"Waves," "Objects" and Special Relativity

Abstract

Field theories such as Maxwell's equations for electrodynamics describe a continuum and predict the existence of waves. A contact to mechanics and its classical kinematical concepts of localized objects, or particles, moving at certain group velocities along well-defined trajectories, is achieved by defining wave packets and developing the Hamilton ray equations governing their motion. The wave packets contain many oscillations which are modulated by a slowly tapering envelope defining the extent of this packet in space and time. It is shown here that the solitary waves, i.e., waves which are characterized by very sharp localized pulses consisting of a small number of oscillations, are described differently. The two entities are analogous when we compare the behavior of one in configuration space-time with the other in representation space. This analogy is remarkable and leads to new concepts that might prove useful for many applications. Thus it is shown that the solitary wave can be described in terms of analog rays in representation space, on which a group slowness function is defined. In space-time the solitary wave is characterized by a wave surface evolving according to the relevant slowness function at each point. For objects and therefore for wave packets, there exists a relativistic formalism in terms of four-vectors. Thus the Hamilton ray equations can be represented in terms of the four-velocity, involving a differentiation of the position four-vector with respect to the proper time. The analog treatment leads to a differentiation of the representation four-vector with respect to the new concept: the proper frequency. Finally, the analogy provides an explanation for the stability of solitary waves in the presence of nonlinear media, without the need for discussing specific differential equations: It is argued that self defocusing of the analog rays in representation space provides the "glue" which counteracts dispersive effects in configuration space, and thus obviates the fissioning of the solitary wave.