E_1^3 3-Boyutlu Lorentz Uzayda Spinor Q-Denklemleri

Year 2022, , 294 - 300, 24.03.2022

Doğan Ünal

https://doi.org/10.17798/bitlisfen.1022461

Abstract

Bu çalışmada, E_1^3 3-boyutlu Lorentz uzayda q-çatıya göre uzay eğrilerinin hiperbolik spinor temsilleri incelenmiştir. E_1^3’te eğrilerin spacelike ve timelike teğet vektörlü olma durumlarına göre q-çatı için eğrilerin spinor formülasyonları hesaplanmıştır. Ayrıca 3- boyutlu Lorentz uzayda q-çatı ve Frenet çatıya ait spinor denklemleri arasındaki ilişkiler de ifade edilmiştir. Sonuçlar bazı teoremlerle desteklenmiştir.

Keywords

Clifford Cebiri, Hiperbolik Spinorlar, Q-Çatı, Lorentz Uzayı

Spinor Q-Equations in Lorentzian 3-space E_1^3

Abstract

In this paper, hyperbolic spinor representations of space curves are studied according to the q-frame in E_1^3. The spinor formulations of curves are calculated for the q-frame according to the spacelike and timelike tangent vector cases of the curves in E_1^3. Moreover, the relationships of spinor equations between q-frame and Frenet frame in Lorentz space are expressed. The results are supported with some theorems.

Keywords

Clifford Algebras, Hyperbolic Spinors, Q-Frame, Lorentz Space

References

• [1] Cartan E. 1966. The Theory of Spinors. Hermann, Paris, 1-192. (Dover, New York, reprinted 1981).
• [2] Misner C. W., Thorne K. S., Wheeler J. A. 1973. Gravitation. W. H. Freeman and Company, San Francisco CA, 1-1304 .
• [3] Montague B. W. 1981. Elemenatry Spinor Algebra for Polarized Beams in Strage Rings. Particle Accelerators, 11: 219-231.
• [4] Torres del Castillo G. F., Barrales G. S. 2004. Spinor Formulation of the Differential Geometry of Curve. Rev. Colombiana Mat., 38: 27-34.
• [5] Unal D., Kisi I., Tosun M. 2013. Spinor Bishop Equation of Curves in Euclidean 3-Space. Adv. Appl. Cliff. Algebr., 23: 757–765.
• [6] Kisi I., Tosun M. 2015. Spinor Darboux Equations of Curves in Euclidean 3-Space. Math. Morav, 19: 87-93.
• [7] Erisir T., Güngör M. A., Tosun M. 2015. Geometry of the Hyperbolic Spinors Corresponding to Alternative Frame. Adv. Appl. Cliff. Algebr, 25: 799-810.
• [8] Balci Y., Erisir T., Güngör M. 2015. Hyperbolic Spinor Darboux Equations of Spacelike Curves in Minkowski 3-Space. J. Chungcheong Math. Soc., 28: 525-535.
• [9] Coquillart S. 1987. Computing Offsets of B-spline Curves. Computer-Aided Design, 19: 305-309.
• [10] Shin H., Yoo S.K., Cho S. K., Chung W. H. 2003. Directional Offset of A Spatial Curve for Practical Engineering Design. International Conference on Computational Science and its Applications, Montreal, 711-720.
• [11] Dede M., Ekici C., Görgülü A. 2015. Directional q-Frame along A Space Curve. IJARCSSE, 5: 775-780.
• [12] Ekici C., Göksel M. B., Dede M. 2019. Smarandache Curves According to q-Frame in Minkowski 3-Space. Conference Proceedings of Science and Technology, Erzincan, 110-118.
• [13] O’Neill B. 1983. Semi-Riemannian Geometry, with Applications to Relativity. Academic Press, New York, 1-488.
• [14] Kuhnel W. 1999. Differential Geometry: Curves – Surfaces – Manifolds. Braunscheweig, Weisbaden, 1-403.
• [15] Otsuki T. 1961. Differential Geometry (Japanese). Asakura Shoten, Tokyo.
• [16] Bonnor W. B. 1969. Null Curves in a Minkowski Space-Time. Tensor, 20: 229-242.
• [17] Ikawa T. 1985. On Curves and Submanifolds in an Indefinite-Riemannian Manifold. Tsukuba J. Math. 9: 353–371.
• [18] Ekici C., Dede M., Tozak H. 2017. Timelike Directional Tubular Surfaces. Journal of Mathematical Analysis, 8: 1-11.
• [19] Birman G.S., Nomizu, K. 1984. Trigonometry in Lorentzian Geometry. Ann. Math. Mont., 91: 534–549.
• [20] Sobczyk G. 1995. The Hyperbolic Number Plane. College Math. J., 26: 268–280.
• [21] Antonuccio F. 1998. Hyperbolic Numbers and the Dirac Spinor. (arXiv:hep-th/9812036v1)
• [22] Carmel, M. 1977. Group Theory and General Relativity, Representations of the Lorentz Group and their Applications to the Gravitational Field. McGraw- Hill, Imperial College Press, New York, 1-412.
• [23] Sattinger D. H., Weaver O. L. 1986. Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics. Springer, New York, 1-207.
• [24] Payne W. T. 1952. Elementary Spinor Theory. Am. J. of Phys, 20: 253.
• [25] Torres del Castillo G. F. 2003. 3-D Spinors, Spin-Weighted Functions and their Applications. Birkhauser, Boston, 1-234.
• [26] Ketenci Z., Erisir T., Gungor M. A. 2014. Spinor Equations of Curves in Minkowski Space. V. Congress of the Turkic World Mathematicians, Issık, 41pp.
• [27] Saad M. K., Abdel-Baky R. A. 2020. On Ruled Surfaces According to Quasi-Frame in Euclidean 3-Space. Aust. J. Math. Anal. Appl., 17: 16pp.