Discretization of the Lorentz Transformation at the Planck Scale and the Derivation of the Photon's Emergent Inertial Character

This treatise addresses the foundational incompatibility between the continuous spacetime fabric of General Relativity and the discrete energy landscapes of quantum mechanics. Within classical relativistic kinematics, a photon traversing at the absolute speed of light (v = c) yields a singular Lorentz factor (γ → ∞), effectively paralyzing its proper time (dτ = 0) and rendering the definition of an intrinsic co-moving reference frame mathematically impossible. To reconcile this deep-seated paradox, the author posits a fundamental discretization of spacetime at the Planck scale, introducing the Planck time (τP ≈ 5.39 × 10⁻⁴⁴ s) as a non-zero strict lower bound for any physically measurable proper time interval (∆τ ≥ τP). By enforcing this quantum mechanical boundary condition, the unphysical divergence of the standard Lorentz factor is systematically eliminated, giving rise to an effective Lorentz factor (γ_eff = ∆t/τP) that remains strictly finite and bounded by the ratio of the Planck energy to the photon’s total energy (Ep/E). The framework reinterprets the local absorption and transmission of electromagnetic radiation not as instantaneous point-like events (∆t → 0), but as finite, scale-limited quantum phase transitions. During the localization process, the wave function collapses within a duration restricted by the Planck time, during which the photon’s spatial propagation drops to zero (v → 0). Through the lens of Einsteinian mass-energy equivalence, this sudden deceleration manifests as an emergent, transient effective rest mass character defined as mrel = hν/c². Consequently, evaluating the momentum transfer (∆p) via generalized Newtonian mechanics yields an extraordinary, spike-like maximum mechanical force (Fmax = ∆p/τP) and an instantaneous peak power output (Pinst = ∆E/τP) localized strictly within the ultrashort Planckian temporal window. The model undergoes empirical cross-examination by applying these formal derivations across varying spectral energy regimes. The author explains the absence of macroscopic or atomic destruction via a spatial damping mechanism: because the Planck length (ℓP) is twenty-four orders of magnitude smaller than the atomic Bohr radius, the localized energy spike is distributed non-locally across the electron cloud, translating into a smooth quantum leap rather than mechanical shearing. When scaled up to extreme 2 MeV cosmic gamma-rays, the calculated force approaches values capable of inducing vacuum polarization breakdown, driving the transition of emergent mass into tangible electron-positron pairs. Extending this discretization globally to early-universe cosmology eliminates the classical Big Bang singularity. By converting the estimated total mass of the universe into an initial field energy, the global power spike integrated over the Planck time yields a finite value (Puniv ≈ 1.67 × 10¹¹³ W), ensuring a smooth, singularity-free bottom-up hydrogen genesis. Furthermore, the model provides an elegant solution to the Greisen–Zatsepin–Kuzmin (GZK) limit anomaly, suggesting that ultra-high-energy cosmic photons bypass interactions with the cosmic microwave background due to the strict information-bandwidth limits of the discrete spatial lattice. To counter critical objections regarding Lorentz Invariance Violation (LIV) and observer-dependent metrics, the author incorporates the κ-Poincaré algebraic formalism. By establishing a non-commutative spacetime geometry where coordinate operators fail to commute ([xi, t] = iτP xi), the effective metric is governed intrinsically by the underlying energy-time scale rather than the reference frame of the laboratory observer, preserving mathematical consistency. Experimental verification remains tied to statistical time-delay analyses of varying energy photons originating from distant gamma-ray bursts (GRBs). More information: gabor.czira-tau.ph@proton.me