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 {T,N,B} and Darboux frame {T,NM ,G} , 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 {T,N,B} 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.
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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 ∈ E1 and 3 the time one. We should care that the axis of time Euclidean unit normal vector field NE ∈ 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 {NE,NM } 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 ∈ 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 ∈ 3 E1 , the frame {TM ,NM ,BM } 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 ∈E1 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 ∈ E1 is de- 3 2 2 2 u = (x1, x2, x3) ∈E1 : 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 ∈E3 is a timelike vector, then the v ∈ 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 ∈ 3 q n 3 o E1 is defined as kukM = |hu, uiM |, which equals Ct (u) = v ∈ Γ ⊂ E1 : hu, viM < 0 , (2) zero iff u is lightlike vector or u =0. Furthermore, 3 3 any two vectors u, v ∈ 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 ∈E3 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 ϕ (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 ϕ. (3) of E1 . The timelike vector v = (v1, v2, v3) ∈ 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 {u, v}. mannian metric, Lorentz metric, respectively. Therefore, the induced metric should be Riemannian,
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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 T 1 0 0 T E2 which is divided into four sections. Two of them 1 N = 0 − sinh θ cosh θ N , are future and past timelike cone, the others are two M spacelike components. Furthermore, we can’t define G 0 cosh θ − sinh θ B the angle between two unit spacelike (timelike) vec- where the hyperbolic angle θ = 6 H (N,G). 2 tors of E1 that nor belong to the same component, nei- Therefore, we get ther the angle between a spacelike and timelike vec- 0 tors. The definition of angle is discussed in details in T 0 κg κn T 0 E2 NM = −κg 0 τg NM , 1 by F. Catoni [18, 19]. The third case appears when 0 the plane containing both lightlike vectors. Therefore, G κn τg 0 G the vectors shouldn’t be timelike. So, we can’t define with κg = κ sinh θ, κn = κ cosh θ and τg = τ − the angle between two spacelike vectors. dθ ds . In this case, we have two subcases according to 2 Corollary 7 Let u, v ∈E1 be two spacelike vectors the type of the relative tangent vector TM . u v such that and lie in the same spacelike On one hand, when TM is a spacelike vector. kukM kvkM section. Then, there exists a unique number ϕ (the Also, T and TM lie in the same spacelike component. hyperbolic angle between u and v) such that Since BM should be a timelike vector. Then, we get T cosh α 0 sinh α TM hu, viM = kukM kvkM cosh ϕ. (4) NM = 0 1 0 NM , G sinh α 0 cosh α BM 3 Frame Construction on Timelike and Surface in E3 TM cosh α 0 − sinh α T 1 NM = 0 1 0 NM , Consider a regular spacelike curve γ(s) lies on a time- BM − sinh α 0 cosh α G like surface Φ with a unit spacelike normal NM ∈ 3 where α = 6 H (T,TM ), so we have E1 . We have a moving orthonormal Frenet frame {T,N,B} on γ(s) with T ⊥M N,N ⊥M B and TM B ⊥ T . From the definition of spacelike curve γ, M NM the tangent vector T is spacelike vector and one of N BM or B is timelike vector. Additionally, we have Darboux frame cosh α − cosh θ sinh α sinh θ sinh α {T,NM ,G}, where G = −T ×M NM is a = 0 − sinh θ cosh θ timelike vector the curve γ(s). We don’t know the − sinh α cosh θ cosh α − sinh θ cosh α behavior of the principal normal vector N. Therefore, T we should discuss the cases of spacelike or timelike N . (6) principal normal vector N. B M Consider a frame {TP ,NM ,BM } at a point P ∈ Differentiating (6) with respect to s and using Frenet Φ or simply {TM ,NM ,BM }, where TM is a relative equations, we obtain tangent vector lies in the plane span {N ,N } . So, 0 E M T we have the following cases: M 0 NM B0 Case 8 When the principal normal vector N is a M 1 2 timelike. Then, the Frenet equations are given in ma- 0 κg − τg −κn 2 1 1 2 trix form as follows = τg − κg 0 τg − κg 1 2 −κn τg − κg 0 0 T 0 κ 0 T 0 TM N = κ 0 τ N . (5) 0 NM , (7) B 0 τ 0 B BM
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1 2 1 where κg = κg cosh α, κg = κg sinh α, τg = which is a new Bishop frame of the first type relative κ 2 dα to the surface Φ. Furthermore, if g = tanh α then we τg cosh α, τg = τg sinh α, and −κn = κn − ds . τg 1 2 On the other hand, if TM is a timelike vector. Then, have κg = τg . Therefore, we can write (7) as follows BM should be spacelike vector. Therefore, we get 0 T 0 0 −κn TM T sinh β 0 cosh β TM M 0 = 0 0 τg N , N = 0 1 0 N , NM cosh α M M M 0 τg B −κn 0 BM G cosh β 0 sinh β BM M cosh α (9) where β = 6 H (G, TM ). Then, we have which is a new Bishop frame of the second type relative to the surface Φ. Similarly, if TM is a timelike T vector, we can get the same results. ut N B sinh β 0 cosh β Case 11 When the principal normal vector N is a = cosh θ cosh β sinh θ cosh θ sinh β spacelike vector. So, the vector B should be timelike vector. Then, the Frenet equations is given as follows sinh θ cosh β cosh θ sinh θ sinh β TM 0 T 0 κ 0 T NM . 0 BM N = −κ 0 τ N . B0 0 τ 0 B Using (7), by replacing the angle α by β, we obtain
0 If N and NM are lie in the same spacelike sector with TM 0 the hyperbolic angle θ = 6 H (N,NM ). Then, we have NM 0 the usual transformation between Frenet and Darboux B M frames; 0 κ2 − τ 1 −κ T g g n M 1 2 2 1 T 1 0 0 T = τg − κg 0 τg − κg NM , 2 1 N = 0 cosh θ − sinh θ N , −κn τg − κg 0 BM M G 0 − sinh θ cosh θ B 1 2 1 where κg = κg cosh β, κg = κg sinh β, τg = 2 dβ and then, we get τg cosh β, τg = τg sinh β, and −κn = κn − ds . 3 0 Definition 9 Let Φ be a timelike surface in E1 and T 0 κn κg T γ(s) be a regular spacelike curve lies on Φ with time- 0 NM = −κn 0 τg NM . like principal normal vector N. The curve γ(s) is 0 G κg τg 0 G called a spaelike S-curve of first or second type, if τg = tanh ψ or κg = tanh ψ , ψ = α or β, ψ 6= 0, κg τg In this case, we have two subcases according to the respectively. type of the relative tangent vector TM . Theorem 10 Let Φ be a timelike surface in E3 and 1 Firstly, when T is a spacelike vector and also T γ(s) a regular spacelike S-curve of first or second kind M and T lie in the same spacelike component. Then, with a timelike principal normal vector N. Then, the M BM should be a timelike vector. Therefore, we get S- frame {TM ,NM ,BM } of γ(s) is Bishop frame of first or second type, respectively. T cosh α 0 sinh α TM Proof: Without loss of generality, consider a case NM = 0 1 0 NM , of T as a spacelike vector. If τg = tanh α, then G sinh α 0 cosh α B M κg M 1 2 we have τg = κg. Therefore, we can rewrite (7) as follows and
0 κg TM 0 cosh α −κn TM TM cosh α 0 − sinh α T 0 κg NM = − cosh α 0 0 NM , NM = 0 1 0 NM , 0 BM −κn 0 0 BM BM − sinh −α 0 cosh −α G (8) (10)
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6 1 2 where α = H (T,TM ). Then, we obtain where κn = κn cosh β, κn = κn sinh β, and −κg = dβ κg − ds . TM NM 3 Definition 12 Let Φ be a timelike surface in E1 and BM γ(s) be a regular spacelike curve lies on Φ with a cosh α sinh θ sinh α − cosh θ sinh α spacelike principal normal vector N. The curve γ(s) = 0 cosh θ − sinh θ is called a spaelike S-curve of third or fourth type if τg = tanh ψ or κn = tanh ψ ; ψ = α or β, ψ 6= 0, − sinh α − sinh θ cosh α cosh θ cosh α κn τg respectively. T 3 N . Theorem 13 Let Φ be a timelike surface in E1 and B γ(s) be a regular spacelike S-curve of third or fourth type with spacelike principal normal vector N. Then, Using Frenet and Darboux formulas lead to the S- frame {TM ,NM ,BM } of γ(s) is Bishop frame 0 of the first or second type, respectively. TM 0 NM Proof: Without loss of generality, consider a case of 0 B T as a spacelike vector. If τg = tanh α, then we M M κn 1 2 have κ2 = τ 1 . Therefore, we can rewrite (11) as 0 κn − τg −κg TM n g 2 1 1 2 follows = τg − κn 0 τg − κn NM (11) , 1 2 0 −κg τ − κ 0 BM κn g n TM 0 −κg TM 0 cosh α N = − κn 0 0 N , where κ1 = κ cosh α, κ2 = κ sinh α , and M cosh α M n n n n 0 −κ 0 0 B −κ = κ − dα . BM g M g g ds (12) which is a new Bishop frame of the first type relative Secondly, if TM is a timelike vector. Then, BM to the surface Φ. Furthermore, if κn = tanh α, then should be a spacelike vector. Therefore, we get τg 2 1 we have τg = κn . So, we can rewrite (11) as follows T sinh β 0 cosh β TM N = 0 1 0 N , 0 M M TM 0 0 −κg TM G cosh β 0 sinh β B 0 τg M NM = 0 0 cosh α NM , 0 τ B −κ g 0 B where β = 6 (G, T ) and then we have M g cosh α M H M (13) T which is a new Bishop frame of the second type N relative to the surface Φ. Similarly, if TM is a timelike vector, we obtain the same results. ut B − sinh β − cosh β sinh θ cosh β cosh θ Corollary 14 Let Φ be a timelike surface in E3. = 0 cosh θ − sinh θ 1 cosh β sinh β sinh θ − sinh β cosh θ Then, the S-frame {TM ,NM ,BM } and Darboux P frame are identical iff ζ ∩ TΦ = T, where P is a TM point lies on any regular curve γ (s) ⊂ Φ. NM . P BM Proof: Since ζ ∩ TΦ = T iff the angle α = 0, then we have β = 0 and hence TM ≡ T for all pervious Similarly, as (10) by replacing the angle α by β, we cases. ut get 0 TM 0 NM 4 Conclusion 0 BM 3 In the three-dimensional Minkowski space E1 , 2 1 0 κn − τg −κg TM a suitable differential equations of an S-frame of a 1 2 2 1 = τg − κn 0 τg − κn NM , regular spacelike curve γ (s) lies on a timelike surface 2 1 3 −κg τg − κn 0 BM Φ relative to E1 according to a relative Euclidean and
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Minkowski normal vectors of Φ have been obtained. [10] F. Manhart, Eigentliche Relativspharen,¨ die Also, some new concepts associated to S-frame Regelflachen¨ oder Ruckungsfl¨ achen¨ sind. Anz. in addition to new invariants have been defined. Osterr.¨ Akad. Wiss. Math.-Naturwiss. Kl., 1988, Moreover, we have generated a so-called S-curve for pp. 37-40. some special values of hyperbolic angles. Finally, [11] F. Manhart, Relativgeometrische Kennzeichnun- we have found that Bishop frame and S-frame are gen euklidischer Hyperspharen.¨ Geom. Dedi- identical for S-curve of different types. cata, 29, 1989, pp. 193-207. [12] E. Calabi, P. J. Olver, C. Shakiban, A. Tannen- In the future studies, we will try to apply this baum, and S. Haker, Differential and numeri- frame on some special curves of a regular spacelike cally invariant signature curves applied to object 3 curve lying on spacelike and timelike surfaces in E1 . recognition, Int. J. Computer Vision 26, 1998, Hopefully, S-frame will be helpful to reconstruct and pp. 107-135. modify most of Minkowski geometry research ideas. [13] I.A. Kogan and P.J. Olver, Invariant Euler- Acknowledgements: We would like to thank Prof. Lagrange equations and the invariant variational Acta Appl. Math. Gunter Weiss, University of Technology, Vienna, bicomplex, 76, 2003, pp. 137- Austria for his guidance in this study. Also, we are 193 very grateful to referees for the useful suggestions and [14] V. Itskov, Orbit reduction of exterior differential remarks for the revised version. systems and group-invariant variational prob- lems, Contemp. Math. 285, 2001, pp. 171-181. [15] M. P. Do Carmo, Differential Geometry of References Curves and Surface. Prentice-Hall, Englewood Cliffs, NJ, 1976. [1] S. Izumiya, K. Saji and N. Takeuchi. Circular [16] J. Suk Ro and D. Won Yoon, Tubes of surfaces, Advances in Geometry 7(2), 2007, pp. Weingarten types in a Euclidean 3-space, 295–313. J. Chungcheong Mathematical Society 22(3), [2] A. A. Ergin, On the generalized Darboux curves, 2009, pp. 359-366. Commun. Fac. Sci. Univ. Ank. Series A1 41, [17] R. Lopez,´ Differential geometry of curves and 1992, pp. 73-77. surfacces in Lorentz-Minkowski space. Int. Elec. [3] A. A. Ergin, Timelike Darboux curves on a time- Journ. Geom. 3(2), 2010, pp. 67-101. 3 like surface M ⊂ M1 , Hadronic Journal, 2001, [18] F. Catoni, et al., The Mathematics of Minkowski pp. 701-712. Space-Time, Birkhauser¨ Verlag AG., 2008, [4] M. Ozdemir¨ and A. A. Ergin, Spacelike Darboux ISBN 978-3-7643-8613-9. curves in Minkowski 3-space, Differential Ge- [19] F. Catoni, et al., Geometry of Minkowski Space- ometry - Dynamical Systems 9, 2007, pp. 131- Time, Springer Science & Business Media, 137. 2011, ISBN 978-3-642-17977-8. [5] S. S¸enyurt and A. C¸alıs¸kan, Spinor Formulation of Sabban Frame of Curve on S2, Pure Mathe- matical Sciences 4 (1), 2015, pp. 37-42. [6] S. Yılmaz, Bishop spherical images of a space- like curve in Minkowski 3-space, Int. Jour. Phys. Scien. 5(6), 2010, pp. 898-905. [7] M. K. Karacan and B. Bukc¨ u,¨ Bishop frame of the timelike curve in Minkowski 3-space, Fen Derg. 3(1), 2008, pp. 80-90. [8] O. J. Garay and A. Pampano,´ Binormal Evolu- tion of Curves with Prescribed Velocity, WSEAS Transactions on Fluid Mechanics 11, 2016, pp. 112-120. [9] F. Manhart, Uneigentliche Relativspharen,¨ die Regelflachen¨ oder Ruckungsfl¨ achen¨ sind. Pro- ceedings of the Congress of Geometry, Thessa- loniki, 1987, pp. 106–113.
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