• gravitational waves
• general relativity
• dissipative horizons
• Bose-Einstein condensate
• black holes
• analogue gravity
• acoustic metric
• acoustic black holes
• Hawking radiation
• hydrodynamics/gravity analogy
• perturbed acoustic horizon

In 1981 Bill Unruh established an analogy between hydrodynamic flow with a supersonic region and a black hole, initiating the research field of Analogue Gravity. One possibility to exploit this hydrodynamics/gravity analogy is to create analogue black holes within Bose-Einstein condensates. The Bose-Einstein condensation is a quantum phenomenon that occurs in systems of identical bosons cooled below a critical temperature, when a large number of particles simultaneously occupies the ground state. The Bose gas can then be described by a mean field approximation where a quantum field embodies the fluctuations around the macroscopic classical wavefunction. At sufficiently low temperatures these fluctuations, called phonons, are the low-energy collective excitations that can be described as Nambu-Goldstone bosons. Phonons propagate as a massless scalar field on an emergent acoustic metric tensor, which in turn is determined by the condensate characteristics. In particular, an acoustic black hole is created by a transonic fluid, and concepts like the event horizon are applicable. At this acoustic horizon, quantum fluctuations result in a thermal radiation of phonons: the acoustic equivalent of Hawking radiation. Remarkably, this emission near the acoustic horizon has been simulated numerically and verified experimentally with atomic Bose-Einstein condensates.
This thesis builds upon the above field of research in Analogue Gravity: the goal is to design a system where an acoustic horizon is excited by a gravitational wave-like perturbation.
As a first step, we have reproduced a gravitational wave perturbation on top of a flat background acoustic metric emergent from a Bose-Einstein condensate. To achieve this result, we have taken the general form of the acoustic metric and cast it as a Minkowski background plus a perturbation. Then, exploiting the gauge symmetry of General Relativity, i.e. the invariance under coordinate transformations, we have written a gravitational wave metric in a new gauge, such that it can be compared to the analogue perturbation metric. In this way, we find the properties that a condensate should have in order to be in a regime where phonons satisfy the equations of motion of a massless scalar field in a Minkowski plus gravitational wave spacetime. At the same time, we checked that those condensate quantities satisfy hydrodynamic equations, to ensure that the obtained system is physical. Noticeably, since in the analogue model we are reproducing the same background and perturbation metric of a gravitational wave, we have that the analogue gravitational wave satisfies Einstein's equations in vacuum. This is something that has never be done before: none of the previously studied acoustic models were able to reproduce the dynamics of General Relativity, but they represented only static solutions. The possibility of simulating a gravitational wave in the analogue system is the first original result of this thesis work. It immediately leads to envision whether the analogue of a gravitational wave can be simulated over a background metric featuring an event horizon. Indeed, this would open the possibility to investigate the response of the acoustic horizon to a gravitational wave-like perturbation.
The second result of the thesis comes straightforwardly from the first one: we have expanded the analogue gravitational wave solution to realize an impinging gravitational wave-like perturbation to an acoustic horizon. We have chosen a cylindrical acoustic black hole rather than a simpler two-dimensional plane horizon to facilitate a more general calculation. Moreover, this geometry is experimentally feasible in the laboratory, as we have confirmed the validity of hydrodynamic equations. Certainly, this configuration is unrelated to observed astrophysical black holes. We have studied how the horizon of the above system is perturbed by such analogue gravitational wave, also computing the horizon's generators.
Our analogue model indeed paves the way for the theoretical and experimental determination of the reflectivity, the shear viscosity and the entropy of an acoustic horizon determined by an external perturbation akin to a gravitational wave. Hence, it may allow to test the conjectured Kovtun-Son-Starinets lower bound for the shear viscosity coefficient to entropy density ratio in an experiment performed e.g. in ultra-cold quantum gases platforms.