On the Fermi–Walker Derivative for Inextensible Flows

1 Introduction

A large class of reference frames in space-time experience inertial forces and therefore constitute noninertial frames. For instance, the frame adapted to an observer “at rest” on the surface of the (rotating) Earth is noninertial. Observers that follow arbitrary timelike trajectories in space-time will regard as natural a reference frame in which they are at rest, and their spatial axes do not rotate. Fermi–Walker transported frames are important in several investigations. A frame that undergoes linear and rotational acceleration may be described by the Frenet–Serret frame. In [14], authors considered tetrad fields as reference frames adapted to observers that move along arbitrary timelike trajectories in space-time. By means of a local Lorentz transformation, they transformed these frames into Fermi–Walker transported frames. Moreover, they presented a simple prescription for the construction of Fermi–Walker transported frames out of an arbitrary set of tetrad fields.

The Frenet–Serret formalism championed by Synge for timelike world lines [17] is a powerful tool for studying the motion of nonzero rest mass test particles in a given gravitational field. This approach depends only on the geometrical (i.e., intrinsic) properties of the timelike world line of the test particle, independent of any particular coordinate system or observer. In particular, it has been proven to be very useful in understanding the properties of timelike circular orbits in stationary axisymmetric space-times and of Fermi–Walker transport along them, and it helps visualise the geometry of this family of orbits. The Frenet–Serret curve analysis has been extended from nonnull to null trajectories in a generic space-time using the Newman–Penrose formalism associated Fermi–Walker transport [4].

The theory of relativity implies that the set of all Lorentz transformations (which form a transformation group) classify all physical quantities as vectors or tensors transforming as representations of this group. This means that physical quantities occur in subsets or ‘multiplets’. This unification of space-time and consequent ordering of all physical quantities as four-dimensional vectors or tensors (and later spinors in quantum theory) was the single most important conceptual advance in physics after Newton’s mechanics and the Faraday–Maxwell theory of fields [15].

Flows have received considerable recent attention because of its relatively simple geometry and its relevance to a large variety of engineering applications [19]. It enables direct insight into fundamental turbulence physics, as well as direct verification by local (such as profiles of mean velocity, Reynolds stress, fluctuation intensities, etc.) and integral measurements (such as skin friction and heat transfer). The fundamental challenge is to predict the mean flow properties, including the mean velocity profile (MVP), the Reynolds stress, the kinetic energy, etc.; however, a deductive theory of this kind is still missing [16, 20].

In the literature, there are many studies about the Fermi–Walker derivative and inextensible flows. However, these two concepts are unsought together until now. In this study, we give the relationship between these two concepts with figures.

The structure of the paper is as follows: First, we construct a new characterisation for inextensible flows of curves by using the Fermi–Walker derivative and the Fermi–Walker parallelism in space. Using the Frenet frame of the given curve, we present partial differential equations. Finally, we give some characterisations for the nonrotating frame of a curve in space.

2 Preliminaries

In this section, we will adopt the notation of [15] for the vector quantities. Let C: I → M be a curve, that is, a differentiable mapping from an interval I on the real line into space M. Choose natural coordinates and basis. Let T(s)=T^μ∂/∂x^μ|_C(s) be the tangent vector to the curve.

Let p=C(s) be a point on the curve, and let q=C(s + Δs) be a neighbouring point on it. Let v^μ be four real numbers chosen as the components of a vector at p. If x^μ and x^μ + Δx^μ are coordinates of p and q, respectively, define v||μ = vμ − ΓνσμΔxνvσ as the components of the vector parallel transported to q [15].

By repeating this process, starting with a single vector at a point p on the curve, we can define vectors all along the points on the image of the curve by parallel transporting [15]. Let us write v^μ(s) as components of the parallel displaced vectors. Then as Δs → 0, the ratio Δx^μ/Δs tends to T^μ(s); hence,

dvμds + ΓνσμTν(s)vσ(s) = 0.

Thus, the parallel transport components are determined by the ordinary differential equation, which has a unique solution given the initial value of v^μ at a fixed value of s.

The vector fields V=v^μ(s)∂/∂x^μ|_C(s) and T=T^μ(s)∂/∂x^μ|_C(s) are defined only on the point of the curve. If the fields were defined everywhere in the neighbourhood, we would have written the left-hand side of the above equation as

DTV = dvμds + ΓνσμTν(s)vσ(s), = Tν(∂∂xνvμ(x(s)) + Γνσμvσ(s)), = TνV;νμ.

We will write the above equation for parallel transport as D_TV=0 with this understanding, that the fields in question are defined only along the curve [15].

A geodesic curve is one for which the parallel transport of the tangent vector is proportional to itself

DTT = f(s)T,

and if we choose the parameter s defining the curve as an affine parameter, then

DTT = 0 for  a  geodesic  with  an  affine  parameter  [15].

3 Construction of the Fermi–Walker Derivative for Inextensible Flows of Curves

In this section, we study the relationship between the Fermi–Walker derivative and the inextensible flows of curves. Moreover, we obtain some characterisations and example of the curve.

Let α: I ⊂ R → E³ be a curve in space and V be a vector field along the curve α. One can take a variation of α in the direction of V, say, a map: Γ: I×(−ε, ε) → E³, which satisfies Γ(s, 0) = α(s), (∂Γ∂s(s, t)) = V(s).

In this setting [2], we have the following functions:

1. The speed function v(s, t) = ‖∂Γ∂s(s, t)‖.

2. The curvature function κ(s, t) of α_t(s).

3. The torsion function τ(s, t) of α_t(s).

The variations of those functions at t=0 are

V(v) = (∂v∂t(s, t))|t = 0 = g(DTV, T)v,V(κ) = (∂κ∂t(s, t))|t = 0 = g(DT2V, N) − 2κg(DTV, T) + g(R(V, T)T, N),V(τ) = (∂τ∂t(s, t))|t = 0 = [1κg(DT2V + R(V, T)T, B]s + κ(DTV, B)+τg(DTV, T) + g(R(V, T)N, B),

where R is the curvature tensor of 𝔼³.

Any flow of α_t(s) can be represented as

∂αt∂t = ϖ1T + ϖ2N + ϖ3B,

where ϖ₁, ϖ₂, and ϖ₃ are smooth functions [8–13].

Definition 3.1.Xis any vector field and α_t(s) is the unit speed of any curve in space.

(1)∇˜X∇˜s = dXds − 〈T, X〉A + 〈A, X〉T, (1)

defined as∇˜X∇˜sderivative is called the Fermi–Walker derivative. HereT = dαds,A = dTds [1, 3, 5, 6].

α_t(s) is the unit speed on any curve and {T, N, B} is the Frenet frame of α_t(s). Then

∇˜T∇˜s = ω∗ ∧ T,∇˜N∇˜s = ω∗ ∧ N,∇˜B∇˜s = ω∗ ∧ B,

where ω* is the Fermi–Walker term Darboux vector according to the Frenet frame [1, 3].

X is any vector field along the α_t(s) space curve. If the Fermi–Walker derivative of the vector field X

(2)∇˜X∇˜s = 0, (2)

the vector field X along the curve, parallel to the Fermi–Walker terms, is called [1, 3].

Definition 3.2.The flow∂αt∂tin space are said to be inextensible if [12]

∂v∂t = 0.

Lemma 3.3.

∂T∂t = [ϖ1κ + ∂ϖ2∂s − ϖ3τ]N + [∂ϖ3∂s + ϖ2τ]B,∂N∂t = −[ϖ1κ + ∂ϖ2∂s − ϖ3τ]T + 1κ[∂∂s[∂ϖ3∂s + ϖ2τ]+τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]B,∂B∂t = 1τ[∂κ∂t − ∂∂s[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]T − [[1κ[∂∂s[∂ϖ3∂s + ϖ2τ]+τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]]N,

whereϖ₁, ϖ₂, and ϖ₃are smooth functions of time and arc length [7].

Theorem 3.4.

∇˜∇˜s∂T∂t = [κ[ϖ1κ + ∂ϖ2∂s − ϖ3τ] − κ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]T+[∂∂s[ϖ1κ + ∂ϖ2∂s − ϖ3τ] − τ[∂ϖ3∂s + ϖ2τ]]N+[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]B,∇˜∇˜s∂N∂t = −∂∂s[ϖ1κ + ∂ϖ2∂s − ϖ3τ]T−[τ[1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]]N+∂∂s[1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]B,

∇˜∇˜s∂B∂t = [−κ[[1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]]+∂∂s[1τ[∂κ∂t − ∂∂s[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]+κ[[1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]]]T+ [−κτ[∂κ∂t − ∂∂s[ϖ1κ + ∂ϖ2∂s−ϖ3τ]] + κ[1τ[∂κ∂t − ∂∂s[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]−∂∂s[[1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + [τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]]]N−τ[[1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]]B,

where ϖ₁, ϖ₂, and ϖ₃are smooth functions of time and arc length.

Proof. By using

(3)∇˜X∇˜s = dXds−κ(B ∧ X). (3)

Equations (1) and (3) give us theorem. This completes the proof.

Using above theorem, we get the following corollaries by straightforward computations.

Corollary 3.5.If∂T∂talong the curve is parallel to the Fermi–Walker terms, then

[τ[1κ[∂∂s[∂ϖ3∂s+ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]] = 0,∂∂s[1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]] = 0.

Corollary 3.6.If∂N∂talong the curve is parallel to the Fermi–Walker terms, then

∂∂s[ϖ1κ + ∂ϖ2∂s − ϖ3τ] = 0,[τ[1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]] = 0,∂∂s[1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]] = 0.

Corollary 3.7.If∂B∂talong the curve is parallel to the Fermi–Walker terms, then

[−κ[ [1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]] + ∂∂s[1τ[∂κ∂t − ∂∂s[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]] + κ[[1κ[∂∂s[∂ϖ3∂s + ϖ2τ]+τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]]] = 0,[−κτ[[∂κ∂t − ∂∂s[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]+κ[1τ[∂κ∂t − ∂∂s[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]−∂∂s[[1κ[∂∂s[∂ϖ3∂s + ϖ2τ]+τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]]] = 0,τ[[1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]] = 0.

Theorem 3.8. (Main Theorem)

(4)∂∂t∇˜X∇˜s = ∇˜∇˜s∂X∂t + ∂∂tκ(B ∧ X) + κ(∂B∂t ∧ X). (4)

Proof. By using (1), we have above equality. This completes the proof.

As an immediate consequence of (4), we have

Corollary 3.9.

∂κ∂t = −∂∂s[ϖ1κ + ∂ϖ2∂s − ϖ3τ] + τ[∂ϖ3∂s + ϖ2τ]∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ] = −κ[[1κ[∂∂s[∂ϖ3∂s + ϖ2τ]+τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]].

Proof. From the main theorem, we have

∇˜∇˜s∂T∂t + ∂κ∂tN + κ[ [ 1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ] ] ] ]B = 0.

Also, by using (4), we obtain

[κ[ϖ1κ + ∂ϖ2∂s − ϖ3τ] − κ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]T+[∂∂s[ϖ1κ + ∂ϖ2∂s − ϖ3τ] − τ[∂ϖ3∂s + ϖ2τ] + ∂κ∂t]N+[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]+κ[[1κ[∂∂s[∂ϖ3∂s + ϖ2τ] + τ[ϖ1κ + ∂ϖ2∂s − ϖ3τ]]]]]B = 0.

Combining (2) and (4) yields proof.

Example 3.10.The helix is parameterised by

αt(s) = (A(t)cos(s), A(t)sin(s), B(t)s),

where A and B are functions only of time.

Now, we draw some pictures corresponding to different cases A(t).

In Figures 1 and 2, the time helix is illustrated in magenta, cyan, and green at the time t=1.2, t=1.8, and t=2.2, respectively.

Figure 1:

A(t)=t.

Figure 2:

A(t)=t².

4 Applications to Electrodynamics

In this section, we construct the Fermi–Walker derivative in the motion of a charged particle under the action of only electric or only magnetic fields.

The equation of motion of a charged particle of mass m and electric charge q under the electric field Ɛ and magnetic field Ɓ is given by the Lorentz equation. In Gaussian system of units [17], we have

mdvds = qℰ + qv × ℬ.

Case I. Only in a magnetic induction B (no electric field 𝔼), the equation of motion is

mdTds = qT × ℬ.

From the above-mentioned equation and the Frenet frame, we easily choose

ℬ = − κmqB.

Therefore, we can write

∇˜ℬ∇˜s = κmτqN − dds(κmq)B,

which implies that

κmτq = 0, κ = constant.

Case II. Only in an electric induction 𝔼 (no magnetic field B), the equation of motion is

ℰ = mκqN.

Then we easily have

∇˜ℰ∇˜s = dds(mκq)N + mκτqB,

which implies that

κmτq = 0, κ = constant.

5 Conclusions

The Fermi–Walker transport and inextensible flows play an important role in geometric design and theoretical physics. A recent problem in the field of classic differential geometry consists in the study of the Fermi–Walker transport and inextensible flows.

In this paper, we have studied the Fermi–Walker derivative and the Fermi–Walker parallelism in space. This work is mainly geared to show that inextensible flows for the Fermi–Walker derivative by the curvatures of curves.

Furthermore, using the Frenet frame of the given curve, we present some partial differential equations. We have illustrated this study with examples through which we have compared flows of the Frenet frame in space as Figures 3 and 4. Finally, we construct the Fermi–Walker derivative in the motion of a charged particle.

Figure 3:

A(t)=1.

Figure 4:

A(t)=−1.

In our future work under this theme, we propose to study the conditions on the Fermi–Walker derivative and the Fermi–Walker parallelism in Minkowski space.