Derivation of Relativistic law of Addition of Velocities from Superposition of Eigenfunctions and Discreteness

TL;DR

This paper derives the relativistic velocity addition law by analyzing the superposition of eigenfunctions within a discrete reciprocal symmetric number system, linking quantum-like eigenfunction superpositions to relativistic velocity composition.

Contribution

It introduces a novel approach connecting eigenfunction superpositions and a discrete reciprocal symmetric number system to derive relativistic velocity addition.

Findings

Superposition of eigenfunctions yields Galilean velocity addition.

Replacing derivatives with finite differences leads to relativistic addition law.

A reciprocal symmetric number system underpins the relation between superposition and relativistic velocities.

Abstract

We have defined a slowness, s, as the reciprocal conjugate of velocity, v. s = -ih/v. We have shown that Einstein's postulate (v has an upper limit) implies that s is discrete. A velocity operator is defined as the derivative with respect to s. Superposition of corresponding Eigenfunctions give Galilean law of addition of velocities. We have replaced the differential operator by the corresponding finite difference symmetric operator. We have shown that superposition of corresponding discrete Eigenfunctions give relativistic law of addition of velocities. A reciprocal symmetric number system is developed and with the help of this number system we have shown the relation between superposition and relativistic law of addition of velocities.