A discrete and finite approach to past proper time

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.