Darboux Vector in Four-Dimensional Space-Time

Hu, Na; Zhang, Tingting; Jiang, Yang

doi:https://doi.org/10.1155/2022/9044567

Advances in Mathematical Physics

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Theory and Applications of Riemannian Submersions

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Research Article | Open Access

Volume 2022 | Article ID 9044567 | https://doi.org/10.1155/2022/9044567

Darboux Vector in Four-Dimensional Space-Time

Na Hu,¹Tingting Zhang,¹and Yang Jiang²

Academic Editor: Mehmet Atçeken

Received08 Feb 2022

Accepted25 Mar 2022

Published12 Apr 2022

Abstract

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 fields in four-dimensional Minkowski space. Using these vector fields, we define some new planes and curves. We find 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 define some new space curves in four-dimensional space-time and describe them with curvature functions. Finally, we give some examples.

1. 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 space-time is also called Minkowski 4-space. The metric tensor in is given by where is a standard rectangular coordinate system in . Minkowski space is not only closely related to physics but also provides theoretical and methodological support for the study of astrophysics and cosmology [1–4]. 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 . We refer the reader to [7–17] 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 [19–26] 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 Nešovi 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 four-dimensional Euclidean space and showed that the determined new planes play the role of the Darboux vector. 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 -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 four-dimensional 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.

2. Preliminaries

Four-dimensional space-time is the real four-dimensional vector space equipped with the standard flat metric given by

For any three vectors , , and in , their exterior product is given by where is an orthogonal basis in , that is,

A vector in is called spacelike, timelike, or null (lightlike), if or , , respectively. In particular, the vector is said to be spacelike. A curve is called spacelike, timelike, or null (lightlike) if all of its velocity vectors satisfy , , or , respectively. The norm of a vector in is given by [15].

Definition 1 (see [30]). Let be a null curve parameterized by null arc length (i.e., ) in . Then, can be framed by a Cartan Frenet frame such that where

In sequence, are called the tangent, principal normal, first binormal, and second binormal vector field of and and are first curvature and second curvature of the curve , respectively.

Definition 2 (see [16]). Let be a pseudo null curve parameterized by arc length (i.e., ) in . Then, the Frenet equation is defined by where and and are first curvature and second curvature of the curve , respectively.

3. Darboux Helix and Planes of Null Curve

When the Frenet frame of a nongeodesic null curve makes an instantaneous helix motion in , 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 null curves in . The Darboux vectors satisfy the Darboux equations

From (10), we know that Frenet vectors and rotate around the plane, and Frenet vectors and rotate around the plane. We find that the plane and plane play the role of Darboux vector in three-dimensional space. We also note that and are Frenet vectors of the null curve, is linearly independent, and is orthogonal to and . We are going to use the subspace spanned by and to represent plane and plane, respectively.

Inspired by [10, 28], we discuss the situation when the curve lies in and planes.

Theorem 3. Let be a null curve parameterized by null arc length s. are the curvature functions of the null curve . If lies in plane, then the curvature functions satisfy and in addition, the curve can be expressed as where is nonzero constant.

Proof. We may assume that and we take the derivative of (13) according to , 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 can be denoted as From the third equation of (15), we get that is, This ends the proof.

Corollary 4. In particular, when , we have , and the curve can be expressed as where and are constants.

Theorem 5. Let be a null curve parameterized by null arc length s. are the curvature functions of the pseudonull curve . If the lies in plane, then the curvature functions satisfy and in addition, the curve can be expressed as where , , and are constants.

Proof. Assume that Differentiating equation (24) with respect to , we have 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 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 be a null curve with parameterized by null arc length . If there exists a fixed direction such that then the null curve is called the null Darboux helix, and the fixed direction is called an axis of the null Darboux helix.

Theorem 7. Let be a null curve with parameterized by null arc length . If is a null Darboux helix in whose fixed direction V satisfies then is given by and the curvature functions satisfy where and .

Proof. Let be a null Darboux helix with parameterized by null arc length . Then, for a fixed direction satisfying we can assume By using (5), we can obtain Taking the derivative of equation (39) according to , we obtain Differentiating equation (38) and using the Frenet equation (5), we have 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 Thus, From the second equation of (42), the relationship between and can be expressed as where is given by the relation (44), and if , the axis , which is a contradiction. Hence, , which completes the proof.

Corollary 8. In particular, when , we have and the curvature functions satisfy where

Some examples of null Darboux helix in are given below.

Example 1. Let be a null curve with the arc length and the curvature and then, is a Darboux helix whose fixed direction is given by

Example 2. Let be a null curve with the arc length and the curvature and then, is a Darboux helix whose fixed direction is given by

Example 3. Let be a null curve with the arc length and the curvature and then, is a Darboux helix whose fixed direction is given by

4. Darboux Helix and Planes of Pseudo null Curve

When the Frenet frame of a nongeodesic pseudonull curve makes an instantaneous helix motion in , 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 . The Darboux vectors satisfy the Darboux equations

From (58), we know that Frenet vectors and rotate around the plane, and Frenet vectors and rotate around the plane. We find that the plane and plane play the role of Darboux vector in three-dimensional space. We also note that and are Frenet vectors of the pseudonull curve, is linearly independent, and is orthogonal to and . We are going to use the subspace spanned by and to represent plane and plane, respectively.

Similar to Section 3, we discuss the situation when the curve is in and planes.

Theorem 9. Let be a pseudonull curve with parameterized by arc length . are the curvature functions of the pseudonull curve . If lies in plane, then the curvature functions satisfy and in addition, the curve can be expressed as where and are nonzero constants.

Proof. We may assume that and we take the derivative of (61) according to , and we obtain Hence, From the second and the third equations of (63), we get where and are nonzero constants.
Then, the curve 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.

Theorem 10. Let be a pseudonull curve with parameterized by arc length s. are the curvature functions of the pseudonull curve . If the lies in plane, then the curvature functions satisfy and in addition, the curve can be expressed as where are constants.

Proof. Assume that Differentiating equation (70) with respect to , we have So we obtain the system of differential equations From the first and the third equations of (72), we get Then, the curve 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 are constants.

Corollary 11. In particular, when , we have Let and . Then, the curve can be expressed as

Definition 12. Let be a pseudo null curve with parameterized by arc length . If there exists a fixed direction such that then the pseudo null curve is called the pseudo null Darboux helix, and the fixed direction is called an axis of the pseudo null Darboux helix.

Theorem 13. Let be a pseudonull curve with parameterized by arc length . If is a pseudonull Darboux helix in whose fixed direction satisfies then is given by and the curvature functions satisfy where and .

Proof. Let be a pseudo null Darboux helix with parameterized by arc length . Then, for a fixed direction satisfying we can assume By using (84), we can obtain Taking the derivative of equation (86) according to , we obtain Differentiating equation (85) and using the Frenet equation (7), we have 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 and can be expressed as This ends the proof.

Corollary 14. In particular, when , we have and the curvature functions satisfy where

Some examples of pseudo null Darboux helix in are given below.

Example 4. Let be a pseudo null curve, and is the pseudoarc length. The curvature function satisfies and then, is a Darboux helix whose fixed direction is given by

Example 5. Let be a pseudo null curve with the arc length and the curvature and then, is a Darboux helix whose fixed direction is given by

Example 6. Let be a pseudo null curve with the arc length and the curvature and then, is a Darboux helix whose fixed direction is given by

5. Conclusion

In this paper, we discuss some new space curves and planes in four-dimensional space-time and give characterizations of them in terms of the curvature functions. Before this study, most researchers studied Darboux vector in three-dimensional space and four-dimensional Euclidean space. In this paper, the Darboux vector fields in three-dimensional space are extended to four-dimensional space-time by mathematical method. By defining Darboux vector fields in four-dimensional space-time in the form of vector products, we find that the Frenet vectors rotate around a plane spanned by two new vector fields, and this plane plays the role that the Darboux vector plays in three-dimensional space. This paper gives a new description of Darboux vector in four-dimensional space-time, which promotes the further development of angular momentum vector in physics and geometry.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

The authors have made the same contribution.

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Copyright © 2022 Na Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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