Crossed spacetime brackets and a Bell inequality for proper-time observables in ADM gravity

This paper completes the classical algebra of geometric observables in linearized ADM gravity and explores its first quantum implication. The starting point is a recent breakthrough by Assanioussi et al. (Class. Quantum Grav. 43, 2026), who showed that the length of a rod measured by two different Lorentz observers does not Poisson-commute. In a previous work (https://zenodo.org/records/20652610) we extended this result in two directions: (i) the length bracket was generalised to two arbitrary inertial observers, revealing a new kinematic factor \((1-2β_1^2)(1-2β_2^2)\) that traces its origin to the strictly one-dimensional regularisation of the rod; (ii) the same framework was applied for the first time to proper-time observables, yielding \(\{T_1,T_2\}≠0\) and showing that the proper times measured by two moving clocks are classically incompatible. The present paper takes the next two logical steps. First, we compute the crossed Poisson bracket between a length measured by one inertial observer and a proper time measured by another, \(\{L_1,T_2\}\). The bracket does not vanish, entailing that the spatial measurement of one observer and the temporal measurement of another are algebraically intertwined. Together with the previously established brackets, this completes the non-commutative algebra: all kinematic observables — lengths and proper times — form a mutually non-commuting set already in the classical phase space of general relativity. Second, we promote the proper-time commutator to a quantum operator algebra. The resulting Weyl algebra is used to construct a Bell test for two gravitationally coupled quantum clocks. As a proof of principle, we show that two clocks prepared in an entangled state violate the CHSH inequality for any non-vanishing squeezing parameter \(r\). The violation is a direct mathematical consequence of the proper-time non-commutativity, while the squeezing parameter is expected to be linked to the gravitational coupling \(G\) through vacuum fluctuations. The analysis is a model-dependent illustration of the non-classical correlations already encoded in the classical symplectic structure of spacetime, rather than a realistic experimental prediction. All calculations are performed within standard linearized ADM gravity with a transparent transverse smearing regularization. The results are verified in all relevant limits and satisfy dimensional consistency. **Additional notes:** This paper is the sequel to "Classical non-commutativity of length and proper-time observables for arbitrary inertial observers in ADM gravity" (https://zenodo.org/records/20652610) Together with the previous work, it establishes a complete classical algebra of spacetime observables and provides a first proof-of-principle connection between gravitational non-commutativity and Bell-type correlations. This manuscript is current in Official Peer Review. Not final version.Copyright©2026 Alex De Giuseppe.All rights reserved. This work is protected by copyright. Any form of plagiarism, unauthorized reproduction, or misappropriation of ideas, mathematically results, or text without proper citation constitutes a violation of academic and intellectual property standards and common laws. No commercial use, adaptation, or derivative works are permitted without explicit written permission from the author. For correspondence, citations, collaboration inquiries, or feedback please contact:degiuseppealex@gmail.com The hash files that determine ownership have been created