The Pond and the Black Hole: Acoustic Metrics, Superradiance, and a Torsional Reading

State the object before the pond

The full object is a Lorentzian metric with a causal structure: a region from which no future-directed causal curve escapes, and, for a rotating solution, an outer region in which no observer can remain static relative to infinity. Horizon and ergoregion are defined by the metric, not by matter. That is the whole of what a black hole contributes to this chapter, and it is stated first so that the water can be compared against something fixed.

The borrowed object is narrower and exact. Linearise the equations of an irrotational, barotropic, inviscid fluid and the perturbation field obeys a wave equation that is formally the curved-space wave equation for a massless scalar on an effective metric built from the background flow and the local sound speed. Unruh (1981) wrote this down; it is not an analogy imposed on the fluid, it is a property of the linearised equations. Where the flow speed exceeds the sound speed, that effective metric has a horizon for sound.

What the tank actually establishes

A draining vortex is the rotating case. The circulating, inflowing background gives the effective metric an ergoregion and, inside it, a sonic horizon. Torrential and Weinfurtner's group measured rotational superradiance in exactly this configuration: surface waves sent onto a draining vortex returned with more energy than they carried in, at the amplification the analogue calculation predicts, with the excess drawn from the vortex's rotation. Later work in the same apparatus reported the characteristic ringdown of the effective geometry. These are real measurements of a real effective metric.

What they establish is that the kinematics of wave propagation on a curved effective metric are correct and testable in a fluid. That is a substantial result: it means horizon-related phenomena which are unobservable in astrophysics — superradiant amplification, quasinormal ringing, in principle a Hawking-like spectrum — can be checked where the background is under experimental control.

What they do not establish, and no practitioner in the field claims they do, is dynamics. The fluid has no Einstein equation. Its background flow is set by a pump and a drain, not by a stress-energy tensor curving anything. The analogue reproduces how fields move on a geometry; it says nothing about what makes the geometry. This is the volume's standard divergence point, and here it is unusually clean.

Pricing the torsional reading

The project's own figure, stated plainly: gravity read as downward torsional vorticity at the positions where the non-trivial zeros sit. Every term in that sentence has a respectable home somewhere. Torsion is the antisymmetric part of a connection and is sourced by spin in Einstein-Cartan theory. Vorticity is the curl of a velocity field and is exactly what the draining tank exhibits. The non-trivial zeros are a spectrum with measured statistics. The figure assembles three real objects.

The assembly is where the price falls due. There is no map from a complex coordinate on the critical line to a spatial position at which a vorticity could be evaluated, so 'at the position of the zeros' has no referent. There is no coupling constant, which means no prediction and no dimensional check. And the direction — 'downward' — imports a Newtonian intuition into a picture that is supposed to be replacing one. The figure is not wrong in the way a miscalculation is wrong; it is not yet the kind of thing that can be wrong, which is a worse condition.

So it stays where the volume keeps such things: an internal figure, admitted for the questions it generates, forbidden from borrowing credibility from the water tank standing next to it. The resemblance between the two is the whole reason the discount has to be written down.

The open baton

The answerable question is not about the zeros. It is whether an analogue system can be built in which the background geometry responds to the perturbations propagating on it — a fluid, condensate or metamaterial with a measurable back-reaction, however weak. That is the missing half of every analogue-gravity result in this volume, and it is the difference between reproducing propagation and reproducing gravity.

The question the figure hands on is narrower and harder: exhibit any map from the critical line to a physical configuration space. Until that exists, the torsional reading is a picture about a picture.