The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space
Abstract
:1. Introduction
2. Basic Geometric Concepts of Timelike Curves in Minkowski Space
- Let X and Y be spacelike vectors ; if X and Y span a timelike vector subspace, then , and there is a unique positive real number θ such that
- Let X and Y be spacelike vectors in that span a spacelike vector subspace; then, there is a unique real number , such that
- Let X and Y be future pointing (past pointing) timelike vectors in ; then, there is a unique nonnegative real number θ, such that
- Let X be a spacelike vector and Y a future pointing timelike vector in ; then, there is a unique nonnegative real number , such that
2.1. Frenet Frame for Timelike Curves in
- ,, , , and
- , and .
- where k and τ are the curvature and torsion for the timelike curve, respectively.
2.2. Quasi-Frame for Timelike Curves in
3. Main Results and Discussion
Time Evolution Equations for the Quasi-Timelike Curves (QTIC) in
4. An Application for the Motion of the QTIC by Velocity Fields
5. The Flows of the QTIC via the Acceleration Fields
An Application for the Motion of the QTIC by the Acceleration Fields
6. Discussion
7. Conclusions
- 1.
- We studied the motion of the QTIC by the velocity fields , and , with the equation of motion , where , and represented the velocity functions in the direction of the q- frame .
- 2.
- The time evolution equations (TEEs) for the q-frame of the QTIC in Minkowski space were derived, and the TEEs for the quasi-curvatures and were obtained as a system of PDEs (Theorem 2).
- 3.
- 4.
- We studied the motion of the QTIC described by the acceleration fields with the equation of motion .
- 5.
- 6.
- Through the given applications, we presented the description of the graphs, which indicated the flows of the quasi-timelike curves and their first and second quasi-curvatures.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
(2-D) | Two dimensions. |
(3-D) | Three dimensions. |
IFC | Inextensible flows of curves. |
PDE(s) | Partial Differential Equation(s) |
q-frame | Quasi-frame. |
QTIC | Quasi-timelike curve. |
TEEs | Time evolution equations. |
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Gaber, S.; Sorour, A.H.
The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space
Gaber S, Sorour AH.
The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space
Gaber, Samah, and Adel H. Sorour.
2023. "The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space
Gaber, S., & Sorour, A. H.
(2023). The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space