Same maths, different names · Audit 01
Under specified assumptions, linear sound perturbations in a fluid obey the equation of a minimally coupled massless scalar field on an effective curved spacetime. Related geometric descriptions occur elsewhere, sometimes at the level of field equations and sometimes only for rays. This audit records the mappings, their assumptions, their historical relationships, and what they do, and do not, establish about gravity.
In plain terms
When a wave travels through something that is itself moving, the motion of the medium changes how the wave goes. Sound in a wind, ripples on a river, light in flowing glass: in each case the wave is carried along, slowed against the flow, sped up with it. In 1981 William Unruh showed that, under four conditions (no friction, one speed of sound, no swirl, small waves), the equation for sound in a moving fluid is not merely similar to the equation for a wave in Einstein’s curved spacetime. It is the same equation. The flow of the fluid plays the part that the curvature of space and time plays in gravity.
That has a consequence you can build. Where a fluid flows faster than sound can travel through it, sound from the far side can never come back upstream. That boundary is, in the mathematics, a horizon, the same object as the edge of a black hole. So laboratories have made “black holes” out of water, light in optical fibre and clouds of ultracold atoms, and have looked at them for the effect Stephen Hawking predicted in 1974: a faint glow of radiation coming off a horizon. The water and fibre experiments have seen the classical, driven version of that effect. The atom experiments claim the quantum version, and that claim is still argued over.
What none of this shows is that gravity is a fluid. The moving medium supplies the shape of the spacetime but not the law that shapes it. In real gravity, Einstein’s equations say how matter curves space; in a fluid, the ordinary equations of fluid flow decide the flow, and Einstein’s equations never appear. Anyone who wants to argue that gravity comes from some underlying medium, as I do, gets the geometry for free and still owes the dynamics. This page is largely a careful record of exactly where that line falls.
The rest of the page is a ledger. It lists nine areas of physics that have this structure, what each calls it, when each first had it, what each assumes, and a verdict on how exact the match is: exact when it is the same equation, limited when it holds only in some approximation, analogy when only some features line up. Every claim carries a note on how far the source was actually read.
What is being audited
Take a fluid with density ρ, pressure p and velocity v. Assume it is inviscid, barotropic (p depends on ρ alone, so there is one sound speed cs² = dp/dρ), and irrotational, so v = ∇Φ. Linearise about a background flow and write the perturbation of the velocity potential as φ. Then, with no further approximation, φ satisfies
gμν = (ρ/cs) · [ −(cs² − v²) , −vj ; −vi , δij ]
ds² = (ρ/cs) [ −cs² dt² + (dx − v dt)² ]
That is the equation of a minimally coupled massless scalar field on a Lorentzian manifold whose metric gμν (inverse gμν) is built from the background flow alone. Nothing about gravity went in. An effective spacetime came out. The four assumptions, inviscid, barotropic, irrotational, linearised, mark exactly where this identity holds, and each row of the ledger is read against them.
Two columns in the ledger do the real work. The dictionary column says what plays the role of what, so that a verdict can be checked rather than believed. The direction column says which way the implication runs: whether a medium was shown to produce a geometry, or a geometry was rewritten to look like a medium. These are different claims and most of the confusion in popular accounts comes from running them together.
A shared equation is also not a shared problem. The notes by row say where boundary conditions, admissible solutions and the observables each field actually measures differ even when the equation does not, and an effective spacetime for wave propagation is distinguished throughout from a derivation of gravitational dynamics.
The ledger
Each verdict applies to the correspondence named in that row’s dictionary column, not to the field as a whole.
| Field | Name used · earliest source located | Dictionary | Direction of implication | Assumptions added or dropped · where it breaks | Verdict |
|---|---|---|---|---|---|
| Optics | Gordon metric; optical metric; Fresnel dragGordon 1923 · revived by Leonhardt & Piwnicki 1999 · Philbin et al. 2008 (fibre horizon) | cs ↔ c/n · v ↔ medium 4-velocity uμ · ĝμν = gμν + (1 − 1/n²) uμuν · Fresnel coefficient (1 − 1/n²) ↔ off-diagonal term at first order in v/c | Medium → geometry. Derived from Maxwell’s equations in a moving dielectric; the medium is physically present. | adds isotropic, non-dispersive, impedance-matched medium; fully relativistic from the startbreaks real media are dispersive, so the metric holds one frequency band at a time; light is a vector field, so the scalar equation governs the eikonal and the polarisation-independent part, not the full field | limited |
| Geometric acoustics | Ray tracing in moving inhomogeneous fluids; Hamiltonian ray theoryWhite 1973 · Anderson & Spiegel 1975 (radiative transfer in a flowing medium, extending Gordon) | Ray Hamiltonian ↔ null geodesic equation · intrinsic frequency ω − k·v ↔ frequency in the comoving frame · no metric written | Medium → rays only. The null cones are found; the manifold they belong to is never named. | adds eikonal (high-frequency) limit; freely allows stratification and mild vorticitybreaks once v is not a gradient the potential φ does not exist; the rays survive but the field equation does not | limited |
| Relativistic hydrodynamics | No name; an effective metric for perturbations of accretion flowMoncrief 1980 · priority credited by Barceló–Liberati–Visser; not all later authors read it that way | Perturbation of velocity potential ↔ φ · background Schwarzschild + flow ↔ effective metric on top of the true one · sonic horizon ↔ boundary of the stability analysis | Medium → geometry, inside GR. Derived as a tool for a stability proof; not recognised as general. | adds relativistic perfect fluid; curved background; WKB for travelling wavesbreaks nothing breaks within the analysis; the structure was not noticed as the same one Gordon had written | exact |
| Analogue gravity | Acoustic metric; sonic (“dumb”) hole; acoustic black holeUnruh 1981 · geometry made explicit by Visser 1998 · Barceló, Liberati & Visser 2005/2011 (survey) | v⊥ = cs surface (with the causal conditions) ↔ horizon · |v| = cs ↔ ergosurface, |v| > cs ↔ ergoregion · gH = ½ ∂(cs² − v⊥²)/∂n ↔ surface gravity (dimensions of acceleration) · kBTH = ħgH/2πcs ↔ Hawking temperature · ρ/cs ↔ conformal factor | Medium → geometry, then QFT on that geometry → phonon Hawking radiation. The metric inherits its dynamics from the fluid equations, not from Einstein’s. | adds the four assumptions, imposed deliberately; a quantised phonon field in a specified statebreaks Einstein’s field equations are not derived; the conformal factor is physical in 3+1 for a minimally coupled scalar, so a chosen GR metric cannot be reproduced at will (Visser: the PG form of Schwarzschild is reachable only up to a conformal factor, because continuity and the needed velocity profile conflict) | exact |
| Superfluid ³He | Effective metric for quasiparticles; “the Universe in a helium droplet”Jacobson & Volovik 1998 (moving planar soliton in bulk ³He-A; thin-film wall noted as a simpler variant) · Volovik 2003 | Fermi-point quasiparticle dispersion ↔ massless relativistic fermion · order-parameter texture and superflow ↔ metric and gauge field · Lorentz violation above a momentum scale ↔ trans-Planckian physics | Medium → geometry, and a stated hypothesis that the vacuum is such a medium. The hypothesis is Volovik’s, labelled as one. | adds quantum two-fluid medium; fermionic quasiparticles as well as phononsbreaks holds only near the Fermi point / at long wavelength; the emergent gravitational dynamics are not Einstein–Hilbert, and Volovik says so | limited |
| Bose–Einstein condensates | Sonic black hole; Bogoliubov phonons on an acoustic metricGaray, Anglin, Cirac & Zoller 2000 · Lahav et al. 2010 (first horizon) · Steinhauer 2016 · Muñoz de Nova et al. 2019 | Gross–Pitaevskii + Madelung ↔ continuity + Euler + quantum pressure · healing length ↔ scale below which the acoustic description fails · Bogoliubov dispersion ↔ superluminal modified dispersion · two-component BEC ↔ bi-metric geometry | Medium → geometry, in the laboratory. The only platform so far reporting spontaneous (quantum) emission and its correlations. | adds the acoustic description applies in the long-wavelength regime where quantum-pressure corrections are negligiblebreaks below the healing length the dispersion is superluminal and the metric description dissolves; this is deliberate, it is where the trans-Planckian question is studied | limited |
| Water waves | Wave blocking; current-induced refraction; later “white-hole horizon”phenomenon: Unna 1942, Johnson 1947, Longuet-Higgins & Stewart 1961, Peregrine 1976 · metric: Schützhold & Unruh 2002 · experiments: Rousseaux et al. 2008, Weinfurtner et al. 2011, Euvé et al. 2016 (all classical or stimulated) | cs ↔ √(gh) · blocking point (current = group velocity) ↔ horizon · negative intrinsic frequency ↔ negative-norm mode · wave action ↔ Klein–Gordon norm | Medium → geometry, written in 2002. The phenomenon was known decades earlier without the geometry. | adds shallow-water long-wave limit λ ≫ h; surface tension and depth dispersion added afterwards as correctionsbreaks deep-water dispersion is subluminal, so the horizon is soft and blocked waves convert to other modes; real tanks have vorticity, viscosity and finite depth together; the experiments measure stimulated mode conversion and classical correlations, not spontaneous emission | limited |
| Atmospheric and ocean dynamics | Critical layer; wave action; Doppler-shifted intrinsic frequencyBooker & Bretherton 1967 · Bretherton & Garrett 1968 | Critical layer (background speed equals the wave’s horizontal phase speed; intrinsic frequency → 0) ↔ a horizon-like surface for rays, not the group-velocity blocking condition · wave action E/ω̂ ↔ conserved norm · no metric written | Medium → rays and conservation laws only. No field-level geometry. | adds stratification, shear, WKBbreaks stratification means there is no single scalar wave equation to be a d’Alembertian; the critical-layer condition is a phase-speed condition and is not generally the blocking condition; the correspondence is at the level of rays and of the conserved quantity | analogy |
| General relativity (i) | Painlevé–Gullstrand coordinates; the river model (Schwarzschild)Painlevé 1921, Gullstrand 1922 · Hamilton & Lisle 2008 | Schwarzschild ↔ ds² = −c²dt² + (dx − vdt)² with flat spatial slices, v = −√(2GM/r) r̂ | Geometry → medium picture. A coordinate rewrite of a known metric, not a derivation from a fluid. | adds a coordinate choice; ρ/cs set to one by handbreaks the “river” carries no density and satisfies no continuity equation (my gloss; Hamilton & Lisle call the flat background “a fictitious construct”). It becomes a fluid only if dynamics are supplied, which GR does not do | exact |
| General relativity (ii) | Doran coordinates; the river model (Kerr)Hamilton & Lisle 2008 · limits: Visser & Liberati 2022 | Kerr ↔ unit lapse with a twisting “river” in Doran coordinates; spatial slices are not flat | Geometry → medium picture, with a weaker dictionary than the Schwarzschild case. | adds unit lapse; a factorised, non-flat 3-metricbreaks Kerr admits no Painlevé–Gullstrand form with flat spatial slices (Visser & Liberati 2022); the acoustic line element, which has flat slices, cannot represent it exactly | limited |
| General relativity (iii) | The gravitational field as an optical medium; constitutive relationsPlebanski 1960 · Landau & Lifshitz, Classical Theory of Fields, 4th ed., problem after §90 | Vacuum Maxwell equations on a curved background ↔ flat-space Maxwell equations in a medium with εij = μij and a magneto-electric vector built from g0i | Geometry → medium picture. An exact rewriting of the field equations, in the opposite direction to Gordon. | adds nothing; the identity is formal and exact for the equationsbreaks the “medium” has no rest frame, no dispersion and no sources of its own; the correspondence is of equations, not of the complete problem | exact |
Related descriptions, with the same caveats, are used in slow light (Leonhardt & Piwnicki 2000), polariton fluids (Nguyen et al. 2015), ion rings and graphene. Not separately audited.
Historical relationships
A structure appearing in many literatures is not many discoveries, and the record here does not yet support a count. Reading the first-appearance column as the earliest sources located rather than as settled priority, there are several apparently independent antecedents to Unruh’s 1981 paper: Gordon in 1923, working from Maxwell’s equations in a moving dielectric; the oceanographers of the 1940s, who had wave blocking as a phenomenon but wrote no metric, which is a different historical event from deriving one; White in 1973 in acoustics; and Moncrief in 1980 inside relativity, who derived the structure as a tool and did not name it. Anderson and Spiegel in 1975 built explicitly on Gordon and are not independent of him. Whether Unruh knew of Moncrief or White I have not established.
The later history is not a simple chain of transplants either. Visser’s 1998 paper states that the connection “has been independently rediscovered several times over the ensuing decade and a half” after 1981, without naming the cases. The condensate, water-tank and optical-fibre programmes of the 2000s do cite Unruh and set out to build his metric in a new medium. So the honest summary is: at least two independent derivations of an effective metric before 1981 (Gordon, Moncrief), a ray-level version in acoustics, an unrelated phenomenon in oceanography later recognised as the same horizon, and an unspecified number of independent rediscoveries after 1981. A documented citation history, which would settle this, is future work for this page.
Notes by row
Optics limited
Gordon is the earliest entry by half a century. Light in a moving dielectric sees the metric gμν + (1 − 1/n²) uμuν. Leonhardt and Piwnicki solve the geodesic equation of that metric for slowly moving media and recover Fresnel’s drag formula, which Fizeau measured in 1851. That is a result about light in a material medium; it says nothing by itself about whether a medium underlies propagation in vacuum, and this page does not use it that way.
The limit is dispersion. Every real material has n(ω), so the Gordon metric holds band by band. The fibre-optic horizon experiments live on this: a moving pulse raises the index by the Kerr effect, and probe light of a nearby frequency is slowed until its group velocity matches the pulse and it cannot enter. That is a white-hole horizon built from dispersion rather than in spite of it, and what was measured there is a stimulated, classical frequency shift.
Geometric acoustics limited
Acousticians had the ray equations of a moving medium, which are the null geodesics of the acoustic metric, by 1973, and Anderson and Spiegel extended Gordon’s metric to a flowing fluid for radiative transfer in 1975. Both wrote Hamiltonian ray equations because that is how their fields think. The step not taken was to ask what manifold the rays were geodesics of. This literature also allows vorticity and stratification more freely than analogue gravity does, and that is why it stops at rays: once v is not a gradient the field equation for φ ceases to exist, but rays survive.
Relativistic hydrodynamics exact
Moncrief wanted to know whether spherical accretion onto a Schwarzschild hole was stable. He perturbed the velocity potential and found no unstable modes outside the sound horizon. The Living Review credits this as the first derivation of a version of the acoustic metric; some later authors describe the paper only as a perturbation analysis and do not credit it with the geometry. I have read the abstract and the secondary accounts, not the full text, so I record the priority claim as Barceló, Liberati and Visser’s reading rather than my own.
Analogue gravity exact
Unruh named the structure and asked what it implied. Two points matter for anyone thinking about gravity as a medium. First, the conformal factor. A minimally coupled massless scalar in 3+1 dimensions is not conformally invariant, so the prefactor ρ/cs is physical. Visser shows that forcing the exact Painlevé–Gullstrand form with v = √(2GM/r) and constant cs is incompatible with the continuity equation; with ρ ∝ r−3/2 one gets a geometry conformal to Painlevé–Gullstrand but not identical to it. What survives a conformal rescaling: null trajectories, as unparameterised curves, and therefore the horizon’s location. What survives under further conditions: Jacobson and Kang show that for stationary black holes the surface gravity and Hawking temperature are unchanged by conformal factors that are regular at the horizon and tend to one at infinity. Greybody factors and anything involving backscattering do not survive, and there the density matters.
Second, and more important: the correspondence is kinematic. Fields on the acoustic spacetime behave exactly as fields on a curved spacetime. The acoustic spacetime itself obeys the Euler equations, and Einstein’s field equations are not derived; no fluid is known whose bulk equations reproduce them for its own acoustic metric. Visser has said this plainly since 1998. Anyone proposing a hydrodynamic origin for gravity has to supply the dynamics from somewhere else.
Superfluid ³He limited
Jacobson and Volovik’s system is a planar soliton moving through bulk ³He-A, which gives quasiparticles both a horizon and an ergoregion; a domain wall in a thin film is mentioned as a simpler case without one. Volovik’s book then reads the whole Standard Model plus gravity as the low-energy structure of a fermionic superfluid vacuum, with the metric emerging from the order-parameter texture near Fermi points. The metric part is the same mathematics as the other rows. The extra claim, that our vacuum is such a medium, is his hypothesis and he labels it as one. He is also explicit that the emergent gravitational dynamics in helium are not Einstein’s, which is the same debt as the row above, seen from the condensed-matter side.
Bose–Einstein condensates limited
The route to the metric is the Madelung transform: write the condensate wavefunction as √ρ eiθ and the Gross–Pitaevskii equation splits into continuity plus an Euler equation with one extra term, the quantum pressure (Barceló, Liberati and Visser 2001 set this out for the analogue-gravity case). The extra term contains ħ and fixes a length, the healing length. The acoustic description applies in the long-wavelength regime where quantum-pressure corrections are negligible; below the healing length phonons disperse superluminally and the metric description dissolves. This is the trans-Planckian problem made tangible, and it is why Jacobson (1991), Unruh (1995) and Corley and Jacobson (1996) used such systems to argue that Hawking radiation does not depend on physics at arbitrarily short wavelengths.
The condensate is also the one platform where the claimed measurement is of spontaneous, quantum emission rather than a stimulated classical effect: Steinhauer’s 2016 correlation measurement and the 2019 thermal-spectrum measurement. Those results have been debated in the literature and the debate is not audited here; the point for this page is that they are a different kind of evidence from the water and fibre experiments.
Two-component condensates give two sound speeds and therefore two metrics: Fischer and Schützhold (2004) for cosmological analogues, Visser and Weinfurtner (2005) for a massive Klein–Gordon field, and Weinfurtner, Liberati and Visser (2007) for the full hierarchy from pseudo-Finsler through bi-metric to mono-metric geometry.
Water waves, oceanography, atmospheric dynamics limited analogy
The oldest phenomenon in the ledger and the latest to be given a geometry. Waves running against a current steepen and stop where the current speed equals their group velocity. Unna described the sea-state change when a tide turns against the wind in 1942; Johnson is usually credited with the deep-water stopping condition in 1947 (I have not read the paper and record the attribution as secondhand); Longuet-Higgins and Stewart gave the radiation-stress treatment of amplitudes near the stopping point in 1961; Peregrine reviewed the field in 1976. Rousseaux’s 2008 title, with its question mark, is the moment the two literatures met. What the tank experiments measure should be stated plainly: Rousseaux 2008 and Weinfurtner 2011 measured stimulated conversion of an incoming wave into negative-frequency partners; Euvé 2016 measured classical noise correlations. None of these is spontaneous emission, and none needs a quantum state.
The atmospheric critical layer is a different object and is kept as an analogy. It is defined by the background wind matching the wave’s horizontal phase speed, so that the intrinsic frequency goes to zero; the blocking condition above involves the group velocity. The two coincide only in special dispersion relations, and no mapping is offered here. What the atmospheric literature does share exactly is the conserved quantity: wave action, energy divided by intrinsic frequency. Rousseaux (2013) proves that wave action density is the conserved Klein–Gordon norm of the analogue field, and that a negative intrinsic frequency is a negative-norm mode, which is what makes Hawking pairs possible. The main quantum-field-theory reviews of analogue Hawking radiation (Robertson 2012; Coutant and Weinfurtner 2016) discuss the norm and its sign without mentioning wave action or Bretherton and Garrett.
General relativity, three correspondences exact limited exact
Painlevé in 1921 and Gullstrand in 1922 wrote Schwarzschild with flat spatial slices and a radial term that reads as an inward flow at the Newtonian escape velocity. Painlevé’s own purpose was to argue that coordinate freedom undermined the uniqueness of GR’s predictions; Einstein replied by letter in December 1921 that the coordinates had no meaning, and rejected the solution again at the Collège de France debate in April 1922. Hamilton and Lisle’s river model is the modern reading: space falls into the hole at √(2GM/r) and light and matter move through the falling space at their ordinary speeds. This is the acoustic line element with v given and ρ/cs set to one, and the correspondence of equations is exact.
Kerr is weaker. Hamilton and Lisle extend the river with a twist using Doran coordinates, but Visser and Liberati (2022) show that Kerr admits no Painlevé–Gullstrand form with flat spatial slices; the best available has unit lapse and a factorised, non-flat 3-metric. Since the acoustic line element has flat slices, it cannot represent Kerr exactly, and this row is marked limited.
The third correspondence runs Maxwell’s equations on a curved background into flat-space Maxwell equations in a medium with equal permittivity and permeability tensors and a magneto-electric term from g0i (Plebanski 1960; Landau and Lifshitz). As an identity between equations it is exact. As a physical medium it is empty: no rest frame, no dispersion, no sources. Whether any of these “rivers” or “media” could be made a real fluid, with an equation of state and dynamics that return Einstein’s equations, is the open question this series circles, and these rows are where its boundary conditions live.
The dictionary
Terms that name the same object across rows, under the four assumptions, except where an entry says otherwise.
- Horizon
- Sonic point (accretion, nozzle flow) · wave-blocking line (oceanography, group-velocity condition) · the surface where v⊥ = cs, with the causal conditions that make it a horizon rather than a local surface
- Ergosurface and ergoregion
- Sonic surface |v| = cs and the supersonic region |v| > cs · coincide with the horizon only where the flow is purely normal to it
- Critical layer
- Not the blocking line: a phase-speed condition, U = cphase · horizon-like for rays only · analogy
- Surface gravity
- Normal gradient at the horizon, gH = ½ ∂(cs² − v⊥²)/∂n (an acceleration) · κ = gH/cs · Hawking temperature kBTH = ħgH/2πcs = ħκ/2π
- Gravitational redshift
- Doppler shift between laboratory frequency ω and intrinsic frequency ω − k·v
- Klein–Gordon norm
- Wave action E/ω̂ (Bretherton–Garrett; identity proved by Rousseaux 2013) · negative norm = negative intrinsic frequency
- Painlevé–Gullstrand form
- Acoustic line element with ρ/cs = 1 · the river model · Schwarzschild only; Kerr has no flat-sliced form
- Frame dragging
- Azimuthal flow (draining vortex) · field-level only while the flow stays irrotational; rays otherwise
- Fresnel drag coefficient
- Off-diagonal term of the Gordon metric at first order in v/c
- Trans-Planckian cutoff
- Healing length (BEC) · capillary length and depth (water) · interatomic spacing (helium)
- Conformal factor
- ρ/cs · leaves null curves and horizon location unchanged; leaves temperature unchanged under the Jacobson–Kang conditions; enters scattering, curvature and back-reaction; degenerate in 1+1
- Bi-metric geometry
- Two-component condensate with two sound speeds (Fischer–Schützhold 2004; Visser–Weinfurtner 2005)
- Semiclassical back-reaction
- Phonon stress tensor sourcing continuity and Bernoulli (Balbinot et al. 2005), explicitly not the Einstein equations
Verdict
Three claims are usually run together. The ledger supports the first two, with qualifications, and is silent on the third.
First: the acoustic theorem establishes an exact correspondence between linearised potential perturbations of a fluid, under the stated assumptions, and a minimally coupled massless scalar field on an effective spacetime. Other rows share related effective geometries, limiting equations or ray structures, with the distinctions recorded in the ledger. “Exact” belongs to the specific correspondences marked exact, not to the set.
Second: consequences derived from that equation carry across, and some have been measured. The evidence differs in kind. Water tanks and optical fibre have measured stimulated, classical effects: mode conversion into negative-frequency partners, frequency shifts at a horizon, classical noise correlations. Condensates have reported spontaneous quantum emission and its correlations, which is the Hawking effect proper, and those reports carry their own debate. The shared classical wave equation does not by itself fix a quantum state or establish spontaneous emission; that needs the additional assumptions of quantum field theory on the effective background, and the condensate experiments are the test of those. What the programme as a whole has shown, and general relativity could not have shown alone, is that the horizon physics survives the breakdown of the metric at short wavelengths.
Third: that gravity itself is such a medium. Nothing in the ledger supports or refutes it. Every “medium → geometry” row produces a metric that inherits its dynamics from the medium’s equations, and Einstein’s field equations are not derived in any of them. Every “geometry → medium” row is a rewrite with no dynamics attached. Unruh, Visser, Volovik and Barceló all say this in print. When a popular account slides from the second claim to the third, that is where the translation failure this series is about actually happens.
What this means for a hydrodynamic theory of gravity. The kinematics come free, which is why the medium picture is not crankery. The debt is dynamics. A theory of this kind must say what fluid, with what equation of state; why its acoustic metric obeys something Einstein-like in the right limit, where Sakharov’s induced gravity (1967) and Jacobson’s thermodynamic derivation (1995) are two approaches considered in this series, as audits 30 and 31; and how the conformal factor ρ/cs is fixed or shown not to matter. Anyone who can answer those three has done something new. Anyone who cannot is still in Unruh’s row, which is respectable but is not a theory of gravity.
What this page does not claim
It does not claim a count of independent discoveries; the historical relationships are recorded as far as located and marked uncertain where they are. It does not claim that analogue experiments test general relativity; they test wave propagation, and in one platform quantum field theory, on a fixed effective background. It does not claim the acoustic metric has anything to say about gravitational dynamics. It does not claim that the four assumptions can be relaxed without cost; each relaxation has its own literature and its own losses, listed below. It does not claim that a result about light in a material dielectric bears on the nature of the vacuum. And it does not claim completeness: the slow-light, polariton, ion-ring and graphene programmes are noted but not audited.
Extensions of the theorem, and what is known about each
Every entry below was searched before being written. Where an earlier version of this page described something as unexplored and the search found otherwise, the correction is in the change log.
Next in the series
Audit 02 is the Madelung transform, because every condensed-matter row above passes through it and it is the cleanest statement of what “quantum mechanics written as a fluid” does and does not give you. Audit 03 is the Hamilton–Jacobi equation and the eikonal, a calibration case with well-established connections, used to test the method before the contested ones.
References
Bibliographic details for every entry (authors, title, year, journal, volume, page) were verified against the publisher’s record. The flag beside each says how far the content itself was consulted.
Almeida, C. R. & Jacquet, M. J. (2023). Analogue gravity and the Hawking effect: historical perspective and literature review. Eur. Phys. J. H 48, 15. full text
Anderson, J. L. & Spiegel, E. A. (1975). Radiative transfer through a flowing refractive medium. ApJ 202, 454. secondary
Balbinot, R., Fagnocchi, S., Fabbri, A. & Procopio, G. P. (2005). Backreaction in acoustic black holes. Phys. Rev. Lett. 94, 161302. abstract
Balbinot, R., Fabbri, A., Fagnocchi, S. & Parentani, R. (2005). Hawking radiation from acoustic black holes, short distance and back-reaction effects. Riv. Nuovo Cimento 28(3), 1. full text
Barceló, C., Liberati, S. & Visser, M. (2001). Analogue gravity from Bose–Einstein condensates. Class. Quantum Grav. 18, 1137. full text
Barceló, C., Liberati, S. & Visser, M. (2001). Analogue gravity from field theory normal modes? Class. Quantum Grav. 18, 3595; (2002) Refringence, field theory, and normal modes. Class. Quantum Grav. 19, 2961. abstract
Barceló, C., Liberati, S. & Visser, M. (2005; 2011). Analogue gravity. Living Rev. Relativ. 8, 12; 14, 3. full text
Booker, J. R. & Bretherton, F. P. (1967). The critical layer for internal gravity waves in a shear flow. J. Fluid Mech. 27, 513. abstract
Bretherton, F. P. & Garrett, C. J. R. (1968). Wavetrains in inhomogeneous moving media. Proc. R. Soc. A 302, 529. abstract
Cadoni, M. (2005). Acoustic analogues of two-dimensional black holes. Class. Quantum Grav. 22, 409. abstract
Corley, S. & Jacobson, T. (1996). Hawking spectrum and high frequency dispersion. Phys. Rev. D 54, 1568. abstract
Coutant, A. & Weinfurtner, S. (2016). The imprint of the analogue Hawking effect in subcritical flows. Phys. Rev. D 94, 064026. full text
Cropp, B., Liberati, S. & Turcati, R. (2016). Vorticity in analogue gravity. Class. Quantum Grav. 33, 125009. abstract
Euvé, L.-P., Michel, F., Parentani, R., Philbin, T. G. & Rousseaux, G. (2016). Observation of noise correlated by the Hawking effect in a water tank. Phys. Rev. Lett. 117, 121301. abstract
Fischer, U. R. & Schützhold, R. (2004). Quantum simulation of cosmic inflation in two-component Bose–Einstein condensates. Phys. Rev. A 70, 063615. full text
Fischer, U. R. & Visser, M. (2002). Riemannian geometry of irrotational vortex acoustics. Phys. Rev. Lett. 88, 110201. abstract
Fizeau, H. (1851). Sur les hypothèses relatives à l’éther lumineux… C. R. Acad. Sci. 33, 349. record
Garay, L. J., Anglin, J. R., Cirac, J. I. & Zoller, P. (2000). Sonic analog of gravitational black holes in Bose–Einstein condensates. Phys. Rev. Lett. 85, 4643. abstract
Garcia de Andrade, L. C. (2004). Non-Riemannian geometry of vortex acoustics. Phys. Rev. D 70, 064004; (2005) Phys. Lett. A 346, 327. abstract
Gordon, W. (1923). Zur Lichtfortpflanzung nach der Relativitätstheorie. Ann. Phys. 72 (ser. 4; continuous vol. 377), 421. record (form of the metric taken from Leonhardt & Piwnicki 1999)
Hamilton, A. J. S. & Lisle, J. P. (2008). The river model of black holes. Am. J. Phys. 76, 519. full text
Jacobson, T. (1991). Black-hole evaporation and ultrashort distances. Phys. Rev. D 44, 1731. abstract
Jacobson, T. (1995). Thermodynamics of spacetime: the Einstein equation of state. Phys. Rev. Lett. 75, 1260. abstract
Jacobson, T. & Kang, G. (1993). Conformal invariance of black hole temperature. Class. Quantum Grav. 10, L201. abstract
Jacobson, T. A. & Volovik, G. E. (1998). Event horizons and ergoregions in ³He. Phys. Rev. D 58, 064021. full text
Johnson, J. W. (1947). The refraction of surface waves by currents. Trans. AGU 28, 867. record
Lahav, O. et al. (2010). Realization of a sonic black hole analog in a Bose–Einstein condensate. Phys. Rev. Lett. 105, 240401. abstract
Landau, L. D. & Lifshitz, E. M. The Classical Theory of Fields, 4th rev. English ed., §90 and the problem following it. secondary
Leonhardt, U. & Piwnicki, P. (1999). Optics of nonuniformly moving media. Phys. Rev. A 60, 4301. full text
Leonhardt, U. & Piwnicki, P. (2000). Relativistic effects of light in moving media with extremely low group velocity. Phys. Rev. Lett. 84, 822. abstract
Liberati, S., Visser, M. & Weinfurtner, S. (2006). Analogue quantum gravity phenomenology from a two-component Bose–Einstein condensate. Class. Quantum Grav. 23, 3129. abstract
Longuet-Higgins, M. S. & Stewart, R. W. (1961). The changes in amplitude of short gravity waves on steady non-uniform currents. J. Fluid Mech. 10, 529. abstract
Madelung, E. (1927). Quantentheorie in hydrodynamischer Form. Z. Phys. 40, 322. record
Moncrief, V. (1980). Stability of stationary, spherical accretion onto a Schwarzschild black hole. ApJ 235, 1038. abstract
Muñoz de Nova, J. R., Golubkov, K., Kolobov, V. I. & Steinhauer, J. (2019). Observation of thermal Hawking radiation and its temperature in an analogue black hole. Nature 569, 688. abstract
Nguyen, H. S. et al. (2015). Acoustic black hole in a stationary hydrodynamic flow of microcavity polaritons. Phys. Rev. Lett. 114, 036402. abstract
Painlevé, P. (1921). La mécanique classique et la théorie de la relativité. C. R. Acad. Sci. 173, 677. Gullstrand, A. (1922). Allgemeine Lösung des statischen Einkörperproblems in der Einsteinschen Gravitationstheorie. Ark. Mat. Astron. Fys. 16(8), 1. record
Patrick, S., Goodhew, H., Gooding, C. & Weinfurtner, S. (2021). Backreaction in an analogue black hole experiment. Phys. Rev. Lett. 126, 041105. abstract
Peregrine, D. H. (1976). Interaction of water waves and currents. Adv. Appl. Mech. 16, 9. record
Perez Bergliaffa, S. E., Hibberd, K., Stone, M. & Visser, M. (2004). Wave equation for sound in fluids with vorticity. Physica D 191, 121. abstract
Philbin, T. G., Kuklewicz, C., Robertson, S., Hill, S., König, F. & Leonhardt, U. (2008). Fiber-optical analog of the event horizon. Science 319, 1367. full text
Plebanski, J. (1960). Electromagnetic waves in gravitational fields. Phys. Rev. 118, 1396. secondary
Procopio, L. M., Aguero-Santacruz, R., Bermudez, D. & Leonhardt, U. (2026). Backreaction of stimulated Hawking radiation in an optical analogue. Nature 655, 336. abstract
Robertson, S. J. (2012). The theory of Hawking radiation in laboratory analogues. J. Phys. B 45, 163001. full text
Rousseaux, G., Mathis, C., Maïssa, P., Philbin, T. G. & Leonhardt, U. (2008). Observation of negative-frequency waves in a water tank: a classical analogue to the Hawking effect? New J. Phys. 10, 053015. full text
Rousseaux, G. (2013). The basics of water waves theory for analogue gravity. Lect. Notes Phys. 870, 81. full text
Sakharov, A. D. (1967). Vacuum quantum fluctuations in curved space and the theory of gravitation. Dokl. Akad. Nauk SSSR 177, 70; Sov. Phys. Dokl. 12, 1040 (1968); reprinted Gen. Relativ. Gravit. 32, 365 (2000). abstract
Schützhold, R. & Unruh, W. G. (2002). Gravity wave analogues of black holes. Phys. Rev. D 66, 044019. full text
Schützhold, R., Uhlmann, M., Xu, Y. & Fischer, U. R. (2005). Quantum backreaction in dilute Bose–Einstein condensates. Phys. Rev. D 72, 105005. abstract
Steinhauer, J. (2016). Observation of quantum Hawking radiation and its entanglement in an analogue black hole. Nat. Phys. 12, 959. abstract
Unna, P. J. H. (1942). Waves and tidal streams. Nature 149, 219. abstract
Unruh, W. G. (1981). Experimental black-hole evaporation? Phys. Rev. Lett. 46, 1351. abstract
Unruh, W. G. (1995). Sonic analogue of black holes and the effects of high frequencies on black hole evaporation. Phys. Rev. D 51, 2827. abstract
Visser, M. (1998). Acoustic black holes: horizons, ergospheres and Hawking radiation. Class. Quantum Grav. 15, 1767. full text
Visser, M. & Liberati, S. (2022). Painlevé–Gullstrand coordinates versus Kerr spacetime geometry. Gen. Relativ. Gravit. 54, doi:10.1007/s10714-022-03025-z. abstract
Visser, M. & Weinfurtner, S. (2005). Massive Klein–Gordon equation from a BEC-based analogue spacetime. Phys. Rev. D 72, 044020. full text
Volovik, G. E. (2003). The Universe in a Helium Droplet. Oxford University Press. secondary
Weinfurtner, S., Liberati, S. & Visser, M. (2007). Analogue space-time based on 2-component Bose–Einstein condensates. Lect. Notes Phys. 718, 115. full text
Weinfurtner, S., Tedford, E. W., Penrice, M. C. J., Unruh, W. G. & Lawrence, G. A. (2011). Measurement of stimulated Hawking emission in an analogue system. Phys. Rev. Lett. 106, 021302. full text
White, R. W. (1973). Acoustic ray tracing in moving inhomogeneous fluids. J. Acoust. Soc. Am. 53, 1700. abstract
Change log
0.4 · 9 Sep 2026 Corrected the Hawking-temperature formula, which was missing a factor of the sound speed (kBTH = ħgH/2πcs). Separated horizon (v⊥ = cs) from ergosurface (|v| = cs) and both from the atmospheric critical layer, which is a phase-speed condition and is now kept as an analogy. Split the general-relativity row into Schwarzschild in Painlevé–Gullstrand form (exact), Kerr in Doran coordinates (limited: no flat-sliced form, per Visser & Liberati 2022) and the Maxwell-to-medium constitutive rewriting (exact as equations). Replaced the independent-discovery count with a statement of antecedents and uncertainties; removed the “everything after 1981 is a transplant” and forty-year-lag claims, which Visser’s 1998 introduction contradicts. Qualified the conformal-invariance statement with the Jacobson–Kang conditions and specified “minimally coupled”. Distinguished stimulated classical measurements (water, fibre) from spontaneous quantum emission (condensates) in the ledger and the verdict. Rewrote the opening and the first verdict so that “exact” attaches only to the correspondences marked exact. Removed the aether aside. Replaced “the metric is exact above the healing length” with the long-wavelength wording. Reference flags now distinguish full text, abstract, secondary and record only, and several entries were downgraded accordingly. Corrected the first author of the 2026 optical back-reaction paper to L. M. Procopio. Earlier versions’ self-corrections moved here from the main text.
0.3 · 9 Sep 2026 Verdict labels changed to exact / limited / analogy and stated as applying per correspondence. First-series count corrected to eleven. Priority language changed to “earliest source located”. Bibliographic verification separated from reading depth. Added the shared-equation / shared-problem requirement.
0.2 · 9 Sep 2026 Rebuilt for publication with dictionary and direction-of-implication columns. Corrected three claims from 0.1 that the literature search contradicted: two-sound-speed (bi-metric) geometry had been described as unexplored (it is well developed, see Extensions); back-reaction had been described as open (first-order theory and two experiments exist); the conformal factor had been described as treated only as a nuisance (it enters scattering, curvature and back-reaction). Corrected Jacobson–Volovik’s system from “thin film” to a planar soliton in bulk ³He-A, the Anderson–Spiegel year to 1975, and the Hamilton–Lisle “no fluid equation” line to a gloss. Added the pre-1961 wave-blocking history.
0.1 · 9 Sep 2026 First pass, private working notes.
Method note. This page was drafted with AI assistance (Anthropic’s Claude) working from my brief and revised with me through two rounds of external review. Bibliographic details were verified against publisher records for every reference; the flag beside each says how far the content itself was consulted. Claims that a topic was unexplored were searched before being kept. The verdicts are mine and I expect some of them to be argued with; corrections are welcome and will be logged. Content on this site is licensed CC BY-SA 4.0.