Proposal for observing the Unruh effect with classical electrodynamics

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by  Gabriel Cozzella

Virtual observation of the Unruh effect
Gabriel Cozzella,1,∗ Andr´e G. S. Landulfo,2,† George E. A. Matsas,1,‡ and Daniel A. T. Vanzella3,§
1Instituto de F´ısica Teo´rica, Universidade Estadual Paulista,
Rua Dr. Bento Teobaldo Ferraz, 271, 01140-070, S˜ao Paulo, Sa˜o Paulo, Brazil
2Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC,
Avenida dos Estados, 5001, 09210-580, Santo Andr´e, Sa˜o Paulo, Brazil
3Instituto de F´ısica de Sa˜o Carlos, Universidade de S˜ao Paulo,
Caixa Postal 369, 13560-970, Sa˜o Carlos, S˜ao Paulo, Brazil
(Dated: January 13, 2017)
The Unruh effect – according to which linearly accelerated observers with proper acceleration
a = constant in the (no-particle) vacuum state of inertial observers experience a thermal bath of
particles with temperature T =a(cid:126)/(2πk c) – has just completed its 40th anniversary. A “direct”
U B
experimentalconfirmationoftheUnruheffecthasbeenseenwithconcernbecausethelinearaccel-
erationneededtoreachatemperatureof1Kisoforder1020 m/s2. AlthoughtheUnruheffectcan
7 be rigorously considered as well tested as free quantum field theory itself, it would be satisfying to
1
observesomelabphenomenonwhichcouldevidenceitsexistence. Here,weproposeasimpleexper-
0
iment reachable under present technology whose result may be directly interpreted in terms of the
2
Unruh thermal bath. Then, instead of waiting for experimentalists to perform the experiment, we

n usestandardclassicalelectrodynamicstoanticipateitsoutputandshowthatitrevealsthepresence
a of a thermal bath with temperature T in the accelerated frame. Unless one is willing to question
U
J the validity of classical electrodynamics, this must be seen as a virtual observation of the Unruh

2 effect.
1

] Introduction: In1976BillUnruhunveiledoneofthe confirmation have motivated the quest for experimen-
c most interesting effects of quantum field theory accord- tal evidences which could settle the issue. This is not
q
ing to which linearly accelerated observers with proper easy, however, because the linear acceleration needed to
-
r accelerationa=constantintheMinkowskivacuum(i.e., reach a temperature of 1 K is of order 1020 m/s2 (see,
g
no-particle state for inertial observers) detect a thermal e.g., Ref. [11] for a compact recent review). John Bell
[
bath of particles at a temperature [1] (see also note [2]) and Jon Leinaas were the first to go into this by trying

1 to explain the electron depolarization in storage rings
v TU =a(cid:126)/(2πkBc). (1) in terms of the Unruh effect [12]. They achieved par-
6
tial success because the Unruh effect is derived for uni-
4 ThiswasthecompletionofSteveFulling’sdiscoverythat
formly accelerated observers who are associated with a
4 inertial and uniformly accelerated (Rindler) observers
3 would extract distinct particle contents from the same time-translation symmetry, namely, the boost isometry,
0 rather than for circularly moving observers who cannot
field theory [4] and came to clarify Paul Davies’ 1975
. be connected to any analogous global time-translation
1 result [5]. The rather nonintuitive content carried by
symmetry. Another proposal relied on the decay of ac-
0 the Unruh effect (for a curious historical note, see [6]),
7 celerated protons [13]. It was shown that Rindler ob-
namely, that inertial observers in Minkowski vacuum
1 servers need the Unruh effect to understand the decay
would freeze at 0 K while accelerated ones would burn
: ofuniformlyacceleratedprotons[14]-[15]. Unfortunately
v at high enough proper accelerations, was missed at first
(for us – particle physicists may disagree), the proton
i by many, including Joseph Bisognano and Eyvind Wich-
X lifetime in actual accelerators is too long, rendering this
mann, who obtained it independently [8] but seemingly
r observation (on Earth) virtually impossible [16]. Under
a did not realize it up to 1982, when Geoffrey Sewell con-
typical accelerations at the LHC/CERN, the proton life-
nected their theorem to the Unruh effect [9]. By 1984
time would be 103×108 yr! A more promising strategy
(after the publication of Bill Unruh and Bob Wald’s
consists of seeking for fingerprints of the Unruh effect
Ref. [10]), it should have become clear that the Unruh
intheradiationemittedbyacceleratedcharges. Acceler-
effect is necessary to keep the consistency of field the-
atedchargesshouldbackreactduetoradiationemission,
oryinuniformlyacceleratedframesanddoesnotrequire
quivering accordingly. Such a quivering would be natu-
any more experimental confirmation than free quantum
rally interpreted by Rindler observers as a consequence
field theory does. But sporadic claims that the Unruh
of the charge interaction with the photons of the Unruh
effect does not exist or, more often, lacks observational
thermalbath[17]-[18]. ThescatteringofRindlerphotons
by the charge in the accelerated frame would correspond
intheinertialframetotheemissionofpairsofcorrelated
∗Electronicaddress: [email protected] photons [19]. The observation of such a signal could be
†Electronicaddress: [email protected] assignedtotheexistenceoftheUnruhthermalbath. The
‡Electronicaddress: [email protected] difficulty with these proposals lies on the dependence on
§Electronicaddress: [email protected]
ultraintense lasers and they have never been realized. It gruenceatthe(right)Rindlerwedge,i.e.,thez >|t|por-
happens,however,thattheusualLarmorradiationwhich tionoftheMinkowskispacetime,where(t,z,r,φ)arethe
doesnotrequireparamountaccelerations,foritisrelated usual cylindrical coordinates. By covering the Rindler
to the emission probability of single photons, is already wedge with (λ,ξ,r,φ) coordinates, the line element can
enough to unveil the existence of the Unruh effect as fol- be written as
lows[20]: eachphotonemittedbyauniformlyaccelerated
charge,asdescribedbyinertialobservers,correspondsto ds2 =e2aξ(cid:0)dλ2−dξ2(cid:1)−dr2−r2dφ2, (2)
eithertheemissionorabsorptionofazero-energyRindler
where λ,ξ ∈(−∞,∞) are given by
photon to or from the Unruh thermal bath, respectively.
Thus, the very observation of the Larmor radiation can
t=a−1eaξsinhaλ, z =a−1eaξcoshaλ, (3)
be seen as a signal of the Unruh effect. The fact that
a quantum effect [note the (cid:126) in Eq. (1)] may be verified
and a=constant>0. Each Rindler observer will be la-
through a classical phenomenon might sound strange at
beledbyconstantvaluesofξ,r,andφwithcorresponding
first but there is no reason for preoccupation once one proper acceleration ae−aξ.
notes that the (cid:126) in the thermal factor
Acircularlymovingchargeq withmassm andworld-
q
line ξ = 0, r = R, and φ = Ωλ, with R,Ω = constants,
exp[(cid:126)ω /(k T )]=exp(2πω c/a),
R B U R has 4-velocity components
associated with the Unruh thermal bath of Rindler par- (cid:112)
ticles with energy (cid:126)ω , cancels out (see Ref. [21] for a uµ =γ(1,0,0,Ω), γ =1/ 1−R2Ω2, (4)
R
more comprehensive discussion). For some reason, how-
giving rise to the electric 4-current
ever – perhaps because the reasoning above involves the
unfamiliarconceptofzero-energyparticlesorbecausethe quµ
calculationrequiredacertainregularization–,thisresult jµ = δ(ξ)δ(φ−Ωλ)δ(r−R). (5)
Ru0
didnotturnoutconvincingenoughtosettletheissueand
papers disputing the existence of the Unruh effect can Thus, the only free parameters are
still be seen (see, e.g., Ref. [22] and references therein).
R,Ω, anda, (6)
In the present paper we suggest a simple laboratory
experimentwhichshouldbeenoughtomakeitclearthat
where a is the proper acceleration of the Rindler ob-
the Unruh effect lives among us. The idea is to consider
servers at the plane ξ =0.
a phenomenon as simple and technically feasible as in
TheLagrangiandensityoftheelectromagneticfieldA
Ref. [20] and, at the same time, free of unfamiliar con- µ
will be written as in Ref. [20]:
ceptsandtechnicalsubtleties, thusavoidingunnecessary
√
concerns. In order to make our strategy clear, we state L=− −g(cid:2)(1/4)F Fµν +(2α)−1(∇ Aµ)2(cid:3) (7)
µν µ
the experiment in the uniformly accelerated frame and
analyze it assuming we are Rindler observers immersed which leads to the following field equations:
in a thermal bath of Rindler particles with temperature
T. Then, we use our results to guide experimentalists ∇ ∇µA =0 (8)
µ ν
(in inertial laboratories) about what they should seek to
allege the observation of the Unruh effect. According to in the Feynman gauge, α = 1. The four independent
theUnruheffect,T mustequalT giveninEq.(1)butwe solutions A((cid:15),m,k⊥,ωR) of Eq. (8) comprise the two phys-
U µ
shall leave T as a free parameter to be measured by the ical modes labeled by (cid:15) = 1,2 and the pure gauge and
inertial experimentalists by fitting the data. However, nonphysical ones labeled by (cid:15) = 3 and 4, respectively,
rather than sitting back and waiting for experimental- with k ,ω ∈ [0,+∞), and m ∈ Z being the remaining
⊥ R
ists to confirm the prediction T = T , we proceed to quantum numbers.
U
a straightforward calculation in the inertial frame, us- The physical modes, which must satisfy the Lorenz
ing standard electrodynamics, to confirm it by ourselves. condition ∇µA =0 and not be pure gauge, are
µ
This must be seen as a virtual observation of the Unruh
effect unless one doubts standard electrodynamics. Aµ(1,m,k⊥,ωR) =k⊥−1(∂ξfmk⊥ωR,∂λfmk⊥ωR,0,0), (9)
We use metric signature (+,−,−,−) and adopt natu-
ral units in which G = c = (cid:126) = k = 1, unless stated Aµ(2,m,k⊥,ωR) =k⊥−1(0,0,−mfmk⊥ωR/r,−ir∂rfmk⊥ωR),
B (10)
otherwise.
The physical problem: The goal posed by Rindler
where
observers will be to calculate the photon emission rate
fromacircularlymovingchargewithconstantangularve- f =C K (cid:0)k eaξ/a(cid:1)J (k r)eimφe−iωRλ
locityasdefinedbythem,assumingthattheelectromag- mk⊥ωR ωR iωR/a ⊥ m ⊥ (11)
netic (radiation) field is in the Minkowski vacuum, |0M(cid:105), satisfies the scalar wave equation
which they perceive as a thermal state due to the Unruh
effect. Our Rindler observers will be chosen to be a con- (cid:2)e−2aξ(∂2−∂2)−∇2(cid:3)f =0 (12)
λ ξ ⊥ mk⊥ωR
2
with ∇2 ≡ r−1∂ (r∂ ) + r−2∂2 and we have cho-
⊥ r r φ
sen C = [sinh(πω /a)/(2π3a)]1/2 to guarantee that v
ωR R
Eqs. (9) and (10) are properly Klein-Gordon orthonor-
malized. (We recall that pure gauge and nonphysical
modes can be chosen orthogonal to the physical ones.)

Let us define the electromagnetic field operator as
z
Aˆ = (cid:88)∞ (cid:90) ∞dk k (cid:90) ∞dω (cid:88)4 (cid:110)aˆR A(i)+H.c.(cid:111), a
µ ⊥ ⊥ R (i) µ
m=−∞ 0 0 (cid:15)=1
(13)
where we have used the shortcut (i) ≡ ((cid:15),m,k ,ω ).
⊥ R
The annihilation aˆR and creation aˆR† operators satisfy
(i) (i)
the usual commutation relations
[aˆR(i),aˆR(i†(cid:48))]=k⊥−1δ(cid:15)(cid:15)(cid:48)δmm(cid:48)δ(k⊥−k⊥(cid:48) )δ(ωR−ωR(cid:48) ) (14) zE zB
and [aˆR ,aˆR ] = [aˆR†,aˆR†] = 0 for physical modes
(i) (i(cid:48)) (i) (i(cid:48))
FIG. 1: Electrons are injected with velocity (cid:126)v in a cylin-
(cid:15)=1,2. The electromagnetic field is coupled to the cur-
dercontainingsuitableelectric,Ez,andmagnetic,Bz,fields.
rent through the interaction Lagrangian density L =
√ int Radiation is released near the center from an open window,
−g jµAˆ . The current will couple to both physical po-
µ which is surrounded by electromagnetic detectors lying on a
larizations. Theemissionandabsorptionphotonnumber
sphere. This allows us to obtain the radiation spectral de-
dpiesrtrRibinudtiloenr ofobrsefirxveerds’mpraonpdertrtaimnsevienrtseervmaolm∆eτnt,umatkth⊥e, composition from which dNkM⊥/dk⊥ is calculated.
R
tree level, are
where we have used Eqs. (5), (9)-(10), and (13) in
1 dΓRem k−1 dNRem
k⊥m ≡ ⊥ k⊥m Eqs. (15) and (16), “(cid:48)” means derivative with respect to
k⊥ dk⊥ ∆τR dk⊥ the argument and Θ(m) ≡ 0,1/2, and 1 for m < 0,m =
(cid:88) (cid:90) +∞ |AR(i)em|2 (cid:18) 1 (cid:19) 0, and m>0, respectively.
= (cid:15)=1,2 0 dωR ∆τR 1+ eωR/T −1 (15) ratNooryw,exRpienrdimlerenotbtsoerbveerrsunarbeyreinaderytitaol epxrpoeproimseenatalalibsots-
andpredictitsoutputasafunctionofthefreeparameter
and
T. The confirmation of the equality T = T should be
U
1 dΓRabs k−1 dNRabs seen as an as-direct-as-possible verification of the Unruh
k⊥m ≡ ⊥ k⊥m
k dk ∆τ dk effect by an inertial-lab-based experiment.
⊥ ⊥ R ⊥
The inertial-laboratory experiment: Let us set
= (cid:88) (cid:90) +∞dω |AR(i)abs|2 1 , (16) a pair of homogeneous and constant electric, Ez =
(cid:15)=1,2 0 R ∆τR eωR/T −1 mqγa/q, and magnetic, Bz = −mqΩγ/q, fields defined
by the free parameters (6) along the z direction. Then,
where |ARem| = |ARabs| = (cid:12)(cid:12)(cid:82) d4x√−gjµ(cid:104)0 |Aˆ |i(cid:105)(cid:12)(cid:12) ∝ a charge q is injected with transverse and longitudinal
(i) (i) (cid:12) R µ (cid:12)
velocity components, v and v , respectively, in such a
δ(ω − mΩ) and the Bose-Einstein thermal factors in ⊥ (cid:107)
R way that its 4-velocity uα – satisfying the Lorentz law
Eqs. (15) and (16) are present because of the thermal
of force m uβ∇ uα = qFαuβ – is given by Eq. (4) [23].
bath in the accelerated frame. However, instead of set- q β β
In the usual cylindrical coordinates, (t,z,r,φ) with the
ting T = T ≡ a/(2π), as would be enforced by the
U
z axis aligned with the 3-acceleration of the Rindler ob-
Unruh effect, here we leave T as a free, independent pa-
servers, the Minkowski line element is ds2 =dt2−dz2−
rameter to be set by fitting the data measured in the
dr2−r2dφ2, uα =γ(cosh(aλ),sinh(aλ),0,Ω), and
inertial laboratory. (Note, from the amplitudes above,
thatwerethechargelinearlyaccelerated,Ω=0,thecur-  0 Ez 0 0 
rent would only interact with zero-energy Rindler pho-
−Ez 0 0 0
tons ωR = 0 [20].) The corresponding total distribution Fαβ = 0 0 0 −rBz. (18)
rate (i.e., emission plus absorption) is computed to be
0 0 rBz 0
dΓRtot q2k (cid:104)
k⊥m = ⊥Θ(m) |K(cid:48) (k /a)|2|J (k R)|2 AprototypeexperimentalapparatusisshowninFig.1,
dk⊥ π2a imΩ/a ⊥ m ⊥ where a sub-picosecond charged bunch containing ∼107
+(RΩ)2|K (k /a)|2|J(cid:48) (k R)|2(cid:105) electrons [24] is injected in a cylinder containing Ez and
imΩ/a ⊥ m ⊥ Bz. The radiation released near the center (where the
(cid:18) (cid:19) (cid:18) (cid:19)
πmΩ mΩ charges are assumed to make the U turn) through an
×sinh coth , (17)
a 2T open window of length L is collected by detectors set
3
on a sphere with radius R (cid:29) L. Since the charges
S
emit radiation at typical wavelengths λ ∼ 1/a , where
√ tot
a = γ2 a2+R2Ω4 is the charge total proper ac-
tot
celeration, we should require a ∼ 1/λ (cid:29) 1/L ≈
tot
1017 m/s2×(1 m/L) in order to avoid finite-size effects
coming from the window. We note that magnetic and
electric fields Bz ≈ 10−1 T and Ez ≈ 1 MV/m, respec-
tively, achievable under present technology [26], produce
accelerationsa∼1017 m/s2 anda ∼1019 m/s2,where
tot
we have assumed R ∼ 10−1 m. We also note that the
radiation backreaction on the charge trajectory is negli-
gible [27].
The relevant quantity to be measured by the inertial
experimentalists is the spectral-angular distribution FIG.2: Forthesakeofillustration,weplotEq.(17)summed
inmfordifferentvaluesofT assumingEz =1MV/m,Bz =
dE(ω,θ,φ) 10−1 T, R = 10−1 m, and injection energy 3.5 MeV. The
I(ω,θ,φ)≡
right-handsideofEq.(21)correspondstothesolid-linecurve.
dωd(cosθ)dφ
of the emitted radiation energy E. From that, one can
calculate the corresponding number of photons by calculation to prove it – to the extent one trusts classi-
cal electromagnetism. The spectral-angular distribution
dNM = dE = I(ω,θ,φ)dωd(cosθ)dφ, (19) I(ω,θ,φ)isexpressedintermsoftheangulardistribution
ωθφ ω ω of the radiated electric field E(cid:126) (t,θ,φ) [28]:
rad
which leads to the k -distribution of radiated photons
dNM (cid:90) 2π ⊥(cid:90) +∞ dk I(ω,θ,φ)= RπS2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) +∞dt E(cid:126)rad(t,θ,φ)e−iωt(cid:12)(cid:12)(cid:12)(cid:12)2. (22)
k⊥ =k dφ z I(ω,θ,φ) (20) −∞
dk ⊥ (k2 +k2)3/2
⊥ 0 −∞ ⊥ z For our accelerated point-like charge, Eq. (22) can be
(recallthatω2 =k2 +k2,k =ωsinθ,anddωd(cosθ)= written as (see, e.g., Ref. [28])
⊥ z ⊥
ω−2k⊥dkzdk⊥). This is the quantity for which the uni- q2ω2 (cid:12) (cid:12)2
formly accelerated observers can make a prediction, for, I(ω,θ,φ)= (cid:12)F(cid:126)(ω,θ,φ)(cid:12) , (23)
4π2 (cid:12) (cid:12)
according to the Unruh effect [10], each emission of a
Minkowski photon according to inertial observers corre- with
sponds to the absorption or emission of a Rindler pho- (cid:90) +∞ d(cid:126)r
ton from or to the Unruh thermal bath, respectively [29]. F(cid:126)(ω,θ,φ)=rˆ× dλ qf(λ), (24)
dλ
Therefore, the validity of the Unruh effect demands −∞
(cid:12) where (cid:126)r (λ) = Rcos(Ωλ)ˆi+Rsin(Ωλ)ˆj+a−1cosh(aλ)kˆ
dNkM⊥ ∝ (cid:88)+∞ dΓRk⊥tmot(cid:12)(cid:12) is the chqarge trajectory ({ˆi,ˆj,kˆ} being the usual Carte-
dk dk (cid:12)
⊥ m=−∞ ⊥ (cid:12)T=TU sfi(aλn) v≡erseoxrps)(cid:2),−irˆω(cid:0)griˆv·e(cid:126)rs (tλh)e−oab−se1rsvianthio(anλ)d(cid:1)i(cid:3)r.ectTiohne, inatned-
= q2k⊥ (cid:88)+∞ Θ(m)(cid:104)|K(cid:48) (k /a)|2|J (k R)|2 grals in Eq. (24) can beq solved by using
π2a imΩ/a ⊥ m ⊥
m=−∞ (cid:90) ∞
(cid:105) (cid:18)πmΩ(cid:19) dζζ−imΩ/a+α−1exp[(iω/(2a))(Ξ−ζ−Ξ+/ζ)]
+(RΩ)2|KimΩ/a(k⊥/a)|2|Jm(cid:48) (k⊥R)|2 cosh a . 0
=2iαexp[πmΩ/(2a)][tan(θ/2)]−(imΩ/a−α)/2
(21)
×K (ωsinθ/a) (25)
imΩ/a−α
for α = 0,±1 and Ξ ≡ 1±cosθ [30], combined with
The proportionality factor appears because the total ±
relations [32]
number of photons depends on how long the experiment
is run. In Fig. 2, we plot the right-hand side of Eq. (17) cos(φ−Ωλ)exp[−ik Rcos(φ−Ωλ)]
⊥
summed in m for different values of T. The prediction (cid:88)
=− im+1J(cid:48) (−k R)exp[im(φ−Ωλ)] (26)
given in Eq. (21) is represented by the solid-line curve m ⊥
(T = T ). We must keep in mind that due to finite- m
U
size effects coming from the window, greater experimen- and
tal care must be taken in the region k (cid:46)1/L.
⊥
VirtualconfirmationoftheUnruheffect: Rather Rk⊥sin(φ−Ωλ)exp[−ik⊥Rcos(φ−Ωλ)]
than waiting for experimentalists to confirm the predic- =(cid:88)mimJ (−k R)exp[im(φ−Ωλ)], (27)
m ⊥
tion (21), here we perform a classical-electrodynamics
m
4
where, again, “(cid:48)” means derivative with respect to the field operator is expanded in terms of inertial normal
argument. Then, by using Eq. (23) into Eq. (19), we modes (30)-(31):
obtain
d2NkM⊥kz = 2q2k⊥ (cid:88)+∞ exp(πmΩ/a) Aˆα = (cid:88)∞ (cid:90) ∞dk⊥k⊥(cid:90) ∞ dkz (cid:88) (cid:110)aˆ(j)A(αj)+H.c.(cid:111).
dk dk πa2ω m=−∞ 0 −∞ (cid:15)=I,II
⊥ z
m=−∞
×(|Jm(k⊥R)|2|Ki(cid:48)mΩ/a(k⊥/a)|2 WenotethatEq.(32)[incontrasttoEqs.(15)-(16)]does
+(RΩ)2|J(cid:48) (k R)|2|K (k /a)|2). (28) not carry any thermal factor because the Minkowski vac-
m ⊥ imΩ/a ⊥
uum is a no-particle state according to inertial observers.
Finally, by integrating it in k and performing the redef- Then, from Eq. (32) we obtain explicitly
z
inition k →κ ≡k /k , we obtain
z z z ⊥
ddNkkM⊥ = (cid:18)4aπ (cid:90) +∞ (1+dκκ2z)1/2(cid:19) (cid:88)+∞ dΓdkRk⊥tmot(cid:12)(cid:12)(cid:12)(cid:12) . ddNkk⊥M⊥ = ∆τ2Rπq22ak⊥ m(cid:88)=∞−∞exp(πmΩ/a)
⊥ −∞ z m=−∞ ⊥ (cid:12)T=TU ×(|Jm(k⊥R)|2|Ki(cid:48)mΩ/a(k⊥/a)|2
(29)
+R2Ω2|J(cid:48) (k R)|2|K (k /a)|2). (33)
m ⊥ imΩ/a ⊥
It is easy to see that Eq. (33) is in agreement with
This concludes our proof of Eq. (21). The fact that the
Eq. (21) and constitutes an inertial quantum field dou-
terminsidetheparenthesisdivergesisbecausethecalcu-
ble check that Rindler-observers’ prediction, T = T , is
lationabovewascarriedoutassumingachargeaccelerat- U
indeed correct.
ing from past to future infinity, in which case an infinite
number of photons is emitted for fixed k element. Of Conclusions: Wehaveproposedasimpleexperiment
⊥
course, in real experiments no divergence appears. where the presence of the Unruh thermal bath is codi-
Double check: As adouble confirmationof Eq.(21), fiedintheLarmorradiationemittedfromanaccelerated
we recover the same result of the previous section using charge. Then, we carried out a straightforward classical-
plain quantum field theory in the inertial frame. In this electrodynamicscalculation(checkedbyaquantum-field-
case, the physical modes of the electromagnetic field in theory one) to confirm it by ourselves. Unless one chal-
cylindrical coordinates (t,z,r,φ) are lenges classical electrodynamics, our results must be vir-
tually considered as an observation of the Unruh effect.
A(αI,m,k⊥,kz) =k⊥−1(kzgmk⊥kz,0,0,−ωgmk⊥kz), (30)
A(αII,m,k⊥,kz) =k⊥−1(0,−mgmk⊥kz/r,−ir∂rgmk⊥kz,0),
(31) Acknowledgments
where gmk⊥kz ≡ (8π2ω)−1/2Jm(k⊥r)eimφeikzze−iωt. Acknowledgments: We are grateful to A. J. Roque
The k⊥-distribution of Minkowski photons per fixed m daSilvaandtheMicrotrongroupattheUniversityofS˜ao
(per Rindler-observers’ proper time interval) is Paulo for explanations on electron accelerators. G. M. is
indebted to A. Higuchi for various discussions. G. C.
1 dΓMk⊥emm = k⊥−1 dNkM⊥m =(cid:88)II (cid:90) +∞dk |AM(j)em|2, and A. L., G. M., D. V. were fully and partially sup-
k dk ∆τ dk z ∆τ portedbyS˜aoPauloResearchFoundation(FAPESP)un-
⊥ ⊥ R ⊥ (cid:15)=I −∞ R derGrants2016/08025-0and2014/26307-8,2015/22482-
(32)
where |AMem| = (cid:12)(cid:12)(cid:82) d4x√−gjα(cid:104)0 |Aˆ |j(cid:105)(cid:12)(cid:12), jα is given 2, 2013/12165-4, respectively. G. M. was also partially
(j) (cid:12) M α (cid:12) supported by Conselho Nacional de Desenvolvimento
in Eq. (5), (j) = ((cid:15),m,k ,k ), (cid:15) = I,II, and here the Cient´ıfico e Tecnol´ogico (CNPq).
⊥ z
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6