On the non-commutativity of geometric observables in different Lorentz frames

TL;DR

This paper investigates whether geometric measurements like length, area, or volume commute between different inertial observers, revealing that they generally do not, which has implications for understanding quantum gravity.

Contribution

It demonstrates that geometric observables measured by different inertial observers do not Poisson commute, even in flat spacetime, highlighting a fundamental non-commutativity in relativistic geometry.

Findings

Geometric observables do not Poisson commute across different observers.

Non-commutativity persists even in Minkowski spacetime.

Implications for quantum gravity and fundamental scales.

Abstract

Our aim is to establish whether geometric observables, such as length, area or volume of a physical object, viewed by different observers Poisson commute or not. To illustrate this, we compute the Poisson bracket of two lengths associated to a rigid rod and measured by two different geodesic (inertial) observers, one of which is at rest while the other is moving with respect to the rod. Our calculation shows that geometric observables measured by different observers generically do not Poisson commute, not even in Minkowski spacetime. This non-trivial result provides interesting insights into questions related to the presence of a fundamental scale in the context of quantum gravity.