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5.2 Super-radiance

Another phenomenon that has been (and is being) analysed from the analogue gravity perspective is super-radiance. The rotational kinetic energy accumulated in a rotating black hole can be extracted from it by scattering into it waves of sufficiently low frequency and high angular momentum. In general, in order that the wave can extract rotational energy from the system it has to satisfy the condition
w < m _O_ (255)
where _O_ is the angular speed of the black hole at the event horizon and m is the harmonic azimuthal number of the wave. This is a purely classical process first considered by Penrose [305]. When dealing with quantum fields, as opposed to classical fields, this process can proceed spontaneously: Quantum mechanically, a rotating black hole will tend to radiate away all of its angular momentum, eventually approaching a non-rotating Schwarzschild black hole [429430]. This process is known as super-radiance. (The term super-radiance was already used in condensed matter to describe processes in which there was some coherent emission of radiation from an otherwise disordered system.)

Again, these processes have a purely kinematical origin so they are perfectly suitable for being reproduced in an analogue model. Regarding these processes, the simplest geometry that one can reproduce, thinking of analogue models based on fluid flows, is that of the draining bathtub of Section 2. Of course, this metric does not exactly correspond to Kerr geometry, nor even to a section of it [401]. However, it is qualitatively similar. It can be used to simulate both Penrose’s classical process and quantum super-radiance as these effects do not depend on the specific multipole decomposition of Kerr’s geometry, but only on its rotating character. A specific experimental setup has been put forward by Schützhold and Unruh using gravity waves in a shallow basin mimicking an ideal draining bathtub [345]. Equivalently to what happens with Kerr black holes, this configuration is classically stable (in the linear regime) [31]. A word of caution is in order here: Interactions of the gravity surface waves with bulk waves (neglected in the analysis) could cause the system to become unstable [415]. This instability has no counterpart in standard general relativity (though it might have one in braneworld theories). Super-resonant scattering of waves in this rotating sink configuration, or in a simple purely rotating vortex, could in principle be observed in this and other analogue models. There are already several articles dealing with this problem [25272664113237].

A related phenomenon one can consider is the black-hole bomb mechanism [312]. One would only have to surround the rotating configuration by a mirror for it to become grossly unstable. What causes the instability is that those in-going waves that are amplified when reflected in the ergosphere would then in turn be reflected back toward the ergoregion, due to the exterior mirror, thus being amplified again, and so on.

An interesting phenomenon that appears in many condensed matter systems is the existence of quantised vortices. The angular momentum of these vortices comes in multiples of some fundamental unit (typically h or something proportional to h). The extraction of rotational energy by a Penrose process in these cases could only proceed via finite-energy transitions. This would supply an additional specific signature to the process. In such a highly quantum configuration it is also important to look for the effect of having high-energy dispersion relations. For example, in BECs, the radius of the ergoregion of a single quantised vortex is of the order of the healing length, so one cannot directly associate an effective Lorentzian geometry to this portion of the configuration. Any analysis that neglects the high energy terms is not going to give any sensible result in these cases.

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