A Geometric Reinterpretation of Causal Inversion and Role Redesignation in General Relativity

This paper presents a geometric reinterpretation of causal inversion and causal-role redesignation in the vicinity of a black hole horizon. Without introducing new physical principles or modifying Einstein’s equations, the model reframes established general-relativistic behaviour as a continuous two-step rotation in causal structure. Spatial and temporal axes progressively exchange operational roles about a closed symmetry direction---the azimuthal Killing vector of a spacetime admitting \(U(1)\) symmetry---which supplies a compact invariant axis allowing the interpolation of causal-role assignments to be represented without discontinuity. Within this geometric perspective, the horizon is reinterpreted as the origin of the infalling observer’s internal causal frame, functioning as a pointlike temporal origin, while the singularity is correspondingly re-expressed as an extended causal boundary toward which the temporal axis is directed. This construction situates horizon crossing within a continuous topological framework, providing a geometric account of smooth causal redesignation consistent with General Relativity.