In this article, we get a innovative interpretation about timelike biharmonic particle by means of Fermi-Walker derivative and parallelism in Heisenberg spacetime

https://doi.org/10.5831/HMJ.2019.41.1.153

ON NEW FERMI WALKER DERIVATIVE OF BIHARMONIC PARTICLES IN HEISENBERG

SPACETIME

Talat K¨orpinar

Abstract. In practical applications play an new important role timelike biharmonic particle by Fermi-Walker derivative. In this article, we get a innovative interpretation about timelike biharmonic particle by means of Fermi-Walker derivative and parallelism in Heisenberg spacetime. With this new representation, we derive necessary and sufficient condition of the given particle to be the inextensible flow. Moreover, we provide several characterizations designed for this particles in Heisenberg spacetime.

1. Introduction

A large number of research on Fermi-Walker derivative and transport have been released. Therefore benefits of this research are all how to find and compute the geodesics on the surface. However, Maluf et al.

studied tetrad fields as basic Fermi Walker parallelism along a particles in spacetime, [21]. Therefore, they presented a new characterization of Fermi Walker transported frames of tetrad vector fields.

Recently, Fermi Walker transports were studied in Minkowski space- time. Thus, Serret-Frenet equations obtained by Synge on world lines is a strong instrument with studying motion of non-zero rest mass for test particles in a assumed gravitational field [27,28]. Also, the Frenet Serret equations have been obtained from non null to null trajectories in a spacetime by new formalism with Fermi Walker transport, [1,4]. On the studies of Fermi-Walker derivative, energy of curves and flows, we refer to papers [4,9-20,26,29-32].

Received July 17, 2018. Accepted August 10, 2018.

2010 Mathematics Subject Classification. 22E70, 53A04, 53C21.

Key words and phrases. Fermi Walker derivative, Energy, Bienergy, Heisenberg spacetime.

This work is organized the following. Firstly, we obtain new charac- terization for timelike biharmonic particle with Fermi-Walker derivative and parallelism in Heisenberg spacetime. Secondly, we derive the nec- essary and sufficient state for the given particle. Finally, we give some characterizations on Frenet fields of this new biharmonic particle on Heisenberg spacetime.

2. New Metric and Biharmonic Particles Heisenberg space-time metric is given by

(2.1) gc= −c²dt²+ dx²+ dy²+ (dz − xdy)². Lie algebra of Heisenberg space time has a basis [14,22]

(2.2) e₁= ∂

∂x, e₂ = ∂

∂y+ x ∂

∂z, e₃ = ∂

∂z, e₄= 1 c

∂

∂t, with

[e₁, e₂] = e₃, [e₃, e₂] = [e₁, e₃] = [e₁, e₄] = [e₂, e₄] = [e₃, e₄] = 0 and

gc(e1, e₁) = 1, gc(e2, e₂) = 1, g_c(e₃, e₃) = 1, g_c(e₄, e₄) = −1.

Let γ be a timelike particle in Heisenberg spacetime. Hence exist three smooth functions k₁, k₂, k₃ on γ with smooth frame field

{T, N, B₁, B2}

along the particle γ. For the this particle [3,5-8,23-25], the following Frenet formula is given by

(2.3)









∇_sT

∇_sN

∇_sB₁

∇_sB2









=









0 k1 0 0

k₁ 0 k₂ 0

0 −k₂ 0 k₃

0 0 −k₃ 0















 T N B₁ B2







 ,

where

g_c(T, T) = −1, g_c(N, N) = g_c(B₁, B₁) = g_c(B₂, B₂) = 1.

Lemma 2.1. ([20]) γ is a timelike biharmonic particle in H⁴₁ iff k₁k₁⁰ = 0,

k²₁− k²₂ = g_c(R, N) , k₂⁰ = g_c(R, B₁) , k2k3 = gc(R, B₂) , where R = R (T, N) T.

3. Fermi Walker Derivative For Biharmonic Particles In The Heisenberg Spacetime H⁴₁

In this important part, we research the association between the Fermi- Walker derivative and the Frenet fields of biharmonic particles. More- over, we obtain some characterizations.

Definition 3.1. ([2]) Fermi–Walker derivative of field X is defined by

(3.1) ∇˜_sX = ∇sX + gc(T, X)∇sT − gc(∇sT, X)T.

Now, we investigate condition of Frenet vectors.

Theorem 3.2. If γ be a timelike biharmonic particle in H⁴₁, then, field N is not Fermi–Walker parallel.

Proof. Suppose that γ be a timelike biharmonic particle in H⁴₁. From (2.1), we have

∇˜_sN = [1

k₁[c²− cosh²φ]¹²($ − sinh φ) cos [$s + $₀] (−$ +1

2sinh φ)

−k₁[c²− cosh²φ]¹² cos [$s + $₀] ]e₁ +[ 1

k1

[c²− cosh²φ]¹²($ − sinh φ) sin [$s + $₀] (−$ +1

2sinh φ)

−k₁[c²− cosh²φ]¹² sin [$s + $0] ]e2

+[ 1

2k₁[c²− cosh²φ](−$ + sinh φ) − k1sinh φ]e3− k₁ce4.

This implies that

[1 k1

[c²− cosh²φ]¹²($ − sinh φ) cos [$s + $₀] (−$ +1

2sinh φ)

−k₁[c²− cosh²φ]¹²cos [$s + $0] ]=0, [1

k₁[c²− cosh²φ]¹²($ − sinh φ) sin [$s + $₀] (−$ +1

2sinh φ)

−k₁[c²− cosh²φ]¹²sin [$s + $₀] ]=0, [ 1

2k₁[c²− cosh²φ](−$ + sinh φ) − k1sinh φ] = 0, k₁c = 0.

This occurs only if k1 = 0. This leads to a contradiction. Hence the vector field N is not Fermi–Walker parallel along the curve.

Theorem 3.3. Let γ be a timelike biharmonic particle in H₁⁴. Then, field B1 is Fermi–Walker parallel iff

(− $ k2k1

[c²− cosh²φ]¹²($ − sinh φ) sin [$s + $₀] (−$ +1

2sinh φ) +$k1

k₂ [c²−cosh²φ]¹² sin [$s + $0] +1

2([c²− cosh²φ]¹² sin [$s + $0] [ 1

2k2k1

[c²−cosh²φ](−$+sinh φ) − k₁ k2

sinh φ] + [ 1 k2k1

[c²−cosh²φ]¹² ($−sinh φ) sinh φ sin [$s+$0] (−$+1

2sinh φ) − k1

k₂k₃[c²−cosh²φ]¹² sinh φ sin [$s+$₀]]) +k₂

k1

[c²−cosh²φ]¹²sin [$s+$₀] (−$+sinh φ)

−k2

k₁[c²− cosh²φ]¹² sin [$s + $0] (−$ + sinh φ))= 0,

+( $ k2k1

[c²− cosh²φ]¹²($ − sinh φ) cos [$s + $₀] (−$ + 1

2sinh φ)

−$k1

k₃k₂[c²− cosh²φ]¹² cos [$s + $0] −1

2([c²− cosh²φ]¹² cos [$s + $0] [ 1

2k2k1

[c²− cosh²φ](−$ + sinh φ) − k₁ k2

sinh φ] + [ 1 k2k1

[c²− cosh²φ]¹² ($ − sinh φ) sinh φ cos [$s + $0] (−$ +1

2sinh φ) − k1

k₂[c²− cosh²φ]¹² sinh φ cos [$s + $₀]]) +k₂

k1

[c²− cosh²φ]¹² cos [$s + $₀] ($ − sinh φ)

−k2

k₁[c²− cosh²φ]¹² cos [$s + $0] ($ − sinh φ))= 0, +1

2([ 1 k2k1

[c²− cosh²φ]($ − sinh φ) sin 2 [$s + $₀] (−$ +1

2sinh φ)

−k1

k₂[c²− cosh²φ] sin 2 [$s + $0]])= 0.

Proof. From definition of Fermi-Walker derivative, we have

∇˜_sB1 = (− $ k2k1

[c²− cosh²φ]¹²($ − sinh φ) sin [$s + $0] (−$ + 1

2sinh φ) + $k₁

k₂ [c²− cosh²φ]¹²sin [$s + $₀] +1

2([c²− cosh²φ]¹² sin [$s + $0] [ 1 2k2k1

[c²− cosh²φ]

(−$ + sinh φ) − k₁

k₂sinh φ] + [ 1

k₂k₁[c²− cosh²φ]¹² ($ − sinh φ) sinh φ sin [$s + $0] (−$ + 1

2sinh φ)

− k1

k₂k₃[c²− cosh²φ]¹² sinh φ sin [$s + $0]]) +k₂

k1

[c²− cosh²φ]¹² sin [$s + $₀] (−$ + sinh φ)

−k2

k₁[c²− cosh²φ]¹² sin [$s + $0] (−$ + sinh φ))e1

+( $

k₂k₁[c²− cosh²φ]¹²($ − sinh φ) cos [$s + $₀] (−$ + 1

2sinh φ) − $k₁ k3k2

[c²− cosh²φ]¹² cos [$s + $₀]

−1

2([c²− cosh²φ]¹² cos [$s + $₀] [ 1

2k₂k₁[c²− cosh²φ]

(−$ + sinh φ) − k1

k2

sinh φ] + [ 1 k2k1

[c²− cosh²φ]¹² ($ − sinh φ) sinh φ cos [$s + $₀] (−$ +1

2sinh φ)

−k₁

k₂[c²− cosh²φ]¹² sinh φ cos [$s + $₀]]) +k2

k1

[c²− cosh²φ]¹² cos [$s + $0] ($ − sinh φ)

−k₂

k₁[c²− cosh²φ]¹² cos [$s + $₀] ($ − sinh φ))e₂ +1

2([ 1

k₂k₁[c²− cosh²φ]($ − sinh φ) sin 2 [$s + $0] (−$ + 1

2sinh φ) − k₁ k2

[c²− cosh²φ] sin 2 [$s + $₀]])e₃.

By using condition of parallelism, we obtain theorem.

Theorem 3.4. If γ be a timelike biharmonic particle in H⁴₁, then, the vector field B₂ is not Fermi–Walker parallel along the curve.

Proof. From definition of Fermi-Walker derivative, we obtain

∇˜_sB₂= [ 1

k₂k₁[c²−cosh²φ]¹²($−sinh φ) cos [$s+$₀] (−$+1

2sinh φ)

−k₁ k2

[c²− cosh²φ]¹² cos [$s + $₀]]e₁ +[ 1

k₂k₁[c²−cosh²φ]¹²($−sinh φ) sin [$s+$₀] (−$+1

2sinh φ)

−k1

k2

[c²−cosh²φ]¹² sin [$s + $0]]e2

+[ 1

2k₂k₁[c²− cosh²φ](−$ + sinh φ) −k1

k₂ sinh φ]e₃− k1

k₂ce₄. Using above equation, we get

[ 1

k₂k₁[c²− cosh²φ]¹²($ − sinh φ) cos [$s + $₀] (−$ +1

2sinh φ)

−k₁ k2

[c²− cosh²φ]¹² cos [$s + $₀]] = 0, [ 1

k₂k₁[c²− cosh²φ]¹²($ − sinh φ) sin [$s + $₀] (−$ +1

2sinh φ)

−k1

k2

[c²− cosh²φ]¹² sin [$s + $0]] = 0, [ 1

2k₂k₁[c²− cosh²φ](−$ + sinh φ) −k1

k₂ sinh φ] = 0 k₁

k2

c = 0.

This occurs only if k1 = 0. This leads to a contradiction. Hence the vector field B₂ is not Fermi–Walker parallel in Heisenberg spacetime.

References

[1] D. Bini, A. Geralico, R.T. Jantzen, Frenet-Serret formalism for null world lines, Class. Quantum Grav. 23 (2006), 3963–3981.

[2] E. Fermi, Atti Accad. Naz, Lincei Cl. Sci. Fiz. Mat. Nat. 31 (1922), 184–306.

[3] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathemat- ica, CRC Press, 1998.

[4] J. Eells, L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1–68.

[5] A. Einstein, Relativity: The Special and General Theory, New York: Henry Holt, 1920.

[6] F.W. Hehl, Y. Obhukov, Foundations of Classical Electrodynamics, Birkhauser, Basel, 2003.

[7] E. Honig, E. Schucking, C. Vishveshwara, Motion of charged particles in homo- geneous electromagnetic fields, J. Math. Phys. 15 (1974), 774–781.

[8] GY. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7(4) (1986), 389–402.

[9] Z. S. K¨orpinar, M. Tuz, T. K¨orpinar, New Electromagnetic Fluids Inextensible Flows of Spacelike Particles and some Wave Solutions in Minkowski Spacetime, Int J Theor Phys 55 (1) (2016), 8–16.

[10] T. K¨orpınar, Bianchi Type-I Cosmological Models for Inextensible Flows of Bi- harmonic Particles by Using Curvature Tensor Field in Spacetime, Int J Theor Phys 54 (2015), 1762–1770.

[11] Z. S. K¨orpinar, E. Turhan, M. Tuz, Bianchi Type-I Cosmological Models for Integral Representation Formula and some Solutions in Spacetime, Int J Theor Phys 54(9) (2015), 3195–3202.

[12] T. K¨orpınar, R. C. Demirkol, Frictional magnetic curves in 3D Riemannian manifolds, International Journal of Geometric Methods in Modern Physics, 15 (2018) 1850020.

[13] T. K¨orpınar, On T-Magnetic Biharmonic Particles with Energy and Angle in the Three Dimensional Heisenberg Group H, Adv. Appl. Clifford Algebras, 28 (1) (2018),15pp.

[14] T. K¨orpınar, New Characterization for Minimizing Energy of Biharmonic Par- ticles in Heisenberg Spacetime, Int J Phys. 53 (2014), 3208-3218.

[15] T. K¨orpınar, R.C. Demirkol, A new approach on the curvature dependent energy for elastic curves in a Lie Group, Honam Mathematical J. 39(4) (2017), 637-647.

[16] T. K¨orpınar, R.C. Demirkol, A New characterization on the energy of elastica with the energy of Bishop vector fields in Minkowski space, Journal of Advanced Physics. 6(4) (2017), 562-569.

[17] T. K¨orpınar, E. Turhan, On characterization of B-canal surfaces in terms of biharmonic B-slant helices according to Bishop frame in Heisenberg group Heis³, J. Math. Anal. Appl. 382 (2011), 57-65.

[18] T. K¨orpınar, E. Turhan, V. Asil, Tangent Bishop spherical images of a bihar- monic B-slant helix in the Heisenberg group Heis³, Iranian Journal of Science and Technology Transaction A: Science 35 (2011), 265-271

[19] T. K¨orpınar, On the Fermi Walker Derivative for Inextensible Flows, Zeitschrift f¨ur Naturforschung A- A Journal of Physical Sciences. 70a (2015), 477-482.

[20] T. K¨orpınar, E. Turhan, Time-Tangent Surfaces Around Biharmonic Particles and Its Lorentz Transformations in Heisenberg Spacetime. Int. J. Theor. Phys.

52 (2013), 4427-4438

[21] J.W. Maluf, F.F. Faria, On the construction of Fermi-Walker transported frames, Ann. Phys. (Berlin) 17(5) (2008), 326 – 335.

[22] S. Rahmani, Metriqus de Lorentz sur les groupes de Lie unimodulaires, de di- mension trois, Journal of Geometry and Physics, 9 (1992), 295-302.

[23] B. O’Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983.

[24] E. Pina, Lorentz transformation and the motion of a charge in a constant elec- tromagnetic field, Rev. Mex. Fis. 16 (1967), 233-236.

[25] H. Ringermacher, Intrinsic geometry of curves and the Minkowski force, Phys.

Lett. A 74 (1979), 381–383.

[26] G.A. Suro˘glu, A modified Fermi-Walker derivative for inextensible flows of bi- normal spherical image, Open Phys., 16 (2018), 14–20.

[27] J.L. Synge, Relativity: The General Theory, North Holland, Amsterdam, 1960.

[28] J.L. Synge, Timelike helices in flat spacetime, Proc. R. Ir. Acad., Sect. A 65 (1967), 27-41.

[29] E. Turhan, T. K¨orpınar, On Characterization Canal Surfaces around Timelike Horizontal Biharmonic Curves in Lorentzian Heisenberg Group Heis³, Z. Natur- forsch. 66a (2011), 441 - 449.

[30] E. Turhan, T. K¨orpınar, On Characterization of Time-Like Horizontal Bihar- monic Curves in the Lorentzian Heisenberg Group Heis³, Z. Naturforsch. 65a (2010), 641-648.

[31] M. Yenero˘glu, On New Characterization of Inextensible Flows of Spacelike Curves in de Sitter Space, Open Mathematics, 14 (2016), 946-954.

[32] C.M. Wood, On the Energy of a Unit Vector Field, Geom. Dedic. 64 (1997), 19-330.

Talat Korpinar

Department of Mathematics, Mus Alparslan University, Mus 49250, Turkey.

E-mail: talatkorpinar@gmail.com