A Time-Free Reframing of the Twin Paradox

A Time-Free Reframing of the Twin Paradox Max Karson November 29, 2025 Abstract The “Twin Paradox” is conventionally resolved by noting the asymmetry of the twins’ worldlines: the traveling twin accelerates and therefore accumulates less proper time. In this note, “time dilation” is reframed without invoking time as an independent dimension. Using a lightclock model, we treat elapsed time operationally as internal interaction density—the frequency at which internal signals complete cycles. Because the speed of the interaction carrier (light) is fixed at c, any external translational velocity vext necessarily reduces the component vint available for internal signaling. The traveling twin returns younger not because a temporal dimension “runs more slowly,” but simply because fewer internal events occurred along that worldline. 1 Self-Interaction Density as Clock Ticks Operationally, a clock is a device that counts repeated internal interactions. For a light clock, a “tick” is one round trip of a photon between mirrors. Let the photon’s velocity be decomposed into an internal component vint (the tick-tock pattern) and an external component vext (the clock’s translational motion). Since the photon always moves at speed c, c2=v2 int +v2 ext.(1) Solving for vint gives vint =cr1− v2 ext c2=c γ.(2) The Lorentz factor γquantifies the trade-off between internal interaction frequency and external motion [1]. 2 Photon Path Geometry We model each twin [2] as a pattern of internal photon paths. Consider the trajectory of one internal photon for each twin during the experiment, from departure to reunion, viewed in the inertial laboratory frame. Its total path length is L=c∆tlab, fixed by the lab’s own clock. •For the inertial (stay-at-home) twin, the entire photon path is spent bouncing between mirrors. •For the traveling twin, part of the path is expended on external translation. Thus, elapsed time operationally equates to the number of photon bounces internal to each the twin’s rest system. 1 3 Conclusion The Twin Paradox can be fully restated without invoking a separate temporal dimension. Each twin allocates the fixed photon speed cdifferently between internal self-interactions and external spatial translations. The inertial twin maximizes internal cycles, whereas the traveling twin diverts photon path length into external motion, inevitably reducing internal event frequency. Thus, the paradox is a purely geometric consequence of redistributing photon paths, eliminating any need for a metaphysical notion of time slowing. AI Disclosure The author used AI language models to assist with drafting, derivations, and algebraic checks. References [1] A. Einstein, “Zur Elektrodynamik bewegter K¨orper,” Annalen der Physik, 17, 891–921 (1905). [2] P. Langevin, “L’´ Evolution de l’espace et du temps,” Scientia, 10, 31–54 (1911). 2