Evading the Boulware-Deser Instability via Decay Catalysis

We present a novel framework for mass-varying gravity featuring a density-dependent effective mass profile for an inherently massless graviton dynamically coupled to a scalar field with a non-Fierz-Pauli parameter (hμν hμν - β h², where β ≠ 1). Traditionally, such frameworks would suffer from Boulware-Deser ghosts, which unlock an unstable sixth degree of freedom, causing the vacuum state to become catastrophically unstable. However, we used this intrinsic instability as a catalyst for a process of constant graviton decay and recombination. This stabilization mechanism fundamentally relies on an important—and more importantly, observationally recognized—postulate: A perfectly flat Minkowski spacetime cannot manifest within the observable universe at any point in time. We utilize the scalar field φ to dynamically modulate the mass of gravitons through a Chameleon-like mechanism for screening, where a graviton's mass in high-density areas (such as the solar system) is reduced to effectively 0. In this first installment of three total papers, we develop the foundational mathematics, decay paths, and boundaries of such a framework. We derive two distinct decay channels: hμν → φφ and hμν → γγ, which all go through identical but reversed recombination paths, and prove that the decay of a graviton occurs before the ghost manifests beyond manageable instabilities that drive decay.