Effects of rotation in the spacetime with a distortion of a vertical line into a vertical spiral
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Effects of rotation in the spacetime with a distortion of a verticalline into a vertical spiral
K. Bakke ∗ Departamento de F´ısica, Universidade Federal da Para´ıba,Caixa Postal 5008, 58051-900, Jo˜ao Pessoa, PB, Brazil.
Abstract
It is investigated the effects of rotation on the scalar field in the spacetime with the distortionof a vertical line into a vertical spiral. By analysing the upper limit of the radial coordinate thatstems from the effects of rotation and the topology of the defect, it is considered this upper limit ofthe radial coordinate as a boundary condition analogous to a hard-wall confining potential. Then,it is obtained a relativistic spectrum of energy in a particular case. In addition, it is analysed arelativistic analogue of the Aharonov-Bohm effect for bound states.
PACS numbers:Keywords: Aharonov-Bohm effect, linear topological defects, screw dislocation, rotating reference frame,radial Mathieu equation ∗ Electronic address: kbakke@fisica.ufpb.br . INTRODUCTION The possibility of arising topological defects from phase transitions during the universeevolution has raised discussions about spacetimes characterized by the presence of torsionand curvature. The best known example is the cosmic string spacetime [1–7]. In connectionwith the description of topological defects in solids [8, 9], it has been considered spacetimeswith a screw dislocation [6, 10, 11] and a spiral dislocation [11–13]. In recent years, the effectsassociated with these topological defect spacetimes have been investigated on the interfacebetween general relativity and quantum mechanics [10–13, 15–48]. Recently, G¨odel-typespacetimes with topological defects have been analysed in this interface [49–57].Another perspective was given by considering a uniformly rotating reference frame. Fromthe classical mechanics point of view, it has been raised that topological defects can deter-mine at least the upper limit of the radial coordinate in this rotating frame [11, 13, 15]. Thischaracteristic of the topological defect spacetimes in a uniformly rotating reference framehas agreed with the point raised by Landau and Lifshitz [14], where they showed that theMinkowski spacetime has a singular behaviour at larges distances in a uniformly rotatingframe. On the other hand, by going through quantum systems in backgrounds with topo-logical defects and rotation, quantum effects have been reported in the literature throughgeometric quantum phases [58]. Besides, the upper limit of the radial coordinate that stemsfrom the uniformly rotating reference frame has played the role of a boundary condition onthe wave function. This has drawn attention to the influence of rotation on the relativisticspectrum of energy from a geometric point of view [11, 13, 59].In this work, we extend the description of a topological defect in a solid to the contextof gravitation. We introduce a spacetime with the distortion of a vertical line into a ver-tical spiral. Then, we consider a uniformly rotating reference frame. Thereby, we start byanalysing the effects of rotation and the topology of the defect on the upper limit of the ra-dial coordinate. Further, we consider this upper limit of the radial coordinate as a boundarycondition analogous to a hard-wall confining potential, and thus, we obtain the relativisticspectrum of energy in a particular case. Besides, we analyse the possibility of finding ananalogue of the Aharonov-Bohm effect for bound states [66].This paper is structured as follows: in section II, we introduce the spacetime with thedistortion of a vertical line into a vertical spiral. We thus consider a uniformly rotating2eference frame and obtain the upper limit of the radial coordinate. Then, by using this upperlimit of the radial coordinate as a boundary condition analogous to a hard-wall confiningpotential, we search for relativistic bound states solutions to the Klein-Gordon equation inthis topological defect spacetime. Finally, we analyse the Aharonov-Bohm-type effect forbound states; in section III, we present our conclusions.
II. EFFECTS OF ROTATION ON THE SCALAR FIELD IN IN THE SPACETIMEWITH A DISTORTION OF A VERTICAL LINE INTO A VERTICAL SPIRAL
In Ref. [60] is shown that an elastic medium with the distortion of a vertical line into avertical spiral can be described by a metric tensor in agreement with the Katanaev-Volovichapproach [9]. In this work, we extend this discussion to the context of gravitation. In thisway, we consider a spacetime with the distortion of a vertical line into a vertical spiral bydescribing it with the line element (with ~ = 1 and c = 1): ds = − dt + dr + r dϕ + 2 β dϕ dz + dz , (1)where 0 < r < ∞ , 0 ≤ ϕ ≤ π and −∞ < z < ∞ . The parameter β is a constant thatcharacterizes the torsion field (dislocation) in the spacetime. It can be defined in the range0 < β <
1. Next, let us consider uniformly rotating frame. Thereby, let us perform thecoordinate transformation: ϕ → ϕ + ω t . Then, the line element (1) becomes ds = − (cid:0) − ω r (cid:1) dt + 2 ω r dϕ dt + 2 ωβ dt dz + dr + r dϕ + 2 β dϕ dz + dz . (2)Hence, from Eq. (2), we have that the line element has a singular behaviour at largesdistances because the radial coordinate is defined in the range:0 ≤ r < /ω. (3)Despite having the presence of torsion in the spacetime with the distortion of a vertical lineinto a vertical spiral, there is no influence of torsion on the range (3). By contrast, it isshown in Ref. [11] that there is the influence of torsion on the range of the possible values ofthe radial coordinate in the spacetime with the distortion of a circular curve into a verticalspiral and also in the spacetime with a spiral dislocation.Our aim is to investigate rotating effects on the scalar field subject to a hard-wall con-fining potential in the spacetime with the distortion of a vertical line into a vertical spiral.3ccording to Refs. [10, 11, 17], the Klein-Gordon equation in the spacetime with a topolog-ical defect can be written in the form: m Φ = 1 √− g ∂ µ (cid:0) g µν √− g ∂ ν (cid:1) Φ , (4)where g µν is the metric tensor, g µν is the inverse of g µν and g = det | g µν | . Observe thatthe indices { i, j } run over the space coordinates. Thereby, with the line element (2), theKlein-Gordon equation has the form: m Φ = − (cid:20) ∂∂t − ω ∂∂ϕ (cid:21) Φ + ∂ Φ ∂r + r ( r − β ) ∂ Φ ∂r + 1( r − β ) (cid:20) ∂∂ϕ − β ∂∂z (cid:21) Φ + ∂ Φ ∂z . (5)Observe that the quantum operators ˆ p z = − i∂ z and ˆ L z = − i∂ ϕ commutes with theHamiltonian operator given in the right-hand side of Eq. (5). Therefore, a possible way ofwriting the solution to Eq. (5) is in terms of the eigenvalues of these operators. In this way,let us write Φ ( t, r, ϕ, z ) = e − i E t + ilϕ + ikz f ( r ), where k = const and l = 0 , ± , ± , ± . . . arethe eigenvalues of the operators ˆ p z = − i∂ z and ˆ L z = − i∂ ϕ , respectively. With this solution,we obtain the following radial equation: f ′′ + r ( r − β ) f ′ − γ ( r − β ) f + (cid:2) ( E + ω l ) − m − k (cid:3) f = 0 , (6)where γ = ( l − β k ). Let us proceed our discussion by defining a dimensionless parameter y as cosh y = rβ , (7)therefore, the radial equation (6) becomes f ′′ + (cid:2) q cosh (2 y ) − λ (cid:3) f = 0 , (8)where we have defined the following parameters in Eq. (8): λ = γ + β (cid:2) ( E + ω l ) − m − k (cid:3) ; (9) q = β (cid:2) ( E + ω l ) − m − k (cid:3) . It is worth observing that Eq. (8) is called in the literature as the modified Mathieu equationor radial Mathieu equation [61–64]. With the purpose of simplifying our analysis, let usfollow Refs. [64, 65] and define the parameter: x = 2 q cosh y = β q(cid:2) ( E + ω l ) − m − k (cid:3) cosh y. (10)4hen, by substituting Eq. (10) into Eq. (8), the modified Mathieu equation becomes: f ′′ + 1 x f ′ + f − x (cid:2) λ f + 2 q ( f + 2 f ′′ ) (cid:3) = 0 . (11)Since the parameter β that characterizes the topological defect in the line element (1) or(2) is defined in 0 < β <
1, we can neglect the terms proportional to β , without loss ofgenerality. Thereby, we can write the last term of Eq. (11) as λ f + 2 q ( f + 2 f ′′ ) ≈ γ f. (12)By using the approximation given in Eq. (12), hence, Eq. (11) becomes f ′′ + 1 x f ′ − γ x f + f = 0 , (13)which is the Bessel differential equation [61, 62]. Observe in the line element (2) that thespacetime has the cylindrical symmetry. Thereby, we search for a regular solution at theorigin. In this perspective, a regular solution to Eq. (13), when r = 0 ⇒ x = 0, is given by f ( x ) = A J | γ | ( x ) , (14)where J | γ | ( x ) is the Bessel function of first kind [61, 62] and A is a constant.Returning to the line element (2), we have that the radial coordinate is restricted to therange determined in Eq. (3). Thereby, the radial wave function of the scalar particle mustvanish when r → r = 1 /ω , i.e., when x → x , where x is given by using the relations r = 1 /ω and Eqs. (7) and (10): x = β q(cid:2) ( E + ω l ) − m − k (cid:3) cosh y = 1 ω q(cid:2) ( E + ω l ) − m − k (cid:3) . (15)This means that the radial wave function of the scalar particle must satisfy the boundarycondition: f ( x ) = 0 . (16)Hence, the boundary condition (16) corresponds to confinement of the scalar field to a hard-wall confining potential. Besides, since x is determined by the restriction on the radialcoordinate given in Eq. (3), thus, we have that the geometry of the spacetime with the5istortion of a vertical line into a vertical spiral plays the role of this hard-wall confiningpotential in the uniformly rotating frame even though no influence of torsion exists on theupper limit given in (3). This behaviour of having the geometry of the spacetime playingthe role of a hard-wall confining potential in uniformly rotating frame has been analysed inthe Minkowski spacetime [59] and other topological defect spacetimes [11, 13, 15].Let us go further by considering a particular case of the Bessel function. Let us assumethat x ≫
1. Then, when | γ | is fixed and x ≫
1, the Bessel function can be written as[10, 62]: J | γ | ( x ) → r π x cos (cid:18) x − | γ | π − π (cid:19) . (17)Therefore, by substituting (17) into (14), we obtain with the boundary condition (16): E n, l, k ≈ − ω l ± s m + k + π ω (cid:20) n + 12 | l − β k | + 34 (cid:21) , (18)where n = 0 , , , . . . is the quantum number related to the radial modes and l =0 , ± , ± , . . . is the angular momentum quantum number.The discrete spectrum of energy given in Eq. (18) is obtained from the confinement ofthe scalar field to a hard-wall confining potential. We can observe that there is the influenceof rotation and the topology of the spacetime with the distortion of a vertical line into avertical spiral on the relativistic energy levels (18). The contribution that stems from thetopology of the spacetime is given by the the effective angular momentum γ = ( l − β k ).This shift in the angular momentum quantum number occurs even though no interactionbetween the quantum particle and the topological defect exists. Therefore, it correspondsto an analogue of the Aharonov-Bohm effect for bound states [10, 11, 17, 66, 67]. On theother hand, one of the contributions that stems from the effects of rotation is given by thepresence of the fixed radius r = 1 /ω in the second term of the right-hand side of Eq. (18).The second contribution that stems from the effects of rotation is given by the first term ofthe right-hand side of Eq. (18), i.e., the coupling between the angular velocity ω and theangular momentum quantum number l . It gives rise to a Sagnac-type effect [11, 68–73].Furthermore, by taking β = 0, the relativistic energy levels (18) becomes E n, l, k ≈ − ω l ± s m + k + π ω (cid:20) n + | l | (cid:21) . (19)6herefore, we recover the relativistic energy levels for a scalar field subject to a hard-wallconfining potential in the Minkowski spacetime in a uniformly rotating frame [11]. III. CONCLUSIONS
We have analysed effects of rotation on the scalar field in the spacetime with a distortionof a vertical line into a vertical spiral. We have seen, despite the presence of torsion inthe spacetime, there is no contribution of the topology of the spacetime on the possiblevalues of the radial coordinate in the uniformly rotating frame. The restriction on the radialcoordinate depends only on the angular velocity of the rotating frame as shown in Eq. (3).Besides, in the analysis of the radial equation, we have focused on the case where the termsproportional to β could be neglected. We have seen that the solution to the radial equationcan be written in terms of the Bessel function. Then, we have considered a boundarycondition where the wave function vanishes when r → r = 1 /ω . This is analogous to theconfinement of the scalar field to a hard-wall confining potential. In this case, the geometryof the spacetime with the distortion of a vertical line into a vertical spiral plays the role ofthe hard-wall confining potential in the uniformly rotating frame even though no influence oftorsion exists on the upper limit given in (3). Then, by analysing the asymptotic expressionof the Bessel function, we have obtained a discrete spectrum of energy. We have observed inthese relativistic energy levels that there exists the influence of rotation and the topology ofthe spacetime with the distortion of a vertical line into a vertical spiral. The contribution ofthe topology of the spacetime yields a shift in the angular momentum quantum number thatcorresponds to an analogue of the Aharonov-Bohm effect for bound states. Furthermore, theeffects of rotation yield the presence of the fixed radius r = 1 /ω and the coupling betweenthe angular velocity ω and the angular momentum quantum number l (a Sagnac-type effect).Finally, we have shown that, by taking β = 0, we can recover the relativistic energy levelsfor a scalar field subject to a hard-wall confining potential in the Minkowski spacetime in auniformly rotating frame [11]. 7 cknowledgments The author would like to thank the Brazilian agency CNPq for financial support. [1] A. Vilenkin and E. P. S. Shellard, “
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