Table Of ContentNovember1,2016 0:58 ws-procs961x669 WSPCProceedings-9.61inx6.69in Good˙Reflections˙on˙a˙Black˙Mirror
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Reflections on a Black Mirror
MichaelR.R.Good
PhysicsDepartment, Nazarbayev University,
Astana, Republic of Kazakhstan
E-mail: [email protected]
6
1
0 Ablackmirrorisanacceleratedboundarythatproducesparticlesinanexactcorrespon-
2 dencetoanevaporatingblackhole. Weinvestigatethespectraldynamicsoftheparticle
creationduringtheformationprocess.
t
c
Keywords:blackholeevaporation, movingmirror,dynamicalCasimireffect
O
To understand particle creation from black holes, it is essential to consider the
0 time-dependent formation phase. Recently, it was shown1–3 that there is an exact
3
correspondence between a black hole and a moving mirror. That is, for a massless
] minimally coupled scalar field in a (1+1)D spacetime in which a black hole forms
c
fromthecollapseofanullshell,andforthequantumamplificationofmodesforthis
q
- field which results from an accelerating moving mirror with a particular trajectory
r
g in (1+1)D, the particle creation is the same. The moving mirror in this case is
[ called a black mirror.
2 First,beforeweinvestigatethe blackmirror’searly-timeparticlecreation,allow
v us to take a moment to inquire about a system which emits thermal radiation
3 forever4. In this case there is no time dependence for the particle creation. This
8
6 system, unlike the black mirror, is in equilibrium. The particles residing in each
0 frequency mode are Planckian:
0
1
. N = , (1)
2 ω e2πω/κ 1
0 −
with temperature T = κ/2π. By construction, the immutable constant emission
6
1 of energy (F = κ2/48π) of this Carlitz-Willey4 moving mirror is echoed by the
: eternal constant emission of Planckian distributed particles. The relevant beta
v
i Bogoliubov coefficient for this eternally thermal moving mirror (henceforth called
X
the equilibrium mirror) is
r
a ′ −iω/κ
1 ω ω iω
βωω′ =−2πκrω′e−πω/2κ(cid:18)κ (cid:19) Γ(cid:20) κ (cid:21). (2)
OnecanseetheformofthePlanckiandistributionofEq.1bycomplexconjugating:
1 1
|βωω′|2 = 2πκω′e2πω/κ 1 . (3)
−
′
Albeit, there is a pre-factor of 1/2πκω which is responsible for the total infinite
′
production of particles, residing in mode ω, when one integrates over ω :
∞
Nω = dω′ βωω′ 2 . (4)
h i Z | |
0
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To avoid infinite particle production, and to remove this pre-factor in Eq. 3, one
most localize the production of particles in time and frequency. Following Hawk-
ing5,oneusesorthonormalandcompletewavepacketstopacketizeandlocalizethe
beta Bogoliubov coefficient in Eq. 2. Then, one has packetized beta Bogoliubov
coefficients6–8,
1 (j+1)ǫ 2πiωn
βjnω′ = dω e ǫ βωω′ . (5)
√ǫZ
jǫ
The valuesofnandj designatethe time andfrequencybins respectfully. The total
number of particles in any particular packet is then,
∞
Njn = dω′ βjn,ω′ 2 ,
h i Z | |
0
∞ (j+1)ǫ dω (j+1)ǫ dω
= Z dω′Z √ǫ1 Z √ǫ2e2πi(ω1−ω2)n/ǫβω1ω′βω∗2ω′ ,
0 jǫ jǫ
1 (j+1)ǫ (j+1)ǫ
= dω dω e2πi(ω1−ω2)n/ǫ N , (6)
ǫ Z 1Z 2 h ω1ω2i
jǫ jǫ
where
∞
′ ∗
hNω1ω2i≡Z dω βω1ω′βω2ω′ . (7)
0
The particlesarelocalizedinthe rangeoffrequenciesjǫ6ω 6(j+1)ǫ. They peak
in the range of times (2πn π)/ǫ 6 u 6 (2πn+π)/ǫ. Further details about these
−
wave packets are discussed in Fabbri8 and Good-Anderson-Evans6. In the case of
the equilibrium mirror, the packetized particle count is founda exactly:
1
N = δ(ω ω ). (8)
h ω1ω2i eω2/T 1 2− 1
−
Toseehowthereisnotimedependence,oneintegrateswiththeDiracdeltafunction
bypluggingEq.8inthelastlineofEq.6. Thiscausesthetimedependencetocancel
out because the n time bin values, in the term e2πi(ω1−ω2)n/ǫ, are negated by the
Dirac delta function. Therefore, the particle count, for the equilibrium mirror,
becomes eternally time independent, N N :
jn j
h i→h i
1 (j+1)ǫ 1
N = dω . (9)
h ji ǫ Z 1eω1/T 1
jǫ −
This integral is easy to do and the final result is:
1 1
N = log[sinh(j+1)ǫ)csch(jǫ)] (10)
j
h i 2ǫ − 2
aThis delta function comes about with help from using the integral: R∞dω′ω′i(ω2−ω1)−1 =
0
2πδ(ω2−ω1).
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where ǫ ǫ/2T,or alternatively expressed,
≡
T e(j+1)ǫ/T 1
N = log − 1. (11)
h ji ǫ (cid:20) ejǫ/T 1 (cid:21)−
−
This is the exact Planckian distribution. It is this expression, Eq. 11, which will
allowustonumericallyconfirm,the late-timethermalemissionoftheblackmirror.
This is the reason for investigating the equilibrium mirror first. The equilibrium
emission will help us explore non-equilibrium emission.
It is easy to check, that in an expansion of ǫ, one can obtain the more familiar
Planckianform. Keepingthecenterfrequency, ω (j+1/2)ǫ,constant,anddoing
j
≡
the series expansion, then N N , and Eq. 11 becomes the usual Planckian
h ji→h ωji
distribution:
N = N +O(ǫ2), (12)
h ji h ωji
where
1
N (13)
h ωji≡ eωj/T 1
−
andT =κ/2π. Eq.13isthe discretecounterpartofEq.1andforitto be accurate,
one requires a small ǫ < 1 wave packet parametrization. The exact Planckian
distribution, Eq.11,will be more usefulthan Eq.13,as ananchorfor investigating
the time evolution of non-thermal systems, which eventually evolve into thermal
equilibrium.
Do we have an intuition about the exact Planckian distribution, Eq. 11? What
choice to make for ǫ as an intuitive parametrization for the wave packets? Our
preferenceisthatitshouldbesmall,whereEq.13holds,yetnotsosmall,thattime
resolution is too coarse. Fine-grained frequency and time resolution is required
to analyze the spectral dynamics of the formation to a black hole. To this end,
we ask what is the typical angular frequency, ǫ for a Hawking quanta in thermal
T
equilibrium? Yan9 asserts that the general thermal de Broglie wavelength is:
2√π Γ(d/2+1) 1/d
λ = , (14)
T
T (cid:18) Γ(d+1) (cid:19)
using a massless linear dispersion relation with space dimensions d, and ~ = c =
k =1. Knowingthis value,wecouldpickapacketizationǫthatissmaller,ǫ<ǫ ,
B T
in order to resolve well in frequency, yet not be so small that we can’t also have
reasonably fine-grained time resolution for a non-eternal process. In our 1 + 1
dimensional setting, Eq. 14 gives,
π ǫ π
T
λ = , and f λ =1 =1 ǫ =2T. (15)
T T T T
T → 2πT →
We are then justified in picking ǫ<ǫ =2T, as a wave packet parametrization for
T
ǫ for good frequency resolution. Note that ǫ ǫ/2T =ǫ/ǫ . Let us pick
T
≡
1+√5
ǫ =T ln =T lnφ, (16)
gr
2
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where φ=1.618 is the Golden Ratio, such that T lnφ T/2. This means that our
≈
choice of ǫ is about 4 times smaller than the general thermal de Broglie angular
gr
frequency, Eq. 15, ǫ = 2T. It also means a 5% relative error between Eq. 13 and
T
Eq. 11. That is, 5% error between the approximate familiar Planckian form and
the exact Planckian form, when j =1.
However, the actual intuitive usefulness of our choice of ǫ=ǫ , is that it gives
gr
onaverage,one particle per packet,inthelowestnon-divergingj =1frequencybin,
for N :
j
h i
N =1. (17)
j=1
h i
This is checkedby solvingEq.11,using j =1 andǫ=ǫ . This alsohappens to be
gr
the same assumption as supposing that the total amount of particles detected on
average is exactly 2 particles, that is:
∞
N =1, (18)
j
h i
Xj=2
or, of course,
∞
N =2. (19)
j
h i
Xj=1
In this note, up until now, we have been looking at the equilibrium mirror. We
developedanintuitivetoolsettounderstandthethermalequilibriumwhereparticle
emissionis discrete andone particleper packetis observed. Now we areina better
position to time resolve the early-time, non-thermal history of the black mirror.
The black mirror trajectory is given by1,2
W(2e−2κt)
z(t)= t , (20)
− − 2κ
where W is the Lambert W function, and the horizonis takento be on the t= x
−
null ray. Itis plottedin Fig. (1)and itbegins atrest,at z =+ andatlate times
∞
asymptotically approaches the speed of light. At late times this trajectory is very
similar to the equilibriummirrorb. The blackmirrorhasparticles ina non-thermal
distribution at early times and they only become Planckian at late times.
The black mirror has beta Bogoliubov coefficients,
√ωω′ ω+ω′ −iω/κ iω
βωω′ = 2πκ(ω+ω′)e−πω/2κ(cid:18) κ (cid:19) Γ(cid:20) κ (cid:21), (21)
with different form than equilibrium mirror beta coefficients of Eq. 2. The total
number of particles produced with frequency ω is Eq. 4 and therefore complex
conjugating, one finds
′
ω 1
|βωω′|2 = 2πκ(ω+ω′)2e2πω/κ 1. (22)
−
bSeeGood-Anderson-Evans6 orGood10 fortheexplicitequilibriumz(t)trajectory.
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Fig.1. Theblackmirrorinaspacetimedi- Fig. 2. The two mirrors in a Penrose di-
agram(theblacktrajectory)isoverlaidwith agram. Notice that both start asymptot-
theequilibriummirror(thegraytrajectory). ically inertial. The black mirror starts at
The accelerations are time dependent. The rest, while the equilibrium mirror starts at
horizonisatt=−x. Wehavesetκ=1. thespeedoflight. κ=1.
Like the equilibrium mirror, we also see the explicit Planckian signature. It’s easy
′
to see that in the late time limit where ω ω, then Eq. 22 goes to Eq. 3,
≫
1 1
|βωω′|2 ∼ 2πκω′e2πω/κ 1 , (23)
−
and there is a thermal distribution of particles with temperature T = κ/2π. Late
time thermality was found 40 years ago by Davies-Fulling1112 for particular late-
time behaved mirrors.
The black mirrorbeta canonly be exactly analyticallypacketizedatlate times.
This is because the exact Planckian result, Eq. 11, holds. Early-time resolution is
only tractable numerically by using wave packets of Eq. 5 on Eq. 21. As a check,
onenumericallyconfirmsthelate-timeexactanalyticPlanckiandistribution,Eq.11.
Thereasonforfirststudyingtheequilibriummirrorbeforetacklingtheblackmirror
head-onshouldnowbeclear: sinceweknowapriorithattheblackmirrorproduces
late-time Planckian particles, we can anchor the late-time numerical precision and
accuracy goals against the exact Planckian distribution to gain confidence for the
early-time, non-thermal radiation. A plot of the time dependence of the particle
production for the frequency band ǫ=ǫ and κ=j =1 is given in Fig. (3). The
gr
radiation emission is monotonic, i.e. here there is no black hole birth cry7. Since
blackmirrorradiationisblackholeevaporation,theblackmirrorprovidesawindow
into the early-time formation phase.
Acknowledgments
MRRG thanks Paul Anderson, Yen Chin Ong and Charles Evans for fruitful con-
versations. MRRG is supported by the Nazarbayev University Social Policy grant.
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Fig. 3. Black mirror particle flux in time. Fig. 4. Black mirror particle flux in fre-
The solid black line at the top is the num- quency. The black curve at the top of the
berofparticlesproducedinanexactPlanck- plotisthePlanckiandistributionasafunc-
ian distribution, Eq. 11, hNj=1i = 1. The tionofthefrequency,Eq.11. Notethatthe
dotted-dashedlineisthePlanckiandistribu- packet (j = n = 1) is in both Figs. Here
tion,Eq.13. Herej=1,ǫ=Tlnφ,κ=1. n=1,ǫ=Tlnφ,κ=1.
References
1. P.R. Anderson,M. R.R. Good, and C. R.Evans, “Black Hole - Moving Mirror I:An
Exact Correspondence,” to appear in the Proceedings of the 14th Marcel Grossmann
Meeting, (2015) arXiv:1507.03489 [gr-qc]
2. M.R.R.Good,P.R.AndersonandC.R.Evans,“BlackHole-MovingMirrorII:Par-
ticle Creation,” to appear in the Proceedings of the 14th Marcel Grossmann Meeting,
(2015) arXiv:1507.05048 [gr-qc].
3. M. R. R. Good, P. R. Anderson and C. R. Evans, Phys. Rev. D 94, no. 6, 065010
(2016) [arXiv:1605.06635 [gr-qc]].
4. R. D. Carlitz and R. S. Willey,“Reflections On Moving Mirrors,” Phys. Rev. D 36,
2327 (1987).
5. S.Hawking,“ParticleCreationbyBlackHoles”,Commun.Math.Phys.43(1975)199.
6. M.R.R.Good,P.R.AndersonandC.R.Evans,“TimeDependenceofParticleCreation
from Accelerating Mirrors,” Phys.Rev.D 88,025023 (2013) [arXiv:1303.6756 [gr-qc]].
7. M. R. R.Good and Y. C. Ong, JHEP 1507, 145 (2015) [arXiv:1506.08072 [gr-qc]].
8. A.FabbriandJ.Navarro-Salas,“Modeling blackholeevaporation,”London,UK:Imp.
Coll. Pr. (2005)
9. Z. Yan, “General thermal wavelength and its applications,’ European Journal of
Physics, 21, 6, (2000).
10. M. R. R. Good, “On Spin-Statistics and Bogoliubov Transformations in Flat Space-
time With Acceleration Conditions,” Int. J. Mod. Phys. A 28, 1350008 (2013)
[arXiv:1205.0881 [gr-qc]].
11. P.C.W.Davies,S.A.Fulling,“RadiationfromaMovingMirrorinTwo-Dimensional
Space-Time Conformal Anomaly,” Proc. Roy.Soc. Lond.A 348 (1976) 393.
12. P. C. W. Davies, S. A. Fulling, “Radiation from Moving Mirrors and from Black
Holes,” Proc. Roy.Soc. Lond. A 356 (1977) 237.
