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Constructing a New Frame of Spacelike Curves on Timelike Surfaces in Minkowski 3-Space

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Constructing a New Frame of Spacelike Curves on Timelike Surfaces in Minkowski 3-Space WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2020.19.24 Emad N. Shonoda, M. Khalifa Saad Constructing a New Frame of Spacelike Curves on Timelike Surfaces in Minkowski 3-Space EMAD N. SHONODA M. KHALIFA SAAD Faculty of Science, Port-Said University Faculty of Science, Sohag University Mathematics and Computer Science Department Mathematics Department EGYPT EGYPT Mt. San Jacinto College Faculty of Science, Islamic University of Madinah Mathematics Department Mathematics Department, 170 Al-Madinah California, USA Saudi Arabia [email protected] mohamed [email protected] Abstract: In this paper, we define a relative Minkowski normal plane ζ and relative tangent vector TM : We construct a new relative S-frame (Shnoda-Saad frame) of regular spacelike curves on timelike surfaces. It depends only on the curve lies on the surface, Euclidean and Minkowski unit normal vectors. Also, we define S-curve according to this frame with some related theorems. Key–Words: Spacelike curve, Timelike surface, Frenet and Darboux frames, Minkowski 3-space Received: February 13, 2020. Revised: April 29, 2020. Accepted: May 14, 2020. Published: May 28, 2020. 1 Introduction transformation between Frenet frame fT; N; Bg and Darboux frame fT;NM ;Gg ; where NM and G are At the beginning of the 20th century, the Einstein’s the principal normal and binormal unit vectors of the theory opened a door to new geometries such as surface Φ respectively. Furthermore, they are using Lorentzian Geometry, which is simultaneously the other known frames, e.g., Sabban frame (Spherical geometry of special relativity. After that, researchers Frenet) which is given according to Euclidean unit discovered a bridge between modern differential sphere S2 [5]. Also, Bishop frame can be defined geometry and the mathematical physics of general 4 even if the curvature vanishes at some points on the relativity. In particular, E1 (Minkowski space-time) curve α (t) [6, 7, 8]. has a very strong background in physics. The 3 4 Minkowski 3-space E1 ⊆ E1 has many properties which is different from E3. Some basic concepts Relative differential geometry can be described such as vector, frame, and the motion of point have as the geometry of surfaces in the affine space with 3 qualitative changes. Since the metric in E1 is not a distinguished relative normal vector field which positive definite metric, then the distance function h; i gives us a general form of unit normal vector field can be positive, negative or zero, whereas the distance from Euclidean differential geometry. In [9, 10, 11], function in the Euclidean space can only be positive the authors obtained a relative normal vector fields [1]. of a differentiable (smooth) surface which is defined as a function of two variables. The moving frame 3 In Minkowski space E1 , every point in space and method provides a direct route to the classification time worlds can be determined in terms of two special of joint invariants and joint differential invariants coordinates besides to the coordinate of time. When [12]. There are some applications of moving frames a point is moving along the time axis, it shouldn’t in the construction of invariant numerical algorithms move through space coordinates system. Moreover, and the theory of geometric integration. Most it moves with the speed of light as a function of modern Astronomy physics based on postulating a the time. Many authors tried to construct Darboux symmetry group and formulating field equations with 3 frame of a regular curve α (t) in E1 , which lies on a a group-invariant variational principle. Furthermore, regular surface Φ, by rotating Frenet frame fT; N; Bg every invariant variational problem can be written in around the tangent unit vector T of α (t) [2, 3, 4]. terms of the differential invariants of the symmetry The curve α (t) is called Darboux curve if it moves group. The Euler-Lagrange equations inherit the on the surface Φ such that its curvature and normal symmetry group of the variational problem, which curvature are identical. In other words, there is a can be written in terms of the differential invariants. E-ISSN: 2224-2880 247 Volume 19, 2020 WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2020.19.24 Emad N. Shonoda, M. Khalifa Saad The general group-invariant formula construct the In other words, the normal vector on the spacelike Euler-Lagrange equations from the invariant form of (timelike) surface is a timelike (spacelike) vector [16]. the variational problem [13, 14]. Definition 2 Relative Minkowski normal plane ζ: Let In this paper, we construct a new coordinate Φ be a spacelike (timelike) surface with a unit time- 3 system by rotating the axes of space system about like (spacelike) normal vector field NM 2 E1 and 3 the time one. We should care that the axis of time Euclidean unit normal vector field NE 2 E . If NM rotates differently than the axes of space. Therefore, and NE are non-congurent, then we have a plane ζ = the rotation of the Frenet frame of α (t) which lies span fNE;NM g which is called ”relative Minkowski 3 on a surface Φ in E1 should be depend on the types normal plane” of spacelike (timelike) surface Φ. of the curve, surface and axis of rotation. We call this frame ”Shonoda-Saad frame” or simply S-frame. Definition 3 Relative tangent vector TM : Consider a 3 spacelike (timelike) surface Φ 2 E1 , then the tangent This frame gives us a relative differential geome- unit vector TM which generated from the intersection try between Euclidean and Minkowski invariants of a of the tangent plane and the relative Minkowski nor- chosen curve on a regular timelike surface Φ. We will mal plane is called a relative tangent unit vector. investigate this special frame according to a regular 3 spacelike curve α (t) on Φ in E1 . Definition 4 For a spacelike (timelike) surface Φ 2 3 E1 , the frame fTM ;NM ;BM g is called Shonoda- Saad frame or simply S-frame where BM is a relative 2 Differential Geometry Properties binormal vector field. In this section, we list some notions, formulas, and conclusions for curves and surfaces in Minkowski 2.1 Angle Measure in Lorentz-Minkowski 3-space E3 [15]. We consider the real vector space 3 1 Space E1 E3 which defined the standard flat metric given 2 2 2 The angle measure between any two vectors by h ; i = dx1+dx2−dx3, where (x1; x2; x3) is a 3 3 u; v 2E1 depends on the position of these rectangular coordinate system of E1 . Minkowski 3 vectors with respect to the light-cone C = inner product of two vectors u; v 2 E1 is de- 3 2 2 2 u = (x1; x2; x3) 2E1 : x1 + x2 − x3 = 0; u 6= 0 . fined as hu; viM = u1v1 + u2v2 − u3v3; where It can be determined only if both are timelike vectors u = (u1; u2; u3) and v = (v1; v2; v3). lie in the same timelike cone or in the same spacelike component. To distinguish the usual Euclidean inner prod- uct, we denote it by the notation h; iE . A vector Definition 5 If u 2E3 is a timelike vector, then the v 2 E3 is said to be spacelike, timelike or lightlike 1 1 timelike cone Ct (u) is given as follows if hv; viM > 0 ; hv; viM < 0 or hv; viM = 0, re- spectively. The Minkowski norm of any vector u 2 3 q n 3 o E1 is defined as kukM = jhu; uiM j; which equals Ct (u) = v 2 Γ ⊂ E1 : hu; viM < 0 ; (2) zero iff u is lightlike vector or u =0. Furthermore, 3 3 any two vectors u; v 2 E1 are orthogonal with the no- where Γ is the set of all timelike vectors in E1 : For tation u ?M v iff hu; viM = 0. The Lorentz vector more details see [17]. product of u; v is given by Corollary 6 If u; v 2E3 are two timelike vectors lie e e −e 1 1 2 3 in the same lightlike cone, then there exists a unique u ×M v = u1 u2 u3 : (1) number ' (the hyperbolic angle between u and v) v1 v2 v3 such that The light-cone is the set of all lightlike vectors 3 3 hu; viM = − kukM kvkM cosh ': (3) of E1 . The timelike vector v = (v1; v2; v3) 2 E1 is called future (past) pointing iff v > 0 (v < 0), i.e. it 3 3 Proof: The proof can be found in [17]. ut lies in the future (past) section of the unit ball. Definition 1 A surface Φ in the Minkowski 3-space For any other two vectors, we have three possibil- 3 E1 is said to be spacelike, timelike surface if the in- ities depending on the type and position of the sub- duced metric on the surface is a positive definite Rie- space which contains the two vectors span fu; vg. mannian metric, Lorentz metric, respectively. Therefore, the induced metric should be Riemannian, E-ISSN: 2224-2880 248 Volume 19, 2020 WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2020.19.24 Emad N. Shonoda, M. Khalifa Saad Lorentzian or degenerate. If the plane is Rieman- If N and G lie in the same timelike cone. Then, nian, then the angle measure definition is determined we have the usual transformation between Frenet and as in Euclidean space E3: If the plane is Lorentzian, Darboux frames; then it is isometric to the Lorentz-Minkowski plane 0 T 1 0 1 0 0 1 0 T 1 E2 which is divided into four sections.
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