The Equations of Motion of a Null Curve in Lightlike Cone of 3-Dimensonal Minkowski Space

3 (2), 2009, 161 - 164

©BEYKENT UNIVERSITY

The Equations of Motion of a Null Curve in

Lightlike Cone of 3-Dimensonal Minkowski Space

Mehmet ERDOĞAN, Jeta ALO and Gülşen YILMAZ

Beykent University, Faculty of Science and Letters, Department of Mathematics, Ayazağa/Şişli, Îstanbul,TURKEY

e-mail: erdoganm@beykent.edu.tr

Received: 30.09.2009, Accepted: 15.10.2009

Abstract.

In this article we investigate null curves in the lightlike cone of 3-dimensional Lorentz-Minkowski space and give some characterizations of these curves.We also obtain the equations of motion of a null curve in the lightlike cone. Mathematics Subject Classificiation : 53B30, 53B50, 53C80

Keywords: null curve, the lightlike cone, the motion of a null particle 3-dimensional Lorentz-Minkowski space

Özet.

Bu makalede 3-boyutlu Lorentz-Minkowski uzayının ışık konisindeki ışıksal eğriler incelendi ve bazı karekterizasyonları verildi. Ayrıca ışık konisindeki bir ışıksal eğrinin hareket denklemleri elde edildi.

Anahtar kelimeler: Işıksal eğri, ışık konisi, 3-boyutlu Lorentz-Minkowski uzayındaki bir ışıksal parçacığın hareket denklemleri

1. Introduction

There exist three families of curves of a proper semi-Riemannian manifold

Mn ( that is the index i of the manifold satisfies 1 < i < dim M -1) with a

semi-Riemannian metric g called spacelike, timelike and null (lightlike) depending on their causal characters.When we study null curves it occurs some difficulties because the arc lentgh vanishes so that it is impossible to normalize the tangent vector in the usual manner. Therefore we introduce a new parameter called pseudo-arc which normalize the derivative of the tangent vector [1].

Minkowski Space

g

(

X

,

Y

) = <

x y

> =

x ^

+

x y

2 - x3 y 3, (1.1)

which is called a Lorentz product. Furthermore, a Lorentz cross product

X X Y is given by

X X Y

=

( x3 y2

-

x2 y 3, x3 y1

-

% x^ 2

-

(1.2)

The pseudo- Riemannian lightlike cone in Ej3 is defined by

C2

= { x

e

E? : g ( x, x)

=

0}.

(1.3)

Now, suppose that E13 be the 3-dimensional Minkowski space and C 2 the

lightlike cone in E13. A vector x ^ 0 in E13 is called spacelike, timelike or

null (lightlike), if g(x, x) > 0 , g(x, x) < 0 , g(x, x) = 0 , respectively. A frame field {e1, e2, e3} on E13 is called an asymptotic orthonormal frame

field , if

<e

1

,e

2

> = < e

1

, e

3

> = < e

2

, e

2

> = < e

3

, e

3

> = 0 , <e

1

,e

1

> = < e

2

, e

3

> = 1

A frame field {e1, e2, e3} on E13 is called a pseudo orthonormal frame field ,

if

<e

1

,e

1

> = < e

2

, e

2

> = - < e

3

, e

3

> = 1 , <e

1

,e

2

>=<e

1

,e

3

> = < e

2

, e

3

> = 0.

Definition 1.1. A curve a ( t ) in E3 is called a Cartan curve, if for all t e I,

the vector fields a ( t ), a '(t ), a ''(t ) are linearly independent and the vector fields

a(t),

a

'(t),

a

''(t),

a

'''(t)

are linearly dependent, where

a(")(t) =

^

.

V 7

dt

n

Let a : I — C2 C E13 be a null curve on the 3-dimensional Lorentzian space

E3. Since < a ( t ) , a ( t ) > = 0 and < a ( t ) , da(t) > = 0 , da(t) is

spacelike. Then the induced arc length s of the curve a ( t ) can be defined by

ds2 = < da(t), d a ( t ) > . If we take the arc length s of the curve a ( t ) as

the parameter and denote a ( s ) = a ( t ( s)) we have a ' = a '(s) and < a '(s), a '(s) > = 1, then s is called the pseudo-arc parameter. Hence, the

1/ \ 1/ \

da

curve a (s) is a unit spacelike curve and a (s) = — is a spacelike unit

ds

a '(s) = X = X ( s ) , Y = Y(s), a(s)=Z = Z (s) along the curve a ( s )

satisfying the conditions :

<

Y, Y

>=<

Z, Z

> = <

X, Z

> = <

X, Y

>=

0,

<

X, X

> = <

Y, Z

> =

1.

Therefore, for the vector fields above we have the following Frenet formulas: V X X = Z , VXY = - k ( S ) Z , VXZ = K(S)Z - Y .

A curve defined as above is called a Cartan framed null curve.

Moreover, if the functions k1 (t) and k2 (t) are positive constants on the curve s ( t ) , then we call the curve a Cartan framed null curve with positive

constant curvatures. In this definition, if k2 (t) ° 0 , then the curve is named as a generalized null cubic [2,3 ]. On the other hand, it is well known that for any constant curvature functions k1 (t) and k2 (t) at a point p of the manifold there exists only one Cartan framed null curve with constant curvatures

k1 (t) and k2 (t) passing through p with velocity vector S ' ( p ) = X(p) and satisfying above conditions [4].

2. Lightlike cone C of the Minkowski space M

1

First, we will consider the lightlike cone C2 of the Minkowski space M\

which is, indeed, a lightlike surface of M13. As for any p e C2, TpC2 is a

plane of the M\ , we consider

T

_{_ p \ / I p p p> p' > p p} p

(C

2

)

1

={Vp e TpE

3

: g(Vp,W

p

) = 0, "W

p

e T

p

C

2

}, (2.1)

and

= T

C2

O T '

1

2

RadT

p

C

2

= T

p

C

2

n T

p

( C

2

) \ (2.2)

For C is the lightlike cone of the Minkowski space M j , we have

RadT

p_p

C

2 at any

pe C

2 and the semi-Riemannian metric

g

on

M\ induces on C2 a symmetric tensor field gp ( Xp, Yp ) for any p e C2

In this case we know that g has constant rank 1 on C2 and

T (C y = U eC2 Tp(C ) is a distribution onC , see [4,5].

Now, let us consider a complementary vector bundle S(TC2) of T(C2)1 in TC2, namely

Minkowski Space

As each S(TC2) is a screen subspace of TpC2, we will call S(TC2) the screen distribution on C2. Since C2 is paracompact there always exist

S(TC2). Moreover, for a lightlike vector space W any complementary

subspace to

RadW

is non-degenerate, so it follows that

S(TC

2

)

is a

non-degenerate distribution. Thus along C2 we have the following decomposition

7Mj

3

|

c2

= S (TC

2

) 1 S (TC

2

)

1

, (2.4)

where S(TC2)1 is orthogonal complementary vector bundle to S(TC2) in

ZMj31 2. On the other hand , since C2 is a lighlike surface of M3 then there

exists a unique vector bundle tr (TC2) of rank 1 over C2, such that for any

non-zero section £ of (TC2)1 on a coordinate neighborhood of C2, there

exists a unique section N of tr (TC2) defined on the coordinate neighborhood

of C2 and satisfying

g(N,£) = 1, g(N,N) = g(N,W) = 0, "We T(S(TC

2

)). (2.5)

Then we may show that the following section satisfies ;

N = ' { v - i S ^ U (2.6)

g(x, v) 1 2 g ( £ , v r j '

( J

where, V ^ 0 is a vector which belongs to the complementary vector bundle of T (C2)1 in S (TC2)1. Hence, it follows from (2.5) that tr (TC2) is a

lightlike vector bundle such that

tr(TC

2

) n T

p

C

2

= {0}

for any

pe

C2.

Moreover, from (2.3) and (2.4) we have the following decompositions of

TM

3_{1 IC}

|

2 : ₂

TW

3

|

2

= S (TC

2

) 1 (T(C

2

)

1

© tr (TC

2

))

C V ; (2.7)

=TC

2

© tr (TC

2

)

Therefore, for any screen distribution S(TC2) we have a unique tr (TC2) which is complementary vector bundle to TC2 in Z M3a n d satisfies (2.5).

to S(TC2). It is known that any screen distribution is non-degenerate of

constant index, in particular, any screen distribution on a lightlike hypersurface of a Lorentz manifold is Riemannian, i.e., the induced metric on S(TC2) is

positive definite. So, S (TC2) is also a Riemannian one.

Now , let us consider x e C2 c M\ and x = ( x1, x2, x3) ^ 0 . The lightlike

cone C2 of M13 is given by the equation

2

_

2

_

2

=

0

x3 - x1 - x2 = 0

and it is a lightlike surface. In order to prove this, we define

D = {(u

1

,u

3

)e M

2

: u

32

_u* > 0}

and the local immersion j of C2 from D into M ^ by

V

2

_

2

x3 x1

.

Thus, the tangent bundle TC2 on j(D) is spanned by

d d

x

3

d d d x d

- -

+

and - 1

du

3

dx

3

x

2

dx

2

du

1

ax

1

x

2

dx

2 Then, we may easily have that

a a a a a

£ = x

1

— + x

3

- — = x^ — + x^ — + x

3

—

(2.8)

du

1 3

du

3 1

3x

1 2

dx

2 3

dx

3

is orthogonal to TC2 at any point of j(D) . Another local immersion of D

into M2 for points with x2 < 0 is given by

V

2

_

2

X3

x .

Then, £ given by (2.8) is also orthogonal to TCj2 at all points of j(D) . For

each immersion £ is given as in the (2.8), so (TC2)1 is globally spanned by

the position vector field on C2.

Next, we consider

Ar

1

f a a a 1

N

=

< -x

3

+

x

1

—

+

x

2

\

(2.9)

Minkowski Space

globally defined on C2. As g(N, N ) = 0 and g(N, £) = 1, consider the

lightlike transversal vector bundle tr (TC2) spanned by N .Then the

corresponding screen distribution S(TC2 ) is represented by vector fields

a

X

=

Z

X

satisfying i=1

a x

x3 X3 = 0, x X1 + x2 X2 = 0 (2.10)

at points of C2 and the integral curves of x are open sets of lightlike rays of

C

2

.

By the investigations above, we see that C2 is a lightlike hypersurface whose T ( C2)1 is globally spanned by the position vector field.

Now, let the curve s(t) be a null curve in the lightlike cone C2 in Ej3

which preserves its causal character. Then, all its tangent vectors and all other vectors in C2 are null . Thus, we have £ given by equation (2.8).

Furthermore, we may prove the following result for the lightlike cone.

Theorem 2.1 ,[4]. The lightlike cone C2 of M3 is a totally umbilical lighlike

hypersurface.

Now, let us consider the lightlike cone C2 of M13 given by the equations

X

1 =

f

(

U,

V),

x

2 =

f

2

( u , V

X

X

3 =

f

3

( u , r a n k

where 1 £ i, j £ 2 and ux = u, u2 = v and define

K

du,

= 2,

df

2

df

df

df\

df

df

a

=

du

du

,

D2

=

du du

,

D3

=

du du

1 df2

df

dfx

df

3

df

df

dv

dv

dv

dv

dv

dv

where since C2 is lightlike we have D32 = D12 + D22. In this case the

distribution RadTC2 = (TC2)1 is locally spanned by e

_ a _ a _ a

£

=

D

₃3

—

+

₁

D D

₂ 2

— .

a x ax, ax,,

We may easily see that X belongs to

r ( (TC

2

)

1

)

by taking the natural frame field as

d = f d

i = 1,2.

du

i

du

i

dx. '

and then C2 is lightlike if and only if g(X, X) = 0 , which is equivalent with

D

32

= D

12

+ D

22

.

If we define a local section of TMf by V = _ D3 , we get

a

dx

3 2

g (V ,X)=d

3

.

On the other hand the vector bundle spanned by

- D j H x }

is called the canonical lightlike transversal vector bundle of C2 and

g

f

_I

d - ,

1

] = 0 .

dt

d t )

Lets consider the tangent bundle of s(t) and denote it by Ts, then its normal bundle manifold is defined by

d

(Ts)

1

= {X e r ( C

2

) : g(X, V) = 0}, V ° - (2.11)

\1

and its dimension is

dim(Ts) = 2.

Furthermore, null curves satisfy the followings:

1) ( T s )1 is a null bundle subspace of C2 c E13.

2)

(Ts) n ( T s )

1

= Ts® Ts© (Ts)

1

* TE^ » E

13

.

3. The equations of motion of null curves in the lightlike cone

In this section, we will investigate null curves in the lightlike cone C2 c E3

Now,

s:I ®

C2

c E

3

be a null Cartan curve such that {s',s'',s'''} is positively oriented for all

s e I , s being the pseudo-arc parameter, with Cartan frame

Minkowski Space

<s,s> = < X, X > = < Z, Z > = 0 and <Y, Y > = 1 , < X, Z > = - 1 (3.1)

and therefore, <s, X> = <s, Y > = < X , Y > = <Y, Z > = 0 . The Cartan

equations are given by

X ' = Y,

Y' = -kX + Z, (3.2)

Z' = -kY,

where the prime sign denotes covariant derivative and k is the Cartan curvature of the curve. The fundamental theorem for null curves tells us that k determines completely the null curve parametrized by the pseudo-arc length parameter up to Lorentzian transformations. Then any local geometrical scalar defined along null curves can always be expressed as a function of its curvature and derivatives. For an arbitrary parameter t of the curve s(t), the cone curvature function k is given by, [6],

Ids d

2

s\

2

Id

2

s d

2

s\jds ds\

\ dt ' dt

2

\ dt

2 ,

dt

2

\ dt ' dt

k (t) = ^ ^ ^ - (3.3)

/ds ds\

\ dt ' dt j

In this part we consider mechanical systems with Lagrangians which linearly depend on the curvature of a null curve. This curvature function is sometimes called torsion since it is obtained from the third derivative of the relativistic null path. The space of elementary fields in this theory is the set W of null Cartan curves in the lightlike cone C2 C E13. For the sake of simplicity S

will also denote a variation of s by null curves

s

= s(s,e):[a,b]x(-d,d) ® C

2

C E

3

, (3.4)

where s(s, 0) is the reparametrization of S. Associated with such a variation is the variational vector field V(s) = V(s, 0), where

a s

V = V(s, e) = (s, e). Let h be the differentiable function satisfying

ae

a s

(s, e) = h(s, e)L( s, e). Then we will write down

as

s(s, e), k (s, e), V (s, e) for the corresponding pseudo-arc length parameter. We consider the action F : W ® M given by

F(s) = f (A+mk(s))ds, (3.5)

Js

A and m both being constant. Describing of the simplest action of a particle

is achieved when it is proportional to the pseudo-arc length parameter (i.e.

m = 0 ), which has been studied by Nersesian and Ramos in [7,8].

Now, a null curve S is said to be a critical point of the action F when

d ^ . d

INI

I (A + mk)ds = 0 (3.6)

Js„

de

F

S e ) =

d

de

e=0

for all variation throught null curves of the variation of S. To compute the first-order variation of this action, along the elementary field space W, and so the field equations describing the dynamics of the particle, we use a standard argument involving some integrations by parts. Then by using the Cartan equations we have b

F'(0) = [A]|b - c f ( V , ( m k " ' + 3mkk-Ak')X>ds (3.7)

a where

A = -cmX (h) + 2cm

(V

2X

V, Z > + c(mk + A)<V

X

V, Y > - c(V, V

X

((mk + A)Y )>

+2c ^ 1 mk"- Ak + 2mZ

( X ,

V > + 2c mk

( X ,

Z

>,

V standing for a generic variational vector field along S and h = -(V2XV, Z>, where V is the Levi-Civita connection on El.

We take curves with the same endpoints and having the same Cartan frame in them, so that [A]| vanishes, [9]. Under these conditions, the first-order

a variation is

b

F '(0) = - c f ( V ,(mk"'+ 3mkk '-Ak') X> ds, (3.8)

a

from which we obtain the following theorem

Theorem 3.1 The trajectory S is in the null space of 3-dimensional spacetime

E13 if and only if Y , Z and k are well defined on the whole null space and the following differential equation is satisfied

Minkowski Space

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