Smarandache Curves in Minkowski Space-time

2008·0.14 MB·

by  Melih Turgut

International J.Math. Combin. Vol.3 (2008), 51-55
Smarandache Curves in Minkowski Space-time
Melih Turgut and Su¨ha Yilmaz
(DepartmentofMathematicsofBucaEducationalFacultyofDokuzEylu¨lUniversity,35160Buca-Izmir,Turkey.)
E-mail: [email protected],[email protected]
Abstract: 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 TB2 Curves in the
space E4. Moreover, we compute formulas of its Frenet apparatus according to base curve
1
via the method expressed in [3]. By this way, we obtain an another orthonormal frame of
E4.
1
Key Words: Minkowskispace-time,Smarandachecurves,Frenetapparatusofthecurves.
AMS(2000): 53C50, 51B20.
§1. Introduction
In the case of a differentiable curve, at each point a tetrad of mutually orthogonalunit 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
E4 aregivenin[2]. AregularcurveinMinkowskispace-time,whosepositionvectoriscomposed
1
byFrenetframevectorsonanotherregularcurve,iscalledaSmarandache Curve. Wedealwith
a special Smarandache curves which is defined by the tangent and second binormal vector
fields. We call such curves as Smarandache TB Curves. Additionally, we compute formulas
2
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 E4 are briefly presented. A more complete elementary treatment can be found in
1
the reference [1].
Minkowskispace-time E4 is anEuclideanspace E4 providedwith the standardflatmetric
1
given by
1ReceivedAugust16,2008. Accepted September 2,2008.
52 MelihTurgutandSuhaYilmaz
g = dx2+dx2+dx2+dx2,
− 1 2 3 4
where (x ,x ,x ,x ) is a rectangular coordinate system in E4.
1 2 3 4 1
Since g is an indefinite metric, recall that a vector v E4 can have one of the three
∈ 1
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 E4 can be locally
6 1
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 = g(v,v).
k k | |
Therefore,v is a unit vector if g(v,v)= 1. Next, vectors v,w in E4 are said to be orthogonal
± 1 p
if g(v,w)=0. The velocity of the curve α(s) is given by α′(s) .
k k
Denote by T(s),N(s),B (s),B (s) the moving Frenet frame along the curve α(s) in
1 2
{ }
the space E4. Then T,N,B ,B are, respectively, the tangent, the principal normal, the first
1 1 2
binormaland the secondbinormal vectorfields. 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 E4, parametrized by arclength function s. Then for
1
the unit speed space-like curve α with non-null frame vectors the following Frenet equations
are given in [2]:
T′ 0 κ 0 0 T
 N′   κ 0 τ 0  N 
= − , (1)
 B′   0 τ 0 σ  B 
 1   −  1 
    
 B2′   0 0 σ 0  B2 
    
where T,N,B and B are mutually orthogonalvectors satisfying equations
1 2
g(T,T)=g(N,N)=g(B ,B )=1,g(B ,B )= 1.
1 1 2 2
−
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.
Definition 2.1 Let a = (a ,a ,a ,a ), b = (b ,b ,b ,b ) and c = (c ,c ,c ,c ) be vectors in
1 2 3 4 1 2 3 4 1 2 3 4
E4. The vector product in Minkowski space-time E4 is defined by the determinant
1 1
e e e e
1 2 3 4
−
(cid:12) a a a a (cid:12)
a b c= (cid:12) 1 2 3 4 (cid:12), (2)
(cid:12) (cid:12)
∧ ∧ −(cid:12) b b b b (cid:12)
(cid:12) 1 2 3 4 (cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) c1 c2 c3 c4 (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where e ,e ,e and e are mutually ortho(cid:12)gonal vectors (coord(cid:12)inate direction vectors) satisfying
1 2 3 4
equations
e e e =e , e e e =e , e e e =e , e e e = e .
1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3
∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ −
SmarandacheCurvesinMinkowskiSpace-time 53
Theorem 2.2 Let α=α(t) be an arbitrary space-like curve in Minkowski space-time E4 with
1
above Frenet equations. The Frenet apparatus of α can be written as follows;
α′
T = , (3)
α′
k k
α′ 2.α′′ g(α′,α′′).α′
N = k k − , (4)
α′ 2.α′′ g(α′,α′′).α′
k k −
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
B =µN T B , (5)
1 2
∧ ∧
T N α′′′
B =µ ∧ ∧ , (6)
2 T N α′′′
k ∧ ∧ k
α′ 2.α′′ g(α′,α′′).α′
k k −
κ= (7)
(cid:13)(cid:13) α′ 4 (cid:13)(cid:13)
(cid:13) k k (cid:13)
T N α′′′ . α′
τ = k ∧ ∧ k k k (8)
α′ 2.α′′ g(α′,α′′).α′
k k −
(cid:13) (cid:13)
and (cid:13) (cid:13)
(cid:13) (cid:13)
g(α(IV),B )
2
σ = , (9)
T N α′′′ . α′
k ∧ ∧ k k k
where µ is taken 1 or +1 to make +1 the determinant of [T,N,B ,B ] matrix.
1 2
−
§3. Smarandache Curves in Minkowski Space-time
Definition3.1 AregularcurveinE4, whoseposition vectorisobtainedbyFrenetframevectors
1
on another regular curve, is called a Smarandache Curve.
Remark 3.2 FormulasofallSmarandachecurves’Frenetapparatuscanbe 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 ,B be moving frame on it. Smarandache TB curves are defined
1 2 2
{ }
with
1
X =X(s )= (T(s)+B (s)). (10)
X 2
κ2(s)+σ2(s)
p
Theorem 3.4 Let ξ = ξ(s) be an unit speed space-like curve with constant and nonzero cur-
vatures κ,τ and σ and X = X(s ) be a Smarandache TB curve defined by frame vectors of
X 2
ξ =ξ(s). Then
54 MelihTurgutandSuhaYilmaz
(i) The curve X =X(s ) is a space-like curve.
X
(ii) Frenet apparatus of T ,N B ,B ,κ ,τ ,σ Smarandache TB curve X =
X X, 1X 2X X X X 2
{ }
X(s ) can be formed by Frenet apparatus T,N,B ,B ,κ,τ,σ of ξ =ξ(s).
X 1 2
{ }
Proof Let X = X(s ) be a Smarandache TB curve defined with above statement. Dif-
X 2
ferentiating both sides of (10), we easily have
dX ds 1
X
= (κN +σB ). (11)
1
dsX ds κ2(s)+σ2(s)
The inner product g(X′,X′) follows thapt
g(X′,X′)=1, (12)
where ′ denotes derivative according to s. (12) implies that X = X(s ) is a space-like curve.
X
Thus, the tangent vector is obtained as
1
T = (κN +σB ). (13)
X 1
κ2(s)+σ2(s)
Then considering Theorem 2.1, we cpalculate following derivatives according to s:
1
X′′ = ( κ2T τσN +κτB +σ2B ). (14)
1 2
√κ2+σ2 − −
1
X′′′ = [κτσT +( κ3 κτ2)N +(σ3 τ2σ)B +κτσB ]. (15)
1 2
√κ2+σ2 − − −
1
X(IV) = [(...)T +(...)N +(...)B +(σ4 τ2σ2)B ]. (16)
1 2
√κ2+σ2 −
Then, we form
1
X′ 2.X′′ g(X′,X′′).X′ = [ κ2T τσN +κτB +σB ]. (17)
1 2
k k − √κ2+σ2 − −
Equation (17) yields the principal normal of X as
κ2T τσN +κτB +σB
1 2
N = − − . (18)
X
√ κ4+τ2σ2+κ2τ2+σ2
−
Thereafter, by means of (17) and its norm, we write first curvature
κ4+τ2σ2+κ2τ2+σ2
κ = − . (19)
X κ2+σ2
r
The vector product T N X′′′ follows that
X X
∧ ∧
1 [κσ(κ2+σ2)(τ2 σ)T +κτσ2(κ2+σ)N
T N X′′′ = − , (20)
X X
∧ ∧ A κ2τσ(κ2+σ)B +κτ(κ2+σ2)(κ2+τ2)B ] 
 − 1 2 
where, A = √(−κ4+τ2σ2+κ12τ2+σ2)(κ2+σ2). Shortly, let us denote TX ∧NX ∧X′′′ with l1T +
l N +l B +l B . And therefore, we have the second binormal vector of X =X(s ) as
2 3 1 4 2 X
l T +l N +l B +l B
1 2 3 1 4 2
B =µ . (21)
2X
l2+l2+l2+l2
− 1 2 3 4
p
SmarandacheCurvesinMinkowskiSpace-time 55
Thus, we easily have the second and third curvatures as follows:
( l2+l2+l2+l2)(κ2+σ2)
τ = − 1 2 3 4 , (22)
X s κ4+τ2σ2+κ2τ2+σ2
−
σ2(σ2 τ2)
σ = − . (23)
X
(κ2+σ2) l2+l2+l2+l2
− 1 2 3 4
Finally, the vector product NX TX B2X gpives us the first binormal vector
∧ ∧
1 [(κσl3 σ2l2 τ(κ2+σ2)l4]T σ(κ2l4+σl1)N
B =µ − − − , (24)
1X L +κ(κ2l +σl )B +[κ2(σl κ2l )+τl (κ2+σ2)]B 
 4 1 1 2− 3 1 2 
where L=  1 .  (cid:3)
√(−l2+l2+l2+l2)(κ2+σ2)(−κ4+τ2σ2+κ2τ2+σ2)
1 2 3 4
Thus, we compute Frenet apparatus of Smarandache TB curves.
2
Corollary 3.1 Suffice it to say that T ,N B ,B is an orthonormal frame of E4.
{ X X, 1X 2X} 1
Acknowledgement
The first author would like to thank TUBITAK-BIDEB for their financial supports during his
Ph.D. studies.
References
[1] B. O’Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983.
[2] J. Walrave, Curves and surfaces in Minkowski space. Dissertation, K. U. Leuven, Fac. of
Science, Leuven, 1995.
[3] S.Yilmaz andM.Turgut, Onthe DifferentialGeometryofthe curvesin Minkowskispace-
time I, Int. J. Contemp. Math. Sci. 3(27), 1343-1349,2008.