Research Article Darboux Vector in Four-Dimensional Space-Time

As the space-time model of the theory of relativity, four-dimensional Minkowski space is the basis of the theoretical framework for the development of the theory of relativity. In this paper, we introduce Darboux vector ﬁ elds in four-dimensional Minkowski space. Using these vector ﬁ elds, we de ﬁ ne some new planes and curves. We ﬁ nd that the new planes are the instantaneous rotation planes of rigid body moving in four-dimensional space-time. In addition, according to some characteristics of Darboux vectors in geometry, we de ﬁ ne some new space curves in four-dimensional space-time and describe them with curvature functions. Finally, we give some examples.


Introduction
On the basis of the principle of relativity and Lorentz transformation, in 1907, Minkowski proposed to add a time dimension on the basis of three space dimensions, thus forming a four-dimensional space-time, and this spacetime is also called Minkowski 4-space. The metric tensor g in E 4 1 is given by where ðx 1 , x 2 , x 3 , x 4 Þ is a standard rectangular coordinate system in E 4 1 . Minkowski space is not only closely related to physics but also provides theoretical and methodological support for the study of astrophysics and cosmology [1][2][3][4]. The study of submanifolds in Minkowski space is of interest in relativity theory; therefore, more and more geometers and physicists are committed to the study of submanifolds in Minkowski space. For example, in [5], the authors studied some local properties of slant geometry on spacelike submanifolds of codimension two in Lorentz-Minkowski space and investigate spacelike curves in Lorentz-Minkowski 3-space from different viewpoints as another special case. In [6], the authors studied null helices of 1-dimensional lightlike submanifolds and gave some characterizations of null helices in ℝ 3 1 . We refer the reader to [7][8][9][10][11][12][13][14][15][16][17] and the references therein for more related works.
The Darboux vector is the local speed vector of the Frenet frame of space curves, which was discovered and named after Gaston Darboux [18]. If an object moves along a regular curve, we can use the Frenet frame of space curves to describe the motion of the object in terms of two vectors: the translation vector and the rotation vector, where the rotation vector is the Darboux vector. Because the Darboux vector is directly related to the angular momentum, it is also called the angular momentum vector.
In the past few decades, many researchers have mainly studied Darboux vectors in 3-dimensional space [19][20][21][22][23][24][25][26] and have obtained some interesting conclusions. For example, in 2012, Ziplar introduced and studied Darboux helices in Euclidean 3-space and proved that Darboux helices coincide with slant helices [19]. In [20], Öztürk and Nešovic ′ defined the pseudo null and null Cartan Darboux helices in Minkowski 3-space and obtained the relationship between pseudo null, null Cartan Darboux helices, and slant helices. In [21], the quasi Darboux vector field of null curve in Minkowski 3-space was defined, and some interesting conclusions about osculating developable of null curve which is defined by quasi Darboux vector field of null curve were obtained. Wang and Pei defined the Darboux vector of the null curve in [23] and described the direction of the rotation axis of the Cartan frame in Minkowski 3-space. Later, in 2017, Düldül [27] extended the Darboux frame field to four-dimensional Euclidean space and gave the relationship between the curvature of Frenet frame and Darboux frame. In [28], Düldül defined some new vector fields in fourdimensional Euclidean space and showed that the determined new planes play the role of the Darboux vector. _ I larslan and Yildirim [29] defined the Darboux helices in four-dimensional Euclidean space as a curve whose Darboux vector makes a constant angle with some fixed direction and obtained relation between the curves Darboux helix, general helix, and V 4 -slant helix in a special case.
Motivated by those ideas, in this paper, we construct four new vector fields along the space curve whose curvatures do not disappear in four-dimensional space-time. Based on these vector fields, we define some new planes and helices in fourdimensional space-time. The corresponding curvature functions are given when the position vectors of the curves lie on different planes. Moreover, we define Darboux helices in Minkowski 4-space and give some descriptions of their curvature functions.
Definition 1 (see [30]). Let γðsÞ be a null curve parameterized by null arc length s (i.e., ∥γ ′ ′ðsÞ∥ = 1) in E 4 1 . Then, γðsÞ can be framed by a Cartan Frenet frame fT, N, B 1 , B 2 g such that where In sequence, T, N, B 1 , B 2 are called the tangent, principal normal, first binormal, and second binormal vector field of γðsÞ and k 2 and k 3 are first curvature and second curvature of the curve γðsÞ, respectively.
Definition 2 (see [16]). LetγðsÞ be a pseudo null curve parameterized by arc length s (i.e., ∥γ′′ðsÞ∥ = 0) in E 4 1 . Then, the Frenet equation is defined by where and k 1 and k 2 are first curvature and second curvature of the curve γðsÞ, respectively.

Darboux Helix and Planes of Null Curve
When the Frenet frame fT, N, B 1 , B 2 g of a nongeodesic null curve makes an instantaneous helix motion in E 4 1 , there exists an axis of the frame's rotation. The direction of such axis is given by the vector From (10), we know that Frenet vectors T and N rotate around the D 1 D 2 plane, and Frenet vectors B 1 and B 2 rotate around the D 3 D 4 plane. We find that the D 1 D 2 plane and D 3 D 4 plane play the role of Darboux vector in threedimensional space. We also note that D 2 and D 3 are Frenet vectors of the null curve, fD 1 , D 2 , D 3 , D 4 g is linearly independent, and D 1 is orthogonal to D 2 and D 4 . We are going to use the subspace spanned by fD 1 , D 2 g and fD 1 , D 4 g to represent D 1 D 2 plane and D 1 D 4 plane, respectively.
Inspired by [10,28], we discuss the situation when the curve γðsÞ lies in D 1 D 2 and D 1 D 4 planes.
and in addition, the curve γðsÞ can be expressed as where c is nonzero constant.
Proof. We may assume that and we take the derivative of (13) according to s, and we obtain Hence, From the second equation of (15), we get and substituting (16) into the first equation of (15), we have Then, the curve γðsÞ can be denoted as From the third equation of (15), we get that is, This ends the proof.

Corollary 4.
In particular, when k 3 = a = constant ≠ 0, we have k 2 = ða 2 /2Þs 2 + b, and the curve γðsÞ can be expressed as where a and b are constants.
and in addition, the curve γðsÞ can be expressed as where c 1 , c 2 , and c 3 are constants.
Proof. Assume that Differentiating equation (24) with respect to s, we have

Advances in Mathematical Physics
Then, we obtain the system of differential equations From the second equation of (26), we get Substituting (27) into the third equation of (26), we have Then, the curve γðsÞ can be denoted as Substituting (28) into the first equation of (26), we have Substituting (27) and (28) into the fourth equation of (26), we can calculate that This ends the proof.
Definition 6. Let γðsÞ: I ⟶ E 4 1 be a null curve with parameterized by null arc length s. If there exists a fixed direction V ≠ 0 such that then the null curve γðsÞ is called the null Darboux helix, and the fixed direction V is called an axis of the null Darboux helix.

Theorem 7.
Let γðsÞ: I ⟶ E 4 1 be a null curve with parameterized by null arc length s. If γðsÞ is a null Darboux helix in E 4 1 whose fixed direction V satisfies then V is given by and the curvature functions k 2 , k 3 satisfy where ξ = and b ∈ ℝ 0 , k 3 ≠ 0, k 2 ≠ const.
Proof. Let γðsÞ: I ⟶ E 4 1 be a null Darboux helix with parameterized by null arc length s. Then, for a fixed direction V satisfying we can assume By using (5), we can obtain Taking the derivative of equation (39) according to s, we obtain Differentiating equation (38) and using the Frenet equation (5), we have u 1′ − k 3 u 4 + k 2 u 2 = 0, By (40), we can obtain Substituting (43) into the fourth equation of (42), we can obtain From (39), (44), and the third equation of (42), we have

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Thus, From the second equation of (42), the relationship between k 2 and k 3 can be expressed as where b is given by the relation (44), and if b = 0, the axis V = 0, which is a contradiction. Hence, b ≠ 0, which completes the proof.

Corollary 8.
In particular, when a = 0, b = 1, we have and the curvature functions k 2 , k 3 satisfy Some examples of null Darboux helix in E 4 1 are given below.
Example 1. Let γðsÞ: I ⟶ E 4 1 be a null curve with the arc length s and the curvature and then, γðsÞ is a Darboux helix whose fixed direction is given by Example 2. Let γðsÞ: I ⟶ E 4 1 be a null curve with the arc length s and the curvature and then, γðsÞ is a Darboux helix whose fixed direction is given by and then, γðsÞ is a Darboux helix whose fixed direction is given by

Darboux Helix and Planes of Pseudo null Curve
When the Frenet frame fT, N, B 1 , B 2 g of a nongeodesic pseudonull curve makes an instantaneous helix motion in E 4 1 , there exists an axis of the frame's rotation. The direction of such axis is given by the vector and we call them the Darboux vectors for the pseudo null curves in E 4 1 . The Darboux vectors satisfy the Darboux equations From (58), we know that Frenet vectors T and B 2 rotate around the D 1 D 2 plane, and Frenet vectors N and B 1 rotate around the D 3 D 4 plane. We find that the D 1 D 2 plane and D 3 D 4 plane play the role of Darboux vector in threedimensional space. We also note that D 1 and D 3 are Frenet vectors of the pseudonull curve, fD 1 , D 2 , D 3 , D 4 g is linearly independent, and D 4 is orthogonal to D 2 and D 3 . We are going to use the subspace spanned by fD 2 , D 4 g and fD 3 , D 4 g to represent D 2 D 4 plane and D 3 D 4 plane, respectively. Similar to Section 3, we discuss the situation when the curve γðsÞ is in D 2 D 4 and D 3 D 4 planes.
and in addition, the curve γðsÞ can be expressed as where c 1 and c 2 are nonzero constants.

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Proof. We may assume that and we take the derivative of (61) according to s, and we obtain Hence, From the second and the third equations of (63), we get where c 1 and c 2 are nonzero constants. Then, the curve γðsÞ can be denoted as Substituting (64) into the fourth equation of (63), we have From (66) and the first equation of (63), we get This ends the proof.
and in addition, the curve can be expressed as where c 1 , c 2 , c 3 are constants.
Proof. Assume that Differentiating equation (70) with respect to s, we have So we obtain the system of differential equations From the first and the third equations of (72), we get Then, the curve γðsÞ can be denoted as Substituting (73) and (74) into the second equation of (72), we have From equations (74) and (76), we can obtain where c 1 , c 2 , c 3 are constants.
Corollary 11. In particular, when c 1 = 1/1, c 2 = c 3 = 0, we have k 2 /k 1 = s 2 : Let k 1 = 3s/ðs 2 + 1Þ 2 and k 2 = 3s 3 /ðs 2 + 1Þ 2 . Then, the curve γðsÞ can be expressed as Definition 12. Let γðsÞ: I ⟶ E 4 1 be a pseudo null curve with parameterized by arc length s. If there exists a fixed direction V ≠ 0 such that then the pseudo null curve γðsÞ is called the pseudo null Darboux helix, and the fixed direction V is called an axis of the pseudo null Darboux helix.  Advances in Mathematical Physics helix in E 4 1 whose fixed direction V satisfies then V is given by and the curvature functions k 1 , k 2 satisfy where and a, b ∈ ℝ, k 1 ≠ 0, k 2 ≠ const.
Proof. Let γðsÞ: I ⟶ E 4 1 be a pseudo null Darboux helix with parameterized by arc length s. Then, for a fixed direction V satisfying we can assume By using (84), we can obtain Taking the derivative of equation (86) according to s, we obtain Differentiating equation (85) and using the Frenet equation (7), we have u 1′ − u 4 = 0, u 1 + u 2′ + k 2 u 3 = 0, k 1 u 2 + u 3′ − k 2 u 4 = 0, u 4′ − k 1 u 3 = 0: Substituting (86) and (87) into the second equation of (89), we can obtain By (90), we can obtain where From (87), (88), and the first equation of (89), we have Thus, From the fourth equation of (89), the relationship between k 1 and k 2 can be expressed as This ends the proof.