microwave background picks out a frame in exactly the same way and for exactly the same reason: it is a physical system with a state, the state is isotropic in one frame, and moving relative to it has consequences.

What would contradict relativity is a frame picked out by the laws — the nineteenth-century aether, in which light was supposed to travel at $c$ relative to the medium and at something else relative to anything moving through it. That was tested — by sending light out and bringing it back, which is the only way it has ever been measured — and is not the case. Light travels at $c$ through the microwave background in every frame, and the background’s rest frame has no effect on the propagation of anything.

So the two statements sit together without strain. There is a frame in which the universe’s radiation is isotropic; there is no frame in which the laws are simpler.

Where the same drag decides the answer #

The force is negligible for slow things. It is not negligible for fast ones, and the reason is a power.

For a single electron the cross-section is Thomson’s, and the power radiated into the bath goes as $\sigma_T c u \gamma^2$. The electron’s own energy goes as $\gamma$. So the cooling time goes as $1/\gamma$: the faster the electron, the shorter the time it has.

That is what makes the process selective rather than merely slow. In a population of electrons with a range of energies, the fastest lose their energy first, so the spectrum steepens from the top down. A radio galaxy’s synchrotron spectrum shows exactly that break, and its position gives the time since the electrons were accelerated — which is how the ages of radio lobes are measured.

The same scattering, seen from the other side, is how the radiation itself is changed: an electron that takes energy from a fast electron leaves as a photon of much higher energy, and a cloud of hot electrons sitting in the microwave background distorts its spectrum in a way that is measured toward galaxy clusters and used to find them.

The version that stops the highest-energy particles #

At the top of the energy scale the same mechanism produces a sharp cutoff rather than a gradual cooling.

A proton moving fast enough sees a microwave photon blue-shifted, in its own frame, to an energy above the threshold for producing a pion. Above that threshold the proton loses about a fifth of its energy per interaction and the interactions are frequent, so a proton above roughly $5\times10^{19}$ electronvolts cannot travel more than about fifty megaparsecs.

The mechanism is the drag above with the interaction changed. The proton’s Lorentz factor is about $5\times10^{10}$, which turns a millielectronvolt photon into a hundred-mega-electronvolt one in the proton’s frame — enough to make a pion — and the geometry of the collision is the same forward-weighted arrangement that produces the drag.

What it predicts is a cutoff in the spectrum of cosmic rays arriving at Earth, because anything above that energy from further than fifty megaparsecs has been stopped. The cutoff is observed, at roughly the predicted energy. What is not settled is whether the observed steepening is the cutoff or is the sources simply running out at that energy, and distinguishing the two requires knowing the composition — a heavier nucleus has a different threshold — which is the hardest measurement in the field.

The bath that is not negligible #

The microwave background’s drag is unmeasurable because the microwave background is cold. Nothing about the mechanism is weak.

That is the honest way to read the first figure. The drag is proportional to the energy density, energy density goes as the fourth power of temperature, and 2.725 kelvin is a very small number raised to the fourth power. A body near a star is in a bath at a few thousand kelvin over part of its sky, and the resulting drag is a real force with real consequences.

It has a name in that setting. A dust grain orbiting a star absorbs sunlight arriving radially and re-emits it isotropically in its own frame, and because the grain is moving, the re-emission carries away slightly more momentum in the forward direction than in the backward one. The grain loses angular momentum and spirals inward. A millimetre grain at the Earth’s distance from the Sun falls in within a few million years, which is why the zodiacal dust has to be resupplied continuously and why any dust in the inner solar system is young.

The mechanism is the same, with the bath replaced by a single hot direction and the absorption by absorption-and-re-emission. The drag is proportional to the speed and to the energy density, it vanishes at rest, and it does not depend on what the body is made of — all three statements survive the change of setting unaltered.

The number that decides which regime anything is in #

One ratio settles whether the drag matters for a given object, and it is worth writing down because it collapses every case above into one comparison.

The drag delivers a force $\tfrac43 uA\beta$, so the time to slow a body of mass $m$ by an appreciable fraction of its own speed is

$$t \sim \frac{m c}{\tfrac43 u A},$$

with the speed cancelling out at low $\beta$. It is a mass per unit area divided by an energy density, times the speed of light — and the mass per unit area is the only property of the body in it.

Put numbers on the three cases. A spacecraft is perhaps a hundred kilograms a square metre, so in the microwave background the time is $10^{29}$ seconds, which is $10^{12}$ times the age of the universe. A micrometre dust grain near the Sun is a gram a square metre in a bath of a kilowatt a square metre, giving a million years. An electron is $10^{-27}$ kilograms over a Thomson cross-section of $10^{-28}$ square metres — ten kilograms a square metre, which sounds large until the factor of $\gamma$ from the relativistic rate is included.

Everything about which regime an object is in is in its mass per unit area, and that is the quantity to reach for first. It is why dust is swept out of solar systems and boulders are not, why electrons cool and protons do not, and why nothing built by anybody will ever notice the microwave background.

A perfect absorber, a geometric cross-section, and Thomson scattering #

The drag expression assumes a perfect absorber. A reflecting body feels a different force — larger, because reflection reverses the photon’s momentum rather than merely stopping it — and a partially reflecting one something between. A body that re-emits what it absorbs also recoils from its own emission, which is isotropic in its own frame and therefore contributes nothing net, so the absorbing case is the clean one.

And it assumes the body is small compared with nothing in particular. The expression uses a geometric cross-section, which is right for a macroscopic object and wrong for anything comparable to a wavelength — for which the scattering cross-section is a function of size and the drag is correspondingly different. The microwave background’s wavelength is a millimetre, so the expression holds for objects larger than that.

The single-electron treatment assumes the scattering is Thomson’s, which requires the photon energy in the electron’s rest frame to be far below the electron’s rest energy. For the microwave background that holds up to a Lorentz factor of about $10^9$, above which the cross-section falls and the cooling slows — which is one of the reasons the highest-energy electrons behave differently from the extrapolation.

Nor is the background the only bath. Starlight and infrared from dust have comparable or larger energy densities inside a galaxy, so the cooling of fast electrons there is dominated by those rather than by the microwave background, and the curve drawn applies cleanly only in intergalactic space.

How fast the Earth is going, and how that was found #

The drag is unmeasurable and the anisotropy that produces it is not, and the Earth’s speed through the microwave background is among the better-determined numbers in astronomy.

The measurement is the dipole: the background is hotter in one direction by 3.36 millikelvin and cooler in the opposite one by the same, on a mean of 2.725 kelvin. Dividing gives $\beta = 1.23\times10^{-3}$, or 369 kilometres a second toward a point in the constellation Leo.

Two features of that number are worth stating.

It decomposes. The Sun orbits the Galactic centre at about 220 kilometres a second; the Galaxy moves within the Local Group; the Local Group falls toward a large-scale concentration of mass. The three contributions are known separately from other measurements, they are of comparable size, and their vector sum agrees with the dipole — which is a consistency check across three quite different kinds of astronomy.

And it is not a velocity relative to the universe. The background is isotropic, at each place, for an observer at rest with respect to the matter around them, and the matter is not all moving the same way. What the dipole measures is the Earth’s motion relative to the local matter, and a distant observer measures a different local motion the same way. There is no global frame; there is a local one everywhere, and they do not agree.

The second point is the one that most often goes missing. A preferred frame that varies from place to place is not a preferred frame in the sense relativity forbids. It is a velocity field — the same kind of object the wind is — and asking for the frame of the universe is like asking for the frame of the atmosphere.

The pattern on the sky that the force is made of #

They cannot show the anisotropy that produces the force. Every figure here plots a force or a time against a speed, and the mechanism is a pattern on the sky — a forward hemisphere that is brighter and bluer than a backward one. The drag is an integral over that pattern, and a curve of force against speed has integrated it away.

Nor can they show the recoil on the radiation. Momentum is conserved, so what the body loses the radiation gains, and the bath is very slightly stirred by everything moving through it. For the microwave background that is an utterly negligible perturbation and it is not zero, and the figures draw only one side of the exchange.

And they cannot show that the cooling of an electron is not a smooth process. A single electron scatters a photon every so often and loses a fraction of its energy each time, and the “cooling time” drawn is a mean. At the highest energies the losses are dominated by rare hard scatterings rather than by many soft ones, and the spread about the mean is comparable to the mean.

The drag on something that is not a body #

One more setting, because it is where the mechanism is most consequential and least recognisable.

A cloud of gas falling toward a young star, or a galaxy’s worth of gas settling toward its centre, is slowed by the radiation it is moving through — and unlike a solid body, a gas can be slowed selectively. Radiation drag acts on the electrons far more strongly than on the protons, because the Thomson cross-section goes as the inverse square of the mass, and the electrons drag the protons along electrostatically. The result is a force on a plasma that is proportional to the electron cross-section and carried by the whole mass.

That combination is why radiation pressure matters so much in astrophysics at all. The force is collected by the light particle and paid by the heavy one, so the effective cross-section per unit mass is the Thomson cross-section divided by the proton mass — a factor of eighteen hundred larger than if the electrons had to carry themselves.

It is the same ratio that sets the brightest a star of a given mass can be: above it, the outward force on the plasma exceeds gravity and the outer layers are blown off. So this drag, the ceiling on stellar luminosity, and the slowing of infalling gas are one calculation with three names.

Still open: where the highest-energy cosmic rays are made #

The cutoff this mechanism predicts is a horizon: anything above it must have come from within about fifty megaparsecs, which is a small volume containing a countable number of candidate sources. That should make the sources findable and it has not.

Two things obstruct it. The particles are charged, so intergalactic and galactic magnetic fields deflect them, and the deflection at the relevant energy is a few degrees for a proton and much more for a heavier nucleus. And the flux is tiny — a few particles per square kilometre per century — so even the largest arrays accumulate a few dozen events above the cutoff in a decade.

Correlations between arrival directions and catalogues of nearby active galaxies have been reported and have weakened as more data arrived, which is the usual history of a marginal anisotropy. What would settle it is either a much larger detector or a coincident neutrino, which is uncharged and points back.

The habit worth carrying away is about frames that are picked out. A preferred frame set by the state of something is ordinary and a preferred frame set by the laws would not be, and telling them apart is a matter of asking whether anything about the propagation changes or only something about what is there. The microwave background gives the universe a rest frame, the drag on a body moving through it is computable, and the speed of light is the same in every direction in every frame all the while.