RESEARCH Revista Mexicana de F´ısica 66 (2) 180–186 MARCH-APRIL 2020

The Frenet-Serret description of Born rigidity and its application to the Dirac equation

J.B. Formiga Departamento de F´ısica, Universidade Federal da Para´ıba, Caixa Postal 5008, 58051-970 Joao˜ Pessoa, Pb, Brazil. Received 26 July 2019; accepted 15 October 2019

The role played by non-inertial frames in physics is one of the most interesting subjects that we can study when dealing with a physical theory. This is especially true for and the Dirac theory In the case of special relativity, a problem with the concept of rigidity emerged as soon as Max Born gave a reasonable definition of rigid motion: the Herglotz-Noether theorem imposes a strong restriction on the possible rigid motions. In this paper, the equivalence of this theorem with another one that is formulated with the help of Frenet-Serret formalism is proved, showing the connection between the rigid motion and the of the observer’s trajectory in . Besides, the Dirac equation in the Frenet-Serret frame for an arbitrary observer is obtained and applied to the rotating observers. The solution in the rotating frame is given in terms of that of an inertial one.

Keywords: Born rigidity; accelerated frames; Dirac equation; rotating observers; Frenet-Serret formalism.

PACS: 03.30.+p; 03.65.-w; 02.40.-k. DOI: https://doi.org/10.31349/RevMexFis.66.180

1. Introduction of RM is the idea that two “points” move in such a way that their distance is kept constant by means of a choreographed Reference frames that are constructed out of the worldline motion. But even in this case, relativity is problematic. First, of a particular observer are of great importance in both gen- we have the problem with the measurement of time. The fact eral and special relativity (SR). They play the role of the ob- that in an RM the distance is kept constant does not mean server’s rest frame and can sometimes even simplify the field that the velocity is the same. Think, for example, of two ob- equations. In general, these frames are taken to be orthonor- servers that describe a with the same angular mal, and one of their vectors is chosen to coincide with the velocity. If their distance to the center of the motion is not the observer’s 4-velocity. A well-known procedure to obtain a same, their velocity will be different. In this case, we cannot frame that rotates only in the plane formed by the 4-velocity be sure that they will agree with time measurements. The and the 4- is the so-called Fermi-Walker trans- second problem lies in the fact that if A is at rest with respect port [1]. In some sense, this frame is the closest we can get to B, and C is at rest with respect to A, in general, we cannot to an inertial frame that follows the observer. Nonetheless, guarantee that C is at rest with respect to B. Nevertheles, a there exists another frame that also possesses interesting fea- reasonable definition of RM was given by Born [12] and will tures, namely, the Frenet-Serret framei [3]. This frame not be considered here. only follows the observer but also gives important informa- Although reasonable, Born rigidity imposes a strong re- tion about the observer’s worldline. Because of these prop- striction on the possible motions. This restriction is described erties, it has been used to understand many different types by the Herglotz-Noether theorem (HNT) [13, 14]. Although of systems [4–11]. For example, in Refs. [6, 9, 10], Bini et we can state this theorem in terms of the 4-acceleration and al use the Frenet-Serret tetrads to analyze circular orbits in rotation tensors [15], it is possible to have a better intuition black hole and to study centripetal acceleration of its geometrical properties by using the curvatures of the and centrifugal forces in general relativity. Another interest- observer’s worldline, as is done in Ref. [16]. In this refer- ing example is Ref. [11], where Iyer and Vishveshwara study ence, a similar theorem based on the Serret-Frenet formalism the Frenet-Serret description of gyroscopic precession. Be- was proved. The main purpose of this paper is to prove the cause of the importance of the Frenet-Serret formalism, we equivalence of these two theorems (Sec. 3.1) and apply the will use it in this paper to deal with the rigid motion problem. Frenet-Serret formalism to the Dirac equation. In doing so, Rigidity is an idea that is present in practically all physi- we write the Frenet-Serret tetrad (FST) in the local coordi- cal theories: we use rigid rods as a standard length, the lab- nate system of a general observer (Sec. 2.2), obtain the Dirac oratory frame is always thought of as a rigid frame, etcetera. equation in this frame (Sec. 4) and solve it for the case of ro- The situation in SR is not different. However, unlike what tating observers ( Sec. 4.1). Finally, we present a summary happens in Newtonian mechanics, the concept of rigidity in of the results in Sec. 5. SR is not that simple. The notion of a is incon- We use the following notations and conventions. A Carte- sistent with the fact that no signal can propagate with a speed sian coordinate system that is homogeneous and isotropic greater than that of light. To avoid this problem, one can work and in which Newton’s laws are valid are represented by with the concept of rigid motion (RM). The ordinary concept xµ = (t, x, y, z), where µ = 0, 1, 2, 3. A tetrad field will THE FRENET-SERRET DESCRIPTION OF BORN RIGIDITY AND ITS APPLICATION TO THE DIRAC EQUATION 181 ³ ´ e b (0) b (1) b b be denoted by a, whose components in the Cartesian basis Σa = k1δa δ(1) + δa k1δ(0) + k2δ(2) µ µ ∂µ = ∂/∂x are ea , where a = (0), (1), (2), (3). When ³ ´ (2) b b (3) b convenient, the coordinate basis ∂µ will be represented as a + δ k3δ − k2δ − k3δ δ , (4) (µ) a (3) (1) a (2) coordinate tetrad in the form ˚ea = δa ∂µ or ˚e(µ) = ∂µ.

Latin letters in the middle of the alphabet stand for spatial where the ks are called curvatures (sometimes k2 and k3 are indices only, i.e., i, j, k, . . . = (1), (2), (3). The dual ba- also called the first and second torsions, respectively), and eµ µ a sis of ∂µ is denoted by dx . The metric tensor η takes are the components of the FST written in an inertial frame the form of a diagonal matrix when written in the basis with Cartesian coordinates (t, x, y, z). ∂µ. To be more precise, the elements of η are η00 = 1, To study Born rigidity, we may use the local coordinate η11 = η22 = η33 = −1. The metric components in a general system (τ, ξ, χ, ζ), which, is related to xµ = (t, x, y, z) by tetrad basis also have the same values, i.e., η(0)(0) = 1 and [1, 16, 17] η(1)(1) = η(2)(2) = η(3)(3) = −1. The metric ηµν is used to raise and lower coordinate indices, which are denoted by µ µ j µ x (τ, ξ, χ, ζ) = xn(τ) + r ej (τ), (5) Greek letters, while ηab is used to raise and lower tangent space indices. In this paper, we use c = 1 and ~ = 1. µ where xn(τ) is the worldline of an observer n, which is at the origin of the new coordinate system, and rj = (ξ, χ, ζ). The coordinate time τ corresponds to this observer’s proper 2. Accelerated Frames µ time, while ej are the Cartesian components of the vector 2.1. Fermi-Walker Transport field constructed out of this observer’s FST (by means of par- allel transport). It should be clear that the inertial observers The main idea behind the Fermi-Walker transport is to elim- use the coordinates xµ = (t, x, y, z), while the accelerated inate as many rotations as possible while the transported ones use xµ¯ = (τ, ξ, χ, ζ) (note the overbar). frame follows an observer’s worldline. In general, this frame Since xµ¯ = xµ¯(t, x, y, z), we can use Eq. (5) to obtain will not be inertial and will possess a rotation in the plane a the frame ea and the coframe θ (the dual basis) in the coor- formed by the 4-velocity u and the 4-acceleration a. To un- dinate system xµ¯. Differentiating Eq. (5) with respect to ξ, derstand this idea, think of the non-relativistic case. Given we find that an arbitrary accelerated observer, which rest frame would ap- proach an inertial frame the most? Of course, it would be a ∂t ∂x ∂y ∂z ∂ = ∂ + ∂ + ∂ + ∂ frame that does not rotate with respect to an inertial frame at ξ ∂ξ t ∂ξ x ∂ξ y ∂ξ z all, but whose origin is accelerated in the same way as the = et ∂ + ex ∂ + ey ∂ + ez ∂ = e . (6) observer. In relativity, the situation is more involved because (1) t (1) x (1) y (1) z (1) the variation of u forces the frame to rotate in the mentioned Doing the same thing for ∂ , ∂ , and ∂ , we arrive at plane. Therefore, it is natural to demand no rotation other χ ζ τ than that. h A general rotation can be expressed as (see, e.g., p. 174 e(0) = f(τ, ξ) ∂τ + k2(τ)(χ∂ξ − ξ∂χ) of Ref. [1]) i deµ + k (τ)(ζ∂ − χ∂ ) , e = ∂ , a = Φµν e (1) 3 χ ζ (1) ξ dτ aν e = ∂ , e = ∂ , f(τ, ξ) = 1/(1 + ξk (τ)), (7) with (2) χ (3) ζ 1

Φµν = aµuν − aν uµ + Ωµν , where we have also used Eqs. (3)-(4) to obtain e(0). On the other hand, the dual basisii (coframe) is µν αβµν Ω = uαΩβε , (2) (0) (1) θ = dτ/f(τ, ξ), θ = −χk2(τ)dτ + dξ, where Ωβ are the components of what we might call the 4- (2) angular velocity vector, τ is the proper time, and εαβµν is the θ = [ξk2(τ) − ζk3(τ)] dτ + dχ, Levi-Civita tensor (ε0123 = +1). The Fermi-Walker trans- (3) µ µ θ = χk3(τ)dτ + dζ. (8) port corresponds to the case Ωβ = 0 and e(0) = u . Since ds2 = η θa ⊗ θb, we see that the metric in the new 2.2. Frenet-Serret Tetrad ab coordinates becomes Given the observer’s worldline xµ(τ), where τ is its proper 2 £ 2 2 2 2 2¤ 2 time, we can construct a tetrad basis attached to the observer ds = (1 + k1ξ) − (k3 + k2)χ − (k2ξ − k3ζ) dτ by using the formulas h i + 2 k2 (χdξ − ξdχ) + k3 (ζdχ − χdζ) dτ µ µ µ dx de µ e = , a = Σb e , (3) 2 2 2 (0) dτ dτ a b − dξ − dχ − dζ , (9)

Rev. Mex. F´ıs. 66 (2) 180–186 182 J.B. FORMIGA where say that the system is a rigid body. In fact, this is just an ide- alized motioniv, where the forces acting on the particles are 2 2 2 2 2 (1 + k1ξ) − (k3 + k2)χ − (k2ξ − k3ζ) > 0. (10) such that the strain always vanishes. In this context, a rigid body would be just an approximation of a body whose mo- The affine connection coefficients in the tetrad field e , a tion is very close to an RM regardless of the force acting on denoted by ωa , do not vanish. As is well known, these co- bc it. The problem in SR is that measurements of distance are efficients can be obtained through the relation ³ ´ not invariant, which means that they depend on the reference a a µ¯ λ¯ λ¯ ν¯ frame used. ω bc = e λ¯eb ∂µ¯ec + Γµ¯ν¯ec , (11) To solve this problem, we can assume that the dis- λ¯ a a tance between two infinitesimally separated parts of the body where ea and e λ¯ are the components of ea and θ in µ¯ should remain constant in the rest frame of these parts. This the coordinate bases ∂µ¯ = (∂τ , ∂ξ, ∂χ, ∂ζ ) and dx = λ¯ is known as the “Born rigidity” [12], which can be formally (dτ, dξ, dχ, dζ), respectively. The object Γµ¯ν¯ denotes the Christoffel symbols of the metric (9). After a straightforward, defined as follows: µ but tedious, calculation, we find that Definition 3.1 Let u be the 4-velocity field associated with the motion of the system, gµν the Minkowski metric in an ar- ω(0)(0)(1) = −ω(1)(0)(0) = f(τ, ξ)k1(τ), bitrary coordinate system, and hµν = uµuν − gµν the re- ⊥ striction of −gµν to u . The motion is said to be rigid if ω(1)(0)(2) = −ω(2)(0)(1) = f(τ, ξ)k2(τ),

ω(2)(0)(3) = −ω(3)(0)(2) = f(τ, ξ)k3(τ). (12) Luhµν = −(∇ν uµ + ∇µuν ) α α Note from Eqs. (12) and (7) that, along the worldline of the + u uν ∇αuµ + u uµ∇αuν = 0, (13) a observer n (ξ = χ = ζ = 0), ω bc is given solely by the curvatures of the curve. where Luhµν is the Lie derivative of hµν , and ∇ν uµ stand One of the advantages of using the FST lies on the fol- for the components of the covariant derivative. lowing results [3]: This is the same as saying that the strain rate tensor van- ishes [15, 21] and that Killing motions are rigid [22]. 1. The curvatures determine the observer’s worldline up While the RM in Newtonian spacetime has six degrees of to a Poincare´ transformation. freedom, three translations, and three rotations, which means that we can give any trajectory we want to a particular point 2. When k3 vanishes, the curve lies inside a hyperplane (a three-dimensional volume in Minkowski spacetime). of the body, Born rigidity possesses only three degrees and, In this case, the 3-velocity is inside a plane. therefore, does not allow for an arbitrary motion [13,14]. The possible motions are given by the HNT, which will be stated 3. When k2 vanishes, the curve lies in a plane (3-velocity in the next section. inside a line).

4. For k1 = 0, the is a straight-line (constant 3.1. Possible motions 3-velocity). The HNT restricts the RMs to the following class [15]: It is worth noting that the FST will be Fermi-Walker Theorem 3.1 The only possible RMs in the sense of Born i transported only if the observer’s worldline has k2 = k3 = 0. rigidity are those with Ωij = 0 or (d/dτ)a = 0 and This suggests a relation between these curvatures and the last (d/dτ)Ωij = 0. term in Eq. (2), that is, k2 and k3 are somehow related to Using the Frenet-Serret formalism, it was also proved the spatial rotations with respect to the Fermi-Walker transported following theorem [16]. frame. This relation will be presented in Sec. 3.1 Theorem 3.2 The only possible RMs in the sense of Born rigidity are those with arbitrary k1 and k2 = k3 = 0 or 3. Rigid Motion (d/dτ)k1 = (d/dτ)k2 = (d/dτ)k3 = 0. Of course, these theorems must be equivalent. To prove Despite the fact that Einstein and Born used the expression their equivalency, let us assume that the observer n uses a “rigid body” in their work [12, 19], it is well known now FST ea, which is given by Eqs. (3) and (4). In this basis, iii that the concept of a rigid body in SR is problematic . In Eq. (1) can be written as fact, even in classical mechanics, this concept is also prob- lematic because there is no medium where a pulse can prop- deµ i = −k η eµ + Ωjeµ, (14) agate with a speed greater than the finite value of the speed dτ 1 (1)i (0) i j of sound. But, in classical mechanics, we can at least have µ µ a totally satisfactory definition of an RM by assuming that a where we have used a = k1e(1) and system of particles describes an RM when the distance be- j j αβµν tween any pair of particles remains constant. This is not to Ω i = e(0)αeµeiν ε Ωβ. (15)

Rev. Mex. F´ıs. 66 (2) 180–186 THE FRENET-SERRET DESCRIPTION OF BORN RIGIDITY AND ITS APPLICATION TO THE DIRAC EQUATION 183

By taking i = (1), (2) in Eq. (14) and using Eqs. (3), and (4), with we arrive at 1 £ a b¤ Γµ = ωaµb γ , γ , (20) (2) (3) (3) 8 k2 = Ω (1), Ω (1) = 0, k3 = Ω (2). (16) µ µ a µ where γ = ea γ , ea are the components of ea in the global It is clear that Ω in the FST has only two nonvanishing com- a ij inertial frame ˚ea, and the γ s are the gamma matrices. When µ b ponents. Moreover, we have a = a ∂µ = a eb = k1e(1), i.e., convenient, we will take the matrices γa as the standard Dirac b a = (0, k1, 0, 0). From this we see that if dk1/dτ vanishes, matrices: then so does dai/dτ. Hence, the theorem 3.2 is equivalent µ ¶ µ ¶ I 0 0 σ to 3.1. γ(0) = , γj = j , 0 −I −σj 0 To see that k2 and k3 are nothing but spatial rotations, we a µ ¶ µ ¶ can use the identities (valid for det(eµ) = 1) 0 1 0 −i σ(1) = , σ(2) = , β αβµν 1 0 i 0 e(3) = e(0)αe(1)µe(2)ν ε , (17) µ ¶ 1 0 β αβµν σ(3) = . (21) e(1) = e(0)αe(2)µe(3)ν ε (18) 0 −1 in Eq. (15) to obtain the relations Ω(2) = Ω and Ω(3) = From Eqs. (12) and (20), we discover that (1) (3) (2) µ Ω(1). Comparing these relations with Eq. (16), we conclude 1 Γ = f k γ(0)γ(1) that k2 = Ω(3) and k3 = Ω(1). This means that the c 2 1 k2 is associated with a rotation about the direction defined by ¶ e(3) while k3 is related to a rotation about e(1). For more de- + k γ(1)γ(2) + k γ(2)γ(3) δ(0). (22) tails about the geometrical meaning of these curvatures, see 2 3 c Refs. [6, 9–11] and references therein. Substitution of Eqs. (22) and (7) into Eq. (19) gives The advantage of using the theorem 3.2 rather than 3.1 n lies on the geometrical intuition that it gives. For instance, if (0) γ f [∂τ + k2 (χ∂ξ − ξ∂χ) + k3 (ζ∂χ − χ∂ζ )] you wish to know whether an observer can be part of a Born ³ RM, you can use the results shown at the end of Sec. 2.2.: If 1 + γ(1)∂ + γ(2)∂ + γ(3)∂ + f k γ(1) the observer’s worldline lies in a plane (in terms of space, its ξ χ ζ 2 1 ´ o trajectory would be a straight line), then it can be seen as part (0) (1) (2) (0) (2) (3) + k2γ γ γ + k3γ γ γ + im Ψ = 0, (23) of an RM (k2 and k3 vanish in this case). On the other hand, if its worldline is not inside a plane and its first curvature (k ) 1 where we have multiplied it by −i, and f = 1/(1 + ξk (τ)). is not constant, then it cannot be part of an RM. In addition, 1 Note that this equation holds for any representation of the one can use this approach to easily construct RMs, as done in gamma matrices. Ref. [16]. In Sec. 3.1, we saw that the curvatures k and k are As examples of rigid observers we have Rindler ob- 2 3 related to space rotations of the frame around ∂ and ∂ , re- servers, (a rigid rod), whose torsions vanish, and the rotating √ ζ ξ spectively. If convenient, one may use a = −aµa = k , ones (“rigid disk” rotating with a constant angular velocity), µ 1 Ω = k , and Ω = k to recast Eq. (23) as whose curvatures are all constant. The worldline of a Rindler (1) 3 (3) 2 µ ¶ observer lies in a plane, while the one of a rotating observer n 1 γ(0)f ∂ + ~a · ~α − iΩ~ · J~ lies in a hyperplane. τ 2 o (1) (2) (3) 4. Dirac equation + γ ∂ξ + γ ∂χ + γ ∂ζ + im Ψ = 0, (24) ~ ~ ~ ˆ (1) ˆ (2) (3) ˆ Quantum mechanics in noninertial frames has been exten- where J = L + S, ~a = aξ, ~α = α ξ + α χˆ + α ζ i (0) i ~ ˆ ˆ ˆ ˆ sively studied by many authors [23–28]. The most interesting (α = γ γ ), Ω = Ω(1)ξ + Ω(3)ζ, and the triad (ξ, χ,ˆ ζ) cases studied so far are the Rindler and the rigidly-rotating are the versions of ∂ξ, ∂χ, ∂ζ in the ordinary vector formal- ones. With respect to the latter case, one can find systems ism. The orbital and spin angular momenta are defined as ~ ~ ~ ~ ˆ ˆ that can behave as rigid disks, such as rapidly-rotating neu- L = −iX × ∂/∂X, where X = ξξ + χχˆ + ζζ, and tron stars [27]. In this section, we use the Frenet-Serret for- ³ ´ ~ (2) (3) ˆ (3) (1) (1) (2) ˆ malism to write the Dirac equation in a arbitrary accelerated S = (i/2) γ γ ξ + γ γ χˆ + γ γ ζ frame; then, we apply the result to a rotating motion that gen- µ ¶ ~σ 0 eralizes to some extent those that are generally used in the = (1/2) ; 0 ~σ literature. In a noninertial frame, the Dirac equation can be written the gamma matrices are given by Eq. (21). The Dirac equa- as tion written in this form is clearly compatible with Eqs. (11)- µ iγ (∂µ + Γµ)Ψ − mΨ = 0, (19) (14) in Ref. [17].

Rev. Mex. F´ıs. 66 (2) 180–186 184 J.B. FORMIGA

2 £ 2 2 2 4 2 2 2 ¤ 2 4.1. Dirac equation in a rotating frame ds = (1 + λ Ω r0ξ) − λ Ω (ξ + χ ) dτ 2 2 2 2 When writing the Dirac equation in a rotating frame, one uses + 2λ Ω(χdξ − ξdχ)dτ − dξ − dχ − dz . (34) a frame that is adapted to an observer at the origin of the in- ertial frame (x, y, z = 0). As a result, the observer’s proper In turn, using Eq. (27) in Eq. (22), we find that time coincides with that of the inertial observers. In a real ex- ³ ´ 1 2 (0) (1) (1) (2) (0) periment, however, that is not always possible. For instance, Γc = Ωλ f Ωr0γ γ + γ γ δc . (35) if we perform an experiment on the surface of the Earth and 2 take into account its rotation, we would not be able to use a So, we have clock at the center of rotation. Here we write the Dirac equa- h i tion in this more realistic situation. i iγbΓ = Ωλ2f Ωr γ(1) + γ(0)γ(1)γ(2) . (36) Let us take as the observer n the one with coordinates b 2 0

0 1 2 xn = tn, xn = r0 cos θ, xn = r0 sin θ, Using Eqs. (33) and (36) in Eq. (19), we finally obtain 3 ½ xn = 0, (θ = Ωtn + θ0). (25) (0) £ 2 ¤ (1) γ f∂τ + Ωλ f (χ∂ξ − ξ∂χ) + γ ∂ξ Its FST is given by ³ (2) (3) 1 2 (1) +γ ∂χ + γ ∂z + Ωλ f Ωr0γ e(0) = λ˚e(0) − Ωr0λ sin θ˚e(1) + Ωr0λ cos θ˚e(2), 2 ´ ¾ e = − cos θ˚e − sin θ˚e , (1) (1) (2) +γ(0)γ(1)γ(2) + im Ψ = 0. (37)

e(2) = −Ωr0λ˚e(0) + λ sin θ˚e(1) − λ cos θ˚e(2),

e(3) = ˚e(3), (26) 4.1.1. Solution 2 2 2 k1 = Ω r0λ , k2 = Ωλ , k3 = 0, (27) To obtain the solution of Eq. (37) in terms of the solution in p the inertial frame, we take θ = π so that the frames e and 2 2 0 a where λ = 1/ 1 − Ω r0. Substitution of Eqs. (25)-(27) in ˚ea coincide as Ω → 0. Furthermore, we also use the defini- Eq. (5) results in tions

t = λ(τ − Ωr0χ), (28) 1 λ = p , β = Ωr0, φ = Ωλτ/2. (38) 2 x = (r0 − ξ) cos θ + χλ sin θ, (29) 1 − β

y = (r0 − ξ) sin θ − χλ cos θ, (30) (µ) The components of ea in the inertial frame ˚ea = δa ∂µ z = ζ, θ = Ωλτ + θ0, (31) correspond to the Lorentz matrix given by the relation ea = µ b a ea ∂µ = Λa ˚eb. We can read off the values of Λ b from where we have used tn = λτ in Eq. (28) and in the angle Eqs. (26): θ. From Eq. (10), we see that the new coordinate system is ³ ´ restricted by a a (0) (1) (2) Λ b = λδ(0) δb + Ωr0 sin θδb − Ωr0 cos θδb 2 2 2 2 4 2 2 ³ ´ (1 + Ω λ r0ξ) > Ω λ (χ + ξ ). (32) a (1) (2) − δ(1) cos θδb + sin θδb Note that, since the curvatures of the worldline of the ob- ³ ´ a (0) (1) (2) server n are constant, the set of observers characterized by ξ, + λδ(2) Ωr0δb + sin θδb − cos θδb χ, and ζ constant, the rotating ones, corresponds to an RM. a (3) Note also that, unlike what was done in Ref. [23], these co- + δ(3)δb . (39) ordinates are adapted to an observer that is not necessarily at x = y = z = 0: the new coordinate time τ corresponds to This Lorentz transformation can be split into two interesting the proper time of the observer whose worldline is given by ones: Eq. (25). That is the reason why we have the Lorentz factor ³ ´ a a (0) a (1) (2) λ in the expressions above. The case where this observer is Λ1 b = δ(0)δb + δ(1) cos 2φδb + sin 2φδb at the origin of the inertial frame is obtained by taking the ³ ´ v a (1) (2) a (3) limit r0 → 0 in Eqs. (26)-(31). + δ(2) − sin 2φδb + cos 2φδb + δ(3)δb , (40) Using Eq. (27), we see that Eqs. (7) and (9) become ³ ´ a a (0) (2) a (1) Λ2 b = δ(0) λδ + βλδ + δ(1)δ e = f∂ + Ωλ2f (χ∂ − ξ∂ ) , e = ∂ , b b b (0) τ ξ χ (1) ξ ³ ´ (0) (2) (3) 2 2 + δa βλδ + λδ + δa δ , (41) e(2) = ∂χ, e(3) = ∂z, f = 1/(1 + Ω λ r0ξ), (33) (2) b b (3) b

Rev. Mex. F´ıs. 66 (2) 180–186 THE FRENET-SERRET DESCRIPTION OF BORN RIGIDITY AND ITS APPLICATION TO THE DIRAC EQUATION 185

a a c where Λ b = Λ2 cΛ1 b. Together, they induce the transforma- 5. Summary tion Ψ = SΨ˚, where Ψ˚ is the solution of the Dirac equation in the inertial frame. In this paper, we have seen that the HNT can be formulated in Following the standard procedure for computing S (see, terms of the Frenet-Serret curvatures, which allowed us to use e.g., p. 70 of Ref. [29]), we find that the FST to deal with the Born rigidity. This approach turned ³ ´ out to be very fruitful because of the geometrical meaning of (1) (2) S = λ+ cos φ − sin φγ γ the curvatures of the observer’s worldline. ³ ´ It was shown in Sec. 3.1. that the curvatures (torsions) (0) (1) (0) (2) − λ− sin φγ γ − cos φγ γ , (42) k2 and k3 correspond to the rotations Ω(3) and Ω(1), respec- √ tively. These relations helped us to see the connection be- 1/2 where λ± = (λ ± 1) / 2. The inverse transformation is tween the rotation of the FST with respect to a Fermi-Walker ³ ´ transported frame and the geometrical properties of the ob- −1 (1) (2) S = λ+ cos φ + sin φγ γ server’s motion. ³ ´ We obtained the Dirac equation in the rest frame of a (0) (1) (0) (2) + λ− sin φγ γ − cos φγ γ . (43) particle that describes an arbitrary motion using the Frenet- Serret formalism. We have seen that the final expression can The solution of Eq. (37) is Ψ = SΨ˚ with Ψ˚ satisfying be easily converted to physical parameters such as angular the Dirac equation in a global inertial frame of reference, and spin momenta. The resultant equation was the same as µ ˚ ˚ µ µ a i.e., i˚γ ∂µΨ − mΨ = 0, where ˚γ = ˚ea γ and ∂µ = that of Ref. [17]. As an application, we wrote this equation µ (∂t, ∂x, ∂y, ∂z). Note that the ˚γ s are the ordinary gamma for the case of rotating observers and found its solution in µ (µ) matrices and ˚ea = δa . terms of the solution in an inertial frame.

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Rev. Mex. F´ıs. 66 (2) 180–186