Smarandache Curves in Minkowski Space-time

: A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache Curve . In this paper, we deﬁne a special case of such curves and call it Smarandache TB 2 Curves in the space E 41 . Moreover, we compute formulas of its Frenet apparatus according to base curve via the method expressed in [3]. By this way, we obtain an another orthonormal frame of E 41 .


§1. Introduction
In the case of a differentiable curve, at each point a tetrad of mutually orthogonal unit vectors (called tangent, normal, first binormal and second binormal) was defined and constructed, and the rates of change of these vectors along the curve define the curvatures of the curve in Minkowski space-time [1].It is well-known that the set whose elements are frame vectors and curvatures of a curve, is called Frenet Apparatus of the curves.
The corresponding Frenet's equations for an arbitrary curve in the Minkowski space-time E 4  1 are given in [2].A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache Curve.We deal with a special Smarandache curves which is defined by the tangent and second binormal vector fields.We call such curves as Smarandache TB 2 Curves.Additionally, we compute formulas of this kind curves by the method expressed in [3].We hope these results will be helpful to mathematicians who are specialized on mathematical modeling.§2.Preliminary notes To meet the requirements in the next sections, here, the basic elements of the theory of curves in the space E 4 1 are briefly presented.A more complete elementary treatment can be found in the reference [1].

Minkowski space-time E 4
1 is an Euclidean space E 4 provided with the standard flat metric given by where (x 1 , x 2 , x 3 , x 4 ) is a rectangular coordinate system in E 4 1 .Since g is an indefinite metric, recall that a vector v ∈ E 4 1 can have one of the three causal characters; it can be space-like if g(v, v) > 0 or v = 0, time-like if g(v, v) < 0 and null (light-like) if g(v, v)=0 and v = 0. Similarly, an arbitrary curve α = α(s) in E 4  1 can be locally be space-like, time-like or null (light-like), if all of its velocity vectors α ′ (s) are respectively space-like, time-like or null.Also, recall the norm of a vector v is given by v 1 are said to be orthogonal if g(v, w) = 0.The velocity of the curve α(s) is given by α ′ (s) .
Denote by {T (s), N (s), B 1 (s), B 2 (s)} the moving Frenet frame along the curve α(s) in the space E 4 1 .Then T, N, B 1 , B 2 are, respectively, the tangent, the principal normal, the first binormal and the second binormal vector fields.Space-like or time-like curve α(s) is said to be parametrized by arclength function s, if g(α ′ (s), α ′ (s)) = ±1.Let α(s) be a curve in the space-time E 4 1 , parametrized by arclength function s.Then for the unit speed space-like curve α with non-null frame vectors the following Frenet equations are given in [2]: where T, N, B 1 and B 2 are mutually orthogonal vectors satisfying equations Here κ, τ and σ are, respectively, first, second and third curvature of the space-like curve α.
In the same space, in [3] authors defined a vector product and gave a method to establish the Frenet frame for an arbitrary curve by following definition and theorem.Theorem 2.2 Let α = α(t) be an arbitrary space-like curve in Minkowski space-time E 4 1 with above Frenet equations.The Frenet apparatus of α can be written as follows; and where µ is taken −1 or +1 to make +1 the determinant of [T, N, B 1 , B 2 ] matrix.§3.Smarandache Curves in Minkowski Space-time Definition 3.1 A regular curve in E 4 1 , whose position vector is obtained by Frenet frame vectors on another regular curve, is called a Smarandache Curve.Remark 3.2 Formulas of all Smarandache curves' Frenet apparatus can be determined by the expressed method.Now, let us define a special form of Definition 3.1.Definition 3.3 Let ξ = ξ(s) be an unit space-like curve with constant and nonzero curvatures κ, τ and σ; and {T, N, B 1 , B 2 } be moving frame on it.Smarandache TB 2 curves are defined with Theorem 3.4 Let ξ = ξ(s) be an unit speed space-like curve with constant and nonzero curvatures κ, τ and σ and X = X(s X ) be a Smarandache TB 2 curve defined by frame vectors of ξ = ξ(s).Then (i) The curve X = X(s X ) is a space-like curve.
Proof Let X = X(s X ) be a Smarandache TB 2 curve defined with above statement.Differentiating both sides of (10), we easily have The inner product g(X ′ , X ′ ) follows that where ′ denotes derivative according to s. (12) implies that X = X(s X ) is a space-like curve.Thus, the tangent vector is obtained as Then considering Theorem 2.1, we calculate following derivatives according to s: Then, we form Equation ( 17) yields the principal normal of X as Thereafter, by means of (17) and its norm, we write first curvature The vector product T X ∧ N X ∧ X ′′′ follows that where, A = .Shortly, let us denote T X ∧ N X ∧ X ′′′ with l 1 T + l 2 N + l 3 B 1 + l 4 B 2 .And therefore, we have the second binormal vector of X = X(s X ) as