A Numbers-Based Approach to a Free Particle's Spacetime

Abstract

A possibility is proposed to define the proper spacetime of a free nonzerorest- mass m_0 particle based on the connection of its lasting proper time to an open sequence of natural numbers counting de Broglie time periods (h/c^2)(m^(-1)_ 0 ) [see R. Ferber, A Missing Link: What is Behind de Broglie's" Periodic Phenomenon"?, Foundations of Physics Letters 9, 575 (1996)]. It is suggested to define a set of twodirectional intervals of the particle's proper space (proper distances) following the construction of positive and negative integers from the ordered pairs of the natural numbers, which belong to the sequence 1, 2, ..., n defining the elapsed interval of de Broglie time t_n. Corresponding to rational numbers, the ratios of proper distances to their common time interval tn are referred to as proper velocities, which are expressed by a rational fraction of c without supposing any relation to an external system of coordinates. The particle's proper reference frame appears by resolving the t_n over proper spatial and temporal coordinates, which are connected in a wave-like process. The corresponding connection in energy representation takes the form of relativistic energy-momentum relation of the respective proper quantities, which reveals the existence, besides the rest energy term m_0c^2, of the term attributed to particle's proper space that is mainly determined by large proper distances and approaches with growing tn the finite value of the order of m_0c^2.