The river that sound cannot swim up

Speeds that refuse to add found that relativistic velocities combine by a rule that never lets their sum reach the speed of light. The space that speeds live in read that rule as the geometry of a curved space of velocities. The drag that was only an addition took it back to Fizeau’s moving water, where light is carried along by only a fraction of the water’s speed, and found the fraction to be the first term of the relativistic rule. That essay ended by pointing to composition in a medium whose own motion changes from place to place, where “the drag becomes a field rather than a number”.

This essay takes that step, starting with sound, where the addition is simpler. Sound travels at a fixed speed relative to the air or water that carries it, and there is nothing relativistic about the air, so the ground sees the sound’s speed plus the medium’s. Make the medium’s speed vary from place to place and the addition, applied point by point, turns into a geometry: sound follows paths that are the null lines of an effective spacetime set by the flow. Where the flow passes the speed of sound that geometry has a horizon, as a black hole does. Unruh pointed this out in 1981, and it has become the basis of a laboratory programme for testing the most famous prediction about black holes that no telescope can check.

Sound cones in a flowing fluid #

A sound made at some point spreads out at the speed of sound, $c$, relative to the fluid. If the fluid moves at $v$ along a pipe, the two fronts of the sound move relative to the pipe at $v + c$ downstream and $v - c$ upstream. On a diagram of time against position, the two fronts from each point form a cone, the region that sound made at that point can reach.

The figure draws the cones for a flow that accelerates smoothly through a constriction, from 0.55 times the speed of sound upstream to 1.3 times downstream. Far upstream the flow is slow and each cone straddles the vertical: sound goes both ways. As the flow speeds up the cones lean downstream, because the fluid carries them. At $x = 0.20$ the flow reaches the speed of sound and the cone’s upstream edge stands exactly vertical: sound aimed upstream stays where it was made. Beyond that point both edges point downstream, and sound made there can reach only places downstream of where it was made.

That is exactly the structure of the light cones around a black hole, drawn in the coordinates of an observer far away. Outside the horizon they straddle the radial direction; at the horizon the outgoing edge stands vertical; inside, both edges point inward. The wall of silence behind a rocket found a similar surface behind an observer who accelerates for ever, made by nothing but the observer’s motion. Here the horizon is made by nothing but the fluid’s motion. There is no curvature of spacetime in either case, only the geometry that a speed limit acquires when it is added to a motion that varies.

Unruh made the analogy precise. For small disturbances in a smoothly flowing, inviscid fluid, the equation for the sound can be written as a wave equation in an effective spacetime whose metric is built from the flow’s velocity and the local speed of sound. Sound rays are the null lines of that metric, and wherever the flow’s speed normal to a surface equals the speed of sound, that surface is a horizon of the metric. The fluid does not know about general relativity; general relativity’s description of how waves move near a horizon applies to it anyway, because the mathematics is the same.

Pulses that peel away from the edge #

The horizon’s most important property is how things behave just outside it.

The figure follows sound pulses sent upstream from points on either side of the horizon. A pulse starting well upstream travels steadily away against the flow. A pulse starting just outside the horizon barely moves at first, because its speed against the ground, $v - c$, is nearly zero there. It creeps upstream, reaches slower flow, and then escapes. A pulse starting just inside creeps the other way and is swept downstream.

Near the horizon the pulse’s distance from it grows exponentially, as $e^{\kappa t}$, because the ground speed $v - c$ is proportional to that distance with a constant $\kappa = dv/dx$, the rate at which the flow speeds up at the horizon. In the flow drawn here $\kappa$ is 0.360 in the figure’s units, and a pulse started 0.005 away from the horizon takes about 14.7 units of time to get clear. A pulse started closer takes longer, logarithmically so, and one started exactly at the horizon stays there.

That exponential peeling is the analogue of the redshift of light escaping a black hole. A wave leaving from close to the horizon is stretched as it leaves, by the same exponential, and $\kappa$ plays the role of the surface gravity. Hawking’s calculation of black-hole radiation depends only on this peeling, not on anything else about gravity. It predicts that a horizon with surface gravity $\kappa$ emits thermal radiation at a temperature $T = \hbar\kappa/2\pi k_B$ — the formula the temperature of an acceleration met for an accelerating observer’s horizon. If the argument depends only on the peeling, it should apply to a sonic horizon too, with the quanta of sound, phonons, in place of photons.

The white hole under a tap #

The simplest analogue horizon is in every kitchen.

A jet of water falling onto a flat surface spreads out in a thin, fast film, and at some radius the film abruptly thickens: the circular hydraulic jump visible in any sink. The figure uses a simplified model to compare the water’s speed with the speed of ripples on it. Ripples on shallow water travel at $\sqrt{gh}$, the speed that depends on the depth when the waves are long compared with it. Inside the jump the film is under half a millimetre deep and moves at half a metre a second, while ripples on it travel at only 0.067 metres a second. Outside, the water is three millimetres deep and moving slowly, at 0.059 metres a second, and ripples travel at 0.17.

So inside the jump the flow outruns its own ripples, and outside it does not. A ripple made outside and travelling inward reaches the jump and can go no further, because beyond it the water would carry it back out faster than it could travel. The jump is a horizon of the opposite kind to a black hole’s: nothing can enter it from outside, which makes it the analogue of a white hole. The jump itself is the place where the supercritical flow’s own disturbances pile up and steepen, much as the front that steepens until it cannot found in a shock. Where the jump sits depends on how deep the water downstream is and on how viscosity slows the film, which the simple model does not compute. What it does show is why the jump is a boundary for ripples: the Froude number, the flow speed over the ripple speed, passes through one there.

Experiments with water flowing over obstacles in flumes have used exactly this kind of white-hole horizon, sending long waves against the flow and watching them convert, at the horizon, into short waves of opposite energy. Weinfurtner, Unruh and colleagues in 2011 measured the ratio of the two converted amplitudes and found it matched the thermal form Hawking’s calculation predicts, at an effective temperature set by the flow’s gradient.

The engineer’s name for a horizon #

Long before anyone compared them with black holes, sonic horizons were an everyday fact of engineering, under another name. A gas flowing from a high-pressure tank through a narrowing pipe speeds up as the pipe narrows. If the pressure difference is large enough, the flow reaches the speed of sound at the narrowest point, the throat, and something striking happens: lowering the pressure downstream any further has no effect on how much gas flows. The flow is choked.

The horizon explains why. A change of pressure is carried by sound. To tell the gas upstream of the throat that the downstream pressure has dropped, a pressure signal would have to travel upstream through the throat, and at the throat the gas is moving at exactly the speed of sound. The upstream edge of the sound cone stands vertical there, as in the first figure, and no signal from downstream can pass. The upstream gas never learns what the downstream pressure is, and so its flow rate cannot depend on it. Every pressure relief valve, every gas regulator and the throat of every rocket nozzle works in this choked regime, and flow meters exploit it: a choked orifice passes a mass flow that depends only on the upstream pressure and temperature.

Past the throat, in the widening part of a rocket’s nozzle, the gas keeps accelerating and becomes supersonic. Disturbances there are swept downstream inside the cone that the cone the source leaves behind drew for a source moving faster than its waves — the Mach cone, whose half-angle is the arcsine of the speed of sound over the flow speed. A Mach cone and a tipped-over sound cone are the same geometry seen from different frames: in one the source moves through still air, in the other the air flows past a still source. When the nozzle’s exit pressure does not match the atmosphere’s, the mismatch cannot propagate upstream through the supersonic gas either, and the flow adjusts through shock waves outside the nozzle, visible as the diamond pattern in a rocket’s exhaust.

So the sonic horizon is not an exotic construction borrowed from astrophysics. It is the reason a valve chokes and a nozzle needs a throat. What the analogy with black holes adds is attention to the waves right at the horizon, which engineers had no reason to care about, and to the quantum fluctuations there, which only a system as cold and clean as a Bose–Einstein condensate can reveal.

Two ways to add a wave to a current #

The analogue works for sound because the addition is Galilean. It would be natural to think it fails for light, whose speeds add relativistically.

The figure compares the two. For sound the upstream speed falls in a straight line, $c - v$, and reaches zero when the flow reaches the speed of sound. For light in water the relativistic addition gives a curve: at half the wave speed the upstream speed is 0.696 of its still-water value rather than 0.5, the drag coefficient of the essay on Fizeau at work. But it too reaches zero exactly when the water moves at $c/n$, the speed of light in water, and turns negative beyond.

The two agree where it matters because in both cases the medium’s own rest frame is the one in which the wave’s speed is fixed. Relativity changes how the upstream speed approaches zero, not where it gets there. So a medium moving faster than light travels in it has a horizon for that light. Water cannot be made to flow at three-quarters of the speed of light, but the medium does not have to be matter moving. In an optical fibre an intense pulse of light changes the refractive index where it is, and that change moves at the pulse’s speed. To a weaker probe wave catching up behind it, the pulse is a region where the fibre’s index is higher and its own speed lower, moving along the fibre. If the probe cannot overtake the pulse’s front, the front is a horizon. Philbin, Leonhardt and colleagues used exactly this in 2008 to shift the frequency of probe light at such a moving front, and estimated that the effective temperature of the horizon was about a thousand kelvin.

Temperatures at the edge #

Hawking’s temperature formula makes the case for building analogues plain.

The figure lists five temperatures. A black hole of one solar mass has a Hawking temperature of 62 nanokelvin, and the black hole at the centre of the galaxy, four million times heavier, $1.5\times10^{-14}$ kelvin. Both are far colder than the 2.7-kelvin microwave background they sit in, so both absorb more than they emit and no telescope could ever see their radiation. The hole that outlives everything and then does not followed the consequences over times vastly longer than the age of the universe.

An analogue horizon’s temperature is $\hbar\kappa/2\pi k_B$, with $\kappa$ its flow gradient. A water-tank flow speeding up by a metre per second over a metre has $\kappa$ of one per second and a temperature of $1.2\times10^{-12}$ kelvin, hopelessly small against the thermal noise of room-temperature water, which is why water experiments measure the stimulated version, with waves sent in, rather than the spontaneous emission. A Bose–Einstein condensate, cooled to nanokelvin and flowing through a step in its trapping potential, has a much larger gradient relative to its own temperature. Steinhauer and colleagues reported in 2019 the spontaneous emission of correlated pairs of phonons from such a horizon, with a spectrum matching a thermal one at about 0.35 nanokelvin, the temperature Hawking’s formula predicted from the measured flow.

What the analogy cannot carry #

The analogy between a flowing fluid and a black hole is exact for waves in the regime the effective metric describes, and three limits come with it.

Short wavelengths. The effective metric treats the fluid as continuous. Near a horizon, waves are stretched from ever shorter wavelengths, and at wavelengths approaching the spacing of atoms, or the healing length of a condensate, the fluid stops behaving as one and the speed of sound depends on wavelength. Hawking’s calculation has the same feature — it traces the outgoing radiation back to modes of absurdly short wavelength near the horizon — and one of the analogues’ main results is that the thermal spectrum survives the modification. That is evidence, not proof, that Hawking’s prediction does not depend on physics at the Planck scale.

The back-reaction. A real black hole loses mass as it radiates, and the horizon shrinks. In an analogue the flow is maintained by pumps or lasers, and the radiated phonons do not change it. Anything that depends on how the horizon responds to its own radiation is outside what analogues can test.

Gravity itself. An analogue reproduces the kinematics of waves near a horizon. It does not reproduce the Einstein equations, the entropy of a black hole, or the question of what happens to information that falls in. Analogue experiments test whether the peeling argument is right; they cannot test whether gravity is described by it.

Still open: what an analogue can say about a real horizon #

Whether analogue experiments have confirmed anything about gravitational black holes is disputed on principle rather than on data. One view is that they test the part of Hawking’s argument that does not involve gravity — that a horizon, whatever makes it, emits thermally at a temperature set by its peeling rate — and that this part was the one most doubted, because of the reliance on very short wavelengths. Another is that a confirmation in a condensate says nothing about spacetime, where the short-wavelength physics is unknown and the analogy may fail exactly there. Current work on condensates aims to measure the entanglement between the emitted phonons and their partners inside the horizon, a signature more specific to Hawking’s mechanism than a thermal spectrum alone, and the first reports of it are being scrutinised for alternative explanations.

The habit worth carrying away is to read a horizon as a statement about speeds rather than about gravity. Wherever a medium carries its own waves and flows faster than they travel, those waves have a horizon, and near it they peel away exponentially at a rate set by how fast the flow is changing. Galileo’s addition is enough to make one for sound; Einstein’s addition makes one for light in a moving medium at the same place; and gravity makes one for everything, which is why it is the only one that no wave can escape by being of a different kind.