Sonic Black Hole: A Horizon Made of Flowing Water

Sound Dragged by the Current

Sound is a disturbance of the fluid, not of the laboratory. A wavefront always travels at the local sound speed c relative to the fluid it is in, so in a moving stream the two directions stop being equivalent. A crest launched downstream advances over the ground at c + v; one launched upstream advances at only c − v, where v is the flow speed.

Now narrow the channel. For a steady stream of width w and depth h, mass conservation demands that ρvwh stay constant along the flow, so pinching w or h forces v up. Somewhere the rising v crosses the local c. At that line c − v = 0: an upstream-swimming wave runs as hard as it can and stays exactly where it is, pinned against the current. A centimetre further downstream v > c, and every wave — however loud, however long you wait — is swept backwards. Sound made downstream of that line can never reach a listener upstream of it.

That stall line is an acoustic horizon. It is not a wall, and nothing dramatic happens to a cork drifting through it. It is a purely causal boundary, defined by where signals are able to go — the identical statement to a black hole's event horizon, with light in place of sound and spacetime in place of water.

Unruh's Acoustic Metric and the 1981 Proof

The analogy is not a metaphor. In Physical Review Letters 46, 1351 (1981), William Unruh showed it is an exact identity for a restricted but common class of flows: inviscid, barotropic (pressure a function of density alone) and irrotational, so the velocity is a gradient, v = ∇φ. Linearise the continuity and Euler equations about such a background and the perturbation of φ obeys, exactly, the massless scalar wave equation of curved spacetime, ∂_μ(√−g g^μν∂_νφ) = 0, with an effective line element that in one dimension reads

• ds² ∝ −(c² − v²)dt² − 2v dx dt + dx² — the overall factor ρ/c is conformal, and conformal factors cannot move light cones, so they cannot move the horizon.
• Compare the Painlevé–Gullstrand form of Schwarzschild, written down by Paul Painlevé in 1921 and Allvar Gullstrand in 1922: ds² = −(c² − v_ff²)dt² + 2v_ff dr dt + dr² + r²dΩ², with the Newtonian free-fall speed v_ff = √(2GM/r) — the cross term carries the opposite sign to the acoustic line above only because this river runs inward, along −r, while the stream above runs along +x.

The two are the same object. Space in Painlevé–Gullstrand coordinates behaves like a river falling inward at v_ff while light swims through it at c; the horizon sits where the river wins, v_ff = c, that is r = 2GM/c² — 2.95 km for the Sun, 29.5 km for a 10 M_☉ stellar black hole. Unruh's fluid does the same with c = the speed of sound: sound cones tip over as v grows, and past v = c every cone points downstream. He called it a “dumb hole” — dumb in its old sense of mute. The fluid itself still lives in ordinary Newtonian space and time; the Lorentzian geometry is felt only by the sound.

The Numbers: Froude Numbers, Nozzles, and Sonic Points

Shallow water is the cheapest laboratory — with one caveat: the wave playing the part of “sound” in a flume is a long surface gravity wave, not a pressure wave, so the speed to beat is centimetres-per-second slow rather than the ~1,500 m/s of true sound in water. Long surface waves obey ω² = gk tanh(kh), which for wavelengths much longer than the depth (λ ≫ h) collapses to the non-dispersive c = √(gh). At h = 5 cm that is √(9.81 × 0.05) = 0.70 m/s; at 20 cm, 1.40 m/s. The horizon condition v = c is therefore just

• Froude number Fr = v/√(gh) = 1 — the “critical” flow open-channel hydraulic engineers have worked with since the nineteenth century. Fr < 1 is subcritical (ripples can travel upstream); Fr > 1 is supercritical (they cannot).
• A hydraulic jump — the ring of standing turbulence in a kitchen sink, or the abrupt rise below a sluice gate, analysed by Jean-Baptiste Bélanger in the 1820s — is the supercritical-to-subcritical transition. The fluid crosses it going fast to slow, making it a white-hole horizon: waves outside cannot get in.
• A de Laval nozzle (Gustaf de Laval, 1888) carries a horizon at its throat. Subsonic gas accelerates through the converging section, reaches Mach 1 at the minimum area — exactly so in the ideal quasi-one-dimensional picture, while in a real nozzle the sonic line is curved and sits a little downstream of the geometric throat — then goes supersonic in the bell. This is why a rocket engine is choked: acoustic disturbances at the nozzle exit cannot propagate back into the combustion chamber. Every RS-25 and Merlin firing runs an acoustic horizon.
• Bondi accretion (Hermann Bondi, 1952) has the same structure: spherical inflow onto a star passes through a sonic point where infall speed equals gas sound speed.

Hawking Radiation from a Dumb Hole

Unruh's motive was not the horizon but what should leak out of it. Stephen Hawking's 1974 result gives a real black hole a temperature T_H = ℏκ/(2πk_Bc) = ℏc³/(8πGMk_B) = 6.2 × 10⁻⁸ K × (M_☉/M) — 6.2 nK for a 10 M_☉ hole, 1.4 × 10⁻¹⁴ K for Sagittarius A* at 4.3 × 10⁶ M_☉. Both are hopelessly buried beneath the 2.725 K cosmic microwave background, which is why astronomical Hawking radiation has never been detected.

Carry the derivation into the fluid and the surface gravity becomes the gradient of the escape speed at the horizon, κ = c|d(c−v)/dx|, giving T_H = (ℏ/2πk_B) · |d(c−v)/dx|, where the prefactor ℏ/2πk_B = 1.22 × 10⁻¹² K·s.

The number is brutal. A water flume that changes its effective escape speed by ~1 m/s over ~1 m has a gradient of order 1 s⁻¹, hence T_H ~ 10⁻¹² K — fourteen orders of magnitude below the 293 K thermal and turbulent noise the phonons would have to be dug out of. Cold atoms change the arithmetic, because the gradient scales inversely with a microscopic length. In a ⁸⁷Rb condensate the sound speed is ~0.5 mm/s and the healing length is of order a micron, so the flow can go subsonic to supersonic across a few microns: 5 × 10⁻⁴ m/s over ~2 × 10⁻⁶ m gives ~250 s⁻¹ and T_H ≈ 0.3 nK — comparable to the condensate's own temperature scale rather than fourteen decades below it.

There is a deeper payoff. Hawking's calculation traces the outgoing modes backwards in time and finds them blue-shifted without bound, past the Planck length — the trans-Planckian problem. A fluid has an honest cutoff (the healing length, or the intermolecular spacing) and an honest modified dispersion relation. Unruh integrated the wave equation with such dispersion in 1995 (Phys. Rev. D 51, 2827) and found the thermal spectrum essentially unchanged — strong evidence the Hawking effect does not depend on unknown physics at the smallest scales.

How It Has Actually Been Measured

Water, 2011. Silke Weinfurtner, Edmund Tedford, Matthew Penrice, William Unruh and Gregory Lawrence used a 6.2 m open-channel flume at the University of British Columbia, running water over a streamlined obstacle so the flow went supercritical and back — a white-hole horizon (Phys. Rev. Lett. 106, 021302). They sent long, sub-hertz surface waves upstream against the current and tracked the free surface optically. At the horizon those waves blue-shifted and mode-converted into a pair of outgoing waves whose measured ratio of norms followed the thermal Boltzmann form the Hawking calculation predicts. Because the input was a wave they generated, this was stimulated conversion — the same coefficient measured classically, with the ~10⁻¹² K spontaneous part far out of reach.

Cold atoms, 2016–2021. Jeff Steinhauer's group at the Technion reached the quantum regime with an elongated ⁸⁷Rb condensate of order 10⁴ atoms, sweeping a step potential along the cloud to make one region supersonic. Reading the density–density correlation function G⁽²⁾(x, x′) off absorption images reveals the Hawking phonon and its infalling partner as a correlation band straddling the horizon.

• Nature Physics 12, 959 (2016): entanglement between the outgoing phonon and its partner.
• Nature 569, 688 (2019), with Juan Ramón Muñoz de Nova, Katrine Golubkov and Victor Kolobov: a thermal spectrum at T_H = 0.35 nK, consistent with the surface gravity computed from the measured velocity profile.
• Nature Physics 17, 362 (2021): stationary spontaneous emission and its time evolution.

Other horizons. A draining “bathtub” vortex at Nottingham reproduced rotational superradiance, waves scattering off the ergoregion and returning amplified (Torres, Patrick, Coutant, Richartz, Tedford and Weinfurtner, Nature Physics 13, 833, 2017). In optics, Thomas Philbin and Ulf Leonhardt made a moving fibre-optic horizon from a soliton's nonlinear index pulse (Science 319, 1367, 2008).

What a Dumb Hole Is Not

The commonest overreach is to say analogue experiments “test general relativity.” They test its kinematics — how a field propagates on a curved Lorentzian background — and nothing else. The acoustic metric solves the Euler equations, not Einstein's field equations; G never appears in it. A dumb hole therefore has no mass, no gravitational attraction, no tidal field, no light bending and no singularity, and anything following from the dynamics — the Bekenstein–Hawking entropy A/4ℓ_P², the area theorem, the singularity theorems — is untouched.

• A sonic boom is not a horizon. In the rest frame of still air, sound from a supersonic aircraft spreads isotropically at 343 m/s; the Mach cone is merely the envelope of expanding spheres from a moving source. No stationary surface exists that the fluid crosses, and no region of air is cut off. Only a transition through v = c at a fixed place makes a horizon.
• It swallows nothing. A float, a fish or a stray atom crosses the acoustic horizon without noticing. Only sound is trapped.
• It does not evaporate. A real black hole loses mass to its own radiation; a dumb hole is fed by a pump, the phonon flux has no measurable back-reaction on the flow, and the horizon does not shrink.
• The idealisations bite. Viscosity, vorticity, surface tension and short-wavelength dispersion all break the exact Lorentz invariance of the effective metric, and in a flume they set the practical floor on how sharp a horizon can be made.

Open Questions

Analogue horizons settle the emission mechanism but not the paradox that made it famous. A black hole's information problem turns on whether the interior degrees of freedom are ultimately recoverable; in a condensate the “interior” is ordinary atoms whose quantum state is in principle fully accessible, so unitarity holds by construction and the analogy says nothing about the real case. Live issues include:

• Black-hole lasing. Put a black-hole and a white-hole horizon back to back and modes bounce between them, amplifying exponentially — predicted by Steven Corley and Ted Jacobson in 1999. Steinhauer reported an observation in 2014 (Nature Physics 10, 864); whether the signal was lasing or a self-amplifying trap instability is still argued.
• Back-reaction. Nobody has watched phonon emission deplete the flow that makes it — the analogue of evaporation.
• Emergent gravity. If a Lorentzian metric can emerge from a non-relativistic substrate, might spacetime itself? Jacobson's 1995 derivation of the Einstein equation as a thermodynamic equation of state takes this seriously, while terrestrial Lorentz-invariance tests bound how far it can go.

Horizons across four media: where the flow outruns the wave

System	Wave speed c at the horizon	Horizon condition	Hawking temperature
Schwarzschild black hole, 10 M☉	299,792,458 m/s (light)	free-fall v = c at r_s = 29.5 km	6.2 nK (undetectable under the 2.7 K CMB)
Sagittarius A*, 4.3 × 10⁶ M☉	299,792,458 m/s (light)	r_s = 1.27 × 10¹⁰ m (~0.085 AU)	1.4 × 10⁻¹⁴ K
Shallow-water flume, UBC 2011	c = √(gh) ≈ 0.70 m/s at h = 5 cm	Froude number Fr = 1	~10⁻¹² K — stimulated conversion measured instead
de Laval rocket-nozzle throat	~1,000 m/s in hot exhaust	Mach number M = 1 at minimum area	~10⁻⁸ K (never attempted)
⁸⁷Rb condensate, Technion 2019	~0.5 mm/s	v = c at a swept step potential	0.35 nK, measured and thermal