An Approximation for the Advent of In-System Mechanics in the Theory of Relativity Revised and Extended with a Fractional Calculus Model
Abstract views: 253
/ 
PDF downloads: 95
Keywords:
Fractional Calculus, Caputo Fractional Derivative, Lorentz Transformations, Temporal Nonlocality, Time Dilation, GNSS SynchronizationAbstract
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 < a =< 1.
�
�(t) = vta
T(1+a)
,
�
�(t)=a
ta
T(1+a)
|1-v2
c2
When a=1 , the standard Lorentz transformations return. When a< 1 sublinear dynamics takes place, proving
that time is not local.
Numerical simulations show that small changes in a cause large changes in the
temporal and spatial behavior of entities.
This suggests that a 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.
Downloads
References
A. Einstein, “On the electrodynamics of moving bodies,” 1905.
Xie, W., Huang, G., Fu, W., Du, S., Cui, B., Li, M., & Tan, Y. (2023). Rapid estimation of undifferenced multi-GNSS real-time satellite clock offset using partial observations. Remote Sensing, 15(7), 1776.
Mainardi, F. (2010). An introduction to mathematical models. Fractional calculus and waves in linear viscoelasticity. Imperial College Press, London.
Bagley, R. L., & Torvik, P. J. (1983). A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of rheology, 27(3), 201-210.
Riewe, F. (1997). Mechanics with fractional derivatives. Physical Review E, 55(3), 3581.
Nasrolahpour, H. (2011). Fractional Modified Special Relativity. arXiv preprint arXiv:1104.0030.
Demir, T., & Hajrulla, S. (2023). Discrete Fractional Operators and their applications. International Journal of Advanced Natural Sciences and Engineering Researches (IJANSER).
Hajrulla, S., Uka, A., & Demir, T. (2023). Simulations and results for the heat transfer problem. European Journal of Engineering Science and Technology, 6(1), 1-9.
Demir, T., Hajrulla, S., Ozer, O., Karadeniz, S., & James, O. O. C. (2024). Caputo Fractional Operator on shallow water wave theory. International Journal of Advanced Natural Sciences and Engineering Researches, 8(7), 1-41.
Demir, T., & Samei, M. E. Applications of q-Mittag Leffler function on q-fractional differential operators. In 6th International HYBRID Conference on Mathematical Advances and Applications (p. 191).
DEMİR, T. (2022). Mittag-Leffler Function and Integration of the Mittag-Leffler Function.
Demir, T. (2025, January). Existence and Uniqueness Analysis in q-Calculus-Based Dynamic Systems.
Demir, T. (2024, December). Asymptotic Analysis of Fractional Derivatives and Applications.
Demir, T. (2025, January). Numerical and theoretical Analysis of Existence and Uniqueness Analysis in q-Calculus-Based Dynamic Systems.
Demir, T. (2025). A Novel Fractional Differential Model with ψ-Caputo Operator: Theory, Analysis, and Numerical Simulations.
Demir, T. (2024, December). Delay Differential Equation Approach to Lotka-Volterra Predator-Prey Dynamics: Modelling Ecological Interactions with Time Delays.
Demir, T. (2025, January). The Application of the Wronskian Determinant in Algebraic Structures and Its Role in Mathematical Analysis.
Demir, T., Hajrulla, S., & Doğan, P. Using Special Functions on Grünwald-Letnikov and Riemann Liouville Fractional Derivative and Fractional Integral.
Demir, T. (2022). Solutions of fractional-order differential equation on harmonic waves and linear wave equation. Solutions of fractional-order differential equation on harmonic waves and linear wave equation.
Demir, T. (2025, January). Simulations of Fractional Variational Calculus and Its Interplay with Wronskian Determinants in Solving Advanced Differential Equations.
Demir, T. (2025, January). Fractional Variational Calculus and Its Interplay with Wronskian Determinants in Solving Advanced Differential Equations.
Demir, T. (2025). Hybrid Fractional-Order Models with Data-Driven Parameters for Complex Systems: Theory, Deep Learning Integration, and Multiscale Applications.
Demir, T. Laplace Transforms of the Fractional Derivatives.
Demir, T. (2025, January). The Application of the Wronskian Determinant in Algebraic Structures and Its Role in Mathematical Analysis.
Demir, T., & Hajrulla, S. (2025). Eco-Epidemiological Dynamics Under State-Dependent Delays: A New Delay Differential Approach.
Demir, T., (2025, January). Fractional-order derivatives on generalized functions and their applications.
Hajrulla, S., & Demir, T. Discrete Fractional Operators in Analyzing Digital Financial Inclusion in Banks by using Linear Fractional Differential Equation Model. In 7th International Conference On Mathematical Modelling, Applied Analysis And Computation (ICMMAAC-24).
Demir, T. (2025). Generalized Fractional Symmetry Principles and Conservation Laws in Nonlinear Multiscale Systems: Theory, Algorithms, and Applications.
Demir, T. (2023). Fractional Models on Surface Waves. Academie. edu.
Hajrulla, S., & Demir, T. Numerical Results for Second-Level Banks Data Using Discrete Fractional Calculus. In 7th International Conference On Mathematical Modelling, Applied Analysis And Computation (ICMMAAC-24).
Demir, T. (2025). Fractional-Order Modeling, Control, and Fault Diagnosis in Intelligent Mechatronic Systems: Theory, Algorithms, and Real-Time Applications.
Demir, T. Application of fractional-order Differential Equations on Diffusion and Wave Equations.
Demir, T. Fractional Noether-Type Theorems via ψ-Caputo Derivatives for Systems with Non-Conservative Forces.
Demir, T. Data-Driven Fractional Modeling and Predictive Digital Twins for Cyber-Physical Mechatronic Systems: Theory, Machine Learning, and Experimental Validation.
Demir, T. Unified Theory and Intelligent Simulation Platform for Coupled Multi-Physics, Multi-Scale, and Fractional-Order Systems: Foundations, Algorithms, and Digital Twin Applications.
Demir, T., Cherif, M. H., Hajrulla, S., & Erikçi, G. Local Fractional Differential Operators on Heat Transfer Modelling.
Demir, T. Fractional Stochastic Differential Equations and Deep Learning for Systemic Risk Forecasting in Complex Financial Networks.
Demir, T. Spectral Theory and Well-Posedness for Nonlocal and Variable-Order Fractional Differential Operators on Irregular Domains.
Demir, T. Fractional-Order Modeling and AI-Driven Forecasting of Soil Moisture Dynamics for Sustainable Agriculture and Climate Resilience.
Demir, T. Fractional-Order Networked Control for Large-Scale Cyber-Physical Systems under Adversarial Attacks: Theory, Secure Algorithms, and Smart Grid Applications.
Demir, T. Data-Driven Fractional Partial Differential Equations for Anomalous Transport and Pattern Formation in Living Systems: Theory, Machine Learning Algorithms, and Bioengineering Applications.
Demir, T., Baleanu, D., Heydari, M. H., Jajarmi, A., & Akgül, A. Caputo fractional operator with Proportional-Integral-Derivative controller.
Kilbas, A. A. (2006). Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204.
Podlubny, I. (1998). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Vol. 198). elsevier.
Mainardi, F. (2010). An introduction to mathematical models. Fractional calculus and waves in linear viscoelasticity. Imperial College Press, London.
Tarasov, V. E. (2011). Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. Springer Science & Business Media.
Laskin, N. (2000). Fractional quantum mechanics. Physical Review E, 62(3), 3135.
Machado, J. T. (2015). Fractional order junctions. Communications in Nonlinear Science and Numerical Simulation, 20(1), 1-8.
Garrappa, R., & Kaslik, E. (2020). On initial conditions for fractional delay differential equations. Communications in Nonlinear Science and Numerical Simulation, 90, 105359.
Ortigueira, M. D. (2011). Fractional calculus for scientists and engineers (Vol. 84). Springer Science & Business Media.
Bagley, R. L., & Torvik, P. J. (1983). A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of rheology, 27(3), 201-210.
Nottale, L. (2011). Scale relativity and fractal space-time: a new approach to unifying relativity and quantum mechanics. World Scientific.
Einstein, A. (1905). On the electrodynamics of moving bodies.
Sikoka, Johannes (2025). An approximation for the advent of In-system mechanics in the theory of relativity, DOI: 10.13140/RG.2.2.29471.68009
