Caputo fractional curvature of curves in the Lorentzian plane

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.