Experimental Black-Hole Evaporation?

From the source

1. Take the standard equations of an irrotational fluid — Euler, continuity, and a pressure that depends only on density — and linearise them about any background flow. The equation that governs small disturbances of the velocity potential is exactly the equation for a massless scalar field propagating in a curved geometry, with a metric built from the background density, the background flow velocity and the local sound speed. This is a mathematical identity, not an analogy.Page 1351, the linearised equations, through the acoustic metric displayed at the head of page 1352

2. If the background flow is steady and converging, and somewhere crosses the local speed of sound smoothly, then near that crossing the acoustic metric takes the same form the Schwarzschild metric takes near a black-hole horizon. Unruh writes both metrics side by side to make the correspondence explicit. That surface is a sonic horizon: sound generated inside it cannot climb back out against the flow.Page 1352, the transonic expansion of the radial velocity and the comparison with the Schwarzschild line element

3. Quantise the sound field and the correspondence carries the physics with it. An observer drifting through the sonic horizon with the fluid sees the phonon field in its ordinary vacuum state, and the modes therefore behave, in the co-moving time, exactly as a scalar field does near a Schwarzschild horizon for a freely falling observer. The consequence is that the sonic black hole emits sound with a thermal spectrum, filtered by a greybody factor, at a temperature equal to Planck’s constant over two pi times Boltzmann’s constant, multiplied by the rate at which the flow velocity changes across the horizon.Page 1352 into 1353, the mode expansion and the displayed temperature formula

4. The result does not lean on the idealisations used to derive it. Unruh states in his notes that the conclusions depend only on the existence of a sonic horizon, not on the assumed spherical symmetry or on the particular smooth transonic profile, and that a model of transonic flow through a shaped nozzle gives exactly the same result.Page 1353, notes 5 and 8

5. The system is offered as a working laboratory for the parts of black-hole evaporation nobody can otherwise reach. The fluid equations are known to fail at atomic distances, just as a smooth spacetime is expected to fail at the Planck scale, so the sonic model lets you ask what a short-distance breakdown does to the emission. And the emitted phonons are themselves fluctuations of the flow, so they alter their own propagation in exactly the way graviton emission alters the spacetime that carries it.Page 1353, the paragraph beginning with the theoretical-laboratory argument

6. Unruh gives the number and the honest difficulty in the same breath. Taking the velocity gradient at the horizon as the sound speed divided by the horizon radius, the predicted temperature is roughly three ten-millionths of a kelvin for a sound speed of 300 metres per second and a horizon one millimetre across — low enough that turbulence in a small nozzle would likely swamp it, but, in his words, a much simpler experimental task than creating a tiny black hole.Page 1353, the numerical temperature estimate and the closing paragraph

The way in

How to cite it

William G. Unruh (1981) Experimental Black-Hole Evaporation? doi:10.1103/PhysRevLett.46.1351

Where it sits in the curriculum

• The vacuum as a quantum fluid