A New Method for Inextensible Flows of Timelike Curves in Minkowski Space-Time
Abstract
We construct a new method for inextensible flows of timelike curves in Minkowski space-time . Using the Frenet frame of the given curve, we present partial differential equations. We give some characterizations for curvatures of a timelike curve in Minkowski space-time .
1. Introduction
Numerous processing operations of complex fluids involve free surface deformations; examples include spraying and atomization of fertilizers and pesticides, fiber-spinning operations, paint application, roll-coating of adhesives, and food processing operations such as container- and bottle-filling. Systematically understanding such flows can be extremely difficult because of the large number of different forces that may be involved, including capillarity, viscosity, inertia, gravity, and the additional stresses resulting from the extensional deformation of the microstructure within the fluid. Consequently many free-surface phenomena are described by heuristic and poorly quantified words such as “spinnability,” “tackiness,” and “stringiness.” Additional specialized terms used in other industries include “pituity” in lubricious aqueous coatings, “body” and “length” in the printing ink business, “ropiness” in yogurts, and “long/short textures” in starch processing [1].
The flow of a curve or surface is said to be inextensible if, in the former case, the arc length is preserved, and, in the latter case, if the intrinsic curvature is preserved [2–7]. Physically, inextensible curve and surface flows are characterized by the absence of any strain energy induced from the motion. Kwon investigated inextensible flows of curves and developable surfaces in R3. Necessary and sufficient conditions for an inextensible curve flow first are expressed as a partial differential equation involving the curvature and torsion. Then, they derived the corresponding equations for the inextensible flow of a developable surface and showed that it suffices to describe its evolution in terms of two inextensible curve flows [8]. Additionally, there are many works related with inextensible flows [1, 8–15].
In the past two decades, for the need to explain certain physical phenomena and to solve practical problems, geometers and geometric analysis have begun to deal with curves and surfaces which are subject to various forces and which flow or evolve with time in response to those forces so that the metrics are changing. Now, various geometric flows have become one of the central topics in geometric analysis. Many authors have studied geometric flow problems [1, 12, 16, 17].
This study is organised as follows: firstly, we study inextensible flows of timelike curves in Minkowski space-time. Secondly, using the Frenet frame of the given curve, we present partial differential equations. Finally, we give some characterizations for curvatures of a curve in Minkowski space-time.
2. Preliminaries
A “particle” in special relativity means a curve α with a timelike unitary tangent vector [18, 19].
- (i)
it can be space-like if g(v, v) > 0 or v = 0;
- (ii)
it can be timelike if g(v, v) < 0;
- (iii)
it can be null (light-like) if g(v, v) = 0 and v ≠ 0.
Let α(s) be a timelike curve in the space-time, parameterized by arc length function s.
3. A New Method for Inextensible Flows of Timelike Curves in
Physically, inextensible curve and surface flows give rise to motions in which no strain energy is induced. The swinging motion of a cord of fixed length or, for example, of a piece of paper carried by the wind, can be described by inextensible curve and surface flows. Such motions arise quite naturally in a wide range of physical applications [8, 11, 12].
Let α(u, t) be a one parameter family of smooth timelike curves in .
Definition 1. The flow ∂α/∂t in is said to be inextensible if
Theorem 2. Let ∂α/∂t be a smooth flow of γ. The flow is inextensible if and only if
Proof. Assume that ∂α/∂t is inextensible. Then,
Substituting (8) in (10) completes the proof of the theorem.
We now restrict ourselves to arc length parameterized curves. That is, v = 1 and the local coordinate u corresponds to the curve arc length s. We require the following lemma.
Lemma 3. If the flow is inextensible, then
Proof. Using definition of α, we have
Substituting (9) in (12), we obtain (11). This completes the proof.
Now we give the characterization of evolution of first curvature as below.
Theorem 4. Let ∂α/∂t be inextensible flow of timelike α in . Then, the evolution of k1 is given by
Proof. Assume that ∂α/∂t is inextensible in .
Thus it is easy to obtain that
By the Frenet equations we have
Also,
By the definition of flow, we have
Combining these we have
Thus, we obtain the theorem. This completes the proof.
By this theorem we immediately have the following.
Theorem 5. Consider
Proof. Using Frenet equations, we have
This implies
Theorem 6. Let ∂α/∂t be inextensible flow of α in . Then,
Proof. Assume that ∂α/∂t is inextensible flow of α in . Consider
Thus we compute
Then we can easily see that
From definition of flow, we have
Thus, we obtain the theorem. The proof of theorem is completed.
Now we give the characterization of evolution of second curvature as below.
Theorem 7. The evolution of k2 is given by
Proof. It is obvious from Theorem 6. This completes the proof.
Theorem 8. Let ∂α/∂t be inextensible flow of α in . Then,
Proof. Differentiating (22) with respect to s,
Thus we easily obtain that
Hence, the proof is complete.
Now we give the characterization of evolution of third curvature as below.
Theorem 9. If the flow is inextensible, then
Proof. It is obvious from Theorem 8. This completes the proof.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
