Velocity Effects on an Accelerated Unruh-DeWitt Detector
aa r X i v : . [ g r- q c ] A p r Velocity Effects on an Accelerated Unruh-DeWitt Detector
Shohreh Abdolrahimi ∗ Institut f¨ur Physik, Universit¨at Oldenburg, Postfach 2503 D-26111 Oldenburg, Germany
We analyze the response of an Unruh-DeWitt detector moving along an unbounded spatial tra-jectory in a two-dimensional spatial plane with constant independent magnitudes of both the four-acceleration and of a timelike proper time derivative of the four-accelration. In a Fermi-Walkerframe moving with the detector, the direction of the acceleration rotates at a constant rate arounda great circle. This is the motion of a charge in a uniform electric field when in the frame of thecharge there is both an electric and a magnetic field. We compare the response of this detector toa detector moving with constant velocity in a thermal bath of the corresponding temperature fornon-relativistic velocities, and two regimes ultraviolet and infrared. In infrared regime, the detectorin the Minkowski space-time moving along the spatially two-dimensional trajectory should movewith a higher speed to keep up with the same excitation rate of the inertial detector in a thermalbath. In ultraviolet regime, the dominant modification in the response of this detector compared tothe black body spectrum of Unruh radiation is the same as the dominant modification perceived bya detector moving with constant velocity in a thermal bath.
PACS numbers: 03.70.+k, 04.62.+v
I. INTRODUCTION
A uniformly accelerated observer in Minkowski space-time, i.e. linearly accelerated observer with constantproper acceleration, associates a thermal bath of Rindlerparticles to the no-particle state of inertial observers.This is the
Unruh effect [1–3]. It implies the conceptuallyimportant result that the particle content of a field theoryis observer dependent. The Unruh effect is important inits own right, perhaps having experimental applicationsin particle accelerators [4, 7, 24, 25], electrons in Pen-ning traps [9, 27], atoms in microwave cavities [10, 11],or hadronic collisions [12–14], and as a tool to investigateother phenomena such as the thermal emission of parti-cles from black holes [15, 16] and cosmological horizons[17]. For a review of the Unruh effect and applicationssee [18]. Recently, Mart´ın-Mart´ınez, Fuentes, and Mannhave shown, [19], that a detector acquires a Berry phasedue to its motion in spacetime and this fact can be usedfor the direct detection of the Unruh effect in regimesphysically accessible with current technology.Unruh has introduced a detector model consisting of asmall box containing a non-relativistic particle satisfyingthe Schr¨odinger equation [1]. The system is said to havedetected a quanta if the particle in the box jumps fromthe ground state to some excited state. DeWitt [20] hasintroduced a detector which consists of a two-level pointmonopole. In this paper we use an idealized point de-tector with internal energy levels labelled by energy E and E > E , coupled via a monopole interaction witha scalar field ϕ , known as an Unruh-DeWitt detector inthe literature.An eternal uniformly accelerated Unruh-DeWitt detec-tor, moving along a linear spatial trajectory in Minkowski ∗ [email protected] vacuum with constant magnitude of its four-acceleration a , perceives a radiation rate which is equivalent to a de-tector at rest in a thermal bath of Minkowski particlesof the temperature T = ~ a/ (2 πck B ), where ~ is the re-duced Planck constant, c is the speed of light, and k B isthe Boltzmann constant. This detector will experiencea time-independent situation and hence settle in a sta-tionary state. However, there exist other time-like curvessuch that the geodesic interval between two points alongthe curve depend only on the proper time interval. Thereare called stationary curves, a classification of such curvesinto six categories has been done by Letaw [21]. The vac-uum excitation spectra of detectors on a representativesample of such stationary world lines have been calcu-lated, some of which were presented only numerically.The corresponding vacuum states have also been classi-fied. It was shown by Letaw and Pfautsch [22] that thecorresponding vacuum states are found to be restrictedto two possibilities: Those in coordinate systems with-out event horizons are the Minkowski vacuum; those incoordinate systems with event horizons are the Fullingvacuum. The analog of Unruh effect for spatially circulartrajectories has been discussed in particular with relationto polarization effects of electrons in storage rings and forelectrons circulating in a cavity [23–30]. Gutti, Kulka-rni, and Sriramkumar have shown that the response ofthe rotating detector can be computed exactly (albeit,numerically) even when it is coupled to a field that isgoverned by a nonlinear dispersion relation [31]. Kors-bakkena, and Leinaasa [32] related the excitation spec-trum of a detector moving along planar stationary tra-jectories to the properties of Minkowski vacuum in theaccelerated frame and defined an effective temperaturein terms of the transition rate of a detector into up ordown states. Barbado and Visser analyzed the responsefunction of an Unruh-DeWitt detector moving with time-dependent acceleration along a one-dimensional trajec-tory in Minkowski spacetime [35].In this paper, we consider a special case of a stationarytrajectory. We consider a detector moving along an un-bounded spatial trajectory in a two-dimensional spatialplane with constant independent magnitudes of both thefour-acceleration and of a timelike proper time derivativeof four-accelration, such that in a Fermi-Walker framemoving with the detector, the direction of the accelera-tion rotates at a constant rate around a great circle. Thisis the motion of a charge in a uniform electric field whenin the frame of the charge there is both an electric anda magnetic field. We choose a special coordinate for de-scribing the motion, in which one of the components ofthe 4-velocity, w = dy/dτ ( τ is the proper-time of the de-tector), is constant. We calculate explicitly the responseof an Unruh-DeWitt detector moving along the abovetrajectory in non-relativistic limit . In non-relativisticlimit, the zero order term is of course the thermal spec-trum of Unruh radiation, but we are interested to find thenext dominant term in the response of the detector, pro-portional to the square of the four-velocity component w . On the other hand, we consider a detector mov-ing with constant non-relativistic speed ˜ w in a thermalbath of temperature corresponding to the Unruh temper-ature T = ~ a/ (2 πck B ). The first dominant term in theresponse of this detector is the Plankian spectrum of athermal bath, we find the next dominant term in the re-sponse of this detector, proportional to ˜ w , and comparethis dominant term to the one we find from the accel-erating detector moving in Minkowski space-time in tworegimes, which we call ultraviolet and infrared.We begin this paper with a review of the definitions ofthe physical quantities involved in the description of anUnruh-DeWitt detector, Sec I. In Section II, we describea trajectory of the detector and calculate the response ofan Unruh-DeWitt detector following the described tra-jectory. We compare the response of this Unruh-DeWittdetector with that of a uniformly accelerated detector( w = 0), moving along spatially straight line, and alsowith a detector moving with constant velocity in a ther-mal bath. In this paper, we use the system of units where ~ = c = k B = 1. II. THE DETECTOR
Suppose we have a pointlike two-level system (detec-tor) moving along a worldline described by the functions x µ ( τ ) = ( t ( τ ) , x ( τ )), where τ is the detector’s propertime, and µ labels the coordinates in the space-time. As-sume that this two-level system has two internal energylevels labelled by the energy E and E > E and is cou-pled to a quantum scalar field ϕ via a monopole inter-action, V = m q ( τ ) ϕ [ x ( τ )], where q ( τ ) is the monopole When the acceleration is set to zero this detector corresponds toone moving with constant velocity in Minkowski vacuum; such adetector perceives no temperature. moment operator [1, 20, 33, 34], and m is the interactionconstant. Then, the system, i.e. the two-level detec-tor, and the quantum field is described by the followingHamiltonian ˆ H = ˆ H ( o )0 + ˆ H ( f )0 + ˆ V , (1)where ˆ H ( o )0 is the Hamiltonian of the free two-level sys-tem, ˆ H ( f ) is the Hamiltonian of the free quantum scalarfield, and ˆ V defines the interaction. Assume that | A i ’sare the eigenvectors of the orthonormal basis of theHilbert space of the states of the system without inter-action,ˆ H ( of )0 | A i = E ( of ) A | A i , ˆ H ( of )0 = ˆ H ( o )0 + ˆ H ( f )0 . (2)For a general trajectory, the system of the two-level de-tector and the field will not always remain in its groundstate E ( of )0 , but will undergo a transition to an excitedstate E ( of ) > E ( of )0 . If we assume that the interactionconstant m is small, in the first-order approximation ofthe perturbation theory the probability amplitude of thetransition from the initial state | A i to the final state | B i at the proper time τ is given by A BA = − i m Z τ −∞ dτ ′ V BA ( τ ′ ) , (3) V BA = h B | exp( i ˆ H ( of )0 τ ) ˆ ϕ (0)ˆ q (0) exp( − i ˆ H ( of )0 τ ) | A i . (4)Let the states | n i and | N i to be the eigenstates of thenon-interacting free Hamiltonian of the two level detectorand the non-interacting free Hamiltonian of the free field,ˆ H ( o )0 | n i = E n | n i , ˆ H ( f )0 | N i = ω N | N i , (5)respectively. Then, the states | A i of the free system ofthe two-level detector and field can be written as | A i = | n i | N i . (6)The two-level detector is either in the ground state | > with energy E or in the excited state | i with energy E .The probability amplitude of the transition (3) can bederived using (4) which in the basis (6) can be written as V BA = V Mm Nn = q mn e i ( E m − E n ) τ h M | ˆ ϕ [ x ( τ )] | N i , (7)where q mn = h m | q (0) | n i . Suppose that the field ϕ isinitially in vacuum state | M i , where the subscript M stands for Minkowski vacuum, and the two-level systemis in ground state E . Let us consider mental copies ofthe above two-level Unruh-DeWitt detector, where thesecopies are different only in one sense, the value of theirsecond energy level E is different. Assume that all ofthese detectors are prepared in the same initial state andfollowing identical trajectories (see [35] for a discussionabout physical construction of such a system of detec-tors). The transition probability to all possible | M i and | i ’s (of different value of energy E ) for this ensemble ofdetectors is p M = m X E | q | Z τ −∞ dτ ′ Z τ −∞ dτ ′′ e i ( E − E )∆ τ × G + ( x ( τ ′ ) , x ( τ ′′ )) , (8)where G + is the positive frequency Wightman function, G + ( x ( τ ′ ) , x ( τ ′′ )) = h | ˆ ϕ [ x ( τ ′ )] ˆ ϕ [ x ( τ ′′ )] | i , (9)which for massless scalar field reads G + ( x ( τ ′ ) , x ( τ ′′ )) = − π [( t ′ − t ′′ − iǫ ) − | x ′ − x ′′ | ] . (10)Here, ǫ ≪
0. Note that ( t ′ , x ′ ) and ( t ′′ , x ′′ ) are func-tions of the proper time. If G + ( x ( τ ′ ) , x ( τ ′′ )) can be writ-ten as G + (∆ τ, r ), where r = | x ′ − x ′′ | and ∆ τ = τ ′ − τ ′′ ,the integrand in (8) depends only on ∆ τ , and we canwrite (8) in the following form p M = m X E | q | Z τ −∞ dτ ′ Z ∞−∞ d (∆ τ )e i ( E − E )∆ τ × G + (∆ τ, r ) . (11)The transition probability per unit proper time is p ∆ τ = dp M dτ = m X E | q | F ( E ) , (12)where F ( E ) = Z ∞−∞ d (∆ τ )e i ( E − E )∆ τ G + (∆ τ, r ) , (13)is the response function per unit proper time, and is in-dependent of the detailed structure of the detector. Ifthe quantum scalar field is initially in the thermal staterather than the Minkowski vacuum state then the re-sponse function F has to be replaced by F β ( E ) = Z ∞−∞ d (∆ τ )e i ( E − E )∆ τ G + β (∆ τ, r ) , (14)where G + β is the Wightman thermal Green function,which for the case of massless scalar field is [34] G + β (∆ τ, r )= G + (∆ τ, r ) + 14 π (∆ t − r ) , + coth[ π ( r + ∆ t ) /β ] + coth[ π ( r − ∆ t ) /β ]8 πβr , (15)where β = 1 /T is the inverse temperature, and ∆ t = t ′ − t ′′ . In what follows we shall consider a two-level Unruh-DeWitt detector as described in this section. Theresponse function of this detector per unit proper timecan be calculated using (13) if the detector is moving inMinkowski vacuum or using (14) if the detector is coupledto the thermal quantum scalar field. III. MOTION OF THE DETECTOR
Consider an Unruh-DeWitt detector explained in theprevious section, moving in Minkowski space-time alongan unbounded spatial trajectory in a two-dimensionalspatial plane with constant square of magnitude of four-acceleration a µ a µ = a , where a µ = d x µ /dτ , and con-stant magnitude of a timelike proper-time derivative offour-acceleration ( da µ /dτ )( da µ /dτ ), and having compo-nent of the four-velocity, dy/dτ = w , a constant, namelya detector moving along the following worldline x µ ( τ ) = (cid:18) aα sinh( ατ ) , aα cosh( ατ ) , wτ, (cid:19) , (16) α = a √ w > , (17)where x µ = ( t, x, y, z ) are the Minkowski coordinates.This is the motion of a charge in a uniform electric fieldwhen in the frame of the charge there is both an electricand a magnetic field. We have chosen a special coordinatefor describing the motion, in which one of the componentsof the 4-velocity, w = dy/dτ is constant. The magnitudeof the Fermi-Walker derivative of the acceleration is | Da | = (cid:18) D ( F ) µ [ a ] D µ ( F ) [ a ] (cid:19) = a w √ w , (18)The parameter η = | Da | /a is less than one. In a Fermi-Walker frame moving with the detector, the direction ofthe acceleration rotates at a constant rate η around agreat circle. For a circular trajectory rather than (16),which gives η >
1, one needs to replace the uniformelectric field with a uniform magnetic field and take thecharge moving so that in the frame of the charge there isboth a magnetic and an electric field.For w = 0, the trajectory is x µ = x µ ( τ ) = ( a − sinh( aτ ) , a − cosh( aτ ) , , , (19)which is a trajectory of a detector moving along a spa-tially straight line along the x direction with constantmagnitude of four-acceleration a µ a µ = a .Note that if instead of the trajectory (16), with com-ponent of the four-velocity w = dy/dτ = const. , we haveconsidered the component of the three-velocity dy/dt = v to be constant, the response of the detector would havebeen completely equivalent to that of a detector whichis moving along a spatially straight line with constantmagnitude of four-acceleration, (19), as such observerscan be related to the ones moving along the trajectory(19) by Lorentz transformations. A. Motion of the detector in the Minkowskivacuum
For the trajectory (16), the positive frequency Wight-man Green function (10) for a massless scalar field reads G + (∆ τ ) = − α π a (cid:20) sinh ( α ∆ τ − iǫα a ) − w α a ∆ τ (cid:21) − , (20)Here, we have absorbed a positive function of τ and τ ′ ,[sinh( ατ ) − sinh( ατ ′ )] / [sinh(∆ τ / α ) cosh(∆ τ / α )], into ǫ . Note that for w = 0 we have α = a and (20) immedi-ately converts to the positive Wightman Green functionof a detector moving along a spatially straight line in the x direction (19), with constant magnitude of the four-acceleration (spatially one-dimensional) G + (1 d ) (∆ τ ) = − a π (cid:20) sinh ( a ∆ τ − iǫa ) (cid:21) − . (21)Here, and in what follows the (1 d ) index is used twodistinguish the quantities such as the Wightman Greenfunction, or response function calculated for the spatiallyone-dimensional trajectory (19) as opposed to the index(2 d ) for the quantities associated with the spatially two-dimensional trajectory (16). The transition probabilityper unit proper time (12) for the detector following tra-jectory (19) is p ∆ τ = m X E | q | F (1 d ) ( E ) , (22) F (1 d ) ( E ) = ∆ E π [e π ∆ E/a − , (23)where ∆ E = E − E . This is the usual black body ex-citation rate, indicating that the excitation rate of anaccelerated detector coupled to the field ϕ in the state | M i is the same as that of a detector, unaccelerated,at rest in a bath of thermal radiation at temperature T = 1 /β = a/ (2 π ).In the “infrared limit” ∆ E ≪
1, the black body exci-tation rate (23) has the following dominant behavior: F (1 d ) ( E ) ∼ πβ . (24)In the “ultraviolet limit” ∆ E ≫
1, the black body exci-tation rate (23) has the following dominant behavior: F (1 d ) ( E ) ∼ ∆ E π e − β ∆ E . (25)We calculate the response function per unit proper time(12) of the detector following the trajectory (16) for non-relativistic velocities v y ≪ v y = dydt = w √ w cosh( ατ ) , (26)or w ≪
1. Note that here we are not considering theultra-relativistic limit because for ultra-relativistic veloc-ities v y → w → ∞ the response function of the detector (16) is suppressed, namely the Unruh-effect issuppressed as F (2 d ) ( E ) ∼ ∆ E π [e π ∆ E/a −
1] 1 w . (27)The Wightman Green function (20) in the leading orderfor w ≪ G +(2 d ) (∆ τ )= (1 − w ) G + (1 d ) (∆ τ ) − a π (cid:20) sinh( a ∆ τ − iǫa )( a ∆ τ − iǫa ) + a ∆ τ (cid:21) × (cid:20) sinh ( a ∆ τ − iǫa ) (cid:21) − w + O ( w ) . (28)To calculate the response function per unit proper time(12), we use the following identity:sinh x = x ∞ Y k =1 (1 + x k π ) . (29)Calculating the integral (13), we arrive to F (2 d ) ( E ) = F (1 d ) ( E ) + F a ( E ) w + O ( w ) , (30)where F a ( E ) = − e β ∆ E ∆ E πβ [e β ∆ E − × (cid:20) π ∆ E + 9 β − β ∆ E ( 4 π ∆ E + β ) e β ∆ E + 1 e β ∆ E − (cid:21) . (31)For the “infrared” tail of the spectrum ∆ E ≪
1, theresponse function is F (2 d ) ( E ) = 12 πβ (cid:20) − ( 76 − π w (cid:21) + O (∆ E ) . (32)For the “ultraviolet” tail of the spectrum ∆ E ≫
1, theexcitation rate (31) has the following dominant behavior: F ( a ) ( E ) ∼ ∆ E π e − β ∆ E β . (33) B. Motion of the detector in the thermal bath
We now consider a detector moving along spatiallystraight line with constant component of its four-velocity˜ w x µ ( τ ) = ( p w τ, , ˜ wτ, , (34)in a thermal bath of temperature T , (see [36, 37]), corre-sponding to the temperature that an accelerated Unruh-DeWitt detector moving along a spatially straight linewith constant magnitude of four-acceleration, trajectory(19), in Minkowski vacuum perceives, i.e. T = a/ (2 π ).Here and in what follows by a thermal bath we mean ther-mal quantum scalar field. We are interested to see if thereis any relation between F (2 d ) ( E ), (30), and the responsefunction of the detector moving along a spatial line withconstant non-relativistic speed v = ˜ w/ √ w ≪ w ≪ T . We consider a detector with the same parameters asthat of previous subsection. From (15) the Wightmanthermal Green function for a detector following trajec-tory (19) is G + β (∆ τ ) = − π (∆ τ − iǫ ) + √ − v πβv ∆ τ (cid:20) coth( πγ + ∆ τβ )+ coth( πγ − ∆ τβ ) (cid:21) + 14 π ∆ τ , (35)where γ ± = p (1 ± v ) / (1 ∓ v ), and β = 1 /T . The re-sponse function per unit time of this detector is F th ( E ) = √ − v πβv ln (cid:20) (1 − e − β ∆ Eγ − )(1 − e − β ∆ Eγ + ) (cid:21) , (36)where ∆ E = E − E is the difference between the groundstate and excited state of the detector.For non-relativistic velocities w ≪ F th ( E ) = F (1 d ) ( E ) + F v ( E ) ˜ w + O ( ˜ w ) , (37)where F v ( E ) = − e β ∆ E ∆ E πβ [e β ∆ E − (cid:20) β − β ∆ E (cid:18) e β ∆ E + 1 e β ∆ E − (cid:19)(cid:21) , (38)and F (1 d ) ( E ) is the same as (23).For the infrared tail of the spectrum ∆ E ≪
1, the F th ,(37), is F th ( E ) = 12 πβ [1 −
16 ˜ w ] + O (∆ E ) . (40)The expression (40) can be mapped to the expression(32) if we relate the speed of the detector moving in athermal bath to the one following the trajectory (16) inMinkowski space by the following transformations˜ w = w p − π / √ . w. (41)The detector in the Minkowski space-time moving alongtrajectory (16) should move with a higher speed w =1 .
54 ˜ w to keep up with the same excitation rate of theinertial detector in a thermal bath in the infrared limit.For the ultraviolet tail of the spectrum E ≫
1, theexcitation rate (38) has the following dominant behavior: F ( v ) ( E ) ∼ ∆ E π e − β ∆ E β . (42)This is the same as (33). Therefore, in ultraviolet regime,the dominant modification in the response of the detec-tor following trajectory (16) compared to the black bodyspectrum of Unruh radiation is the same as the domi-nant modification perceived by a detector moving withconstant four-velocity component ˜ w in a thermal bathalong the trajectory (34). IV. CONCLUSION
We have considered the response of an Unruh-DeWittdetector moving along an unbounded spatial trajectory in It is easy to understand (40). Consider two observers immersedin the blackbody radiation, observer O at rest relative to theradiation, thus he/she sees strictly isotropic blackbody radiation,and the other observer, O ′ is moving with speed v along the x-axis of the first observer. The moving observer carries with him adetector with collecting area A , with its normal at angle θ to theaxis. It has been shown, [38–40], that the Lorentz transformationchange the radiation temperature T to an effective directionalradiation temperature T ′ T ′ ( T, v, θ ) = T √ − v − v cos θ , (39)but the observer O ′ looking in the fixed direction θ still wouldmap out a blackbody spectrum. Even though the validity of thisconclusion has been questioned in [37], the author has consid-ered this conclusion to be valid for infrared sector of radiation∆ E ≪ T . An Unruh-DeWitt detector perceives only the radia-tion over the whole angles. Integrating (39) over the solid angle,the average temperature perceived by an Unruh-DeWitt detec-tor is Lorentz transformed according to T ′ = T (1 − v / β ′ = β (1+ v /
6) or β ′ = β [1+ ˜ w/ (6 √ − ˜ w )]. Plugthis transformation of β ′ into the black body excitation rate, andconsider the non-relativistic limit ˜ w ≪ E ≪ T , expression (40) will be reproduced. a two-dimensional spatial plane with constant indepen-dent magnitudes of both the four-acceleration a and ofa timelike proper time derivative of four-accelration, andhaving component of four-velocity w = dy/dτ constant.This is the motion of a charge in a uniform electric fieldwhen in the frame of the charge there is both an electricand a magnetic field. We have compared the responsefunction of this detector (30) and (31) to that of a de-tector moving with constant velocity in a thermal bathof the corresponding temperature T = a/ (2 π ) in non-relativistic limit, in the ultraviolet and in the infraredlimit. The dominant term in the response function isthe Plank distribution, equivalent to a detector movingalong a spatially straight line with constant magnitudeof four-acceleration a . The second dominant term F a inthe response function of this detector, can be mapped tothe second dominant term F v of a detector moving non-relativisticly in a thermal bath via (41) in the infraredlimit. In order to map the response functions of thesetwo detectors in these different situations, the detector in the Minkowski space-time moving along trajectory (16)should move with a higher speed w = 1 .
54 ˜ w to keepup with the same excitation rate of the inertial detectorin a thermal bath in the infrared limit. We also haveshown that in ultraviolet regime the dominant modifica-tion in the response of the detector following trajectory(16) compared to the black body spectrum of Unruh radi-ation is the same as the dominant modification perceivedby a detector moving with constant four-velocity compo-nent ˜ w in a thermal bath along the trajectory (34). ACKNOWLEDGMENTS
The author gratefully acknowledges support by theDFG Research Training Group 1620 “Models of Grav-ity”. I would like to thank professor Don N. Page andDr. Andrey A. Shoom, and Christos Tzounis for valuablesuggestions. I would also like to thank professor ValeriP. Frolov for pointing out the reference [32] to me. [1] W. G. Unruh,
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