On the Differential Geometry of the Curves in Minkowski Space-Time II

Abstract—In the first part of this paper [6], a method to
determine Frenet apparatus of the space-like curves in Minkowski
space-time is presented. In this work, the mentioned method is
developed for the time-like curves in Minkowski space-time.
Additionally, an example of presented method is illustrated.
Keywords—Frenet Apparatus, Time-like Curves, Minkowski
Space-time.
I. INTRODUCTION
N the case of a differentiable curve, at each point, a tetrad
I 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 curvatures of the curve in four dimensional space
[2]. In [3], regular curves in Euclidean 4-space are studied and
to determine Frenet frame and Frenet elements, a way, which
3
is similar to E , is also given.
At the beginning of the twentieth century, Einstein's theory
opened a door to use of degenerate submanifolds. Then, the
researchers treated some of classical differential geometry
topics and extended to Lorentz manifolds. For instance, in [5],
the author introduces a method to calculate Frenet apparatus
of space-like curves in Minkowksi 4-space according to
signature ( + , + , + , −)
. Thereafter, in [6], in an analogous way,
mentioned method is adapted to space-like curves in
Minkowski space-time (with respect to signature
( −, + , + , + ), e.g.).
In this paper, using vector product expressed as in [6], the
method is developed for the time-like curves in Minkowski
4
space-time E 1 . Moreover, an example of this method will be
provided.
II. PRELIMINARIES
To meet the requirements in the next sections, here, the
4
basic elements of the theory of curves in the space E 1 are
Süha Yılmaz and Melih Turgut are with Department of Mathematics, Buca
Educational Faculty, Dokuz Eylül University, 35160, Buca-Izmir, Turkey.
(Corresponding author, +902324208593, E-mail: melih.turgut@gmail.com)
Emin Özyılmaz is with Department of Mathematics, Faculty of Science,
Ege University, Bornova-Izmir, Turkey. (E-mail:
eminozyilmaz@hotmail.com)
The third author would like to thank TUBITAK-BIDEB for their financial
supports during his Ph.D. studies.
World Academy of Science, Engineering and Technology 35 2009
On the Differential Geometry of the Curves in
Minkowski Space-Time II
Süha Yılmaz, Emin Özyılmaz and Melih Turgut
1029
briefly presented (A more complete elementary treatment can
be found in [2].)
Minkowski space-time
4
E 1 is an Euclidean space
provided with the standard flat metric given by
2
4
E 1
g = −dx1
+ dx2
+ dx3
+ dx4
,
(1)
where x , x , x , x ) is a rectangular coordinate system in
4
1
( 1 2 3 4
3
E . Since g is an indefinite metric, recall that a vector
4
v ∈ E1
r
can have one of the three causal characters; it can be
r r
space-like if g(
v,
v)
〉 0 or v = 0,
r
r r
time-like if g(
v,
v)
〈 0 and
r r
null (light-like) if g(
v,
v)
= 0 and v ≠ 0.
r
Similarly, an
r r
4
arbitrary curve α = α(s)
in E1 can be locally be space-like,
time-like or null (light-like), if all of its velocity vectors α (s)
r ′
are respectively space-like, time-like or null. Also, recall the
norm of a vector v r r r r
r
is given by v = g(
v,
v)
. Therefore, v
r r
is a unit vector if g(
v,
v)
= ± 1.
Next, vectors v w
r r 4
, in E 1 are
said to be orthogonal if g ( v,
w)
= 0.
The velocity of the curve
α r is given by α .
r ′ Thus, a space-like or a time-like curve α r
is said to be parametrized by arclength function s , if
( α ′ , α ′ ) = ± 1.
r r
g The Lorentzian hypersphere of center
r
+
m = ( m1,
m2
, m3
, m4
) and radius r ∈ R in the space 4
E 1
defined by
3 r
4 r r r r 2
S = { α = ( α , α , α , α ) ∈ E : g(
α − m,
α − m)
= r }. (2)
1
1
2
r r r r
Denote by { T s),
N(
s),
B ( s),
B ( s)
}
3
4
( 1 2 the moving Frenet
frame along the curve α r 4
in the space E 1 . Then 1 2 , , , B B N T
r r r r
are, respectively, the tangent, the principal normal, the first
binormal and the second binormal vector fields. For a timelike
unit speed curve α r 4
in E 1 the following Frenet equations
are given in [1], [2]
⎡ &r
T ⎤
⎢ ⎥ ⎡0
⎢ &r
N
⎥ ⎢
⎥ = ⎢
κ
⎢
⎢ &r
⎥ ⎢0
⎢
B1
⎥ ⎢
⎢ &r ⎥ ⎣0
⎣B2
⎦
κ
0
−τ
0
1
2
2
r
0 0⎤
⎡T
⎤
0
⎥
⎢ r ⎥
τ
⎥
⎢N
⎥
,
0
⎢ r
σ ⎥
⎥
⎢B1
⎥
⎥
−σ
0⎦
⎢ r ⎥
⎣B2
⎦
(3)

r
r
where T,
N,
B1
satisfying equations
and B2 r are mutually orthogonal vectors
r r
g(
T,
T ) = −1
r r r r r r
g N,
N)
= g(
B , B ) = g(
B , B ) = 1.
( 1 1 2 2
And here, κ ( s), τ ( s)
and σ (s)
are first, second and third
curvature of the curveα r , respectively.
Suffice it to say that, for an arbitrary parametrized time-like
curve the following Frenet equations are hold:
r
⎡T
′ ⎤
⎡ 0
⎢ r ⎥
⎢
⎢N
′ ⎥
⎢
vκ
⎢ r ⎥ =
⎢
⎢B
0
1′
⎥
⎢
⎢ r ⎥ ⎣ 0
⎣B2′
⎦
vκ
0
− vτ
0
0
vτ
0
− vσ
where v denotes speed of the curve.
r
0 ⎤
⎡T
⎤
0
⎥
⎢ r ⎥
⎥
⎢N
⎥
⎢ r ⎥,
vσ
⎥
⎢B1
⎥
⎥
0 ⎦⎢
r ⎥
⎣B2
⎦
In the same space, the authors, in [6], expressed a vector
product with the following definition.
r
r
Definition 1. Let a = ( a1,
a2
, a3
, a4
), b = ( b1
, b2
, b3
, b4
) and
r
4
c = ( c1,
c2
, c3
, c4
) be vectors in the space E 1 . The vector
product in Minkowski space-time is defined with the
determinant
r
− e1
r
e2
r
e3
r
e4
r r r
a ∧ b ∧ c = −
a1
b1
a2
b2
a3
b3
a4
, (5)
b4
where
c1
c2
c3
c4
r r r
e1
, e2
, e3
and e4 r are coordinate direction vectors
satisfying
r r r r r r r r r r r r
e1
∧ e2
∧ e3
= e4
, e2
∧ e3
∧ e4
= e1,
e3
∧ e4
∧ e1
= e2
r r r r
e ∧ e ∧ e = −e
4
1
III. DETERMINATION OF FRENET APPARATUS OF THE TIME-
LIKE CURVES IN MINKOWSKI SPACE-TIME
r r
4
Let ϕ = ϕ(s)
be an arbitrary time-like curve in E 1 with
curvatures κ , τ and σ for each s∈ I ⊂R.
. If we calculate 1 st ,
2 nd , 3 rd and 4 th order derivatives (with respect to s ) of this
time-like curve, we have, respectively,
vT, r r
ϕ ′ =
(6)
r r r
2
ϕ ′ = v′
T + v κN,
(7)
r
r
r r
3 2
2
3
ϕ ′′′ = ( v′
′ + v κ ) T + ( 3vv′
+ v κ ')
N + ( v κτ)
B1,
(8)
r r r r r
( IV )
4
ϕ = (...) T + (...) N + (...) B1
+ ( v κτσ)
B2
. (9)
Considering (6), we immediately arrive
r
r
ϕ ′
T = .
(10)
v
2
World Academy of Science, Engineering and Technology 35 2009
3.
(4)
1030
From definition of velocity, we may express that, due to timelike
tangent vector
2
( ϕ , ϕ ).
r r
v = −g
′ ′
(11)
Differentiating both sides of (11), we have
r r
g(
ϕ ′ , ϕ ′′ )
v′
= − .
(12)
v
Using (12) in (7) and taking the norm of both sides this
expression, we have, respectively,
r 2 r r r r
ϕ ′ . ϕ ′
+ g(
ϕ ′ , ϕ ′
). ϕ ′
κ = r 4
ϕ ′
(13)
and
r
N =
r 2 r r r r
ϕ ′ . ϕ ′
+ g(
ϕ ′ , ϕ ′
). ϕ ′
r
.
2 r r r r
(14)
ϕ ′ . ϕ ′
+ g(
ϕ ′ , ϕ ′
). ϕ ′
The vector product of T N
r r , and ϕ r ′′ ′ gives us
r r r
−T
N B1
r
B2
r r r
T ∧ N ∧ϕ
′
= −
1
0
0
1
0
0
0
0 (15)
3 2
v ′′ + v κ
r
3
= μ v κτB2
2
3vv′
+ v κ '
3
v κτ 0
where μ is taken ± 1 to make + 1 determinant of
[ 1 2 ] , , , B B N T
r r r r
matrix. By this way, Frenet frame provides
positively oriented.
Taking the norm of both sides of (15) and using (13), we have
r r r r
T ∧ N ∧ϕ
′
ϕ ′
τ =
r
.
2 r r r r
(16)
ϕ′
. ϕ ′
+ g(
ϕ′
, ϕ ′
). ϕ ′
Substituting (16) and (13) into (15), we have second binormal
vector field
r r
r
r
T ∧ N ∧ϕ
′′ ′
B 2 = μ r r r .
(17)
T ∧ N ∧ϕ
′′ ′
r r
( IV )
And using inner product g(
ϕ , B2
) and considering
obtained equations, we obtain the third curvature
r r
( IV )
g(
ϕ , B2
)
σ = r r r r .
(18)
T ∧ N ∧ϕ
′′ ′ ϕ ′
Finally, vector product of N T
r r , and B2 r gives us the first
binormal vector field as
r r r r
B1
= μ N ∧ T ∧ B2
.
(19)
Since, we calculate Frenet apparatus
r r r r
{ T ( s),
N(
s),
B1
( s),
B2
( s),
κ ( s),
τ ( s),
σ ( s)
} of an arbitrary time-
r r
like curve ϕ = ϕ(s)
in Minkowski space-time.
IV. AN EXAMPLE
In this section, we illustrate an example of presented method.

4
E 1
Let us consider the following curve in the space
r r
ψ =ψ ( s ) = ( 2s,
1, sin s,
cos s).
(20)
Differentiating both sides of (20) with respect to s , we have
ψ ′ = ( 2,
0, cos s, − sin s).
r
(21)
The inner product ( ψ ′ , ψ ′ )
r r
g follows that ( ψ ′ , ψ ′ ) = −1,
r r
g
which shows ψ (s)
r
is an unit speed time-like curve. Thus,
ψ ′ = v = 1.
r
Then, we decompose tangent vector of ψ r as
follows:
r
T = ( 2,
0, cos s,
− sin s).
(22)
And, considering equation (14), we have principal normal
vector
r
N = ( 0,
0,
− sin s,
− cos s).
(23)
r r r
Writing vector productT ∧ N ∧ψ
′
, we have
∧ ∧ψ
′′′ = ( 0,
− 2,
0,
0).
r r r
T N
(24)
Since, we have the second binormal vector
B2 = ( 0,
−1,
0,
0).
r
(25)
Using vector product of N T
r r , and 2 Br , we express
r
B1 = ( −1,
0,
2 cos s,
− 2 sin s).
(26)
Finally, using fourth order derivative of ψ (s),
r
(13), (16) and
(18), we write curvatures of the curve, respectively,
κ = 1,
(27)
τ = 2,
(28)
σ = 0.
(29)
r r r r
Corollary 1. Suffice it to say that, { T ( s),
N(
s),
B1
( s),
B2
( s)
} is
4
an orthonormal frame of E .
1
V. CONCLUSION AND FURTHER REMARKS
Throughout the presented paper, we present a method to
calculate all Frenet apparatus of a time-like curve which lies
4
fully in E 1 . Here, using vector product, we give formulas of
frame vectors (and therefore curvatures).
Via this method, some of classical differential geometry
topics can be treated. Relations among spherical indicators,
Bertrand curves and Involute-evolute curve couple may be
easily calculated. We hope these results will be helpful to
mathematicians who are specialized on mathematical
modeling.
REFERENCES
[1] C. Camci, K. Ilarslan and E. Sucurovic, “On pseudohyperbolical curves
in Minkowski space-time”. Turk J.Math. vol. 27, pp. 315-328, 2003.
[2] B. O'Neill, Semi-Riemannian Geometry, Academic Press, New York,
1983.
4
[3] A. Mağden, “Characterizations of Some Special Curves in E ” Ph.D.
dissertation, Dept. Math, Atatürk Univ., Erzurum, Turkey, 1990.
[4] J. Walrave, “Curves and surfaces in Minkowski space” Ph.D.
dissertation, K. U. Leuven, Fac. of Science, Leuven 1995.
World Academy of Science, Engineering and Technology 35 2009
1031
[5] S. Yilmaz. “Spherical Indicators of curves and characterizations of some
special curves in four dimensional Lorentzian space
4
L ” Ph.D.
dissertation, Dept. Math, Dokuz Eylül Univ., İzmir, Turkey, 2001.
[6] S. Yilmaz and M. Turgut, “On the differential geometry of the curves in
Minkowski space-time I”. Int. J. Contemp. Math. Sci. vol.3 no. 27 pp.
1343-1349, 2008.