E E ON DARBOUX VECTORS OF A SPECIAL FRENET MOTION FOR A TIME-LIKE CURVE IN Bahaddin BÜKCÜ* İ N İ N DARBOUX VEKTÖRLER İ ÜZER İ NE E ’de TIME-LIKE B İ R E Ğ R İ İ Ç İ N ÖZEL B İ R FRENET HAREKET

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E ’de TIME-LIKE BİR EĞRİ İÇİN ÖZEL BİR FRENET

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HAREKETİNİN DARBOUX VEKTÖRLERİ ÜZERİNE Bahaddin BÜKCÜ*

*Gaziosmanpaşa Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, 60250- Taşlıçiftlik-Tokat,TÜRKİYE; e-mail: bbukcu@yahoo.com

ÖZET

Bir parametreli özel Frenet hareketi, Bottema tarafından 3-boyutlu Öklid uzayı E³ de verilmiştir [1]. Bu çalışmada, E³deki [1] Frenet hareketinin, E₁³, 3-boyutlu Minkowski uzayına bir genelleştirilmesi veriliyor. İlk olarak, E³deki bu hareket, time-like bir eğri için

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E uzayında tanımlandı daha sonra, bu hareketin Darboux vektörleri [2] sabit ve hareketli uzaylar için hesaplandı.

Anahtar Kelimeler: Causal karakter, Darboux vektörleri, Özel Frenet hareketi, Time-like curve

ON DARBOUX VECTORS OF A SPECIAL FRENET MOTION FOR A TIME-LIKE CURVE IN E

₁³

ABSTRACT

A special Frenet motion with an one-parameter in E³has been given by Bottema [1]. In this study, we give a generalization of [1] to E₁³, Minkowski 3-space. Firstly, this motion inE³is defined inE₁³ for a time-like space curve and then Darboux vectors [2] of this motion is calculated for fixed and moving spaces inE₁³ .

Keywords: Causal character, Darboux vector, Frenet motion, Lorentz space, Time-like curve.

1. INTRODUCTION

An one-parameter motion of a body in Minkowski 3-space is generated by the transformation

⎥

⎦

⎢ ⎤

⎣

⎥⎡

⎦

⎢ ⎤

⎣

=⎡

⎥⎦

⎢ ⎤

⎣

⎡

1 1 0 1

x0

c A

X (1.1)

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where A is a semi-orthogonal matrix , A∈SO(3,1), and c is the displacement vector of the origin. A ve c are C^∞ functions of a real parameter s. x₀, X,c are n×1 matrices and

{

³³ ¹

}

(3,1) ^T , det 1, ( 1,1,1) [3].

SO = A∈R A⁻ = εA ε A= ε =diag −

x0 and X respectively correspond to the position vectors of the same point P, with respect to the orthogonal coordinate systems of the moving space H and the fixed spaceH ′. At the unital time

s = s

₀ we consider the coordinate systems of H and H ′are coincident.

We define a special motion in terms of a timelike space curve α given in the fixed spaceH ′. This motion is such that the moving frame O_xyz moves with O along α while rotating so that the x and y axes always coincide with, respectively, the tangent and principal normal of α. This means that as O coincides with a point Q of α, the O_xyz frame coincides with the Frenet trihedron at Q : Q_ξηρ. This trihedron consists of the tangent Q_ξ, the principal normal Q_η, and the binormal Q_ρ, which are three mutually orthogonal axes. Obviously, the geometry of this motion is completely defined by α. For our special motion, because O moves along α, X=c represents the time-like space curveα. We will use the arc length s of α as the motion parameter, and use primes to denote derivatives with respect to s.

The Lorentzian or Minkowski 3-space E₁³ is the Euclidean 3-space provided with the Lorentzian inner product

<x,y>:=−x₁y₁+x₂y₂+x₃y₃

where x=(x₁,x₂,x₃) and y=(y₁,y₂,y₃). An arbitrary vector x=(x₁,x₂,x₃)in E₁³ may have one of three Lorentzian causal characters:

if<x,x>>0 or

x r = 0

, it is space-like, if <x,x><0 it is time-like and

if <x,x>=0 it is null ( light-like) [3].

Similarly, an arbitrary curve α=α(s) in E₁³ is locally space-like, time-like or null, for each s∈I⊂R. Recall that the pseudo-norm of an arbitrary vector xr∈E₁³ is given by

>

<

= x x

x , , and that the velocity v of the curve α is v= α′(s) . Therefore, α is a unit speed curve if and only if <α′(s),α′(s)>=±1.

Denote by

{

T(s),N(s),B(s)

}

the moving Frenet frame along the curve α=α(s) parameterized by a pseudo-arclength parameter s, i.e. <α′(s),α′(s)>=±1. If

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) , , (t₁ t₂ t₃

T = ,N=(n₁,n₂,n₃) and B=(b₁,b₂,b₃)are the unit vectors along Q_ξ, Q_ηand Qρ, Lorentzian differential geometry gives us

T s( )=α′( )s ,N s( )=α′′( ) /s α′′ ,B s( )=T s( )∧N s( ), (1.2) where k= α′′ is the curvature of curveα and “∧” denotes vector product in E³. If

) α(s

α= is a time-like curve, i.e., if T is a time-like vector, then the Frenet formulae read

T′=kN, N′=kT+τB, B′=−τN (1.3) 0

, ,

, , 1 , ,

, 1

, >=− < >=< >= < >=< >=< >=

<T T N N B B T N T B N B ,

here, k the curvature and

τ

the torsion of the timelike curve α[4]. Then we have

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=

3 3 3

2 2 2

1 1 1

b n t

A . (1.4)

It can be defined that an one-parameter special motion of a body in Lorentzian space H ′ is generated by the transformation

X=Ax0+c (1.5) where A∈SO(3,1) and

x

₀

, X , c

are 3x1 real matrices [1].

2. DARBOUX VECTORS (MATRICES)

Theorem: Let the motion H/H′be represented by the equation (1.4). Then the

components of Darboux vectors of the motion H/H′, respectively, are )

, ,

(kb₁ τt₁ kb₂ τt₂ kb₃ τt₃

ω= − − −

and

) , 0 ,

( k

w= −τ . Proof: From (1.1) we obtain

x₀ =A⁻¹(X −c) (2.1)

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and as A⁻¹=ε A^Tε and det A = + 1, we have from (1.4) ve (2.1), eliminating

x

₀

X′=Ω(X −c)+c' (2.2) with Ω=A^′A⁻¹, or explicitly, by means of (1.2),

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡−

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡−

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

′

= Ω

1 0 0

0 1 0

0 0 1

1 0 0

0 1 0

0 0 1

3 2 1

3 3 3

2 2 2

1 1 1

b b b

n n n

t t t

b n t

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

−

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

′

= Ω

3 2 1

3 3 3

2 2 2

1 1 1

b b b

n n n

t t t

b n t

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

+ ′ + ′

− ′ + ′

+ ′

− ′

′

+ ′ + ′

− ′ + ′

+ ′

− ′

′

+ ′ + ′

− ′ + ′

+ ′

− ′

′

= Ω

3 3 3 3 3 3 2 3 2 3 2 3 1 3 1 3 1 3

3 2 3 2 3 2 2 2 2 2 2 2 1 2 1 2 1 2

3 1 3 1 3 1 2 1 2 1 2 1 1 1 1 1 1 1

b b n n t t b b n n t t b b n n t t

.

One can find as follows

T =N×B=(t₁,t₂,t₃)=n₃b₂−n₂b₃,n₃b₁−n₁b₃,n₁b₂−n₂b₁ (2.3)

B=T×N=(b₁,b₂,b₃)=(n₂t₃−n₃t₂,n₁t₃−n₃t₁,n₂t₁−n₁t₂) (2.4)

1 2 3 1 2 3

1 2 3 1 1 2 2 3 3

1 2 3 1 2 3

( , , ) ( , , )

( , , ) ( , , ).

T t t t kn kn kn

N n n n kt b kt b kt b

B b b b n n n

′= ′ ′ ′ = ⎫

′= ′ ′ ′ = + τ + τ + τ ⎪⎬

′= ′ ′ ′ = −τ −τ −τ ⎪⎭

. (2.5)

By using (2.3), (2.4) and (2.5), we get

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

+ ′ + ′

− ′ + ′

+ ′

− ′

′

′ +

′

−

′ +

′

−

′

−

′

−

′

+ ′ + ′

− ′ + ′

+ ′

− ′

′

= Ω

3 3 3 3 3 3 2 3 2 3 2 3 1 3 1 3 1 3

3 2 3 2 3 2 2 2 2 2 2 2 1 2 1 2 1 2

3 1 3 1 3 1 2 1 2 1 2 1 1 1 1 1 1 1

b b n n t t b b n n t t b b n n t t

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⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− +

+ +

−

− +

+ +

−

+

−

− +

+ +

−

− +

+ +

−

+

−

− +

+ +

−

− +

+ +

−

+

−

= Ω

3 3

3 3 3 3 3 2

3

2 3 3 2 3 1

3

1 3 3 1 3

3 2

3 2 2 3 2 2

2

2 2 2 2 2 1

2

1 2 2 1 2

3 1

3 1 1 3 1 2

1

2 1 1 2 1 1

1

1 1 1 1 1

) (

) ( )

(

) (

) ( )

(

) (

) ( )

(

) (

) ( )

(

) (

) ( )

(

) (

) ( )

(

) (

b n

n b kt t kn b

n

n b kt t kn b

n

n b kt t kn

b n

n b kt t kn b

n

n b kt t kn b

n

n b kt t kn

b n

n b kt t kn b

n

n b kt t kn b

n

n b kt t kn

τ

τ τ

τ

τ τ

τ

τ τ

τ

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

− +

+

−

− +

+ + −

−

− +

+

−

− +

+ + −

−

− +

+

−

− +

+

− +

−

− +

= Ω

3 3 3 3

3 3 3 3 2

3 2 3

2 3 2 1 3

3 1 3

1 3 1 3

3 2 3 2

3 2 3 2 2

2 2 2

2 2 2 1 2

2 1 2

1 2 1 2

3 1 3 1

3 1 3 1 2

1 2 1

2 1 2 1 1

1 1 1

1 1 1 1

b n n b

n t k t n k b

n n b

n t k t n b k

n n b

n t k t kn

b n n b

n t k t n k b

n n b

n t k t n b k

n n b

n t k t kn

b n n b

n t k t n k b

n n b

n t k t n k b

n n b

n t k t kn

τ τ

τ τ τ

τ

τ τ

τ τ τ

τ

τ τ τ

τ τ

τ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

− +

−

− +

−

− +

−

− +

= Ω

0 )

(

) (

) 0 (

) (

) 0 (

2 3 2 3

2 3 2 3 1

3 1 3

1 3 1 3

3 2 3 2

3 2 3 2 1

2 1 2

1 2 1 2

3 1 3 1

3 1 3 1 2

1 2 1

2 1 2 1

n b b n

t n n t k n

b b n

n t t n k

b n n b

n t t n k b

n n b

n t t n k

n b b n

n t t n k n

b b n

t n n t k

τ τ

) 6 . 2 ( ).

, ,

( 0

) (

0 )

(

) (

0

3 3 2 2 1 1 1

1 2 2

1 1 3

3

2 2 3

3

t kb t kb t kb t

kb t kb

t kb t

kb

t kb t

kb

τ τ

τ ω

τ τ

−

=

↔

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

−

= Ω

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Ω is a semi skew–symmetric matrix as Ω = − Ωε ^Tε . Its components with respect to the moving frame follow from w=A⁻¹ω [5], and we obtain the vector

1 1

1 2 3

1 2 3 2 2

1 2 3 3 3

w AT

k b t

t t t

n n n k b t

b b b k b t

= ε ε ω

− τ

− − ⎡ ⎤

⎡ ⎤

⎢ ⎥

= −⎢ ⎥⎢ − τ ⎥

⎢ ⎥

⎢− ⎥ − τ

⎣ ⎦ ⎣ ⎦

2 2 2

1 2 3 1 1 2 2 3 3

1 1 2 2 3 3 1 1 2 2 3 3

2 2 2

1 2 3 1 1 2 2 3 3

( ) ( )

t t t k t b t b t b

k n b n b n b n t n t n t

b b b k t b t b t b

⎡τ − + + − − + + ⎤

⎢ ⎥

=⎢ − + + − τ − + + ⎥

⎢τ − + + − − + + ⎥

⎣ ⎦

[ ]

, ,

( 1) 0 ^T

T T k T B

k N B T N

B B k T B

k τ < > − < >

⎡ ⎤

⎢ ⎥

=⎢ < > −τ < >⎥

⎢τ < > − < >⎥

⎣ ⎦

= τ −

and so

w=(−τ,0,k).

REFERENCES

[1] Bottema, O., and Roth, B. ’’Theoretical Kinematics’’, Lauwerier, H.A., and Koiter, W.T., North-Holland Publ. Com., New York, 301-304 (1979).

[2] Uğurlu, H.H., “On the Geometry Of Timelike Surfaces”, Communications, Ankara University, Faculty Of Sciences Dept. Of Math. Seeries A1,Vol.46 pp.211-223 (1997).

[3] O’Neill, B., ”Semi-Riemannian Geometry, Academic Pres”, New York, 278-292, (1983).

[4] Ekmekçi, N., and İlarslan, K., “Higher Curvatures in Lorentzian Space”, Jour. of Ins.

of Math. and Comp. Sci. (Math. Ser.), Vol.11, No.2, p.97-102, (1998).

[5] Bükcü, B., Cayley Formula and Its Applications in Lorentz Space, Ph.D. Thesis, Ankara University Graduate School and The Natural Science, (2003).

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