Abstract
The Einstein cylinder is the first cosmological model for our universe in modern history. Its geometry not only describes a static universe—a universe being invariant under time reversal—but it is also the prototype for a maximally symmetric spacetime with constant positive curvature. As such, it is still of crucial importance in numerous areas of physics and engineering, offering a fruitful playground for simulations and new theories. Here, we focus on the implementation and simulation of acoustic wave propagation on the Einstein cylinder. Engineering such an extraordinary device is the territory of metamaterial science, and we will propose an appropriate tuning of the relevant acoustic parameters in such a way as to mimic the geometric properties of this spacetime in acoustic space. Moreover, for probing such a space, we derive the corresponding wave equation from a variational principle for the underlying curved spacetime manifold and examine some of its solutions. In particular, fully analytical results are obtained for concentric wave propagation. We present predictions for this case and thereby investigate the most significant features of this spacetime. Finally, we produce simulation results for a more sophisticated test model which can only be tackled numerically.
Highlights
After concluding with the formulation of the general theory of relativity in 1916, Einstein moved on to devise relativistic models of the universe, applying his new theory to the realm of physical cosmology
We explore the possibilities of simulating acoustics on the Einstein cylinder with the help of acoustic metamaterials—materials which enable researchers and engineers to contrive extraordinary devices with exceptional properties, exceeding the limits established by nature
We presented an analysis of acoustic wave propagation on the Einstein cylinder
Summary
After concluding with the formulation of the general theory of relativity in 1916, Einstein moved on to devise relativistic models of the universe, applying his new theory to the realm of physical cosmology. In the last part of the same section, the geometric features of the Einstein cylinder are linked to the physical properties and conditions of this spacetime This formalism is required to derive the corresponding wave equation for the acoustic analogue model. In this process of metamaterial tuning, the relevant acoustic parameters (the scalar bulk modulus and the density tensor) have to be selected appropriately in order to engineer in the laboratory a suitable device which mimics the geometric features of the Einstein cylinder acoustically. In order to examine the significant features of this acoustic analogue space, monochromatic (i.e., having fixed frequency) and concentrically traveling test waves are used to probe the metamaterial In this case, fully analytical results are obtained— for the non-trivial radial dependence of the wave propagation.
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