The Illusion of Falling: Observer-Dependent Horizon Completion via the Apparent Completion Index (ACI)

This work introduces the Apparent Completion Index (ACI), a new general-relativistic measure that quantifies how different observers perceive the progression of infalling motion near a black hole horizon. The ACI unifies two contradictory yet self-consistent perceptions: 1. The external observer sees the infalling object shrinking, slowing, and fading, approaching an asymptotic freeze-out. 2. The internal observer experiences a completed fall, while the external universe recedes and becomes unobservable. ACI is derived directly from the gravitational redshift factor governing observable radial dynamics. This formalism offers a unified approach to understanding observer-dependent horizon physics, allowing a consistent quantitative description of these dual interpretations. The paper provides a quantitative structure to describe the relationship between perceived fall progression and the observers' relative positions, and highlights the role of the event horizon in limiting observational access to the ultimate completion of the fall. This work contributes to the ongoing discussion of black hole information and causality by providing a testable metric for understanding the dual interpretations of events near black hole horizons.