Light and Airy: a simple solution for relativistic quantum acceleration radiation
LLight and Airy: a simple solution for relativistic quantum acceleration radiation
Michael R.R. Good , and Eric V. Linder , Physics Department, Nazarbayev University, Nur-Sultan, Kazakhstan Energetic Cosmos Laboratory, Nazarbayev University, Nur-Sultan, Kazakhstan Berkeley Center for Cosmological Physics & Berkeley Lab, University of California, Berkeley, CA, USA (Dated: January 27, 2021)We study the quantum radiation of particle production by vacuum from an ultra-relativistic mov-ing mirror (dynamical Casimir effect) solution that allows (possibly for the first time) analyticallycalculable time evolution of particle creation and an Airy particle spectral distribution. The realityof the beta Bogoliubov coefficients is responsible for the simplicity, and the mirror is asymptoticallyinertial at the speed of light, with finite energy production. We also discuss general relations regard-ing negative energy flux, the transformation to the 1-D Schr¨odinger equation, and the incompletenessof entanglement entropy.
I. INTRODUCTION
Acceleration radiation with finite energy production isphysically well-motivated. In the case of black hole evap-oration, for example, this is a conspicuous sign that theevolution has finished, energetic radiation has stopped,and conservation of energy is upheld. The canonicalmoving mirror model of DeWitt-Davies-Fulling [1–3],for a single perfectly reflecting boundary point in flat(1+1)-D spacetime, has solutions demonstrating in asimple way total finite energy production (e.g. the fourdecade old solution of Walker-Davies which first deriveda finite amount of energy creation [4]). Recently, sev-eral finite energy mirror solutions have been found thatdemonstrate close connections to strong gravitationalsystems. These gravity analog models are called acceler-ated boundary correspondences (ABCs). The finite en-ergy ABC solutions closely characterize interesting well-known curved spacetime end-states, including extremalblack holes (asymptotic uniformly accelerated mirrors[7, 9–12]), black hole remnants (asymptotic constant-velocity mirrors [13–18]) and complete black hole evapo-ration (asymptotic zero-velocity mirrors [4, 19–24]).Despite this progress, it has been very hard to find amirror solution whose particle spectrum is simple. Onlytwo known solutions have analytic forms, one whose spec-trum is an infinite sum of terms [18] and another whichis so lengthy as to be prohibitively cumbersome [19, 20].Consequently, analytic time evolution is impossible tofind for the above spectra. Further investigation of theparticle production at any given moment is hobbled be-cause one must instead resort to numerical analysis andfinite sized frequency-time bins utilizing the discrete na-ture of orthonormal wave packets [22].Motivated by simplicity, we take a step back and con-sider that any Bogoliubov transformation can be brokendown into two types: (1) the trivial unitary transforma-tion with β Bogoliubov coefficient zero, β = 0, indicating The infinite energy ABC solutions correspond to the most well-known spacetimes, e.g. Schwarzschild [5], Reissner-Nordstr¨om(RN) [6], Kerr [7], and de Sitter [8]. no particle production and (2) squeezing transformationswhere the β (cid:54) = 0 is given by a transformation matrix thatis diagonal [25] (see the Bloch-Messiah decomposition orthe theory of singular values). The simplest examples ofthe non-trivial transformations are those where the Bo-goliubov coefficients are real-valued . We therefore lookfor some mirror motion (i.e. ABC) that should lead toa real non-zero beta Bogoliubov coefficient for particlecreation, and anticipate corresponding simplicity in theresulting spectrum.We take the simplest possible choice for global mirrormotion with characteristics leading to the desired realityof the Bogoliubov coefficient, and indeed find a simple so-lution for the particle production spectrum. Remarkably,a transformation to the time domain on this spectrumanalytically gives the particle production at any givenmoment.Our paper is organized as follows: in Sec. II we give avery brief motivation of the connection between the re-ality of the beta Bogoliubov coefficient and the mirrortrajectory properties. We analyze this accelerated tra-jectory in Sec. III, computing the key relativistic dynam-ical properties such as rapidity, speed, and acceleration.In Sec. IV we derive the energy radiated, by analysisof the quantum stress tensor, and in Sec. V we derivethe particle spectrum, finding a unique Airy-Ai form forthe radiation and confirming consistency with the stresstensor results. Finally, in Sec. VI we compute the timeevolution of particle creation analytically. Appendices Aand B discuss some general properties leading to nec-essary negative energy flux, and connecting to the 1-DSchr¨odinger equation, respectively. Appendix C is a noteon the connection between rapidity and entanglement en-tropy. Throughout we use natural units, (cid:126) = c = 1. II. REALITY, ACCELERATION, AND INERTIA
The beta Bogoliubov coefficient controls quantum par-ticle production. In light-cone coordinates ( u, v ), withretarded time u = t − x and advanced time v = t + x ,the moving mirror trajectory f ( v ) gives retarded time a r X i v : . [ g r- q c ] J a n position, and the beta Bogoliubov coefficient is [26] β ωω (cid:48) = − π √ ωω (cid:48) (cid:90) + ∞−∞ d v e − iω (cid:48) v − iωf ( v ) ( ωf (cid:48) ( v ) − ω (cid:48) ) , (1)where ω and ω (cid:48) are the frequencies of the outgoing andincoming modes respectively [27].To maintain finite energy and the simplicity of no in-formation loss, there must not be a horizon at finite time,and the acceleration must vanish at infinity (i.e. the mir-ror motion must be asymptotically inertial). Under theseconditions we can carry out an integration by parts togive β ωω (cid:48) = 12 π (cid:114) ω (cid:48) ω (cid:90) + ∞−∞ d v e − iω (cid:48) v − iωf ( v ) . (2)To guarantee a real-valued beta Bogoliubov coefficient,the mirror trajectory f ( v ) must be an odd function sothat the exponential over the symmetric interval turnsinto a cosine of the argument, i.e. a real valued func-tion. The simplest odd function that accelerates in therequired manner is f ( v ) ∼ v + v . We will find this resultsin not only interesting dynamics, but analytic calculationof particle production spectrum and time evolution. III. TRAJECTORY MOTION
As motivated in the previous section, we expect theaccelerated mirror trajectory f ( v ) = v + κ v , (3)to have interesting physical properties. Here κ is a quan-tity related to the acceleration (and the surface gravityin the black hole case).We can also write the trajectory in spacetime coordi-nates, t = − x + 1 κ ( − κx ) / , (4)taking the real cube root, or x = − t − κ (cid:104) A / A / − + A / A / − (cid:105) , (5)where A ± = 3 κt ± (cid:112) κ t + 8 . (6)Note at late times x → − t + O ( t / ). These forms makeit obvious that asymptotically the mirror travels at thespeed of light.A spacetime plot with time on the vertical axis is givenof the trajectory in Figure 1. A conformal diagram isplotted in Figure 2. We next investigate the dynamics ofthe trajectory Eq. (3). We compute the rapidity η ( v ) by 2 η ( v ) ≡ − ln f (cid:48) ( v )where the prime is a derivative with respect to the argu-ment, η ( v ) = −
12 ln (cid:0) κ v + 1 (cid:1) . (7)From the rapidity we may easily compute the velocity V ≡ tanh η , plugging in Eq. (7), V ( v ) = − tanh (cid:20)
12 ln (cid:0) κ v + 1 (cid:1)(cid:21) = − κ v κ v , (8)and the proper acceleration, which follows from α ( v ) ≡ e η ( v ) η (cid:48) ( v ), α ( v ) = − κ v ( κ v + 1) / . (9)At x = t = 0 = v , the velocity and acceleration arezero. At asymptotic infinity, the velocity is the speed oflight and the acceleration goes to zero. The magnitude ofthe velocity, Eq. (8), along with the proper acceleration,Eq. (9), are plotted in Figure 3. t f ( v ) - - - - x FIG. 1. A spacetime diagram of the mirror trajectory, Eq. (3)with κ = 1. It starts off asymptotically inertial with zero ac-celeration and light-speed velocity and decelerates, eventuallyreaching zero speed (at t = 0), and then accelerates again ap-proaching the speed of light in an asymptotically inertial way.Note that field modes moving to the left will always hit themirror, demonstrating no horizon, despite the mirror acceler-ating to light-speed. I L + I R + I L - I R - i i i + i - FIG. 2. A Penrose diagram of the mirror trajectory, Eq. (3).The mirror is moving at light-speed at v → ±∞ . Since theacceleration is asymptotically zero as v → ±∞ then this mir-ror is asymptotically inertial. The various colors correspondto different maximum accelerations; here κ = 1 , , ,
64 fromred, blue, black, and green.
IV. ENERGY FLUX AND TOTAL ENERGYA. Energy Flux
The quantum stress tensor reveals the energy flux emit-ted by the moving mirror. Typically, one will see [2] F ( u ) = − π { p ( u ) , u } , (10)where the energy flux, F ( u ), is a function of light-conecoordinate retarded time u = t − x [3, 26] and the brack-ets define the Schwarzian derivative. The trajectory inlight-cone coordinates of the mirror is p ( u ) which is theadvanced time position “ v ” as a function of retarded time u . However, since we want advanced time v as the inde-pendent variable, we write the radiated energy flux using f ( v ) [8, 13], F ( v ) = 124 π { f ( v ) , v } f (cid:48) ( v ) − , (11)where the Schwarzian brackets are defined as usual, { f ( v ) , v } ≡ f (cid:48)(cid:48)(cid:48) f (cid:48) − (cid:18) f (cid:48)(cid:48) f (cid:48) (cid:19) . (12)For f ( v ) given by Eq. (3), this yields F ( v ) = κ π − κ v ( κ v + 1) . (13) velocityacceleration - - - - - v FIG. 3. The velocity and proper acceleration as a func-tion of light-cone coordinate advanced time v = t + x forthe mirror trajectory, Eq. (3). At v = 0, the velocity V = 0, but asymptotically | V | → α →
0. The maximum acceleration occursat | α max | = 2 κ/ (3 √
3) = 0 . κ . Here v is in units of 1 /κ and the maximum accelerations happen at advanced time κv = ± / √ . It is clear that asymptotically F ( v ) → v →±∞ . Figure 4 shows the energy flux as a function ofadvanced time v . - - - v F ( v ) FIG. 4. The energy flux, Eq. (13), is asymptotically zero at v = ±∞ . The total energy, as we shall see in Eq. (16), istherefore finite, E = κ/
96. Notice the emission of negativeenergy flux near early and late advanced times. The maxi-mum flux F max = κ / (12 π ) occurs at v = 0 and the minimumflux F min = − κ / (192 π ) occurs at v = ± /κ . The energy fluxcrosses zero, F = 0, at v = ± / ( √ κ ). Here κ = 1. B. Total Energy
The total energy measured by a far away observer at I + R is [28] E = (cid:90) ∞−∞ F ( u ) d u , (14)where integration occurs over retarded time (it takes theenergy time to reach I + R ). Since we are using advancedtime v , we write this with du = d f d v dv to get the Jacobiancorrect, E = (cid:90) + ∞−∞ F ( v ) f (cid:48) ( v ) dv . (15)Plugging in Eq. (3) and Eq. (13) into Eq. (15), with Ja-cobian du/dv = κ v + 1, the simple result is E = κ , (16)which is finite and positive.Physically, the finite value tells us the evaporationprocess stops, similar to the ABC’s of extremal blackholes (asymptotic uniformly accelerated mirrors), blackhole remnants (non-horizon sub-light-speed asymptoticcoasting mirrors), and complete black hole evaporation(asymptotic static moving mirrors). The fact that thetotal energy is positive is consistent with the quantuminterest conjecture [29] as derived from quantum inequal-ities [30]. C. Negative Energy Flux
As seen from Figure 4, there are regions of negativeenergy flux (NEF). This is required by the unitarity sumrule (see Appendices A and B and, e.g., [19]). Theseregions extend for | v | > / ( κ √ E NEF = 2 (cid:90) v =+ ∞ v =+ κ √ F ( v ) f (cid:48) ( v ) d v , (17)which gives an analytic result E NEF = κ (cid:0) − √ π − − √ (cid:1) π = − . κ (18)As a ratio, the emission of NEF to positive energy flux(PEF) is | E NEF | E P EF ≈ . . (19)Note one cannot judge by eye this ratio in Figure 4 dueto the redshift Jacobian f (cid:48) ( v ) in Eq. (17). Another interesting aspect is that because the rapid-ity diverges, so does the entropy flux S = − η/
6. How-ever, since there is no horizon there is no informationloss. This indicates that entanglement entropy is not acomprehensive measure of the unitary, finite energy, in-formation preserving dynamics, due to the inertial lightspeed asymptote (see Appendix C).
V. PARTICLE SPECTRUM
The particle spectrum can be obtained from the betaBogoliubov coefficient, given by Eq. (2) in Sec. II. For theparticular trajectory Eq. (3), as promised the Bogoliubovcoefficient is real, β ωω (cid:48) = − ωκ ) / (cid:114) ω (cid:48) ω Ai (cid:18) ω + ω (cid:48) ( ωκ ) / (cid:19) , (20)which is highly unusual. This corresponds to the Bogoli-ubov transformation being a pure boost without rotation,i.e. there is no phase on the beta coefficient, giving us anatural choice for both field modes and coefficients (andpotentially an action integral whose real part defines thevacuum–vacuum amplitude [31]).To obtain the particle spectrum, we take the modulussquare, N ωω (cid:48) ≡ | β ωω (cid:48) | , which gives N ωω (cid:48) = ω (cid:48) κ / ω / Ai (cid:18) ω + ω (cid:48) κ / ω / (cid:19) . (21)The Airy-Ai function is perhaps most well-known as thesolution to the time-independent Schr¨odinger equationfor a particle confined within a triangular potential welland for a particle in a one-dimensional constant forcefield. The spectrum Eq. (21), | β ωω (cid:48) | , is explicitly non-thermal and plotted as a contour plot in Figure 5. The triangular potential well solution is directly relevant for theunderstanding of electrons trapped in semiconductor heterojunc-tions. ω ' ω FIG. 5. The Airy-Ai spectrum, | β ωω (cid:48) | from Eq. (21), as acontour plot, here with κ = 1. The brighter the contoursthe more particle production. Notice the asymmetry between ω and ω (cid:48) which are uniformly scaled. This asymmetry ulti-mately shows up in the infinite total particle count due tothe infrared divergence of ω in N ω but makes it possible toanalytically integrate N ωω (cid:48) over ω (cid:48) . This demonstrates a new spectrum of radiation ema-nating from a moving mirror trajectory. Eq. (21) can becompared to the late time (equilibrium after formation)spectra of non-extremal black holes (e.g. Schwarzschild,RN, Kerr), N ωω (cid:48) = 12 πκω (cid:48) e πω/κ − , (22)and extremal black holes (e.g. ERN, EK, EKN), N ωω (cid:48) = e − πωc/ A π A (cid:12)(cid:12)(cid:12)(cid:12) K iωc/ A (cid:18) A √ ωω (cid:48) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (23)(For EK, c = √
2; for ERN, c = 2; for EKN, c = A /κ .)Here κ is the surface gravity, i.e. κ = 1 / (4 M ) in the caseof a Schwarzschild black hole, or outer horizon surfacegravity for the RN and Kerr non-extremal black holes. Inaddition, A is the extremal parameter, or the asymptoticuniform acceleration [12] in the case of the mirror system,while K ν is the modified Bessel function of the secondkind with order ν .Furthermore, it is remarkable that the spectrum N ω = (cid:90) ∞ N ωω (cid:48) dω (cid:48) , (24)is analytic, N ω = 2 √ ¯ ω κ Ai (¯ ω ) − Ai(¯ ω )Ai (cid:48) (¯ ω )3 κ ¯ ω / − (cid:48) (¯ ω )3 κ √ ¯ ω , (25) where ¯ ω ≡ ( ω/κ ) / . This analytic N ( ω ) spectrum isplotted in Figure 6 for all κ . - - - ω / κ κ N FIG. 6. The Airy particle spectrum, N ( ω ), Eq. (25). Notethe infrared divergence at ω →
0; the soft particle diver-gence results in infinite total particle count characteristic ofasymptotic coasting mirrors. Larger maximum accelerationas measured by κ results in more particles for a wider rangeof frequencies, i.e. N scales as 1 /κ as seen by the κN curveplotted vs ω/κ . The Airy functions can be reformulated into Besselfunctions using the identitiesAi( x ) = (cid:114) x π K / (cid:18) x / (cid:19) (26)Ai (cid:48) ( x ) = − x √ π K / (cid:18) x / (cid:19) . (27)This turns Eq. (21) into N ωω (cid:48) = 13 π κ q (cid:48) ( q + q (cid:48) ) q K / (cid:18) q + q (cid:48) ) / q / (cid:19) , (28)which has similarities to the extremal black hole expres-sion. Here q = ω/κ , q (cid:48) = ω (cid:48) /κ . For the particle spectrumwe get9 π κ N ω = 2 qK / (2 q/
3) + K / (2 q/ K / (2 q/ − qK / (2 q/ . (29)In the small and large ω limits the leading order termsare, respectively, N ω → √ πω , ω → , (30) → κ πω e − ω/ (3 κ ) , ω → ∞ . (31)The 1 /ω in the small frequency limit (note this is in-dependent of κ ) demonstrates the infrared divergenceleading to an infinite total particle count commonly as-sociated with constant-velocity moving mirror solutions[13–18], that are not asymptotically static (asymptoticzero-velocity [4, 19–24]).To check that the energy is indeed carried away by theparticles, we look for consistency between Eq. (21) andthe total energy, Eq. (16), found from the stress tensor.This is done by quantum summing, E = (cid:90) ∞ (cid:90) ∞ ωN ωω (cid:48) d ω d ω (cid:48) , (32)that is, associating a quantum of energy ω with the par-ticle distribution and integrating over all the frequencies.The result is pleasingly analytic: E = κ . (33)Since this is also the result of Eq. (16), the beta spectrumEq. (21), or Eq. (29), is consistent with the quantumstress tensor, Eq. (13).The time dependence of particle creation can be com-puted via wavepacket analysis treated in Hawking [32],and explicitly numerically computed in [19, 20]. Wavepacket localization, particularly via orthonormal andcomplete sets in the moving mirror model, was first car-ried out in detail in [33]. For completeness, we utilizethe same code to illustrate particle creation in time andpresent the results in Figure 7. The rate of emission ofparticles is finite only in a given time and frequency in-terval which can be seen by these complete orthonormalfamily of wave packets constructed from the beta Bogoli-ubov coefficients, following Hawking’s notation, β jnω (cid:48) = 1 √ (cid:15) (cid:90) ( j +1) (cid:15)j(cid:15) dω e πiωn/(cid:15) β ωω (cid:48) , (34)where j ≥ n are integers. These packets are builtat future right null infinity, I + R , and peak at delayedexterior time, u = 2 πn/(cid:15) , with width 2 π/(cid:15) . Thereforethe vertical axis in Figure 7 has a discrete and intuitivephysical interpretation, giving the counts of a particledetector sensitive to only frequencies within (cid:15) of ω j = j(cid:15) ,for a time 2 π/(cid:15) at u = 2 πn/(cid:15) . Late times correspondto large quantum number n (for the mirror Eq. (5), latetimes have u ≈ t [1 + O ( κt ) − / ]). For excellent timeresolution, only one frequency bin is needed, where theparticles pile up, j = 0, and a relatively large value of (cid:15) resolves the count in time. The text of Fabbri-Navarro-Salas [34] also describes the details needed to reconstructFigure 7 by first packetizing the beta coefficient as donein Eq. (34) and then secondly numerically integratingover ω (cid:48) from 0 to ∞ , and third, computing the results, N jn , N jn = (cid:90) + ∞ dω (cid:48) | β jnω (cid:48) | , (35)for each individual time bin, n , for a set frequency bin, j (in our fine-grained time resolution case, j = 0). Whilethis numerical approach evolves the particle count intime, it is not particularly stream-lined, fast, nor arbi-trarily accurate. In Sec. VI we will find an analytic ap-proach to the evolution process, resolving these issues. n - - - - - - - - - - n N j n ϵ =
10, j = κ = FIG. 7. The particle count in time, via wave packet local-ization. The detector is set with (cid:15) = 10, a relative large value( (cid:15) >
1) in order to get clear time resolution. The scale ofthe system is κ = 1 and the frequency bin is in the lowestpossible j = 0 value, where most of the particle productionoccurs, and finer resolution in time is possible. Notice thereis no plateau, hence indicative of non-thermal radiation. Thisemission includes the ‘phantom radiation’ of soft particles asdescribed in [9]. It is symmetric in delayed time, u , centeredaround time bin n = 0. VI. ANALYTIC TIME EVOLUTION
The spectrum, Eq. (25), is simple enough that analyti-cal time evolution without discrete wave packetization ispossible – possibly uniquely in the literature. Typicallywe would like to employ a Fourier transform convertingfrom frequency to time. Since this does not work out ina straightforward manner, we consider that the Fouriertransform of a radially symmetric function in the planecan be expressed as a Hankel transform. The Hankeltransform, N u = H ( N ω ) / u ranges from −∞ to + ∞ – is analytically tractable forthe spectrum Eq. (25):384 κ N u = 5 F (cid:18) , ,
116 ; 1 , − u κ (cid:19) + 4 F (cid:18) , ,
76 ; 1 , − u κ (cid:19) − F (cid:18) , ,
136 ; 1 , − u κ (cid:19) . (36)The particle spectrum dies off at large times as u − , sothe total number indeed diverges.Turning to the energy, a consistency check can be doneby Hankel transforming the quantum of energy ωN ω , andintegrating over all time. The result for the transform, E u = H ( ωN ω ) /
2, is E u = sinh θ √ πκu − cosh θ √ πu √ κ u + 16 , (37)where θ ≡ sinh − (cid:0) κu (cid:1) . Eq. (37) dies off as u − / for large times, so the total energy is finite. The resultfor the total energy by integrating over all time is alsoanalytic, E = (cid:90) + ∞−∞ E u du = κ , (38)which agrees with the total energy as derived by the stresstensor, Eq. (16), and the total energy as derived by in-tegration of the particle spectrum with respect to fre-quency, Eq. (33).As far as we know, this is the first solution for analytictime evolution of particle production from the quantumvacuum. Notice there is no need to resort to wavepacketdiscreteness as the creation is continuous. Nor have wemade any analytic approximations. A plot of the evolu-tion is given in Figure 8. N u E u - - u FIG. 8. The continuous time evolution of particle creation,Eq. (36), and time evolution of energy quanta, Eq. (37). Here κ = 1 (though N u /κ and E u /κ have invariant forms as afunction of κu ). VII. CONCLUSIONS
An interesting connection between the reality of thebeta Bogoliubov coefficient, asymptotic inertia and finiteenergy, and mirror motion near the speed of light leadsto particle radiation by quantum vacuum that is analyticin the energy flux, simple in the particle spectrum – anAiry function – and, remarkably, analytic expression ofthe time evolution of particle creation.We evaluate the simplest allowed accelerated mirrorwith the needed conditions and derive all these physicalquantities. The Airy mirror is asymptotically inertial, coasting at the speed of light; the total energy radiated isfinite and simply κ/
96 despite a soft particle divergence;the beta Bogoliubov coefficient is given by a real Airy-Aifunction; the particle creation time evolution is analyticand exact.The mirror has no horizon, and so there is no infor-mation loss. The finite energy corresponds to the blackhole analog case where evaporation ceases, related to ex-tremal black holes, remnants, or complete evaporation.The asymptotic inertia is responsible for finite energy,but inertial motions that asymptotically approach thespeed of light do not preserve the interpretation of entan-glement entropy derived from the rapidity as an adequatemeasure of unitarity (see Appendix C).The radiated flux exhibits regions of negative energyflux (NEF); these are required by unitarity for the con-ditions present, and we expand on this “necessity of neg-ativity” in the Appendices, showing it follows directlyfrom the asymptotically inertial (the lack of a horizonensures information conservation ) nature. We furtherconnect it to the 1-D Schr¨odinger equation and interpre-tation of the rapidity as a Lorentz transformation andwavefunction in a potential well defined by the accelera-tion properties.While obtaining a real, and simple, Bogoliubov coeffi-cient is a significant advance, we further derive an ana-lytic particle spectrum (integrating over the beta coeffi-cient squared), time evolution (through a Hankel trans-form), and energy (further integrating over the spectrumtimes frequency). An exact analytic time evolution so-lution for particle production from the quantum vacuummay be unique in the literature. No discrete wave packe-tization is required (although we also show those results,consistent with the analytic one).The techniques of accelerating boundary correspon-dences (ABC) and moving mirrors continue to deliverintriguing insights into connections between acceleration(or surface gravity), particle creation, and information.Furthermore, these lead to interesting directions for re-search in the properties of black holes (for which theyserve as analogs) and quantum information, entangle-ment, and gravity. ACKNOWLEDGMENTS
MG acknowledges funding from state-targeted pro-gram “Center of Excellence for Fundamental and Ap-plied Physics” (BR05236454) by the Ministry of Educa-tion and Science of the Republic of Kazakhstan, and theFY2018-SGP-1-STMM Faculty Development Competi-tive Research Grant No. 090118FD5350 at NazarbayevUniversity. This work is supported in part by the En-ergetic Cosmos Laboratory. EL is supported in part by Information loss occurs from an inertial horizon [35]. the U.S. Department of Energy, Office of Science, Officeof High Energy Physics, under contract no. DE-AC02-05CH11231.
Appendix A: Necessity of Negativity
We emphasize that negative energy flux is a common,and indeed required, component of certain accelerationdynamics. That this follows from unitarity is discussedin [19] and references therein. Here we give two quickderivations.From Eq. (11) and the relations f (cid:48) ( v ) = e − η and α ( v ) = η (cid:48) ( v ) e η , we can write24 πF ( v ) = − e η (cid:2) η (cid:48)(cid:48) + ( η (cid:48) ) (cid:3) = − e η α (cid:48) ( v ) . (A1)This immediately implies − π (cid:90) ∞−∞ dv e − η F ( v ) = (cid:90) ∞−∞ dv dαdv = α (cid:12)(cid:12) ∞−∞ . (A2)Whenever the acceleration α vanishes asymptotically –as it does for any asymptotically inertial dynamics – (orif it is time symmetric), then the left hand side must bezero. Since e − η is positive, then F ( v ) must have negativeregions.This depends only on the conditions mentioned in theprevious paragraph and not on the specific mirror trajec-tory used in this paper. One can also see this even moredirectly in terms of proper time τ :12 πF ( τ ) = − α (cid:48) ( τ ) e η ( τ ) , (A3)so 12 π (cid:90) dτ e − η F ( τ ) = − (cid:90) dτ dαdτ . (A4) Appendix B: Zero-Energy Resonance
The simple harmonic oscillator is the basis of manydiverse physics areas. Here we consider a relation be-tween particle radiation from an accelerated system andthe oscillator equation. Let us adapt the usual form,¨ φ ( t ) + ω ( t ) φ ( t ) = 0 (in the time domain) or φ (cid:48)(cid:48) ( x ) + k ( x ) φ ( x ) = 0 (in the space domain) and write it in termsof the light-cone coordinate retarded time u = t − x , ψ (cid:48)(cid:48) ( u ) + V ( u ) ψ ( u ) = 0 . (B1)We allow the resonance frequency or spring constant tobe spacetime dependent, and write it as V ( u ) for reasonsdiscussed below.We immediately have the consequence that (cid:90) + ∞−∞ du V ( u ) ψ ( u ) = − (cid:90) du dψ (cid:48) du . (B2) This looks quite similar to Eq. (A4). If ψ (cid:48) vanishes atasymptotically early and late times, | u | → ∞ , then wefind that for positive ψ the “potential” V must have neg-ative regions.Let us make the analogy more concrete. If ψ ( u ) = e − η then − ψ (cid:48) ( u ) = α ( u ), the acceleration ; so our constrainton ψ (cid:48) vanishing at infinity is exactly our condition in Ap-pendix A, and the asymptotically inertial case we treatin the main text. Note that indeed ψ is always positive.Now the derivatives of η , and hence ψ , are also relatedthrough the Schwarzian in Eq. (10) to the energy flux F ( u ) – which arises from the acceleration – through V ( u ) ≡ πF ( u ) = − { p ( u ) , u } = η (cid:48) ( u ) − η (cid:48)(cid:48) ( u ) . (B3)Under these definitions, Eq. (B2) is identical to Eq. (A4).Thus again we see the “necessity of negativity”.The derivation in Appendix A relied on acceleratingsystem dynamics while the one here arose from the simpleharmonic oscillator equation. The harmonic oscillatorcan also be related to the 1-D Schr¨odinger equation − (cid:126) m ψ (cid:48)(cid:48) + V ψ = Eψ , (B4)for a spacetime-dependent potential where the “springconstant” k ↔ m ( E − V ) (cid:126) . (B5)Absorbing the (cid:126) and m factors, and taking the zero en-ergy case, we see we can rewrite the Schr¨odinger equationas Eq. (B1). Hence our V ( u ) = 12 πF ( u ) does act like apotential and ψ ( u ) acts like a wave function. The movingmirror differential equation for energy flux, Eq. (B3), andthe zero-energy case with absorption of a negative signinto the definition of the potential, Eq. (B4), correspondsto the physics of resonance transmission for a potential, V ( u ) = V ( − u ), of a 1-D scattering threshold anomaly[36].For the particular trajectory of the main text, we havethe asymptotic condition ψ (cid:48) → x → − x ,on the mirror trajectory f ( v ), Eq. (3), resulting in p ( u ) = u + κ u . (B6)With 2 η ( u ) = ln p (cid:48) ( u ), the rapidity η ( u ) = ln( κ u +1),hence asymptotically + ∞ rather than −∞ without theparity flip, i.e. the mirror approaches an observer locatedat I + R at the speed of light, instead of receding at thespeed of light as is the case with Eq. (3). While ψ (cid:48) ( u ) = − α , it is worth pointing out that ψ ( u ) itself isthe Lorentz transformation (LT) in retarded time from un-tildedto tilded boosted frame ˜ u = e − η u . Here the LT acts like a wavefunction. The wave function form is then ψ ( u ) = 1 √ κ u + 1 , ψ ( ±∞ ) = 0 , (B7)plotted in Figure 9. The wave function is normalized bysetting κ = π so (cid:90) ∞−∞ | ψ ( u ) | d u = 1 . (B8)Plugging Eq. (B6) into the Schwarzian relation, Eq. (10),gives F ( u ) = κ (2 κ u − π ( κ u + 1) . (B9)which is PT symmetric u → − u . Phrasing this as thepotential V ( u ) = 12 πF ( u ) of the Schr¨odinger equationwe see in Figure 9 how the wave function is localizedwithin the potential well. V ( u )| ψ ( u )| - - - - - - - u FIG. 9. The potential Eq. (B3) with Eq. (B9), and the wavefunction Eq. (B7). | ψ | is normalized according to Eq. (B8)where κ = π . The potential maxima occur at u m = ±√ /κ with maximum value V m ( u m ) = κ /
3; the zero crossings areat u = ± / ( κ √ Appendix C: Entanglement Entropy and the Speedof Light
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