Canonical form of a deformed Poisson bracket spacetime

TL;DR

This work recasts a GUP-inspired, deformed Poisson-bracket spacetime into a canonical, covariant Hamiltonian framework by constructing a Hamiltonian constraint whose flow reproduces the GUP metric while preserving the hypersurface-deformation algebra. By solving in static, Gullstrand–Painlevé, and homogeneous gauges, it shows that the same geometry emerges in different coordinate charts, demonstrating covariance across gauges. The formalism permits covariant coupling to scalar matter and dust, enabling analysis of dynamical processes such as collapse and field evolution (Klein–Gordon dynamics) within the GUP-corrected spacetime. Overall, it bridges heuristic GUP spacetime constructions with a robust canonical structure suitable for dynamical studies and backreaction analyses.

Abstract

The general uncertainty principle applied to gravity can be implemented as a set of modified Poisson brackets in the canonical formalism. As such, the theory is not canonical and the resulting equations of motion, diffeomorphism constraint, and Hamiltonian constraint are unlikely to lead to a covariant metric. We construct a Hamiltonian that when applying the usual canonical formalism gives a closed algebra and equations of motion that result in the original metric obtained by using distorted Poisson brackets. The resulting theory is thus rendered canonical and covariant. We then covariantly couple scalar matter and dust to the modified gravity to allow the study of dynamics.