Caputo fractional curvature of curves in the Lorentzian plane

Meltem Ogrenmis; Handan Oztekin; Y. S. Hamed; Muhammad Bilal Riaz; Muhammad Abbas; Meltem Ogrenmis; Handan Oztekin; Y. S. Hamed; Muhammad Bilal Riaz; Muhammad Abbas

doi:10.3934/math.2025923

AIMS Mathematics

2025, Volume 10, Issue 9: 20670-20688. doi: 10.3934/math.2025923

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Research article

Caputo fractional curvature of curves in the Lorentzian plane

1.
Department of Mathematics, Faculty of Science, Firat University, Turkey
2.
Department of Mathematics and Statistics College of Science, Taif University P. O. Box 11099, Taif 21944, Saudi Arabia
3.
IT4Innovations, VSB–Technical University of Ostrava, Ostrava, Czech Republic
4.
Applied Science Research Center, Applied Science Private University, Amman, Jordan
5.
Department of Mathematics, University of Sargodha, 40100 Sargodha, Pakistan

Received: 03 June 2025 Revised: 21 July 2025 Accepted: 28 July 2025 Published: 09 September 2025
MSC : 26A33, 53A04, 53B30

We present a new concept of fractional curvature invariant for regular curves in the Lorentz plane by generalizing the Caputo‐fractional curvature from Euclidean geometry to the pseudo‐Riemannian setting. Our construction projects the integer-order derivative of the Caputo vector of fractional-order derivatives onto the Lorentzian normal direction, yielding a curvature measure that naturally distinguishes timelike and spacelike curves. Explicit formulas for representative model curves are derived, and we illustrate how the Lorentzian metric signature fundamentally changes fractional curvature behavior. This framework extends fractional‐order geometric analysis into relativity, providing new tools for studying memory effects and nonlocal dynamics along curves in relativistic contexts.
- fractional derivative,
- Caputo fractional derivative,
- Lorentz plane,
- timelike and spacelike curves
Citation: Meltem Ogrenmis, Handan Oztekin, Y. S. Hamed, Muhammad Bilal Riaz, Muhammad Abbas. Caputo fractional curvature of curves in the Lorentzian plane[J]. AIMS Mathematics, 2025, 10(9): 20670-20688. doi: 10.3934/math.2025923

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Abstract

We present a new concept of fractional curvature invariant for regular curves in the Lorentz plane by generalizing the Caputo‐fractional curvature from Euclidean geometry to the pseudo‐Riemannian setting. Our construction projects the integer-order derivative of the Caputo vector of fractional-order derivatives onto the Lorentzian normal direction, yielding a curvature measure that naturally distinguishes timelike and spacelike curves. Explicit formulas for representative model curves are derived, and we illustrate how the Lorentzian metric signature fundamentally changes fractional curvature behavior. This framework extends fractional‐order geometric analysis into relativity, providing new tools for studying memory effects and nonlocal dynamics along curves in relativistic contexts.

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