Particle with Torsion in. the Minkowski Space E 3 1

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1 Adv. Studies Theor. Phys., Vol. 4, 1, no. 3, Particle with Torsion in the Minkowski Space E 3 1 Nevin Gürbüz Osmangazi University, Department of Mathematics 648, Eskişehir, Turkey Abstract The motion of the relativistic particle with torsion have some geometric models Minkowski and Euclidean spaces. In this paper, we derived solutions for motion of the relativistic particle with torsion. We found solutions of the equations of motion of the relativistic particle with torsion for a timelike curve on a timelike surface in the Minkowski space E INTRODUCTION In ( + 1) and (3 + 1) Minkowskian spaces and 3 Euclidean spaces, there are exist important geometric models of relativistic particle. With this reason, recently, a lot of mathematicians and physiciants nd interesting to investigate these models. Plyushchay (199) studied the model of relativistic particle of the anyon with torsion. Nessessian () investigated large massive 4d particle with torsion and conformal mechanics. Model of relavistic particle with torsion appears in the Bose-Fermi transmutation mechanism. This model is also investigated by Plyushchay (1991). More recently, Arreaga, Capovilla and Guven (1) studied dynamic of a relativistic particle determined by an action depending on the torsion. Barros, Ferrandez, Javaloyes, Lucas (5) also condireded mechanical systems linearly depending on the curvature and the torsion. Nickerson and Manning (1988) derived intrinsic equations for a relaxed elastic line in R 3. Later Gürbüz (1999) studied same problem in R 3 1. In this paper, we construct solutions of the equation of the model of the relativistic particle with torsion in.e 3 1

2 1 N. Gürbüz The motion of relativistic particle and solution Let M be the pseudo-euclidean surface in E 3 1 and α denote an arc on surface pseudo- Euclidean M. There is exist a second frame{t, Q, N} frame apart from the Frenet frame {T, n, b} at every point of the curve α. let T (s) =α (s) denote the unit tangent vector to α, let N(s) denote the unit normal surface. And N T = εq, ε = ±1. The analogue of the Frenet -Serret formulas is T = ε k g Q+ε 3 k n N,Q = ε 1 k g T +ε 3 τ g N,N = ε 1 k n T ε τ g Q where k g is the geodesic curvature, τ g is the geodesic torsion, k n is the normal curvature. < T,T >= ε 1,<Q,Q>= ε, < N,N >= ε 3 and cross product, T Q = ε N, Q N = ε 1 T, N T = ε 3 Q. The model of the relativisitic particle with torsion is described by the action S = ( m + γτ)ds.where m is a a parameter with the dimension of mass and γ is additional term induced by the Chern-Simons gauge eld (Plyushchay, 1991).Assume that α curve lies in a coordinate patch (u, v) x(u, v) of pseudo-euclidean surface M in E 3 1 and let x u = x u, x v = x. Then, α is expressed as α(s) =x (u(s),v(s)), v s l with T (s) =α (s) = x du + x dv and Q(s) =p(s)x u ds v ds u + q(s)x v for suitable scalar functions p(s) and q(s). In order to obtain variational arcs of length l, it is generally necessary to extend α to an arc α dened for s l, with l >l,but sufciently close to l so that α lies in the coordinate patch. Let μ(s), s l, be a scalar function of class C, not vanishing identically. Dene η(s) =μ(s)p (s), ξ(s) =μ(s)q (s).then, along α η(s)x u + ξ(s)x v = μ(s)q(s). (1) Assume also that Now dene μ() =,μ () =,μ () =. () β(σ; t) =x (u(σ)+tη(σ),v(σ)+tξ(σ)), (3) for σ l. For t <ε(where ε>depends upon the choice of α and of μ), the point β(σ; t) lies in the coordinate patch. For xed t, β(σ; t) gives an arc with the same initial point and initial direction as α, because of (). For t =,β(σ; ) is the same as α and σ is arc length. For t, the parameter σ is not arc length in general. For xed

3 Particle with torsion in Minkowski space 13 t, t <ε, let L (t) denote the length of the arc β(σ; t), σ l. Then with L (t) = β β (σ; t), (σ; t) dσ (4) L () = l >l. (5) It is clear from (3) and (4) that L (t) is continuous. In particular, it follows from (5) that L (t) > l + l >l, ( t <ε 1 ) (6) for a suitable ε 1 satisfying <ε 1 ε. Because of (6), we can restrict β(σ; t), t <ε 1, to an arc of length l by restricting the parameter σ to an interval σ λ(t) l, by requiring λ(t) β, β dσ = l. (7) Note that λ() = l. The function λ(t) need not be determined explicitly, but we shall need dλ l dt = ε 1 μk g ds. (8) The proof of (8) and of other results below will depend on calculations from (3) such as β = T, σ l (9) which gives β = T = ε k g Q + ε 3 k n N. (1) 3 β 3 =( ε 1 ε kg ε 1ε 3 kn )T +(ε k g ε ε 3 k n τ g )Q +(ε ε 3 k g τ g + ε 3 k n )N Also, it follows from (1) that β t = μq. (11) Using (1), the second differentiation of (11) gives β t = ε 1μk g T + μ Q + ε 3 μτ g N (1)

4 14 N. Gürbüz and the third differentiation of (11) gives 3 β t = ( ) ( ) ε 1 μ k g ε 1 μk g ε 1 ε 3 μτ g k n T + μ ε 1 ε μkg ε ε 3 μτg Q + ( ) (13) ε 3 μ τ g ε 1 ε 3 μk g k n + ε 3 μτ g N. and the fourth differentiation of (11) gives 4 β t 3 =( 3ε 1 μ k g 3ε 1 μ k g + ε μk 3 g + ε 1 ε ε 3 μk g τ g ε 1 ε 3 μτ g k n 3ε 1 ε 3 μ τ g k n + ε 3 μk g kn ε 1ε 3 μk n τ g)t + ( μ 3ε 1 ε μ kg 3ε 1ε μk g k g 3ε ε 3 μτ g τ g 3ε ) ε 3 μ τg Q +(3ε 3 μ τ g 3ε 1 ε 3 μ k g k n ε 1 ε 3 μk g k n +3ε 3 μ τ g ε 1μτ g kn ε μτg 3 ε 1ε ε 3 μkg τ g + ε 3 μτ g ε 1ε 3 μk g k n )N. (14) Thus, 3 β t β, 3 β 3 = μ k g τ g ε μ k n + ε 1ε μk 3 g τ g (15) +ε 1 μk g k n + ε ε 3 μk g τ 3 g + ε 3μτ g k n +μ τ g k g ε 3 μ τ g k n ε 1 μk g k gk n + ε 1 ε 3 μk g k nτ g +μk g τ g ε 3μk n τ g τ g, β β t, 3 β 3 = ε 1 μk g k gk n + ε 1 ε 3 μk g knτ g ε 1 ε μkgτ 3 g ε 1 μkgk n, (16) β β, 4 β t 3 = ε 1 μk g k gk n 3μ k g τ g + ε ε 3 μk g τ 3 g +ε 1 ε 3 μk g k nτ g + ε 1 ε μk 3 gτ g + ε 1 μk gk n +ε μ k n 3ε 3 μk n τ g τ g 3ε 3μ τ g k n 3μ τ g k g μk g τ g.(17)

5 Particle with torsion in Minkowski space 15 To prove (8), differentiate (7) with respect to t, remembering that l is constant, and evaluate at t = using (9) and (1), with λ() = l. dλ dt β, β + β, β t β β v u t,, β β ds = For β(σ; t) is σ λ(t), t <ε 1. Since σ is not generally arc length for t, the motion S = λ(t) m + γ β β, 3 β β 3, β 1/ β, β β, β β +, β dσ. (18) In calculating S (t), we give explicitly only those terms which do not vanish for t =. β The omitted terms are those with a factor, β, which vanishes at t =, since T,T =. For arbitrary μ satisfying Eq.(), using (8), (9), (1),(1),(13),(15),(17), we nd S l () = ε 1 +γ { μk g ds ( m + γ( ε kg τ g ε 3 τ g kn k gk n + k nk g )) ε kg + ε 3kn [ε μ k n + μ ( 4k g τ g ε k n )+μ ( 5ε 3 τ g k n +k g τ g 3k g τ g ) +μ(ε ε 3 k g τg 3 + ε 3τg k n ε 1k g k g k n +ε 1 ε 3 k g kn τ g + k g τ g 4ε 3 k n τ g τ g k gτ g )] ε kg + ε 3kn 1 ds γ (μ k g ε 1 ε μk 3 g ε ε 3 μk g τ g +ε 3 μ k n τ g ε 1 ε 3 μk g k n ε 3 μk n τ g )( ε k g τ g ε 3 τ g k n k gk n + k nk g ) ε k g +ε 3k n 1 1} σ=λ() (ε k g +ε 3k n ) ds (19)

6 16 N. Gürbüz.1 The motion of relativistic particle with torsion for timelike arc α on a timelike surface If T is timelike, Q and N are spacelike, then <T,T >= ε 1 = 1,< Q,Q >= ε =1,< N,N >= ε 3 =1 In the case of kg >k n, ε kg + ε 3 kn = k g + kn. () With using (), we nd S () = γ μ{(k g(l)k g + k g τ g (l)k n(l)+k g k g (l)k n(l) k g k n (l)k g (l)) ( k g (l)+k n (l) +(k g τ 3 g + τ g k n +k g k gk n k nk g τ g + k gτ g 4k n τ g τ g k gτ g ) ( k g + k n +(k 3 gτ 3 g + τ 3 g k g k n + k gτ g k n k g τ g k n k g k 3 gk nτ g k g k 4 n τ g k g k n k n + k gk 3 n k g k4 g k n + k nk g k3 g ) ( k g + k n ((k g τ g 5τ g k n 3k g τ g ) ( k g + k n +(k nτ g k g k n τ g k g k n τ g k g k n k 3 nτ g ) ( k g + k n +((( 4k g τ g k n ) ( k g + k n ) ) +(k n k g k g kgτ 3 g k g τ g kn kgk n)) ( ) ( kg + kn) (k n k 1) g + kn) }ds +γμ(l){(k g(l)τ g (l) 5τg (l)k n (l) 3k g (l)τ g(l)) ( kg(l)+k n(l) ) 1 (k n (l)τ g(l)k g (l)+k n(l)τ g (l)k g (l)+k n(l)τ g (l)k g (l)k n (l) +k 3 n(l)τ g (l)) ( k g(l)+k n(l) ) + ((4kg (l)τ g (l)+k n(l)) ( k g(l)+k n(l) ) (k n (l)k g (l)k g (l)+k3 g (l)τ g(l)+k g (l)τ g (l)kn (l) +kg(l)k n(l)) ( kg(l)+k n(l) ) ) +(k n (l) ( kg(l)+k n(l) ) 1 ) } +γμ (l){ (4k g (l)τ g (l)+k n (l)) ( kg (l)+k n (l) +(k n (l)k g (l)k g(l) kg(l)τ 3 g (l) k g (l)τ g (l)kn(l) kg (l)k n (l))) ( kg (l)+k n (l)) (k n (l) ( kg (l)+k n (l) ) } +γμ (l)(k n (l) ( kg(l)+k n(l) ) 1 ) In order that S () = for all choices of the function μ(s) satisfying (), with arbi-

7 Particle with torsion in Minkowski space 17 trary values of μ(l) μ (l) and μ (l) the given timelike arc α must satisfy three boundary conditions (1) (k g (l)τ g(l) 5τ g (l)k n(l) 3k g (l)τ g (l)) ( k g (l)+k n (l) (k n (l)τ g(l)k g (l)+k n(l)τ g (l)k g (l)+k n(l)τ g (l)k g (l)k n (l) +k 3 n (l)τ g (l)) ( k g (l)+k n (l)) +((4k g (l)τ g (l)+k n (l)) ( k g (l)+k n (l) (k n (l)k g (l)k g(l)+k g(l)τ 3 g (l)+k g (l)τ g (l)kn(l) +kg (l)k n (l)) ( kg (l)+k n (l)) ) +(k n (l) ( kg (l)+k n (l) ) = (4k g (l)τ g (l)+k n(l)) ( kg(l)+k n(l) ) 1 () +(k n (l)k g (l)k g (l) k3 g (l)τ g(l) k g (l)τ g (l)kn (l) kg(l)k n(l))) ( kg(l)+k n(l) ) (kn (l) ( kg(l)+k n(l) ) 1 ) = (3) (k n (l) ( kg (l)+k n (l) = and differential equation (k g (l)k g + k g τ g (l)k n (l)+k gk g (l)k n (l) k gk n (l)k g (l)) ( k g (l)+k n (l) +(k g τ 3 g + τ g k n +k g k gk n k nk g τ g + k gτ g 4k n τ g τ g k g τ g ) ( k g + k n (4) +(k 3 g τ 3 g + τ 3 g k gk n + k g τ g k n k gτ g k nk g k3 g k n τ g k g k 4 n τ g k g k n k n + k gk 3 n k g k 4 g k n + k nk g k3 g ) ( k g + k n ) ((k g τ g 5τg k n 3k g τ g ) ( kg + k n +(kn τ gk g k nτg k g k nτ g k g k n k3 n τ g ) ( kg + n) k ) +(( 4k g τ g k n ) ( kg + k n +(k n k g k g k3 g τ g k g τ g kn k g k n ) ( kg + ( n) k ) (k n k g + kn) 1 ) =. REFERENCES Arreaga G.& Capovilla R & J. Guven.1.Frenet-Serret dynamics. Class. Quantum Grav.18 3: Barros M&Ferrandez &Javaloyes &A, Lucas P 5. Relativistic particles rigidity and torsion in D=3 spacetimes. Class. Quantum Grav.: Gürbüz, N On elastic lines in Minkowski space R 3 1. Süleyman Demirel University Nevin Gürbüz and Ali Görgülü, Intrinsic equations for a relaxed elastic line on an oriented surface in the Minkowski space in Rˆ3 1, Hadronic Journal, 3, ()

8 18 N. Gürbüz Nessessian, A.. Large massive 4d particle with torsion and conformal mechanics. Phys.Lett. B 473. : Nickerson H. K& Manning G. M Intrinsic equations for a relaxed elastic line on an oriented surface.geometriae Dedicate 7 : O Neill B Semi-Riemannian Geometry,Academic Press. New York,London,. Plyushchay,M.S The model of the relavistic particle with torsion. Nucl.Phys.B 36, :54-7. Plyushchay,M.S. 199.Relativistic massive particle with higher curvatures as a model for the description of bosons and fermision. Phys.Lett.B 35 : Received: October, 9