A discrete and finite approach to past proper time

TL;DR

This paper explores the connection between the Lorentz factor in relativity and binomial distribution through power series expansion, linking proper time to return probabilities of Bernoulli random walks, including cases with absorbing barriers.

Contribution

It introduces a novel discrete approach to proper time using random walk return probabilities, connecting physics and probability theory.

Findings

Power series of gamma function relates to return probabilities

Proper time is proportional to sum of return probabilities

Symmetric random walk case corresponds to speed of light

Abstract

The function $\gamma(x)=\frac{1}{\sqrt{1-x^2}}$ plays an important role in mathematical physics, e.g. as factor for relativistic time dilation in case of $x=\beta$ with $\beta=\frac{v}{c}$ or $\beta=\frac{pc}{E}$. Due to former considerations it is reasonable to study the power series expansion of $\gamma(x)$. Here its relationship to the binomial distribution is shown, especially the fact, that the summands of the power series correspond to the return probabilities to the starting point (local coordinates, configuration or state) of a Bernoulli random walk. So $\gamma(x)$ and with that also proper time is proportional to the sum of the return probabilities. In case of $x=1$ or $v=c$ the random walk is symmetric. Random walks with absorbing barriers are introduced in the appendix. Here essentially the basic mathematical facts are shown and references are given, most interpretation is left to the reader.