Instability mechanism of the Schrödinger-Newton soliton near the Planck scale

Abstract The Schrödinger–Newton equation predicts that self-gravitating quantum wave packets form stable solitons for arbitrarily large masses. We show that including confinement energy as an active gravitational mass provides a physically well-motivated extension of the Schrödinger–Newton framework, resolving a known limitation of the standard model and correcting an inconsistent truncation of its non-relativistic limit. By performing a systematic 1/ c expansion of the Einstein–Klein–Gordon system, we derive the consistent non-relativistic limit in which the active gravitational mass density includes the kinetic energy density at order c −2 . This relativistic correction explicitly breaks the Schrödinger–Newton scaling symmetry and introduces short-range attractive contributions to the effective potential for the wave packet width, scaling more strongly than the repulsive quantum pressure. As a result, the metastable soliton disappears above a critical mass. Using Gaussian and hyperbolic secant trial functions, we find instability thresholds m c ≈ 0.94 m P and m c ≈ 0.74 m P , respectively, in good agreement with the exact general-relativistic mass limit for mini-boson stars (0.633 m P ). We provide a stability analysis via a generalized virial identity and a dynamical frequency study, identifying a ‘frequency catastrophe’ where gravitational breathing modes outpace the fundamental Zitterbewegung limit. This identifies the Planck scale as a natural dynamical threshold at which confinement energy destabilizes single-particle self-gravitation.