Acoustics in Curved Spacetime
Michael Tung (Polytechnic University Valencia)

In recent years metamaterials have provided researchers and engineers with unprecedented tools for the design and construction of artificial devices with remarkable properties exceeding the possibilities found in nature. While optical metamaterials have been the focus of continued interest for the last decade acoustic metamaterials have only recently drawn the attention of researchers. Nevertheless, the considerable successes in the design of optical analogue models of gravity should also carry over to investigations for physical systems describing sound propagation in acoustic metamaterials—though with major modifications. The simulation of optical and acoustic phenomena with curved background spacetimes not only poses challenges in engineering, but may also raise fundamental questions beyond their possible verification in the laboratory.
This presentation develops and explains a novel technique to model acoustic wave propagation on a curved spacetime based on a fundamental variational principle in combination with powerful differential-geometric methods. Its aim is to completely describe the evolution of acoustic pressure through an appropriate non-dissipative metamedium.
In the proposed framework we derive the partial differential equation for the acoustic potential which governs the wave propagation on the underlying curved spacetime. The time dependence is usually harmonic and a Sturm-Liouville problem emerges for the remaining spatial variables. For some cases it is possible to obtain analytical solutions allowing for an exhaustive analysis of the expected wave predictions. Of major concern is also the design and implementation of a particular acoustic space- time. We derive and use the so-called constitutive equations which directly relate the physical acoustic parameters to a particular spacetime.
As a first example we show how to implement acoustic wave propagation for the Poincare ́ half-plane model. It is the simplest and one of the most thoroughly investigated non-Euclidean models of two- dimensional hyperbolic geometry. This makes it an ideal spacetime candidate for the implementation and study of an acoustic metamaterial. In the second application we demonstrate how to implement the gravitational frequency-shift analogue in acoustics, i.e., the acoustic imitation of a shift in radiation frequency due to a gravitational field. We close this discussion with a brief examination of Maxwell’s fish-eye lens in spacetime acoustics as it might facilitate the construction of perfect acoustic focussing devices.