The Unruh Effect for Eccentric Uniformly Rotating Observers
aa r X i v : . [ g r- q c ] J a n The Unruh Effect for Eccentric Uniformly Rotating Observers H . Ramezani-Aval ∗ Faculty of Engineering Science, University of Gonabad, Gonabad, Iran
Abstract
It is common to use Galilean rotational transformation to investigate the Unruh effect for uni-formly rotating observers. However, the rotating observer in this subject is an eccentric observerwhile Galilean rotational transformation is only valid for centrally rotating observers. Thus, thereliability of the results of applying Galilean rotational transformation to the study of the Unruheffect might be considered as questionable. In this work the rotational analog of the Unruh effect isinvestigated by employing two relativistic rotational transformations corresponding to the eccentricrotating observer, and it is shown that in both cases the detector response function is non-zero. Itis also shown that although consecutive Lorentz transformations can not give a frame within whichthe canonical construction can be carried out, the expectation value of particle number operatorin canonical approach will be zero if we use modified Franklin transformation. These conclusionsreinforce the claim that correspondence between vacuum states defined via canonical field theoryand a detector is broken for rotating observers. Some previous conclusions are commented on andsome controversies are also discussed. ∗ Electronic address: [email protected] . INTRODUCTION The Unruh effect predicts that linearly accelerated observer with constant proper accel-eration in flat spacetime (Rindler observer) associates a thermal spectrum of particles tothe no-particle state (Minkowski vacuum) of inertial observer. In other words, two vacuumstates of these observers are not the same. So the particle content of field theory is observer-dependent and observers with different notions of positive and negative modes will disagreeon the particle content of a given state; what we think of as an inert vacuum actually hasthe character of a thermal state. For a review on this effect and its applications and experi-mental proposals see [1]. First approach to this effect was based on canonical quantization offields [2–4], which we will call ” canonical approach ” and the second was based on excitationof a detector [4–6], which we will refer to it as ” detector approach ”. Despite the controversiesand disagreements in the interpretations and relations between them, both approaches givethe same mathematical result that an uniformly accelerated detector (observer) observes athermal spectrum of particles and behaves as though it were placed in a thermal bath withtemperature T = a/ π , where a is the magnitude of the proper acceleration.A special case of accelerated observers which can have more feasible experimental testsis eccentric uniformly rotating observer. In addition to the usual ambiguities related tothe rotating observers [19], the particle detection due to acceleration of rotational motionis also controversial and the agreement between canonical approach and detector approachseems not to occur for this observer. The problem of finding true rotational transformationbetween the rotating and non-rotating frames is one important aspect of conflict. Whilemost authors have used Galilean rotational transformation (GRT) to investigate the Unruheffect for rotating observer [7–13], some other have tried to use corresponding relativistictransformations [14–17]. As we briefly mention them below, their results are different .All of those who use GRT, in canonical approach, obtain zero expectation value forparticle number operator of rotating observer in vacuum state of inertial observer, but theydo not have agreement on values (zero or nonzero) of detector response function [18]. Thisproblem was named the puzzle of rotating detector [14]. A solution for this problem wasintroduced in [10] which states that ” confining the detector inside the limiting surface andimposing the speed of light restriction for detector, the rotating detector registers the absenceof quanta and has vanishing response. ” Another explanation is that ” the correspondence etween expectation value for particle number operator defined via canonical quantum fieldtheory and detector response function is broken for general stationary motions ”, and we mustconclude that the two definitions are inequivalent [11]. On the other hand, some of thosewho use relativistic transformations conclude the particle detection both for canonical anddetector approaches[14, 15] and the other claim that there is no particle detection becauserotating observe does not have event horizon[16].In this paper we will discuss that GRT is not applicable for eccentric uniformly rotat-ing observers and we must replace it with the correct relativistic transformations betweenlaboratory inertial observer and eccentric rotating observer. We will use two sets of rela-tivistic transformation to investigate unruh effect for eccentric uniformly rotating observerin canonical and detector approach. Also we will discuss other relativistic transformationmentioned above.In section II we briefly mention the limitations and problems of GRT and introduce twotypes of relativistic transformations for eccentric uniformly rotating observer that can bereplaced with GRT. In section III we use two sets of relativistic transformations introducedin section II to investigate the Unruh effect for eccentric uniformly rotating observer bothin canonical and detector approaches. We conclude with a discussion section; comment onsome former papers and discuss some controversies.In this paper we use S ′ for rest frame of laboratory observer and S for acceleratedobserver’s frame. Greek letters take on the values 0,1,2,3. We use the metric with signature(+,-,-,-) and work in natural units. II. RELATIVISTIC TRANSFORMATIONS FOR ECCENTRIC UNIFORMLY RO-TATING DETECTOR
As we have shown in [19, 20], Galilean rotational transformation (GRT) t = t ′ , r = r ′ , φ = φ ′ − Ω t , z = z ′ (1)for relation between centric inertial observer and eccentric uniformly rotating observer whorotates with constant angular velocity Ω at constant radius distance from the center ofrotation is not true; Specially absoluteness of time and not distinguishing between observersat different radii in these transformations cause inconsistent kinematical interpretations3hen we want to explain phenomenon such as transverse Doppler effect and Sagnac effect.This transformation is only applicable for relation between two centric observer that onerotates uniformly and has no translational motion and the other is a non-rotating inertialobserver. So using GRT for an eccentric rotating detector is not true and the results thathas been obtained by these transformations for the Unruh effect in rotating frames are notvalid.We assume a detector on the edge of a rigid disc that rotates uniformly counterclockwisewith angular velocity Ω in the X ′ Y ′ plane around its axis ( Z ′ axis). Such detectors arethe ones which are related to the real experimental setups. As we have shown in [20] thereare two type of relativistic rotational transformations to describe the relation between thisrotating observer (detector) and laboratory inertial observer: A. Special Relativistic Transformation (SRT)
SRTs are based on consecutive Lorentz transformations and Fermi coordinates. In [23,24] these coordinate transformations between inertial laboratory (primed) and eccentricuniformly rotating (unprimed) frames are given as follows t = γ − ( t ′ − R Ω γy ) x = x ′ sin( γ Ω t ) + y ′ cos( γ Ω t ) − Ry = γ − [ x ′ cos( γ Ω t ) + y ′ sin( γ Ω t )] , z = z ′ (2)in which γ = (1 − R Ω ) − / , Ω is the uniform angular velocity of the disk and R is theradius of the circular path. In their setup the origin of the rotating frame is on the rim ofthe circular path. The metric components in such a rotating frame are given as follows ds = − γ [1 − ( R + x ) Ω − Ω γ y ] dt + dx + dy + dz − y Ω dxdt + 2 x Ω dydt (3) B. Modified Franklin Transformations (MFT)
In [19], looking for a consistent relativistic rotational transformation between an inertialobserver and an observer at a non-zero radius (eccentric observer) on a uniformly rotatingdisk, the following transformations (in cylindrical coordinates) were introduced t = cosh(Ω R/c ) t ′ − Rc sinh(Ω R/c ) φ ′ ; ρ = ρ ′ = cosh(Ω R/c ) φ ′ − cR sinh(Ω R/c ) t ′ ; z = z ′ , (4)in which Ω is the uniform angular velocity of the disk and R is the radial position of theobserver on the disk. Note that the origin of the rotating frame S is chosen to be at thecenter of the rotating disk so that both inertial and rotating frames assign the same radialcoordinate to the events. The corresponding metric in the rotating observer’s frame is givenby, ds = − c cosh ( R Ω)(1 − ρ R tanh ( R Ω)) dt + dρ + ρ cosh ( R Ω)(1 − R ρ tanh ( R Ω)) dφ − cR sinh( R Ω) cosh( R Ω)(1 − ρ R ) dtdφ + dz . (5) III. PARTICLE DETECTION BY UNIFORMLY ROTATING ECCENTRIC OB-SERVER
Solving Klein-Gordon equation for a massless scalar field in flat spacetime of laboratoryinertial observer in cylindrical coordinate gives positive modes solution as f = 12 π √ ω exp( − iωt ′ + imφ ′ + ikz ′ ) J m ( qρ ′ ) (6)in which m is an integer, J m is the Bessel function and ω = p q + k . By expanding thefield in term of a complete set of positive modes f i and negative modes f i ∗ and creation andannihilation operators(ˆ a † i and ˆ a i ), we haveΦ = X i (ˆ a i f i + ˆ a † i f ∗ i ) (7)Also we can solve Klein-Gordon equation for a massless scalar field in the rotating observersframe and expand the field in term of a new complete set of positive and negative modes( g i , g ∗ i ) and new creation and annihilation operators (ˆ b i , ˆ b † i )Φ = X i (ˆ b i g i + ˆ b † i g ∗ i ) . (8)If we show the vacuum state of inertial observer by | f i and the rotating observer’s particlenumber operator by ˆ n g then we have h f | ˆ n gi | f i = X j | β ij | (9)5n which the Bogolyubov coefficient β is defined as β ij = − ( g i , f j ∗ ) (10)and definition of inner product is given by( ϕ , ϕ ) = − i Z P ( ϕ ∇ µ ϕ ∗ − ϕ ∗ ∇ µ ϕ ) n µ √ hdx n − (11)where P is hypersurface that we integrate over it, h is the determinant of h ij which isthe induced metric on P and n µ is the normal vector to P . Since the inner product isindependent of the hypersurface over which the integral is taken, we can take the integralover t = 0 hypersurface. According to (9) non-zero value for coefficient β means non-zeroexpectation value of particle number operator of the rotating observer in the vacuum stateof laboratory observer.On the other hand, according to [18] the detector response function is given by F ( E ) = Z ∞−∞ d △ τ e − iE △ τ G + ( x ( τ ) , x ( τ )) (12)where G + is the positive Wightman function defined as G + ( x ′ , x ′ ) = − π [( t ′ − t ′ − iǫ ) − | x ′ − x ′ | ] (13)and x ( τ ) is the worldline of detector and τ is its proper time.Now we calculate Bogolyubov coefficient β and detector response function using two rela-tivistic rotational transformations introduced in previous section. A. Special Relativistic Transformation (SRT)
1. Canonical Approach
By special relativistic transformations (2) and corresponding metric (3) and make twosimplifying assumptions on the values of R and Ω that R = 1 and Ω = (which are notimportant in our discussion)we have g = 1 − x / − x/ − y / , g = 2 y/ , g = − x/ , g = 0 , g ij = − δ ij (14)6nd so g = − ( x − / g µν = 1( x − y − x y − x + 6 x + 4 y − − yx − x − yx x + 2 x −
3) 00 0 0 − (15)and so the corresponding Klein-Gordon equation for massless scalar field( √− g ∂ µ ( √− gg µν ∂ ν Φ) = 0) is given by9( x − ∂ Φ ∂t + 12 y ( x − ∂ Φ ∂t∂x − x ( x − ∂ Φ ∂t∂y − y ( x − ∂ Φ ∂t + 4 y − ( x − ( x − ∂ Φ ∂x + 12 y − x − x − ∂ Φ ∂x + 3( x + 2 x − x − ∂ Φ ∂y = 0 (16)Although it seems necessary to obtain the analytic solution for this partial differential equa-tion to continue and calculate the Bogolyubov coefficient, but it is not possible. As itmentioned very shortly in [26], this is the reason why rotational transformations based onconsecutive Lorentz transformations can not give a coordinate system within which thecanonical approach of a quantum field can be carried out.
2. Detector Approach
With the assumption that the detector is at the origin of rotating frame and using trans-formations (2), the detector’s trajectory in the laboratory frame is given by x ′ = R cos( γ Ω t ) , y ′ = R sin( γ Ω t ) , z ′ = 0 , t ′ = γt (17)and by (13) we have G + ( x ′ , x ′ ) = − π [( γ △ τ − iǫ ) − R (1 − cos( γ Ω △ τ ))] (18)Inserting (18) in (12) the detector response function is given by F ( E ) = Z ∞−∞ d △ τ e − iE △ τ ( γ △ τ − iǫ ) − R sin ( γ Ω △ τ ) . (19)Except in some constant coefficients this is the same as obtained and numerically evaluatedin [8] and so has non-zero value. 7 . Modified Franklin Transformations (MFT)
1. Canonical Approach
For eccentric rotating observer using modified Franklin transformations (4) and corre-sponding metric (5) we have g = − ρ and g µν = cosh ( R Ω) − R sinh ( R Ω) ρ R − ρ Rρ cosh( R Ω) sinh( R Ω) 00 − R − ρ Rρ cosh( R Ω) sinh( R Ω) 0 − cosh ( R Ω) ρ + sinh ( R Ω) R
00 0 0 − (20)So the Klein Gordon equation is as bellow(cosh β − R ρ sinh β ) ∂ Φ ∂t − ρ ∂∂ρ ( ρ ∂ Φ ∂ρ ) − ( cosh βρ − sinh βR ) ∂ Φ ∂φ + 2( R − ρ ) Rρ sinh β cosh β ∂ Φ ∂t∂φ − ∂ Φ ∂z = 0 . (21)Assuming a trial solution as g = exp( − iω ′ t + im ′ φ + ik ′ z ) R ( ρ ) (22)and inserting in (21), the radial part equation is d R ( ρ ) dρ − ρ dR ( ρ ) dρ + [( m ′ + √ ω ′ ) + − ω ′ + 2 m ′ − √ ω ′ m ′ ρ ] R ( ρ ) = 0 (23)in which we set R = 1 and cosh ( β ) = 2 for simplicity. (These assumptions do not affect theresults.) This equation is a cylindrical Bessel equation and so the positive mode solutioncorresponding to it is g = N exp( − iω ′ t + im ′ φ + ik ′ z ) J m ( q ′ ρ ) (24)in which N is a normalization factor. As in the case of GRT [7], we can see that theBogolyubov coefficient β is zero here, so using MFT the canonical approach concludes theabsence of particle in the vacuum state of laboratory observer for the rotating observer.8 . Detector Approach On the other hand the detector’s trajectory in the rotating frame is ρ = R , φ = 0 , z = 0 (25)Using MFT to obtain the trajectory for the laboratory observer we have t ′ − t ′ = cosh( R Ω)∆ τ , ϕ ′ − ϕ ′ = sinh( R Ω) R ∆ τ (26)in which ∆ τ = t − t . Wightman function in cylindrical coordinate is as below G + ( x ′ , x ′ ) = − π t ′ − t ′ − iǫ ) − [ ρ ′ + ρ ′ − ρ ′ ρ ′ cos( ϕ ′ − ϕ ′ ) + ( z ′ − z ′ ) ] (27)so we have G + ( x ′ , x ′ ) = − π [(cosh( β ) △ τ − iǫ ) − R (1 − cos( sinh( β ) R △ τ ))] (28)inserting in (12) the detector response function is given by F ( E ) = 14 π Z ∞−∞ d △ τ e − iE △ τ (cosh( β ) △ τ − iǫ ) − R sin ( sinh( β )2 R △ τ ) (29)so in comparison with (19) it has non-zero value. IV. DISCUSSION AND CONCLUSIONS
Here the Unruh effect for eccentric uniformly rotating observers was investigated by tworelativistic rotational transformations corresponding to the eccentric rotating observer: con-secutive Lorentz transformation and modified Franklin transformation. It was shown thatthe detector response function is non-zero in both cases. We also showed that althoughconsecutive Lorentz transformations lead to calculational problem and give a frame withinwhich the Klein-Gordon equation does not have an analytic solution, but if we use modi-fied Franklin transformation, we obtain that the Bogolyubov coefficient related to numberoperator and so the expectation value of particle number operator is zero. This conclusionsreinforce the claim that correspondence between vacuum states defined via canonical fieldtheory and via a detector is broken for rotating observers [9, 11]. Following our comparativestudy in [20], here we showed that employing MFT instead of the SRT helps to investigate9he Unruh effect in canonical approach. It must be emphasized that in these relativistictransformations the upper limit for the velocity of the disk points (speed of light) is consid-ered and unlike [10] there is no need to confine the detector inside a light cylinder. In orderto answer this question that if particle distribution is characteristic of the thermal blackbodyradiation with the finite temperature, we need analytic solution of coefficient β and detectorresponse function. Then we can judge the claim stated in [16] that the Stationary detectorwill not show an excitation spectrum which can be expressed simply in terms of Boltzmanfactor.There are two important issues we face in investigation of the unruh effect which seemto be the source of this effect: the acceleration and the event horizon. Is the existence ofan event horizon a necessary condition for the Unruh effect? What about acceleration? In[16] following up [9] the existence of horizon is assumed as a necessary condition for creationof the Unruh effect; When there is an event horizon we can define two different Fock spaceand mixing creation and annihilation operators and will expect to have nonzero Bogolyubovcoefficients. Also they argue that ”for a rotating observer there is no event horizon since theorbit is restricted to a bounded region of space, so that a signal from an event anywhere inspace will be abe to reach the spacetime curve and any spacetime point can be reached by alight signal from a point on the curve.” and conclude that for a uniformly rotating observerthere is no corresponding unruh effect. The observer in our special relativistic approachwhich is the same as Mashhoon observer [23], has the a/c < ω condition and is the same asuniformly rotating observer in [16] and so, according to it’s result, should not observe Unruheffect. But as mentioned in [25] if the existence of horizon is necessary then even for linearaccelerating detector the particle detection will be impossible, unless there is a detector withconstant acceleration from the past infinity to future infinity and this situations is practicallyinaccessible. On the other hand if the acceleration is the necessary and sufficient condition,then since the eccentric rotating observer has centripetal acceleration, particle detection canbe expected. The remaining point is that the work done by the centripetal force in the caseof uniform circular motion is zero and it can be an important differentiation between Rindlerand uniformly rotating observer.In [14, 15]using Trocheries-Takeno transformations, which we called Franklin transfor-mations (FT) [19], it is shown that the rotating observer defines a vacuum state which isdifferent from the Minkowski one. But as we have discussed [19], FTs have all kinematical10roblems of GRT and can not be applied for relating eccentric rotating detector to centriclaboratory observer. In addition, in [15] the Klein-Gordon’s solution that has given in rotat-ing frame is coordinate transformed solution of the inertial one. But it is easy to show thatby this assumption, unlike their conclusion, always we will have zero Bogolyubov coefficient.If g ( x ′ ) in (12) is coordinate transformed of f ( x ), when we calculate integral (13) we needto express g and f in the same coordinates and need to apply inverse transformation on g ,so we will have β = − ( g ( x ′ ) , f ∗ ( x )) = − ( f ( x ) , f ∗ ( x )) = i Z P ( f ∇ µ f − f ∇ µ f ) n µ √− hdx n − = 0 (30)and so it is impossible to obtain nonzero coefficient β by that suggested solution. Acknowledgments
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