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

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.