The Spheronic Toy Universe: How Special Relativity may be Visualised to Emerge from a Wave-Nature of Matter

We construct an idealised universe for didactic purposes. This universe is assumed to consist of absolute Euclidean space and to be filled with a classical medium which allows for sound waves. A known solution to the wave equation describing the dynamics of the medium is a standing spherical wave. Although this is a problem of classical mechanics, we demonstrate that the Lorentz transformation is required to generate a moving solution from the stationary one. Both solutions are here collectively referred to as "spherons". These spherons exhibit properties which have analogues in the physical description of matter with rest mass, among them de Broglie like phase waves and at the same time "relativistic" effects such as contraction and a speed limit. This leads to a theory of special relativity by assuming the point of view of an observer made of such spheronic "matter". The argument made here may thus be useful as a visualisation or didactic approach to the real universe, in which matter has wave-like properties and obeys the laws of special relativity.