Three Temporal Relations in the Description of Physical Events

Descriptions of physical events involve temporal quantities that must be distinguishedif occurrence, reference-frame description, and observation are not to be conflated. Inspecial relativity, proper time accumulated along a timelike worldline, coordinate timeassigned within an inertial reference frame, and the reception event at which a signalreaches an observer are related but conceptually different. This paper offers a didacticand mathematically explicit discussion of these three relations, using atmospheric muonsas a case study. It reviews the Minkowski-space definition of proper time, Lorentz trans-formations, the causal propagation of light signals, and the relativistic survival law forunstable particles. Particular attention is paid to dimensional consistency and to theassumptions required for simplified muon-travel estimates. The central conceptual proposalis methodological: distinguish the physical event, its coordinate description, and the laterreception of information about it. The paper does not introduce a new physical time ormodify special relativity. Rather, it organizes established concepts and identifies a researchprogram for representing particle worldlines, detector events, observer worldlines, and signalpropagation within a single spacetime framework.