Illustration of the Relativity Principle Using a Double Breather

Abstract

The purpose of this article is to give more clarity to some relativistic laws, primarily the principle of relativity.These laws have been illustrated using linear waves in several of our previous articles. In this work, solutions of the nonlinear sine-Gordon equation in the form of double breathers are used for the same purposes. The exact two-breather solution and the approximate one that is convenient for analysis are presented. Moving breathers are derived from standing breathers through the active Lorenz transformations. Oscillations in a moving breather become out of phase. For each system, the distance between the amplitude maxima is taken as a unit length, and the oscillation period is taken as a time interval unit. It is shown that the observer associated with a moving breather and an observer associated with a standing breather see the same picture; an observer cannot distinguish the state of movement of his breather from rest. Therefore, in a frame of reference based on a moving breather, its oscillations are considered in-phase. In this case, time is determined as if the clocks were synchronized according to the Einstein method.Space-time coordinates of the same event in different frames of reference turn out to be related by the passive Lorentz transformations.