Influence of the dispersion relation on the Unruh effect according to the relativistic Doppler shift method
aa r X i v : . [ g r- q c ] F e b Influence of the dispersion relation on the Unruh e ff ect according to the relativistic Doppler shiftmethod F. Hammad, ∗ A. Landry, † and D. Dijamco ‡ Department of Physics and Astronomy, Bishop’s University,2600 College Street, Sherbrooke, QC, J1M 1Z7 Canada Physics Department, Champlain College-Lennoxville, 2580 College Street, Sherbrooke, QC, J1M 0C8 Canada D´epartement de Physique, Universit´e de Montr´eal,2900 Boulevard ´Edouard-Montpetit, Montr´eal, QC, H3T 1J4 Canada
We examine the influence of the dispersion relation on the Unruh e ff ect by Lorentz boosting the phase ofMinkowski vacuum fluctuations endowed with an arbitrary dispersion relation. We find that, unlike what hap-pens with a linear dispersion relation exhibited by massless fields, thermality is lost for general dispersionrelations. We show that thermality emerges with a varying “apparent” Davies-Unruh temperature depending onthe acceleration of the observer and on the degree of departure from linearity of the dispersion relation. The ap-proach has the advantage of being intuitive and able to pinpoint why such a loss of thermality occurs and whensuch a deviation from thermality becomes significant. We discuss the link of our results with the well-knownfundamental di ff erence between the thermalization theorem and the concept of Rindler noise. We examine thepossible experimental validation of our results based on a successful setup for testing the classical analogue ofthe Unruh e ff ect recently described in the literature. PACS numbers: 03.30. + p, 03.70. + k, 02.30.Nw, 04.70.Dy I. INTRODUCTION
Concepts like modified dispersion relations and minimallength have been introduced into physics through various ap-proaches to quantum gravity [1–5]. As soon as they appearedin the literature, they have immediately been put to use astools to investigate deviations from what is predicted for phe-nomena relying on Heisenberg’s uncertainty principle or thosephenomena relying on the usual relativistic dispersion rela-tion (see, e.g., Refs. [6–12] and the review papers [13, 14]).In addition, it is well known that these two concepts are ac-tually related to each other. A minimal length concept, asimplied by the generalized uncertainty principle (GUP) [16–19], does yield a modified dispersion relation as well [14] andvice-versa [20]. See Ref. [15] for a thorough discussion of theinter-relationships between the various concepts. Therefore,any result obtained under the assumption of modified disper-sion relations should also shed some light on cases invokingthe generalized uncertainty principle. Our focus in this paperwill thus be restricted to the former for which a more intuitivepicture of the results can be gained as we shall see.One of the very fundamental applications of the conceptof modified dispersion relations (or minimal length) is foundin the investigation of black hole thermodynamics [21–28].In fact, as the Hawking radiation combines quantum fieldtheory with a curved spacetime [29], one naturally hopes tolearn from such an investigation more about semi-classicaland quantum gravity by working in deformed settings: ei-ther on the classical spacetime side or on the quantum fieldtheory side. These two alternatives o ff er thus two di ff erent ∗ [email protected] † [email protected] ‡ [email protected] approaches. The first possibility allows one to work with ablack hole’s metric background by using the classical metricitself and its possible deformations [30] to investigate blackhole thermodynamics [31] and the outgoing Hawking radia-tion. The second possibility is to take full advantage of theequivalence principle [32]. One can then compute, instead,the spectrum of the deformed quantum fields that would bedetected by an accelerated observer in a Minkowski vacuum, i.e. , using the Unruh e ff ect [33–38]. We shall focus in thispaper on the second possibility rather than the first becausemuch less is rigorously known about the e ff ect of modifieddispersion relations on spacetime at the quantum level. Inaddition, the Unruh e ff ect has been suggested, among otherthings, to be a potential alternative for investigating non-localfield theories [39, 40] as well as various quantum gravity pro-posals [41]. Moreover, as we shall see, it is conceptually veryinstructive to mimic the e ff ect of gravity by letting an acceler-ating observer detect the spectrum of vacuum fluctuations thatwould obey an arbitrary dispersion relation.Several authors have already investigated the influence ofminimal length and modified dispersion relations on the Un-ruh e ff ect based on the standard and widely used methods ofBogoliubov transformations and Wightman’s two-point func-tions (for a review, see, e.g., Refs. [42, 43]). The latter emergewithin the Unruh-DeWitt point-like detector approach [44].However, based on these two di ff erent methods, di ff erent re-sults were reported by various authors. Literature on the e ff ectof minimal length / modified dispersion relations on the Unruhe ff ect can indeed be split into two classes. In one class, onefinds reports concluding that the Unruh e ff ect is preserved— in the sense that thermality emerges — but that Unruh’stemperature acquires a correcting factor. In another class,one learns that the Unruh e ff ect gets destroyed altogether asthermality is lost beyond a certain energy / frequency thresh-old determined by the minimal length / cuto ff frequency im-posed. In Ref. [45], for example, it was found, based on Bo-goliubov transformations, that a GUP-inspired modification tothe commutation relations still implies an Unruh temperaturealbeit modified by a factor that is quadratic in the accelera-tion a of the observer. In Ref. [46], on the other hand, Wight-man’s function is combined with the generalized proper-timeproposal of Ref. [47] to arrive at a di ff erent dependence ofUnruh’s temperature on a (from which a constraint on theconcept of maximal acceleration was suggested). Very im-portant also, is that in these references the mass of the de-tected Rindler particles does not seem to a ff ect the results.In Ref. [48], yet another expression for Unruh temperature interms of acceleration a is obtained based on the so-called ex-tended uncertainty principle (EUP), which introduces the no-tion of minimal momentum through a modified time-energyuncertainty relation.Instead of making use of modified dispersion relations, amodified Wightman’s function was used in Ref. [49] to com-pute the power spectrum which was found to be thermal onlyup to a certain frequency scale that depends on the minimallength introduced inside the deformed Wightman’s function(see also Ref. [50]). On the other hand, using the Unruh-DeWitt detector method combined with a specific dispersionrelation for the massless photons, it was found in Ref. [51]that the power spectrum of the detected modes of the Rindlerparticles deviates from the Planck spectrum by a frequency-dependent factor such that only a limited range of the origi-nal vacuum frequencies yields a positive spectrum (see also,Ref. [52]). In Ref. [53], the Unruh e ff ect was examined byusing Wightman’s function extracted from a deformed propa-gator as implied by minimal length. It was found there that theUnruh e ff ect disappears as thermality is also lost, to be recov-ered only for accelerations a of the observer below a thresholdfixed by the minimal length. Similarly, using the particle de-tector approach, it was found in Refs. [54, 55] that the Unruhe ff ect disappears exponentially as the proper time of the de-tector (observer) exceeds a certain threshold fixed by the pa-rameter κ within a κ -Minkowski spacetime [56, 57] in whichthe commutation relations of the quantum field are deformedas well. In Ref. [58], the e ff ect on Unruh radiation of a super-luminal dispersion relation with a very specific form has beeninvestigated using Wightman’s function. It was concluded thatthe Unruh e ff ect remains a low energy phenomenon as the cor-rection to thermality is inversely proportional to the square ofthe cuto ff scale in the modified dispersion relation.Now, it turns out that the approach based on the Unruh-DeWitt point-like detector can also be viewed as a relativis-tic Doppler shift calculation [37, 59–62]. Although such apoint of view is extremely intuitive, it has not been muchmentioned in the literature regarding the Unruh e ff ect, andnever discussed regarding the case of modified dispersion re-lations. Nevertheless, one of the advantages of the approachis to provide an intuitive picture for the Unruh e ff ect by show-ing that the latter is deeply rooted in classical physics as well(see also, Refs. [63–65]). Another advantage of the approachis to naturally provide a description of the e ff ect in terms ofspontaneous and induced emissions of particles by the detec-tor leading to a thermal spectrum thanks to Einstein’s detailedbalance equation for systems in thermal equilibrium [66]. We thus propose in this paper to examine the influence of modi-fied dispersion relations on the Unruh e ff ect by analyzing thespectrum perceived by an accelerated observer as the formeris shaped by the relativistic Doppler shift caused by the accel-erated motion of the latter. The approach has one more advan-tage of being very general as it easily works for an arbitrarydispersion relation of the field under consideration. Further-more, it provides a precise and a very clear intuitive picture ofwhy there might be a loss of thermality in the detected spec-trum. The superiority of the approach over the Bogoliubovtransformations method when dealing with arbitrary disper-sion relations or the minimal length via the GUP consists alsoin the fact that these imply a modification of the equationsof motion of the field and its propagator (see, e.g., Refs. [67–69]), rendering thus Bogoliubov transformations very di ffi cultto extract.The appearance of the Unruh e ff ect as a result of the rel-ativistic Doppler shift of the vacuum fluctuations has beenvery pedagogically exposed, and in much greater detail, inRef. [70]. In the latter reference, the authors started by ap-plying the approach to classical plane waves and then showedhow the procedure easily adapts to quantized fields. For mass-less scalar fields, the method gives back the usual Planck spec-trum and Unruh temperature. For massless spin- fields, themethod gives back the Fermi-Dirac spectrum [42, 70]. Thus,because the approach yields the same results as the Bogoli-ubov transformations and Wightman’s function approachesfor the case of massless fields, it never seemed necessary toapply the approach to the case of massive fields and, moregenerally, to the case of fields obeying arbitrary dispersion re-lations. It is, however, well known that one does not recovera thermal spectrum with massive fields based on the Unruh-DeWitt detector method. This is in contrast to what the Bogol-ubov transformations-based approach seems to suggest. Thefundamental reason behind this fact, as elaborately explainedin Ref. [42], is deeply rooted in the “thermalization theorem”,extracted from the Bogoliubov transformations, as opposedto the “Rindler noise” associated with the point-like detectorapproach. As we shall see, the relativistic Doppler shift calcu-lation readily allows one to clearly see why the Unruh-DeWittdetector method does not yield a thermal spectrum in the mas-sive case. Moreover, the procedure is easier to apply — bothformally and conceptually — to arbitrary dispersion relationsthanks to its very intuitive nature.The remainder of this paper is structured as follows. InSec. II, we expose the intuitive feature of the relativisticDoppler shift approach by applying it to classical plane wavesobeying an arbitrary dispersion relation. We show how andwhy the Planck spectrum is lost when the dispersion relationof the waves departs from the massless case. In Sec. III, wediscuss briefly how to apply the approach to quantized fieldsobeying an arbitrary dispersion relation. In Sec. IV, we ex-amine in detail the experimental implications of our resultsand their possible validation using the recently successful ex-perimental setup described in Ref. [71]. The more involvedcalculations in this paper are gathered in the two appendicesA and B. We conclude this paper with a brief summary. II. CLASSICAL FIELDS
The first goal of this section is to acquire some intuition forthe relativistic Doppler shift method by applying the latter tothe case of classical plane waves. The other goal is to derivethe useful equations that can be adopted to quantum fields andthat we shall adopt in Sec. IV for an eventual experimentaltest. We shall therefore examine here the spectrum detectedby an accelerating observer by working out the phase shiftcaused by the motion of the latter on classical plane wavesobeying an arbitrary dispersion relation.First, recall that for a particle of mass m , of energy E , andof three-momentum p , the usual relativistic dispersion rela-tion reads, E = p c + m c . The simplest example of amodified dispersion relation for massless particles, often en-countered in the literature, has the form E = p c + β f ( p ),where the parameter β determines the energy scale at whichsuch a modification becomes relevant (see Ref. [72] for exam-ples of non-relativistic versions). For the sake of generality,we shall consider here an arbitrary dispersion relation of theform E = f ( P ), inside which a nonzero mass might be in-cluded. In order to be able to work with waves, however, wehave to consider instead an arbitrary relation between the an-gular frequency ω and the wave vector k which might be ex-tracted from the momentum p and energy E thanks to generalexpressions of the form ω = f ( E , p ) and k = f ( E , p ), re-spectively [72] . Thus, a general modified dispersion relationmight be taken to be of the form ω = f ( k ) for a regular andsmooth function f . For the massless non-deformed case, sucha dispersion relation reduces to the linear relation ω = | k | . Asthe generality of our approach is already guaranteed by the ar-bitrary function f , we shall assume isotropy and consider onlythe one-dimensional case for which the results become moretransparent and free from extra unnecessary transverse terms.Let us therefore consider a one-dimensional classical planewave obeying the dispersion relation, ω = ω ( k ), for an ar-bitrary dependence of the angular frequency ω on the wavenumber k . As we require that the modified dispersion relationreduces to a standard one at low energies and still guaranteea positive ω for high energies, we assume that ω ( k ) ≥ k forclassical and quantum fields. Therefore, at any given point inMinkowski spacetime, the phase of a plane wave obeying sucha dispersion relation and moving either in the negative or posi-tive direction, respectively, is of the form, e i φ ± ( t , x ) = e i [ ω ( k ) t ± kx ] .Let us then examine how this phase becomes a ff ected by themotion of an accelerating observer and what phase wouldthe latter observe by following the same steps exposed inRef. [70].First, the Minkowski time t and position x are related tothe proper time τ of an observer moving, say in the posi-tive x -direction, with constant acceleration a , by the followingRindler coordinate transformation: t ( τ ) = sinh a τ a , x ( τ ) = cosh a τ a . (1) For ease of notation, we shall set in this section the Planck constant ~ aswell as the speed of light c to unity. Therefore, substituting these expressions for x and t inside theabove expression of the phase in Minkowski spacetime, wefind the e ff ective phase detected by the observer to be of theform e i φ ± ( τ ) = exp i " ω ( k ) ± k a e a τ − ω ( k ) ∓ k a e − a τ . (2)Before we examine the resulting detected spectrum as it arisesfrom this expression of the e ff ective phase, a couple of impor-tant remarks concerning (i) our use of plane waves and (ii)our assignment of the hyperbolic motion (1) to the accelerat-ing observer are in order here.The first remark concerns our use of plane waves with amodified dispersion relation. Since our plane waves exhibita four-momentum that obeys modified dispersion relations,these plane waves are taken to be solutions of modified waveequations in the case of classical waves and solutions of mod-ified field equations of motion in the case of free quantumfields [40, 67–69, 73–77]. The modified wave equations be-ing linear, the plane wave solutions are always guaranteedto exist provided only that one requires that the contraction p µ x µ between position and momentum remains linear, i.e. , p µ x µ = p x + p i x i [30, 73]. It is, actually, such a require-ment that allows one to have linear modified Lorentz transfor-mations in position space even though such transformationsare nonlinear in momentum space [73, 78, 79]. In addition, itis such a requirement that leads to the elegant interpretationof the modified dispersion relations in terms of an energy-dependent spacetime metric as “seen” by a quantum parti-cle thanks to the existence of a modified quadratic invariant[73, 78], i.e. , the so-called gravity’s rainbow [30].On the other hand, minimal length, as implied by a non-commutativity of the position and momenta operators, stillallows one to expand a quantum field as usual in terms ofplane waves weighted by creation and annihilation operators[80]. If, however, one decides to replace plane waves by theso-called “maximal localization states” of the GUP-modifiedcommutation relations [18, 81, 82], then one would lose anytrace of the Planck spectrum in this approach because of theloss of a meaningful separation between the position x of theobserver and the position expectation value h ˆ x i in the field ex-pansion. Indeed, one would then associate to the field phasesof the form [83]: exp (cid:26) i (cid:20) ω ( k ) t ± h ˆ x i√ β tan − ( √ β k ) (cid:21)(cid:27) . This ex-pression of the phase does not allow the emergence of thePlanck spectrum because of the lack in it of a symmetry be-tween the position and time coordinates. Such a symmetry isindeed required to give rise to the crucial term e − π Ω / a , as weshall see in detail shortly.The second remark concerns our assignment of a hyper-bolic motion of the form (1) to the observer. In fact, oneshould recall that such a description of the motion in termsof the proper time τ of the observer is obtained based onthe usual Lorentz transformations. Being here interested in-stead in waves and fields that obey modified dispersion re-lations, which automatically violate Lorentz invariance, onemight wonder what allows us to keep using Lorentz boosts.The reason behind assigning such a hyperbolic motion to ourobserver is that we assume the latter’s trajectory to be inde-pendent of the fields and particles he / she is supposed to detect.The observer —like a detector— is taken here to be a macro-scopic object that is not altered by the quantum fluctuationsof the background spacetime inside which it propagates. Thisis unlike the quantum particles and fields whose modified dis-persion relations are precisely due to their interaction with thebackground spacetime [30, 73] (see also Ref. [72] for an in-terpretation in terms of an induced particle species-dependentpseudo-Finslerian geometry). It must be noted in this regardthat such modified Lorentz transformations cannot actually beconsistently applied to bound systems of particles as the lattermay exceed the Planck mass leading to the so-called “soccer-ball problem” [84]. Therefore, while the waves / fields obeymodified Lorentz transformations, the observer’s position andtime coordinates ( x , t ) displayed in Eq. (1) —at which the “on-board sensor” is located, independently of the waves / fieldshitting it— still obey the usual Lorentz transformations. Thisis the natural approach which is consistent with a Lorentztransformation that applies to a passive observer moving alongthe usual macroscopic trajectory while being hit by randomwaves / fields, regardless of when or where the latter have beencreated and where they are coming from.If, instead, one is interested in finding the spectrum of thewaves (or the vacuum fluctuations) as seen by the observerunder the influence of the waves / fields themselves — whichare thus used as probes for the spacetime location of the ob-server — then one has to use the deformed Lorentz transfor-mations, not the linear ones. In other words, one takes in thiscase Lorentz transformations to be active transformations inthe sense that the detected waves / fields are monitored by theobserver from the moment of their creation to the moment oftheir detection. While such an approach does not square wellwith the picture of an observer moving independently of thebackground and randomly hit by these waves / fields, it is, nev-ertheless, very instructive to examine such a possibility, as wedo it in detail in Appendix B. Although it is much easier toanticipate for such a case that the detected spectrum wouldnever be Planckian — due to the combination of highly non-linear modified Lorentz transformations with modified disper-sion relations back-reacting on the transformations — the ap-proach based on a macroscopic observer obeying the usualLorentz transformations provides results that are physicallymuch richer as we shall see now.To get the shape of the spectrum as detected by the accel-erating observer, we Fourier transform the τ -dependent phase(2) using an arbitrarily chosen angular frequency Ω amongthe continuous spectrum of frequencies accessible to the ob-server. As the transform consists in evaluating the integral, g ± ( Ω ) = R + ∞−∞ e i Ω τ e i φ ± ( τ ) d τ , we are going to perform the changeof variable e a τ = y . Such a Fourier transform then simplifiesgreatly and reduces to g ± ( Ω ) = a Z ∞ y ± ν − e ± i (cid:16) ξ y − η y (cid:17) d y . (3)Here, we have distinguished the spectrum g + ( Ω ) of the left-moving modes from the spectrum g − ( Ω ) of the right-moving ones. We have also set, for convenience, ν = i Ω a , ξ = ω ( k ) + k a , η = ω ( k ) − k a . (4)To evaluate integral (3), it is actually more practical to splitthe exponential function into the complex sum of a cosine anda sine function (see Eq. (A1) of Appendix A). In fact, the inte-gral then becomes easier to evaluate by using the tables of in-tegrals given in Ref. [85]. Thus, the expressions of the Fourieramplitudes g ± ( Ω ) in each wave are given in terms of the mod-ified Bessel function K ν ( z ) [85] in the following form, g ± ( Ω ) = e i πν a ηξ ! ± ν K ± ν (cid:16) p ξη (cid:17) . (5)The amplitudes g ± ( Ω ) have thus been found in terms of themodified Bessel function K ν ( z ), but could also be expressed interms of the first Hankel’s function H (1) ν ( z ) by using the well-known link (A2) between these two functions [85]. The ex-pression in terms of the modified Bessel function K ν ( z ) willallow us to easily find an approximation for g ± ( Ω ) when ξη isvery large, whereas the expression in terms of Hankel’s func-tion H (1) ν ( iz ) will allow us to find an approximation for g ± ( Ω )when ξη is very small.Now, we already see from this result that, in contrast towhat one finds when the simple dispersion relation ω = k holds (valid for massless particles), the spectrum that emergesfor an arbitrary dispersion relation cannot be Planckian any-more. In fact, for the latter to show up the final expression ofthe spectrum (5) should display the term Γ ( ν ) which leads tothe famous denominator ( e π Ω / a −
1) which is characteristic ofthe Planck spectrum .In order to search for any hidden Planckian spectrum insideour result (5), we are going to dissect the latter by examiningthe two extreme cases of ξη ≫ ξη ≪
1. These wouldrepresent, respectively, cases of small and large accelerations a compared to the angular frequency ω . However, we shouldkeep in mind that the case ξη ≫ a of the observer as long as the dispersionrelation of the plane wave departs greatly from the linear dis-persion relation ω = k of massless particles. In other words,the case ξη ≫ a but with very large deformations of the dispersion relation ω = k . Similarly, the case ξη ≪ a with small deformations of the dispersionrelation, i.e. , as long as the dispersion relation of the wave be-comes very close —but not identical— to the linear dispersionrelation ω = k .We shall discuss now the two cases separately by using thegeneral infinite series expansion of Hankel’s function H (1) ν ( z )valid for any complex argument z . When the dispersion relation reduces to the linear one ω = k , expression(5) is, of course, not valid anymore for in this case η = g ± ( Ω ) = R ∞ y ± ν − e ± i ξ y d y . This is, in fact, the integral that leads tothe usual Planckian spectrum as it is proportional to Γ ( ν ) which gives rise,thanks to the property (A6), to 1 / sinh( i πν ) from which, in turn, one obtainsthe crucial term 1 / ( e π Ω / a −
1) by using the definition (4) of ν . A. Small accelerations and / or large deformations For small accelerations a compared to the angular fre-quency ω of the wave and / or for large departures from thelinear dispersion relation ω = k , we have ξη ≫
1. Usingthe large-argument expansion of the modified Bessel function(A3), we find the following approximations at the lowest orderin 1 /ξη , g ± ( Ω ) ≈ e − π Ω a a r π √ ξη ηξ ! ± i π Ω a e − √ ξη . (6)We clearly see from this expression that there is no way for thePlanck spectrum, i.e. , for the factor Γ ( ν ), to be recovered bysquaring the amplitudes g ± ( Ω ) and their complex conjugates.In fact, taking the squared magnitude of g ± ( Ω ) we find thefollowing unique result for both amplitudes g ± ( Ω ), | g ± ( Ω ) | ≈ π a √ ξη e − π Ω a e − √ ξη . (7)These amplitudes are exponentially decreasing and the usualdenominator ( e π Ω / a − η .The natural physical interpretation of this result is as fol-lows. For small accelerations of the observer and / or large de-formations of the dispersion relation, the relativistic Dopplershift that a ff ects the original frequencies ω of the plane wavesis not su ffi cient to give the latter the shape of the Planck dis-tribution. In other words, in contrast to the linear case ω = k ,the relativistic Doppler shift of the original spectrum of theplane waves becomes in this case overwhelmed and com-pletely veiled behind the nonlinearity of the deformed disper-sion relation. B. Large accelerations and / or small deformations For large accelerations a compared to the specific angularfrequency ω and / or for small deformations of the dispersionrelation, we have ξη ≪
1. First, using the relation (A2) be-tween the modified Bessel function K ν ( z ) and Hankel’s func-tion H (1) ν ( iz ) we easily re-express the amplitudes (5) in termsof the latter. Then, using the small-argument expansion (A5)of Hankel’s function, we arrive at the following approxima-tion at the leading order in ξη , g ± ( Ω ) = i π a ηξ ! ± ν e i π ( ν ± ν ) H (1) ± ν (2 i p ξη ) ≈ e i πν a (cid:2) Γ ( ∓ ν ) η ± ν + Γ ( ± ν ) ξ ∓ ν (cid:3) . (8)In the first line, the factor e i π ( ν ± ν ) cancels of course from theamplitude g − ( Ω ). By computing the square of the magnitudesfrom this expression, we find the following unique result forboth amplitudes g ± ( Ω ), | g ± ( Ω ) | ≈ e − π Ω a a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ i Ω a !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + cos " θ − Ω a ln( ξη ) . (9) To arrive at the expression in the square brackets we have usedthe fact that for the purely imaginary parameter ν , we havethe identity Re (cid:16) [ Γ ( ν )] ( ξη ) − ν (cid:17) = | Γ ( ν ) | cos h θ − Ω a ln( ξη ) i ,where θ = arg Γ ( ν ). Therefore, for large accelerations of theobserver and / or large deformations of the dispersion relationthe spectrum reads, | g ± ( Ω ) | ≈ π a Ω (cid:16) e π Ω a − (cid:17) cos " θ − Ω a ln( ξη ) . (10)The Planck spectrum is thus recovered with a specific correct-ing factor. If we were to interpret this last expression in termsof the Unruh e ff ect, we would conclude that the observershould detect a slightly deformed thermal spectrum with anUnruh temperature given by T = a / π .This can be interpreted physically as follows. For large ac-celerations of the observer and / or small deformations of thedispersion relation the relativistic Doppler shift of each orig-inal angular frequency ω of the plane waves is large enoughthat deviations of the plane waves from the linear dispersionrelation ω = k have no noticeable e ff ect on the global shapeof the resulting spectrum. The latter then takes the same formas the one obtained for the case of the linear dispersion re-lation except for a minor correcting factor which is closer tounity towards the higher-frequency side of the spectrum. Yet,it is clear that the multiplicative factor cos [ θ − Ω ln( ξη ) / a ]in Eq. (10) is frequency-dependent and thus does deform theglobal shape of the detected spectrum towards the lower-frequency side of the latter. Nevertheless, we can still extractan “apparent” Unruh temperature with a specific frequency-dependent correction as follows.For large accelerations and low frequencies making Ω ≪ a ,the argument θ of the function Γ ( ν ) can be approximated to thethird order in Ω / a by θ ≈ π − Ω γ a − Ω γ a ! + π γ ! , (11)where γ is the Euler-Mascheroni constant [85]. Therefore, themultiplicative factor cos [ θ − Ω ln( ξη ) / a ] in formula (10) canbe approximated up to the third order in Ω / a as well. Then,factoring out from such an approximation the Ω -independentterm, the deviation from the usual Planck spectrum takes thefollowing form, | g ± ( Ω ) | ≈ π Ω a (cid:16) e Ω T − (cid:17) " γ + ln ω − k a ! , (12)where we have introduced in the denominator the followingapparent Unruh temperature, T ≈ a π + Ω a ! π γ − γ ln ω − k a − γ ln ω − k a − ln ω − k a γ +
12 ln ω − k a . (13) For convenience, we also set in the rest of this paper the Boltzmann con-stant k B equal to unity. The power spectrum is clearly frequency-dependent and ex-pression (13) could be identified with a genuine temperatureonly for the low frequencies Ω of the spectrum. It does notonly depend on the probed frequency Ω , but it depends evenon the frequency ω ( k ) of the particular wave that has beenDoppler shifted. As we shall see in the next two sections,these results and interpretations still hold when the procedureis correctly applied to quantized fields and to waves on a watersurface. III. QUANTIZED FIELDS
Let us now discuss the relativistic Doppler shift calculationapproach for an observer accelerating in a Minkowski vac-uum, i.e. , by taking into account the fluctuations of a quantumfield, which we shall take here to be a scalar and neutral quan-tum field for simplicity.As mentioned in the Introduction, the approach based onthe linear dispersion relation ω = k has also been successfullyapplied to the case of a quantum fermion field in Ref. [70].The conclusion drawn in that reference was that the Fermi-Dirac distribution arises naturally as a relativistic Dopplershift e ff ect provided that one takes into account the behav-ior of spinors under Lorentz boosts. Indeed, the only di ff er-ence from the scalar field case is the additional Fermi-Walkertransport of the fermion field that one has to perform to takeinto account the e ff ect of the di ff erent observer’s instanta-neous velocities on the phase of the detected spinor field dur-ing motion. The crucial multiplicative factor 1 / cosh( i πν ) thatgives rise to the Fermi-Dirac distribution, rather than the fac-tor 1 / sinh( i πν ), then emerges naturally [70]. Given that suchan extra factor is added as a multiplicative factor that is in-dependent of the dispersion relation, our conclusions in thissection concerning the scalar field will remain valid for thecase of the fermion field. In other words, the conditions wefind for the appearance of the Planck distribution in the caseof the scalar field will also be valid for the appearance of theFermi-Dirac distribution in the case of a fermion field.Now, the intuition and the physical picture we gainedin the previous section concerning the distinction betweensmall / large accelerations and deformations of the linear dis-persion relation by dealing with classical waves will consti-tute a great input here. In addition, however, a richer physicalpicture and interpretation are now involved as the procedureinvokes not just a varying relativistic Doppler shift, but alsocreation of quanta caused by the accelerated motion of the ob-server.Indeed, in order to be consistent with what we just did forclassical waves, we need to Fourier transform only the phasesthat accompany the operators a k and a † k in the mode expansionof quantum fields, rather than Fourier transforming the wholequantum field operator itself. As a consequence, the randomphases of the Minkowski vacuum fluctuations become auto-matically Doppler-shifted from the point of view of the accel-erating observer. One then only needs to compute the squaredmagnitudes of the resulting Fourier transformed phases. Withthis way of proceeding, one is guaranteed to recover the re- sults of Sec. II where we dealt with classical waves. It is in-deed clear that this way of applying the approach will just takeone through all the steps taken in Sec. II, starting from Eq. (2)and all the way to the very last results (10) and (13).Thus, with the result (7) valid also for the case of quan-tized fields, we conclude that the observer would not detectany thermal spectrum of particles for small accelerations ofhis / her motion and / or large departures of the dispersion rela-tion of such detected particles from the massless case. Sim-ilarly, from Eq. (10) we conclude that a thermal spectrum ofparticles would be detected for large accelerations of the ob-server and / or small departures of the dispersion relations ofthe detected particles from the massless case. Finally, withthe result (13) we conclude that the detected spectrum of par-ticles would look like a deformed Planck spectrum to whichan apparent Unruh temperature can be associated. Such anapparent temperature is, in turn, di ff erent from the familiarUnruh temperature found for massless particles and can onlybe interpreted as an apparent temperature with a correctionterm that is frequency-dependent.A more elaborate and pedagogical presentation of the fun-damental di ff erence between Fourier transforming only thephases that accompany the operators a k and a † k in the mode ex-pansion of quantum fields and Fourier transforming the wholequantum field operator itself will be presented elsewhere. Alink of the present approach with the Unruh-Dewitt detectormethod will then be presented there as well. IV. IMPLICATIONS ON THE EXPERIMENTALLYACCESSIBLE CLASSICAL ANALOGUE OF THE UNRUHEFFECT
Until very recently, no experimental observation of the Un-ruh e ff ect was possible. The obvious reason being that the Un-ruh temperature, as given by Eq. (13) after setting in the latter ω = k and restoring to it the fundamental constants, becomes T = ~ a / (2 π ck B ). Such an expression implies that even an ac-celeration as large as 10 m / s would only produce a temper-ature which is even smaller than the 2 . ff ect with ex-perimentally accessible accelerations one can never hope forbeing able to test the deviations from thermality we derivedhere.Fortunately, there have been many proposals in the lit-erature to get around the technological limitations imposedon any attempt to observe the Unruh e ff ect by focusing, in-stead, on attempts to observe a classical analogue of the e ff ect[71, 86–90]. Accessible laboratory accelerations of the ob-server / detector are indeed su ffi cient for the e ff ect to arise inthis case. As it is already well known [42, 64, 91–93], the Un-ruh e ff ect (and its deviation from thermality as we have seen inSec. II) are not limited to the quantum fields of the Minkowskivacuum. The e ff ect, and the deviations therefrom we derivedhere, extend to classical waves as well. All we would needthen to test experimentally the results we derived here is anykind of waves with a specific dispersion relation that departsfrom linearity. The setup proposed and realized in Ref. [71]is, in this regard, what would best suit our needs.The principle behind such a setup is to replace the vacuumfluctuations of Minkowski spacetime by the gravity waves onthe surface of water subject to white noise. A laser beam isto be emitted perpendicularly toward the surface of the waterto detect the ripples that play the role of vacuum fluctuations[71]. By making such a laser beam translate horizontally witha constant acceleration, the laser spot traveling along the wa-ter surface, together with a camera recording the height of theilluminated spot of the water surface, would play the role of aRindler observer (see Ref. [71] for a description of the actualexperiment). What makes such an experimental setup idealfor our present investigation is that one can easily modify thedispersion relation of the waves simply by adjusting the depthof the water in the container. For a gravity wave of wave-length λ in shallow water of depth h , such that h < . λ , thedispersion relation of the wave becomes linear and takes theform ω ( k ) = k p h , where is the gravitational acceleration atthe location of the experiment [94]. However, for deep water,such that h > λ/
2, the dispersion relation of the wave is non-linear and it takes the form ω ( k ) = p k . The waves’ phasevelocity c p and group velocity c g in the case of deep water arethen given by c p = p / k and c g = c p /
2; whereas for the caseof shallow water both velocities become identical and reduceto the constant p h . Therefore, to achieve an analogue of aRindler observer in deep water, we only need to take c p to bethe limiting speed in lieu of the speed of light c in vacuum.Given that the experiment is performed with standing waves,the smallest wavenumber is k = π/ L , where L is the size ofthe container [71]. Therefore, the limiting speed of the laserbeam will be taken here to be c = p L /π .Let us denote by A ( x , t ) the amplitude of the ripples travel-ing on the water surface along the x -axis. We shall now adaptto our case the analysis presented in Ref. [71] for the case of alinear dispersion relation of the water waves. However, beforeintroducing noise into the water ripples as done in Ref. [71],let us first consider a pure monochromatic plane wave trav-eling on the water surface with a wavenumber k . For such avibration mode, the boundary conditions imposed by the wa-ter container give rise to a standing wave that we can write inthe form [71], A k = √ ω ( k ) e − i ω ( k ) t sin ( kx ) . (14)In accordance with Ref. [71], we have normalized this stand-ing wave so that ( A k , A k ) = δ ( k − k ), where ( A k , A k ) = i R (cid:16) A ∗ k ∂ t A k − A k ∂ t A ∗ k (cid:17) d x is the time-invariant scalar prod-uct for modes obeying the wave equation. Expressing thephase in the standing wave (14) in terms of the proper time τ of the observer using the expression (2) we derived above,and then Fourier transforming the result with an arbitrary an-gular frequency Ω using the prescription (3), we arrive at the following expression, h A k ( Ω ) A ∗ k ( Ω ) i = ω ( k ) | g + ( Ω ) − g − ( Ω ) | = ω ( k ) h | g + ( Ω ) | + | g − ( Ω ) | − g ∗ + ( Ω ) g − ( Ω ) − g + ( Ω ) g ∗− ( Ω ) i . (15)Here, the functions g ± ( Ω ) are now given by g ± ( Ω ) = ca Z ∞ y ± ν − e ± i (cid:16) ξ y + η y (cid:17) d y , (16)and the parameters ν , ξ and η are given by ν = i Ω ca , ξ = kc − ω ( k ) c a , η = kc + ω ( k ) c a . (17)For a later convenience, we have restored here the constant c to our parameters. Note also the slight di ff erence in forms.The exponential in integrals (16) involves the sum ξ y + η y ,whereas the exponential in integrals (3) involves the di ff er-ence ξ y − η y . These di ff erences are due to having assigned,in accordance with Ref. [71], the phase factor e − i ω t instead of e i ω t . Also, the parameters ξ and η are here switched. Nev-ertheless, both parameters are still positive as can be seen byplugging into the parameter ξ our expressions of ω ( k ) for thewater waves.Integrals (16) are evaluated in Eqs. (A8) and (A9) of Ap-pendix A. It is already evident from those expressions that,as with expression (5), the term Γ ( ν ) which would lead to( e π Ω c / a −
1) in the denominator is missing. The result (15)clearly shows that, even for a monochromatic plane wave onthe water surface, there is no trace of a Planck spectrum forarbitrary accelerations a and dispersion relations ω ( k ). More-over, we see that now there are, in addition, the interferenceterms g ∗ + ( Ω ) g − ( Ω ) and g + ( Ω ) g ∗− ( Ω ) due to the reflected wavefrom the boundary. This interference will further a ff ect thedetected spectrum. However, as we saw in Sec. II, one canstill probe limiting cases in the search for traces of the Planckspectrum. Therefore, in analogy with Eq. (9) of Sec. II, we ex-pect that the only way to recover the Planck spectrum wouldbe to have ξη ≪
1, which is equivalent to a large acceleration a and / or a small departure from linearity of the dispersion re-lation. The latter condition can be realised experimentally byadjusting the depth of the water in the container. For ξη ≫ a of the laser beam than to adjust the depth of thewater and the degree of departure from linearity of ω ( k ),we shall consider here the case of large accelerations. Forthat to be achieved, we need and acceleration such that a ≫ √ k c − ω c . Given that the wavenumber is conditionedby k = m π/ L [71], for any positive integer m , we deduce thatfor the case of deep water the acceleration of the laser beamneed only satisfy a ≫ √ m ( m − m = ξ = ω = kc .Inserting now the result (A8) for the first integral inEq. (16), and then using the expansion (A11) of H (1) ν ( z ) for z ≪
1, we easily obtain the amplitude g + ( Ω ) for ξη ≪ H (2) ν ( z ) for z ≪
1, we obtain the amplitude g − ( Ω ) for ξη ≪
1. The resultat the leading order in ξη is the following unified expression, g ± ( Ω ) ≈ ca h e − i πν Γ ( ∓ ν ) η ± ν + e i πν Γ ( ± ν ) ξ ∓ ν i . (18)This result allows us to compute the various terms inside thesquare brackets in Eq. (15). We find, h A k ( Ω ) A ∗ k ( Ω ) i ≈ π c ω a Ω (cid:16) e π Ω ca − (cid:17) × " e π Ω ca sin θ − Ω ca ln η ! − sin θ − Ω ca ln ξ ! . (19)This result is similar to expression (10), but with a correctionterm in which the parameters η and ξ are are now ‘disentan-gled’, and an overall multiplying factor of 1 /ω emerges. Thisis due to the extra deformation of the Planck spectrum causedby the interference with the reflected wave.Similarly to what we did for expression (10), we can probethe low-frequency region of the spectrum. For large acceler-ations and low frequencies such that Ω c ≪ a , the correctionterm in Eq. (19) can be expanded in the ratio Ω c / a . As weare not looking for any apparent Unruh temperature here, weshall keep only terms up to the second order in Ω c / a insidethe correcting factor. We arrive at the following approximateexpression, h A k ( Ω ) A ∗ k ( Ω ) i ≈ π Ω a (cid:16) e π Ω ca − (cid:17) L √ m . (20)This correction depends on the probed frequency Ω and caneasily be measured experimentally in the laboratory as the in-teger m just counts the number of harmonics of the monochro-matic wave traveling on the water surface.We are now going to examine the case of water waves per-meated with white noise and follow step by step the analysisconducted in Ref. [71]. Therefore, we should now express theamplitude A ( x , t ) as a superposition of modes A k with coe ffi -cients α k encoding the noise in the ripples, A ( x , t ) = Z ∞ (cid:16) α k A k + α ∗ k A ∗ k (cid:17) d k . (21)The mode amplitudes α k represent Gaussian noise of uni-form strength I , with the following averages: h α k i = h α k α ∗ k i = I δ ( k − k ) [71]. This is specifically what guar-antees the emergence of a delta function in the frequency forthe detected Doppler shifted noise. Now, inserting the expres-sion (14) into the expansion (21) and using the result (16), weextract the Fourier transformed mode expansion as follows,˜ A ( Ω ) = Z ∞ d k i √ ω ( k ) (cid:16) α k [ g + ( Ω ) − g − ( Ω )] + α ∗ k [ ˜ g − ( Ω ) − ˜ g + ( Ω )] (cid:17) . (22) Here, the amplitudes ˜ g ± ( Ω ) stand for the expressions (5) de-rived in Sec. II. Of course, in those expressions, the parameter η should be taken with its absolute value | η | since for the waterwaves we are considering here, we have kc > ω ( k ) c .The total detected noise can be found by computing h ˜ A ( Ω ) ˜ A ∗ ( Ω ) i [71]. For the case of a linear dispersion re-lation, such a calculation yields a Planckian spectrum accom-panied by δ ( Ω − Ω ). This is rendered possible for two rea-sons. The first is the linearity in the relation ω ( k ) = k p h .The second, is that the amplitudes g ± ( Ω ) yield simply a fac-tor of sin[ θ − Ω ca ln( kc / a )] [71]. These two facts lead to theappearance of the following transformed α ( Ω ) in the integral(22), α ( Ω ) ∼ Z ∞ α k d kk sin " θ − Ω ca ln kc a ! . (23)This integrated mode amplitude has been shown in detail inRef. [71] to be a Gaussian as well, in the sense that one stillhas the averages h α Ω i = h α Ω α ∗ Ω i = I δ ( Ω − Ω ).In our case, however, what we have is not only a nonlin-ear dependence on k in the denominator because of ω ( k ), butalso a nonlinear dependence on k inside the functions g ± ( Ω )which do not yield a simple sine function of ln k . Moreover,this holds even for the case of large accelerations and / or smalldeviations of the dispersion relation from linearity. We seethis by plugging the expression (18) inside the first term mul-tiplying α k in the integral (22). We get, α ( Ω ) ∼ Z ∞ α k d k √ k h e π Ω c a sin (cid:16) θ − Ω ca ln η (cid:17) + e − π Ω c a sin (cid:16) θ − Ω ca ln ξ (cid:17)i . (24)When recalling that inside the parameters η and ξ there hidenonlinear functions of k as well, it is evident that even theGaussian structure guaranteed in the linear case by the trans-formed α ( Ω ) is here lost.We thus clearly see the e ff ect of noise on the detected spec-trum when dispersion relations are allowed. For the case oflarge accelerations and / or small deviations of the dispersionrelation from linearity, each monochromatic mode leads to aslight deviation from the Planck spectrum as given in Eq. (19).But, when all possible modes are combined into a Gaussiannoise, thermalization is simply destroyed. Neither the large-acceleration regime nor the small deviations from linearitycould then help restore the thermal spectrum. V. SUMMARY
We have adapted the relativistic Doppler shift derivation ofthe Unruh e ff ect to the case of a classical plane wave as wellas to the case of a quantized field when both obey a modi-fied dispersion relation of the form ω = ω ( k ). The larger thedi ff erence ω ( k ) − k , the larger the deviation of the dispersionrelation from linearity one witnesses. We saw that the result-ing general power spectrum of the detected waves / particlesdoes not display any Planck-like pattern. As a result, we hadto take into account the acceleration of the observer and thedegree of deviation from linearity of the dispersion relationby distinguishing the two di ff erent cases of (i) small accelera-tions and / or large deviations and (ii) large accelerations and / orsmall deviations. In addition, we found that the approach ap-plies successfully to the classical case and to the quantum casealike.Among the many advantages of the approach is that oneeasily gains an intuitive explanation for the disappearance ofthe Planck spectrum for small accelerations and / or large de-viations of the dispersion relation from linearity. The reasonis that the relativistic Doppler shift in that case gives rise to asmothered Planck spectrum due to the great deviation of thedispersion relation from linearity. In contrast, for large accel-erations and / or small deviations from linearity of the disper-sion relation, an asymptotic Planckian-like spectrum emerges,to which one might associate an apparent Unruh temperature.The intuitive reason being that, although not a purely Planck-ian spectrum, one can regard the latter as so provided that oneaccepts to assign to it a frequency-dependent temperature, thatone might thus call an apparent Unruh temperature.Another advantage of this approach is that it is so flexiblethat it easily accommodates the use of modified Lorentz trans-formations. Indeed, as we discussed in Sec. II and as we sawin detail in Appendix B, it is possible to combine within thisapproach waves / fields with modified dispersion relations to-gether with modified Lorentz transformations. The result isagain physically very transparent and very intuitive in that itshows clearly how the nonlinearity of the dispersion relationsspoils the Planck spectrum.Yet, another advantage of the approach, mathematical innature, is twofold. First, it is clear that to use Wightman’sfunctions one needs first to relate the dispersion relation ofthe field to the deformation of the two-point functions of thetime and space parameters ∆ τ and ∆ x , respectively [49]. Sec-ond, as we argued in the Introduction, the Bogoliubov trans-formations approach cannot be relied on either as (i) it heavilydepends on the equations of motion of the deformed field and(ii) it only leads to the thermalization theorem which is al-ready incapable of distinguishing even the massless case fromthe massive case [42].Finally, as a prospect for an experimental test of our results,we have examined in detail the possibility of using gravitywaves as an analogue substitute for the vacuum fluctuationsof Minkowski spacetime. The role of the detector would beplayed by a light spot made by a laser beam emitted down-wards perpendicularly towards the water surface. We exam-ined two cases, the case of a pure monochromatic standingwave and the case of standing waves permeated with whitenoise. In the first case, thermality emerges corrected for largeaccelerations of the detector. In the second case, thermality isdestroyed no matter what acceleration the detector has as longas the dispersion relation deviates from linearity. Our analy-sis thus showed great promise for experimentally testing thedeviations from thermality we predicted here for waves witha dispersion relation. As we argued in Sec. IV, that goal iseasily achievable when using waves on a water surface, forboth the acceleration of the detector and the dispersion rela-tion of such waves are easily accessible and easily adjustableexperimentally. Appendix A: Evaluating integrals (3) and (15) and finding theirvarious expansions
In this appendix we gather the main identities that were use-ful in the text and we give the detailed calculations leading tothe various formulas found in the text.We start by displaying the two useful integrals involving apower function and a trigonometric function [85], Z ∞ y ν − cos ξ y − η y ! d y = ηξ ! ν K ν (cid:16) p ξη (cid:17) cos πν , Z ∞ y ν − sin ξ y − η y ! d y = ηξ ! ν K ν (cid:16) p ξη (cid:17) sin πν . (A1)In these integrals, both parameters ξ and η are assumed to bepositive, which is the case in Sec. II where we deal with clas-sical and quantum fields. Multiplying the second line by i andadding it to the first, yields the result (5) displayed in Sec. II.Next, we have the following relation between the modifiedBessel function K ν ( z ) and Hankel’s function H (1) ν ( z ) (valid fora complex number z such that − π < arg z ≤ π/ K ν ( z ) = i π e i πν H (1) ν ( iz ) . (A2)On the other hand, the useful series expansions for thefunctions K ν ( z ) and H (1) ν ( z ) are given as follows. For large-magnitude arguments, | z | ≫
1, we use the series expansionfor the Bessel function [85] and then terminate the series atthe zeroth order in z as follows, K ν ( z ) = r π z e − z n − X m = (2 z ) − m Γ (cid:16) ν + m + (cid:17) m ! Γ (cid:16) ν − m + (cid:17) + O (cid:0) z − n (cid:1) ≈ r π z e − z . (A3)For small-magnitude arguments, | z | ≪
1, we find the series ex-pansion for Hankel’s function H (1) ν ( z ) by combining the seriesexpansions of Bessel’s first and second kind functions J ν ( z )and Y ν ( z ), respectively, and then using H (1) ν ( z ) = J ν ( z ) + iY ν ( z )[85]. From such a combination, we easily deduce indeed thefollowing infinite series, H (1) ν ( z ) = ∞ X m = ( − m z m + ν m ! 2 m + ν + i cot( πν ) Γ ( m + ν + − i (cid:16) z (cid:17) ν csc( πν ) Γ ( m − ν + . (A4)For small-magnitude arguments, | z | ≪
1, we may terminatethis infinite series in m at the zeroth order to arrive at the fol-lowing approximation for H (1) ν ( iz ): H (1) ν ( iz ) ≈ e i πν z ν ν " + i cot( πν ) Γ (1 + ν ) − i (cid:16) iz (cid:17) ν csc( πν ) Γ (1 − ν ) ≈ − ie − i πν π (cid:20) Γ ( − ν ) (cid:18) z (cid:19) ν + Γ ( ν ) (cid:18) z (cid:19) − ν (cid:21) . (A5)In the second step we have used the trigonometric identitiescos( ix ) = cosh x and − i sin( ix ) = sinh x which hold for any0real number x . In addition, we have also used the followingtwo properties of the gamma function for a complex argument z and a purely imaginary argument ix , respectively [85]: Γ (1 + z ) = z Γ ( z ) , | Γ ( ix ) | = π x sinh( π x ) . (A6)We need now to evaluate integrals similar to those inEq. (A1), but which involve cos( ξ y + η/ y ) and sin( ξ y + η/ y ),respectively. Such integrals can also be evaluated using thetable of integrals in Ref. [85] (see, p. 480): Z ∞ y ν − cos ξ y + η y ! d y = − π ηξ ! ν h J ν ( z ) sin πν + Y ν ( z ) cos πν i , Z ∞ y ν − sin ξ y + η y ! d y = π ηξ ! ν h J ν ( z ) cos πν − Y ν ( z ) sin πν i . (A7)Here, z stands for 2 √ ξη and the functions J ν ( z ) and Y ν ( z ) are,respectively, Bessel’s first and second kind functions. Mul-tiplying the second line in Eq. (A7) by i and adding it tothe first line, and then using H (1) ν ( z ) = J ν ( z ) + iY ν ( z ) and H (2) ν ( z ) = J ν ( z ) − iY ν ( z ) [85], yields, Z ∞ y ν − e i (cid:16) ξ y + η y (cid:17) d y = i π e i πν ηξ ! ν H (1) ν (cid:16) p ξη (cid:17) , (A8) Z ∞ y − ν − e − i (cid:16) ξ y + η y (cid:17) d y = − i π e i πν ηξ ! − ν H (2) − ν (cid:16) p ξη (cid:17) . (A9)As these integrals directly involve Hankel’s functions H (1) ν ( z )and H (2) ν ( z ) rather than Bessel’s function K ν ( z ), we need tofind the series expansions of the former for both | z | ≪ | z | ≫
1. Using the series expansions for J ν ( z ) and Y ν ( z ) asgiven in Ref. [85], we find the following series expansions for | z | ≫ H (1) ν ( z ) = r π z e i ( z − πν − π ) n − X m = i m Γ (cid:16) ν + m + (cid:17) (2 z ) m m ! Γ (cid:16) ν − m + (cid:17) + O (cid:0) z − n (cid:1) ≈ r π z e i ( z − πν − π ) , H (2) ν ( z ) = r π z e i ( πν + π − z ) n − X m = ( − i ) m Γ (cid:16) ν + m + (cid:17) (2 z ) m m ! Γ (cid:16) ν − m + (cid:17) + O (cid:0) z − n (cid:1) ≈ r π z e i ( πν + π − z ) . (A10)For | z | ≪
1, we already have the expansion (A5) for H (1) ν ( iz )which for a real argument z , leads to H (1) ν ( z ) ≈ − ie − i πν π (cid:20) e − i πν Γ ( − ν ) (cid:18) z (cid:19) ν + e i πν Γ ( ν ) (cid:18) z (cid:19) − ν (cid:21) . (A11)For H (2) ν ( z ), we find the following expansion when z ≪ H (2) ν ( z ) = ∞ X m = ( − m z m + ν m ! 2 m + ν − i cot( πν ) Γ ( m + ν + + i (cid:16) z (cid:17) ν csc( πν ) Γ ( m − ν + ≈ ie i πν π (cid:20) e i πν Γ ( − ν ) (cid:18) z (cid:19) ν + e − i πν Γ ( ν ) (cid:18) z (cid:19) − ν (cid:21) . (A12) Appendix B: Working with deformed Lorentz transformations
In this appendix we examine the case of observers (detec-tors) using the waves / fields they are hit by as probes of theirspacetime location, requiring, as a consequence, the use ofdeformed Lorentz transformations when computing the per-ceived phase φ ( τ ). Now, given the multitude of proposalsintroduced in the literature for modified dispersion relationsand their corresponding modified Lorentz transformations, weshall not consider here every single model introduced in theliterature, but focus instead on the very general model reportedin Ref. [79].For a particle of angular frequency ω ( k ) and wave number k relative to an inertial frame, the modified Lorentz transfor-mation for time in 1 + t ′ = f ( p ) f ( p ′ ) t − kk ′ ωω ′ x q − k ′ ω ′ . (B1)Here, p is the momentum of the particle in the original frameand f ( p ) is an arbitrary function that relates the angular fre-quency to the energy of the particle: ω = E f ( p ). The trans-formed quantities p ′ , ω ′ and k ′ represent the properties of theparticle in a frame moving with instantaneous velocity ( t ).From this expression we extract the relation between the ele-ment of the proper time d τ in terms of d t as follows:d τ = f ( p ′ ) f ( p ) r − k ′ ω ′ d t . (B2)Next, by using the transformation of the position [79], x ′ = f ( p ) f ( p ′ ) x − ω k ′ k ω ′ t q − k ′ ω ′ , (B3)we easily find the formula for the transformation of veloci-ties, from which, in turn, we deduce the relation between theacceleration d / d t in the rest frame of the laboratory and theconstant proper acceleration a in the instantaneous rest frameof the observer. The result is, a = f ( p ) f ( p ′ ) ω k ′ k ω ′ d / d t (cid:16) − k ′ ω ′ (cid:17) / . (B4)By integrating this equation after taking the initial condition = t =
0, we extract the velocity in terms of the properacceleration a and the time t as follows: ( t ) = f ( p ′ ) f ( p ) k ω ′ ω k ′ at q + f ( p ′ ) f ( p ) k ω a t . (B5)Substituting this result into Eq. (B2) and integrating, after tak-ing the initial condition τ = t =
0, allows us to find thetime t in terms of the proper time τ and then the position x interms of the proper time τ as well. We find, t ( τ ) = ω f ( p ) k f ( p ′ ) sinh ( ak τ/ω ) a , x ( τ ) = ωω ′ f ( p ) kk ′ f ( p ′ ) cosh ( ak τ/ω ) a . (B6)1Finally, substituting these expressions of t ( τ ) and x ( τ ) insidethe phase e i φ ± ( t , x ) = e i [ ω ( k ) t ± kx ] of Minkowski spacetime, wefind the e ff ective phase detected by the observer to be of theform, e i φ ± ( τ ) = exp i f ( p ) ω f ( p ′ ) " ω k ′ ± ω ′ k akk ′ e ak τ/ω − ω k ′ ∓ ω ′ k akk ′ e − ak τ/ω . (B7)Let us keep in mind here that both angular frequencies ω ( k )and ω ′ ( k ′ ) in this expression depend nonlinearly on their re-spective wave numbers k and k ′ in the original and the newframes, respectively.To get the shape of the spectrum as detected by the accel-erating observer, we Fourier transform the τ -dependent phase(B7) using an arbitrarily chosen angular frequency Ω amongthe continuous spectrum of frequencies available to the ob-server. Once again, as the transform consists in evaluating theintegral, g ± ( Ω ) = R + ∞−∞ e i Ω τ e i φ ± ( τ ) d τ , we are going to performthe change of variable e ak τ/ω = y . The Fourier transform thentakes the form, g ± ( Ω ) = a Z ∞ y ± ν − e ± i (cid:16) ξ y − η y (cid:17) d y . (B8)Here, we have distinguished the spectrum g + ( Ω ) of the left-moving modes from the spectrum g − ( Ω ) of the right-moving ones. We have also set, for convenience, ν = i ω Ω ak , ξ = ω f ( p ) f ( p ′ ) ω k ′ + ω ′ k akk ′ , η = ω f ( p ) f ( p ′ ) ω k ′ − ω ′ k akk ′ . (B9)To evaluate integral (B8) we follow the same steps describedin Section III. As this integral is identical to integral (3), wefind again that the amplitudes are given in terms of the modi-fied Bessel function K ν ( z ) by, g ± ( Ω ) = e i πν a ηξ ! ± ν K ± ν (cid:16) p ξη (cid:17) . (B10)The important di ff erence, however, is that now the exponent ν that is responsible for giving rise to the Planck spectrumvia the crucial term e − π Ω / a (emerging from the factor e i πν/ inthis expression) is here replaced by the term e − πω Ω / ak . Thedetected angular frequency Ω is thus never isolated from thewaves’ phase velocity ω ( k ) / k . As a consequence, the Planckspectrum can never be “purified” from the e ff ect of the nonlin-ear dispersion relation of the detected waves —no matter howlarge the acceleration a is or how close to linearity the disper-sion relation is— as long as the latter is not exactly linear asin the massless case. ACKNOWLEDGMENTS
The authors are grateful to the anonymous referee for thepertinent comments and insightful remarks that improved ourmanuscript. This work is supported by the Natural Sciencesand Engineering Research Council of Canada (NSERC) Dis-covery Grant (No. RGPIN-2017-05388). [1] D. J. Gross and P. F. Mende, “The high-energy behavior ofstring scattering amplitudes” Phys. Lett. B , 129 (1987).[2] D. J. Gross and P. F. Mende, “String theory beyond the Planckscale”, Nucl. Phys. B , 407 (1988);[3] D. Amati, M. Ciafaloni and G. Veneziano, “Can spacetime beprobed below the string size?”, Phys. Lett. B , 41 (1989).[4] K. Konishi, G. Pa ff uti and P. Provero, “Minimum physicallength and the generalized uncertainty principle in string the-ory”, Phys. Lett. B , 276 (1990).[5] M. Maggiore, “A generalized uncertainty principle in quantumgravity”, Phys. Lett. B , 65 (1993) [arXiv:hep-th / , 044013 (2011)[arXiv:1107.3164].[7] F. Hammad, “ f ( R )-modified gravity, Wald entropy, and the gen-eralized uncertainty principle”, Phys. Rev. D , 044004 (2015)[arXiv:1508.05126].[8] T. Foughali and A. Bouda, “From Fock’s transformation to deSitter space”, Can. J. Phys. , 734 (2015) [arXiv:1605.01943].[9] T. Foughali and A. Bouda, “Dirac’s equation in R-Minkowski spacetime”, Int. J. Theo. Phys. , 2247 (2016)[arXiv:1605.04080].[10] M. Salah, F. Hammad, M. Faizal and A. F. Ali, “Non-singularand cyclic universe from the modified GUP”, JCAP , 035 (2017) [arXiv:1608.00560].[11] N. Takka, A. Bouda and T. Foughali, “Maxwell’s equationsin the context of the Fock transformation and the magneticmonopole”, Can. J. Phys. , 987 (2017) [arXiv:1902.11040].[12] N. Takka and A. Bouda, “Exact form of Maxwell’s equationsand Dirac’s magnetic monopole in Fock’s nonlinear relativity”,Mod. Phys. Lett. A , 1850173 (2018) [arXiv:1902.09440].[13] A. N. Tawfik, and A. M. Diab, “Generalized uncertainty prin-ciple: Approaches and applications”, Int. J. Mod. Phys. D ,1430025 (2014) [arXiv:1410.0206].[14] A. N. Tawfik and A. M. Diab, “Review on generalized un-certainty principle”, Rep. Prog. Phys. , 126001 (2015)[arXiv:1509.02436v2].[15] S. Hossenfelder, “Self-consistency in theories with a min-imal length”, Class. Quantum Grav. , 1815 (2006)[arXiv:hep-th / , 83 (1993)[arXiv:hep-th / , 5182 (1994)[arXiv:hep-th / Rev. D , 1108 (1995) [arXiv:hep-th / , 1687 (2003) [arXiv:gr-qc / ,127 (2018) [arXiv:1807.11552].[21] L. Xiang, “Black hole entropy without brick walls”, Phys. Lett.B , 9 (2002).[22] G. Amelino-Camelia, M. Arzano, Y. Ling and G. Mandanici,“Black-hole thermodynamics with modified dispersion rela-tions and generalized uncertainty principles”, Class. QuantumGrav. , 2585 (2006) [arXiv:gr-qc / , 63 (2007)[arXiv:0704.1261].[24] M-I. Park, “The generalized uncertainty principle in (A)dSspace and the modification of Hawking temperature fromthe minimal length”, Phys. Lett. B , 698 (2008)[arXiv:0709.2307].[25] D. Mania and M. Maziashvili, “Corrections to the black bodyradiation due to minimum-length deformed quantum mechan-ics”, Phys. Lett. B , 521 (2011) [arXiv:0911.1197].[26] Y. Sabri and K. Nouicer, “Phase transitions of a GUP-correctedSchwarzschild black hole within isothermal cavities”, Class.Quant. Grav. , 215015 (2012).[27] B. Vakili and M. A. Gorji, “Thermostatistics with minimallength uncertainty relation”, J. Stat. Mech., P10013 (2012)[arXiv:1207.1049].[28] E. Maghsoodi, H. Hassanabadi and W. S. Chung, “Black holethermodynamics under the generalized uncertainty principleand doubly special relativity”, Prog. Theor. Exp. Phys. ,083E03 (2019).[29] S. W. Hawking, “Particle creation by black holes”, Commun.Math. Phys. , 199 (1975).[30] J. Magueijo and L. Smolin, “Gravity’s rainbow”, Class. Quan-tum. Grav. , 1725 (2004) [arXiv:gr-qc / , 329 (2010) [arXiv:0902.3927v2].[32] A. Ahmadzadegan, E. Martin-Martinez and R. B. Mann, “Cav-ities in curved spacetimes: The response of particle detectors”,Phys. Rev. D , 024013 (2014) [arXiv:1310.5097][33] S. A. Fulling, “Nonuniqueness of canonical field quantizationin Riemannian space-time”, Phys. Rev. D , 2850 (1973).[34] P .C. W. Davies, “Scalar production in Schwarzschild andRindler metrics”, J. Phys. A: Math. Gen. , 609 (1975).[35] W. G. Unruh, “Notes on black-hole evaporation”, Phys. Rev. D , 870 (1976).[36] W. G. Unruh, in Proceedings of the 1st Marcel GrossmannMeeting on General Relativity , edited by R. Ru ffi ni (North-Holland, Amsterdam, 1977).[37] V. L. Ginzburg and V. P. Frolov, “Vacuum in a homogeneousgravitational field and excitation of a uniformly accelerated de-tector”, Sov. Phys. Usp. , 1073 (1987).[38] U. H. Gerlach, “Quantum states of a field partitioned by an ac-celerated frame”, Phys. Rev. D , 1037 (1989).[39] A. Belenchia el al ., “Low energy signatures of nonlo-cal field theories”, Phys. Rev. D , 061902(R) (2016)[arXiv:1605.03973].[40] Y. Gim, H. Um and W. Kim, “Unruh e ff ect of nonlocal fieldtheories with a minimal length”, Phys. Lett. B , 206 (2018).[41] N. Alkofer, G. D’Odorico, F. Saueressig and F. Versteegen,“Quantum Gravity signatures in the Unruh e ff ect”, Phys. Rev.D , 104055 (2016) [arXiv:1605.08015]. [42] S. Takagi, “Vacuum Noise and Stress Induced by Uniform Ac-celeration: Hawking-Unruh E ff ect in Rindler Manifold of Ar-bitrary Dimension”, Prog. Theor. Phys. Supp. , 1 (1986).[43] L. C. B. Crispino, A. Higuchi and G E. A. Matsas, “The Unruhe ff ect and its applications”, Rev. Mod. Phys. , 787 (2008)[arXiv:0710.5373].[44] N. D. Birrell, P. C. W. Davies, Quantum Fields in Curved Space ,(Cambridge University Press, Cambridge, 1982).[45] F. Scardigli, M. Blasone, G. Luciano and R. Casadio, “Modi-fied Unruh e ff ect from generalized uncertainty principle”, Eur.Phys. J. C , 728 (2018) [arXiv:1804.05282].[46] G. G. Luciano and L. Petruzziello, “GUP parameter from max-imal acceleration”, Eur. Phys. J. C , 283 (2019).[47] E. R. Caianiello et al. , “Quantum corrections to the spacetimemetric from geometric phase space quantization”, Int. J. Theor.Phys. , 131 (1990).[48] W. S. Chung and H. Hassanabadi, “Black hole temperature andUnruh e ff ect from the extended uncertainty principle”, Phys.Lett. B ,451 (2019).[49] I. Agullo, J. Navarro-Salas, G. J. Olmo and L. Parker, “Two-point functions with an invariant Planck scale and thermal ef-fects”, Phys. Rev. D , 124032 (2008) [arXiv:0804.0513].[50] I. Agullo al. , “Acceleration radiation and the Planck scale”,Phys. Rev. D , 104034 (2008) [arXiv:0802.3920].[51] B .R. Majhi, E. C. Vagenas, “Modified dispersion relation, pho-ton’s velocity, and Unruh e ff ect”, Phys. Lett. B , 477 (2013)[arXiv:1307.4195v2].[52] V. Husain and J. Louko, “Low Energy Lorentz Violation fromModified Dispersion at High Energies”, Phys. Rev. Lett. ,061301 (2016) [arXiv:1508.05338].[53] P. Nicolini and M. Rinaldi, “A minimal length versus the Unruhe ff ect”, Phys. Lett. B , 303 (2011) [arXiv:0910.2860].[54] H-C. Kim, J. H. Yee and C. Rim, “ κ -Minkowski spacetime anda uniformly accelerating observer”, Phys. Rev. D , 045017(2007) [arXiv:hep-th / κ -deformed space-time”, Phys. Rev. D ,045022 (2012) [arXiv:1206.6179v2].[56] A. Nowicki, E. Sorace and M. Tarlini, “The quantum deformedDirac equation from the κ -Poincar´e algebra”, Phys. Lett. B ,419 (1993) [arXiv:hep-th / κ -relativistic systems”, Ann. Phys. , 90 (1995) [arXiv:hep-th / ff ect”, Phys. Rev. D , 124029 (2008) [arXiv:0802.0618v3].[59] L N. Pringle, “Rindler observers, correlated states, boundaryconditions, and the meaning of the thermal spectrum”, Phys.Rev. D , 2178 (1989).[60] H. Kolbenstvedt, “The principle of equivalence and quantumdetectors”, Eur. J. Phys. , 119 (1991).[61] K. Srinivasan, L. Sriramkumar and T. Padmanabhan, “Planewaves viewed from an accelerated frame: Quantum physics ina classical setting”, Phys. Rev. D , 6692 (1997).[62] K. Srinivasan, L. Sriramkumar and T. Padmanabhan, “PossibleQuantum Interpretation of Certain Power Spectra in ClassicalField Theory”, Int. J. Mod. Phys. D , 607 (1997).[63] G. W. Ford and R. F. O’Connell, “Is there Unruh radiation?”,Phys. Lett. A , 17 (2006) [arXiv:quant-ph / ff ects”, Found. Phys. , 1499 (1999)[arXiv:gr-qc / J. Phys. , 154 (2006) [arXiv:physics / , 095017 (2010) [arXiv:1010.4004].[67] R. C. Myers, M. Pospelov, “Ultraviolet Modifications of Dis-persion Relations in E ff ective Field Theory”, Phys. Rev. Lett. , 211601 (2003) [arXiv:hep-ph / ,025014 (2013) [arXiv:1208.5761].[69] M. Faizal and B. Majumder, “Incorporation of generalized un-certainty principle into Lifshitz field theories”, Ann. Phys. ,49 (2015) [arXiv:1408.3795].[70] P. M. Alsing and P. W. Milonni, “Simplified derivation of theHawking-Unruh temperature for an accelerated observer in vac-uum”, Am. J. Phys. , 1524 (2004) [arXiv:quant-ph / et al. , “Classical analogue of the Unruh e ff ect”,Phys. Rev. A , 022118 (2018) [arXiv:1709.02200].[72] S. Hossenfelder, M. Bleicher, S. Hofmann, J. Ruppert, S.Scherer and H. St¨ocker, “Signatures in the Planck regime”,Phys. Lett. B , 85 (2003) [arXiv:hep-th / , 084007 (2004)[arXiv:gr-qc / , 469 (2010)[arXiv:0711.4851].[75] M. M. Ferreira Jr., J.A.A.S. Reis and M. Schreck, “Dimen-sional reduction of the electromagnetic sector of the nonmin-imal standard-model-extension”, Phys. Rev. D , 095026(2019) [arXiv:1905.04401].[76] M. D. C. Torri, V. Antonelli and L. Miramonti, “Homoge-neously modified special relativity (HMSR): A new possibleway to introduce an isotropic Lorentz invariance violation inparticle standard model”, Eur. Phys. J. C , 808 (2019).[77] D. Park, “Generalized uncertainty principle and D -dimensionalquantum mechanics”, Phys. Rev. D , 106013 (2020)[arXiv:2003.13856].[78] S. Liberati, S. Sonego and M. Visser, “Interpreting doubly spe-cial relativity as a modified theory of measurement”, Phys. Rev.D , 045001 (2005) [arXiv:gr-qc / , 310 (2007).[80] M. Kober and P. Nicolini, “Minimal scales from an extendedHilbert space”, Class. Quantum Grav. , 245024 (2010).[81] T. Matsuo and S. Yuuichirou, “Quantization of fields basedon Generalized Uncertainty Principle”, Mod. Phys. Lett. A ,1285 (2006) [arXiv:hep-th / , 104029 (2012) [arXiv:1205.0158].[83] K. Nozari, F. Moafi and F. Rezaee Balef, “Quantization of FreeScalar Fields in the Presence of Natural Cuto ff s”, Adv. High E.Phys. , 201856 (2012).[84] S. Hossenfelder, “The Soccer-Ball Problem”, SIGMA , 074(2014) [arXiv:1403.2080].[85] I S. Gradshteyn and I. M Ryzhik, Table of Integrals, Series, andProducts , Eds. A. Je ff rey and D. Zwillinger. 7th Edition, (Aca-demic Press, Amsterdam, 2007).[86] G. L. Comer, “Superfluid analog of the Davies-Unruh e ff ect”,arXiv:gr-qc / el al. , “Methods for Detecting Acceleration Radi-ation in a Bose-Einstein Condensate”, Phys. Rev. Lett. ,110402 (2008).[88] A. Iorio and G. Lambiase, “The Hawking-Unruh phe-nomenon on graphene”, Phys. Lett. B , 334 (2012)[arXiv:1108.2340].[89] J. Rodriguez-Laguna et al. , “Synthetic Unruh e ff ect in coldatoms”, Phys. Rev. A , 013627 (2017) [arXiv:1606.09505].[90] G. B. Barros et al. , “Traces of the Unruh e ff ect in surfacewaves”, Phys. Rev. D , 065015 (2020) [arXiv:2001.04429].[91] T. H. Boyer, “Thermal e ff ects of acceleration through randomclassical radiation”, Phys. Rev. D , 2137 (1980).[92] A. Higuchi and G. E. A. Matsas, “Fulling-Davies-Unruh e ff ectin classical field theory”, Phys. Rev. D , 689 (1993).[93] M. Fink and J. Garnier, “How a moving passive observer canperceive its environment ? The Unruh e ff ect revisited”, WaveMotion , 102462 (2020) [arXiv:1901.06575].[94] M. W. Dingemans, Water Wave Propagation over Uneven Bot-toms: Part 1-Linear Wave Propagation , in Advanced Serieson Ocean Engineering Vol.13