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

Mehdi Assanioussi, Jerzy Kowalski-Glikman, Ilkka Mäkinen, Ludovic Varrin

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.

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

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.
Paper Structure (5 sections, 43 equations, 1 figure)

This paper contains 5 sections, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Context and general considerations
  3. Simultaneity surface and induced metric
  4. Lengths in different frames and their Poisson brackets
  5. Summary and discussion

Figures (1)

  • Figure 1: Depicting the world sheet $R$ of a rod (grey) and the process of measuring its length, via light signals (red) emitted at the point $p$, by two geodesic observers: One at rest with respect to the rod (black) and one in motion with respect to it (blue).