Reconstructing Special Relativity from Ze
DOI:
https://doi.org/10.65649/1sdtpd07Keywords:
Special Relativity, Emergent Spacetime, Discrete Physics, Time Dilation, Minkowski Metric, Lagrangian Dynamics, Spacetime FunctionalismAbstract
This paper presents a reconstruction of special relativity from a fundamentally discrete framework called Ze, in which physical reality is composed of discrete events eₖ and local counters Cᵢ with increments ΔCᵢ. Two primitive modes of change are distinguished: temporal mode (sequential updates) and spatial mode (configurational changes across channels). From these elements, a fundamental invariant I = Σᵢ (ΔCᵢˢᵖᵃᵗⁱᵃˡ)² – γ Σᵢ (ΔCᵢᵗᵉᵐᵖᵒʳᵃˡ)² is constructed, which serves as the discrete precursor to the Minkowski interval. The discrete Ze Lagrangian Lₖᶻᵉ emerges naturally from the counter dynamics, with spatial updates contributing positively and temporal updates negatively to the invariant. In the continuous limit, where counter increments become field derivatives, the Lagrangian takes the form Lᶻᵉ = (∂C/∂x)² – c²(∂C/∂t)² with γ identified as c². For a single central counter representing a particle, this yields L_Ze-particle = –(1/2) m c² (1 – v²/c²)—exactly the relativistic free particle Lagrangian. Velocity is interpreted operationally as the fraction of counter activity allocated to spatial versus temporal modes: v² = (∂C/∂x)²/(∂C/∂t)². From this, proper time is derived as dτ = dt √(1 – v²/c²), corresponding to the count of temporal mode increments accumulated by a moving clock. Time dilation and Lorentz invariance thus emerge as statistical features of the discrete counter distribution rather than imposed postulates. The Minkowski metric arises from the relative weighting of spatial and temporal counter modes, with the light cone structure reflecting the balance between stabilizing (spatial) and destabilizing (temporal) updates. The Ze framework provides an operational, discrete foundation for special relativity, demonstrating that relativistic spacetime is not fundamental but emerges from the collective dynamics of primitive counters and their increments. This approach aligns with spacetime functionalism and offers new perspectives on the nature of time, velocity, and the speed of light.
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