On the Uncertainty Principle for Proper Time and Mass

TL;DR

This paper presents an alternative derivation of the uncertainty relation between proper time and mass in relativistic physics, using the geodesic equation from general relativity.

Contribution

It offers a new derivation of the uncertainty principle for proper time and mass based on the geodesic equation, complementing previous work.

Findings

Derivation based on time-like geodesic equation

Supports the uncertainty relation $c^2 riangle m riangle au \\geq \\hbar/2$

Provides an alternative mathematical approach

Abstract

In [J. Math. Phys. {\bf 40}, 1237 (1999)] Kudaka and Matsumoto derive the uncertainty relation $c^2 \Delta m \Delta \tau \geq \hbar/2$ between the rest mass $m$ and the proper time $\tau$, by considering the Lagrangian $M(\dot \tau - c^{-1} \sqrt{-g_{\mu \nu} \dot x^\mu \dot x^\nu}) + e A_\mu (x) \dot x^\mu$. In this note we give an alternative derivation based on a special case of the time-like geodesic equation obtained using the general relativistic Lagrangian $g_{\mu \nu} \frac{dx^\mu}{d \tau}\frac{dx^\nu}{d \tau}$.