An Approximation for the Advent of In-System Mechanics in the Theory of Relativity Revised and Extended with a Fractional Calculus Model

This study suggests a fractional extension of special relativity by adding Caputo fractional derivatives to the Lorentz transformation framework. Experimental observations in viscoelastic media, ultrafast optical systems, and global navigation satellite systems (GNSS) show small but measurable time-dependent delays, which is different from classical relativity, which assumes instantaneous rod contraction and clock synchronization. We use a Caputo fractional derivative of order \( 0 <\alpha\leq 1 \). \( L(t)=\frac{vt^{\alpha}}{\Gamma(1+\alpha)} \), \( \tau(t)=\sqrt{1-\frac{v^{2}}{c^{2}}} \). When, \( \alpha=1 \), the standard Lorentz transformations return. When \( \alpha<1 \) sublinear dynamics takes place, proving that time is not local. Numerical simulations show that small changes in \( \alpha \) cause large changes in the temporal and spatial behavior of entities. This suggests that \( \alpha \) could be used as a measurable metric to investigate memory effects. The suggested framework has an impact on applications such as GNSS, which demand a high degree of precision and where even small temporal variations can affect positioning accuracy. Additionally, this method opens the door for fractional calculus to be applied in curved spacetime, which could lead to improved fractional general relativity formulations.