Hawking radiation on a falling lattice

11 Pages·0.61 MB·

by  Ted Jacobson

Hawking radiation on a falling lattice
Ted Jacobsona,b∗ and David Mattinglya†
aDepartment of Physics, University of Maryland, College Park, MD 20742-4111, USA
bInstitute for Theoretical Physics, UCSB, Santa Barbara, CA 93106
[4,5]). Quantizingthe soundfieldUnruharguedthatthe
Scalarfieldtheoryonalatticefallingfreely intoa1+1di-
horizon would radiate thermal phonons, in analogy with
mensionalblack holeisstudiedusingbothWKBandnumer-
theHawkingeffect,atatemperaturegivenbyh¯/2πtimes
icalapproaches. Theoutgoingmodesareshowntoarisefrom
the gradient of the flow velocity at the horizon. Since
incomingmodesbyaprocessanalogoustoaBlochoscillation,
the sound field has an atomic cutoff, and the physics of
withanadmixtureofnegativefrequencymodescorresponding
the fluid is completely understood in principle, the fluid
to the Hawking radiation. Numerical calculations show that
model may provide a source of insight into the Hawking
the Hawking effect is reproduced to within 0.5% on a lattice
0 process.
0 whose proper spacing where the wavepacket turns around at
The atomic cutoff produces dispersion of sound waves
0 thehorizon is 0.08 in unitswhere thesurface gravity is 1.
∼ leading to subsonic propagation at high frequencies. It
2
turns out that this dispersionis all that is needed to ob-

n
viate the need for a trans-Bohrian reservoir. In the lin-
a
earized theory, an outgoing short wavelength mode can
J
be draggedin by the fluid flow and redshifted enough so
2
I. INTRODUCTION that its increased group velocity overcomes the flow and
1
itescapestoinfinity. Investigationsofanumberoflinear

2 If nothing can emerge from a black hole then where fieldtheorymodels[6–11]andlatticemodels[12]incorpo-
v doestheHawkingradiationcomefrom? Relativisticfield ratingsuchhighfrequencydispersionhavedemonstrated
9 theorytellsusthatitcomesfromatrans-Planckianreser- this mechanism of “mode conversion” and have shown
9 voir of outgoing modes at the horizon [1]. The existence thattheHawkingradiationisrecovered.2 Thedispersion
0
ofsuchatrans-Planckianreservoirisdoubtfulongeneral in these models is not locally Lorentz-invariant since a
8
physical grounds. It is the cause of the divergences in frameis pickedoutinwhichto specify whichfrequencies
0
9 quantum field theory which are expected to be removed are“high”. Inthe fluidmodelthis isthe localrestframe
9 inaquantumtheoryofgravity,andinparticularitwould of the fluid. A real black hole also defines a preferred
/ seem to imply an infinite black hole entropy unless can- frame but it does so in a non-local way. Perhaps quan-
h
celed by an infinite negative “bare entropy”. tum gravity produces a dispersion effect related to this
t
- String theory has produced an account of the Hawk- non-local notion of a preferred frame, or perhaps micro-
p
e ing radiation and black hole entropy which requires no physicsis just notlocally Lorentzinvariant. Inany case,
h trans-Planckianreservoir,howeverinthatpictureinstead it seems worthwhileto understand the mechanismof the
: of a black hole one has a stringy object in flat spacetime outgoing modes and Hawking radiation in these models
v
i that arises from a black hole as the string coupling con- for the hints it may provide about a correct quantum
X stant tends to zero [2]. Thus gravitationalredshift plays gravity account.
r no role in the string calculations and nothing is learned In [12] a falling lattice model was introduced to ad-
a
about the origin of the outgoing modes in a black hole dress the “stationaritypuzzle” [17,8]: how cana low fre-
spacetime.1 If we could follow the theory as the string quency outgoing mode arise from a high frequency ingo-
couplingbecomesstrongwewouldpresumablylearnhow ing mode when it propagates only in a stationary back-
it is that string theory produces the outgoing black hole ground spacetime and hence has a conservedKilling fre-
modes. quency? Inthecontinuumbaseddispersivemodelsthere
Absent a quantum gravity description of the Hawking is nosatisfactoryresolutionofthis puzzle. As the outgo-
process,Unruhinventeda fluidanalogofablackhole [3] ing modes are traced backwards in time their incoming
in which an inhomogeneous flow exceeds the long wave- progenitors are squeezed into the short distance regime
length speed of sound creating a sonic horizon (see also
2 Dispersion can also produce superluminal propagation,
∗[email protected] whichallowsoutgoingmodestoemergefrombehindthehori-
†[email protected] zon. Thesuperluminalcasedoesnotseemveryhealthyfroma
1The fact that redshifting plays no role in producing the fundamentalpointofview,thoughitissurelyrelevantinsome
stringyHawkingradiationispresumablytiedtothefactthat condensedmatteranaloguesofblackholehorizons[13–15]and
thestringcalculationsapplyinthe“near-extremal”limitthat is capable of producingthe usual Hawking spectrum [16].
thesurface gravity tends to zero.
1
of the model which is not physically sensible. horizon.
Alatticeprovidesasimple modelforimposingaphys- The precise choice of worldlines for the lattice points
ically sensible short distance cutoff. It might seem at should not be important to the leading order Hawking
first that the most natural choice would be to preserve effectaslongasthe groundstateevolvesadiabaticallyin
the time translation symmetry of the spacetime with a the lattice theory. Our numerical results are consistent
static lattice whose points follow accelerated worldlines. with this expectation, in that the particular worldlines
Onastaticlattice,however,theKillingfrequencyiscon- depend on the lattice spacing yet the results convergeto
served,sooutgoingmodesarisefromingoingmodeswith the continuum as the lattice spacing is decreased. More-
the same frequency, and there is no Hawking radiation. over it was shown recently [10], in a continuum model
Thein-vacuumthereforeevolvestoasingularstateatthe with high frequency dispersion, that when the preferred
horizon. If the lattice points are instead freely falling, a frame is changed from the free-fall frame of the black
discrete remnant of time translation symmetry can still hole to that of a conformally related metric there is no
be preserved. On such a lattice there is still a frequency leadingorderchangeintheHawkingradiationunlessthe
conservationlaw,andtheunsatisfactoryresolutionofthe acceleration of the preferred frame is very large. Pre-
stationaritypuzzle is thatthe lattice spacingin this case sumablyallthatmattersisthatthepreferredworldlines
goes to zero at infinity (if the lattice points are asymp- flow smoothly across the horizon. From a fundamental
totically at rest) [12]. point of view, if there really is a cutoff in some preferred
If we insist that the lattice spacing asymptotically ap- frame,itis plausible thatthis frame wouldcoincide with
proaches a fixed constant and that the lattice points are the cosmic rest frame and would fall across the event
at rest at infinity, we find a fascinating resolution of the horizons of any black holes that form by collapse.
stationarity puzzle [12]: the lattice cannot have even a Theremainderofthispaperisorganizedasfollows. In
discretetimetranslationsymmetry,sincethereisagrad- Section II the falling lattice model is set up, in Section
ualspreadingofthe lattice pointsasthey falltowardthe IIIsomeresultsoftheWKBmethodareshown,illustrat-
horizon. The timescale of this spreading is of order 1/κ ing the frequency shift and the mode conversion at the
where κ is the surface gravity. This time dependence of horizon,and the role of the negative frequency branchis
the lattice is invisible to long wavelength modes which explained. In Section IV the results of the full numer-
senseonlythe stationarybackgroundmetric ofthe black ical evolution are presented, and Section V contains a
hole, but it is quite apparentto modes with wavelengths discussion or the results and directions for further work.
of order the lattice spacing. On such a lattice the long TheAppendixdescribesthefinitedifferencingofthetime
wavelengthoutgoing modes come from short wavelength derivates used in the numerical calculations.
ingoingmodes which startout witha high(lattice scale) We use units with h¯ = c = κ = 1, where κ is the
Killingfrequencywhichisexponentiallyredshiftedasthe surface gravity.
wavepacketpropagates to the horizonand turns around.
This behavior of wavepackets in the (time-dependent)
fallinglatticemodelwasdeterminedin[12]withthehelp II. FALLING LATTICE MODEL
of the WKB approximation in which the frequency is
usedasaHamiltoniantosolveforthewavepackettrajec- In this section we set up the field theory on a lattice
tory. It was also argued that, since the time dependence falling into a black hole in two spacetime dimensions.
of the lattice is adiabatic for those modes with wave-
length short enough to know they are on a lattice, these
modes will remainunexcited ontheir wayto the horizon A. Black hole spacetime
and the ground state condition for the Hawking effect
will be met. However, since the wavelength is of order We assume the spacetime metric is static, so [18] co-
the lattice spacing and the wavepacket can vary wildly ordinates can be chosen (at least locally) such that the
onthe lattice nearthe horizon,itis not obviousthatthe line element takes the form
WKB approximation is reliable nor that the adiabatic
argument really applies. ds2 =dt2 dx v(x)dt 2. (2.1)
− −
The primary motivation for the work reported here (cid:0) (cid:1)
In these coordinates the Killing vector is given by
was to check the WKB and adiabatic assumptions by
carrying out the exact calculation numerically using the
χ=∂ , (2.2)
t
latticewaveequation. Wefoundthatindeedthefieldbe-
haves this way, and the Hawking radiation is recovered with squared norm
to within half a percentfora lattice spacingδ =0.002/κ
which corresponds to a proper spacing 0.08/κ where χ2 =1 v2(x). (2.3)
∼ −
the wavepacket turns around at the horizon. We also
For v(x) we choose
studiedthedeviationsfromtheHawkingeffectthatarise
as κδ is increased, and our simulations reveal an inter- vmax
v(x)= , (2.4)
estingpictureofhowthewavepacketsturnaroundatthe −cosh(βx)
2
w+her,ev(vxm)axvaannisdheβs, saorethpeosliintieveelecmonesnttanbtesc.omAesst|hxa|t→of ∂ta/a= βvmcoasxhs2i(nβhx(β)x) (2.13)
∞
Minkowski spacetime. Provided v >1 there is an er-
max
goregion centered on x = 0 in which the Killing vector whose maximum outside the horizon (cosh(βx) > vmax)
is spacelike, and the boundaries of this region are black isκ(atthehorizon)ifv √2andκv2 /2 v2 1
max ≥ max max−
and white hole horizons at if v √2. Thus ∂ a/a < O(κ) everywhepre outside
max t
≤ | |
the horizon as long as v >∼1 is not too close to unity.
x = β−1cosh−1(v ). (2.5) max
H max
±
The surface gravity of the horizons is given by
B. Lattice
κ=v′(x )=β 1 v−2 . (2.6)
H max
− The lattice is defined by discretizing the z coordinate
p
The surface gravity sets the length scale for our space- with spacing ∆z =δ. That is, we have only the discrete
time, andwe will keepit fixed. Hence it willbe useful to values zm = mδ. We will consider values of κδ which
employ units for which κ=1. rangefrom0.002to0.128. Togiveafeelingforthelattice
We now introduce a freely falling “Gaussian” coordi- we plot in Fig. 1 the (t,x) coordinates of lattice point
natez whichwillbediscretizedtodefineafallinglattice. worldlines at intervals of 50 lattice points.
The worldlines with dx = v(x)dt are geodesics of the
metric (2.1) with proper time t, at rest as x ,
| | → ∞
and they are orthogonal to the surfaces of constant
t. On these curves the quantity t dx/v is con-
− 30
stant, so if z is constant on these curvesRwe must have
W(z(x,t))=t dx/v forsomefunctionW. Wechoose
−
this function so tRhat z = x at t = 0, which implies that
z
W(z)= dx/v =sinh(βz)/βv , hence
max
− 20
R
sinh(βz)=sinh(βx)+βv t. (2.7)
max
In terms of z, the line element (2.1) takes the form
10
ds2 =dt2 a2(z,t)dz2, (2.8)
−
with
cosh(βz)
a(z,t)= . (2.9) 0
1+(sinh(βz) βv t)2
max
−
p
We calla(z,t)the “scalefactor”,eventhoughitdepends
0 1 2 3 4 5 6
on z as well as t. It can also be expressed as
FIG. 1. Every 50th lattice point, t vs. x, for the case
v(x(z,t))
a(z,t)= . (2.10) δ = 0.004 and vmax = √2. In the black region of the plot
v(z) thepointsaretooclose toresolve. Thehorizon isat x 0.6.
≃
In (t,z) coordinates the Killing vector (2.2) is given by
Since dx = a(z,t)dz at constant t, the separation ∆x
ofthelatticepointsinFig. 1givesadirectrepresentation
v(x(z,t))
χ=∂ v(z)∂ =∂ ∂ . (2.11) ofthescalefactor. Theproperlatticespacingisgivenby
t z t z
− − a(z,t)
a(z,t)δ which, according to (2.12), grows approximately
linearly with time at the horizon.
Thescalefactorisunityatt=0,andoutsidetheblack
Note that there is a region where the lattice spacing
holeitgrowstowardsthefuture. Itisimportanttoknow
becomes very small. This happens because we chose the
how rapidly the scale factor is changing in time, since in
spacing to be uniform at t = 0. If the points did not
the lattice theory that time dependence is not merely a
bunch up outside the black hole before t=0, they could
coordinate effect and can therefore excite the quantum
not wind up equally spaced at t = 0, since they are all
field. At the horizon the scale factor takes the value
on unit energy free-fall trajectories. (The line element is
a(zH,t)=q1+2κt+κ2t2/(1−vm−a2x), (2.12) suypmamlseotroiuctusinddeetrh(et,wxh)it→e h(o−let,h−oxr)iz,osnoatfhteerptoi=nt0s.)bunch
To get a better idea of how sparse the lattice is near
so a(z ,t) κt for κt > 1. Thus the fractional rate of
H ∼ the horizonatthe turning point ofa typicalwavepacket,
change ∂ a/a is of order∼κ. In general, we have
t
3
we plot in Fig. 2 the velocity function v (2.4) for every hence there is no wave scattering. The discretization
lattice point near the horizon at time t = 27, for the breaks this conformal invariance, however there is still
case v = √2 and κδ = 0.004. In this case there essentially no scattering on the lattice providing the lat-
max
are only 6 points in the region between v(x) = 0.5 and ticespacingismuchsmallerthantheradiusofcurvature.
the horizon, and the proper spacing at the horizon is Thereasonisthatmodeswithwavelengthlongcompared
√2κδt 0.15/κ. tothe lattice spacingcannottellthey arenotinthe con-
∼ ∼
tinuum, while modeswithwavelengthcomparableto the
-2 -1 1 2 3 lattice spacing cannot tell that the spacetime is curved.
Thus, when we compute the Hawking occupation num-
-0.2
bers, there is no “greybody factor”.
-0.4
-0.6 D. Quantization and the Hawking effect
-0.8
Thefieldisquantizedasacollectionofself-adjointop-
-1 erators ϕˆm(t) satisfying the equation of motion (2.16)
and canonical commutation relations. For each complex
-1.2 solution to the equations of motion we define the oper-
ator a(f) = (f,ϕˆ) using the inner product (2.17). The
-1.4
commutation relations are equivalent to the relations
FIG.2. v(x) for every lattice point at t=27, for the case
vmax =√2 and δ=0.004. [a(f),a†(g)]=(f,g) (2.18)
for all f and g.
C. Scalar field Forapositivenormsolutionp,a(p)actslikealowering
operator, while for a negative norm solution n, the con-
We consider a massless,minimally coupled scalarfield jugate n∗ has positive norm, so a(n)= a†(n∗) acts like
−
on the lattice. In the continuum the action would be a raising operator. The Hilbert space of ingoing modes
is just the Fock space built from solutions with positive
1
S = d2x√ ggµν∂ ϕ∂ ϕ frequency with respect to t (hence positive norm), and
cont µ ν
2Z −
similarly for the outgoing Hilbert space. We assume the
= dtdz (a(z,t)(∂ ϕ)2 1 (∂ ϕ)2 . (2.14) boundary condition that the ingoing positive frequency
t z
Z (cid:18) − a(z,t) (cid:19) field modes are all in their ground state,
where we substituted the metric (2.8) in the secondline.
a(p )0 =0. (2.19)
in
On the lattice we adopt the same action except that the | i
partial derivative ∂zϕ is replaced by the finite difference The occupation number 0a†(p)a(p)0 of an outgo-
Dϕm := (ϕm+1 ϕm)/δ and am(t) is replaced by its ing normalized positive freqhu|ency wave|piacket p in the
−
average over the relevant lattice sites: in-vacuum 0 is is just minus the norm of the negative
| i
frequency part of the ingoing wavepacket that gives rise
1 2(Dϕ (t))2
S = dt a (t)(∂ ϕ (t))2 m top. Toseethis,supposetheingoingwavepacketq +q
m t m + −
2Z (cid:18) − a (t)+a (t)(cid:19)
Xm m+1 m evolvesto p accordingto the fieldequation(2.16),where
q haspositivefrequencyandq hasnegativefrequency.
(2.15) + −
Then a(p)=a(q )+a(q )=a(q ) a†(q∗), hence
+ − + − −
Varying the action (2.15) gives the equation of motion
for ϕ (t), 0a†(p)a(p)0 = (q ,q ), (2.20)
m − −
h | | i −
2Dϕm−1(t) where we used the in-vacuum condition (2.19) and the
∂ (a (t)∂ ϕ (t)) D =0. (2.16)
t m t m
− (cid:18)a (t)+a (t)(cid:19) commutationrelations(2.18). Ifpisnotnormalizedthen
m+1 m
toobtaintheoccupationnumberwemustdivideby(p,p).
Thereisaconserved(notpositivedefinite)innerproduct
Hawking’sresult[19]instandardfieldtheoryyields,fora
between complex solutions:
wavepacketp= dωc exp( iω(k)t+ikz), the thermal
ω
−
(ψ,ϕ)=i a (t)(ψ∗ (t)∂ ϕ (t) ϕ (t)∂ ψ∗ (t)). occupation numbRer
m m t m − m t m
Xm ω c 2
ω
N = dω | | , (2.21)
(2.17) Hawking Z eω/TH 1
−
In two dimensions the continuum action (2.14) is con-
where T =κ/2π is the Hawking temperature.
H
formally invariant, and all metrics are conformally flat,
4
III. WKB ANALYSIS (inthefreefallframe)changessignandthewavevectoris
equivalentto +π/δ. Fromthere it continues to decrease,
The motionofwavepacketsin the falling lattice model winding up small and positive. This reversal of group
canbe studied using the WKB approximation[12]. This velocity by a monotonic change of the wavevector on a
amounts to using Hamilton’s equations for the position latticeisanalogoustotheBlochoscillationofanelectron
z and momentum k with a Hamiltonian determined by in a crystal acted on by a uniform electric field.
the dispersion relation Thustheinterval( π/δ, kc )ismappedontothein-
− −| |
terval (0,π/δ). The map is onto since every outgoing
2/δ wavevector arises from some ingoing wavevector. This
H(z,k,t)=ω = sin(kδ/2) (3.1)
±a(z,t) dynamical stretching of the k-space interval of length
π/δ k to one of length π/δ is not problematic in a
c
which is obtained by inserting a mode of the form parti−cle| p|hase flow, but if we think about the wavepack-
exp( iωt+ikz)into the field equationand keeping only etsthatarebeingapproximatedbythisWKBanalysisit
−
thetermswiththehighestderivativesorfinitedifferences is disturbing since there really are more outgoing modes
ofthefield[20]. Aplotofthisdispersionrelationisshown than ingoing modes which produced them. It seems the
in Fig. 3 for a(z,t) = 1. Wavevectors differing by 2π/δ waveevolutionissomehownotconservingthe numberof
are equivalent, so only the range ( π/δ,π/δ) (the “Bril- wave degrees of freedom.
−
louin zone”) is plotted. TheresolutionofthispuzzleistheessenceoftheHawk-
ing effect. The wave evolution mixes positive and nega-
2 tivefrequencymodes,soapurelypositivefrequencyout-
going wavepacket arises from a mixture of positive and
negative frequency ingoing wavepackets. WKB evolu-
1
tionbreaksdownattheturingpoint,henceitmissesthis
mode mixing.
AnexampleofapositivefrequencyWKBtrajectoryis
-3 -2 -1 1 2 3
showninFig. 4. Thegraphshowsseveraldifferentquan-
tities as functions of t (in units with κ = 1): the static
-1
position coordinate of the wavepacketx and the horizon
x (solid lines), the free-fall frequency ω (3.1) (dashed
H
line), the Killing frequency ω (3.2) (dot-dashed line),
-2 K
and 50sin(kδ/2) (sparsely-dashed line). The positions
FIG.3. Dispersion relation ωδ = 2sin(kδ/2) plotted vs. are scaled up by a factor of 10 so they can be seen on
±
kδ. the same graph. In this example the lattice spacing is
δ = 0.004, v = √2, and the trajectory has z(34)= 6
max
The frequency ω is not conserved, even when kδ 1, and k(34)=1.
≪
simply because it is not the Killing frequency. Using
(2.11)weseethatinthecontinuumtheKillingfrequency
50
would be given by
40
v(x(z,t))
ωK =ω+ k, (3.2) sin(kd /2)
a(z,t)
30 free-fall frequency
with which the dispersion relation can be expressed as
20 Killing frequency
packet trajectory
a(z,t)ω v(x)k =(2/δ)sin(kδ/2). (3.3)
K
− 10 horizon
Note however that this relation is only valid when kδ
is not too large, otherwise the fact that the Killing fre-
20 22.5 25 27.5 30 32.5
quency involvesa partialderivative in a directionthat is Time
not along a lattice point worldline renders the relation
meaningless. FIG.4. WKBevolution.
Itwasshownin[12]thattheWKBtrajectoryofanin-
goingpositivefrequencywavepacketbouncesoffthehori- Notice first the behavior of sin(kδ/2). Backwards in
zon and comes back out provided the ingoing wavevec- time this starts out very small, rises to the maximum,
tor is greater in magnitude than some critical value k . and comes back down but not to zero. The group veloc-
c
(This critical value depends on the time and place f|rom| ity in the frame falling with the lattice vanishes at the
which the packet is launched.) The wavevector evolves top, when kδ/2 = π/2, corresponding to a wavelength
fromnegativevalues to positivevalues bydecreasingun- equal to two lattice units. This rise and fall happens
tilitislessthan π/δ,atwhichpointthegroupvelocity via a monotonicchange ofk, andis the Blochoscillation
−
5
mentioned above. B. Wavepackets and parameters
The wavepacket that ends up with ω = ω = 1 at
K
late times starts out at early times with ω = ωK 43. If the Killing frequency were conserved as it is in
∼
The maximum possible lattice frequency is 2/δ = 500 the dispersive continuum models the entire Hawking
in this case,so the ingoing frequency is certainly“high”. spectrum could be probed just by propagating one
Whereasthefree-fallfrequencycontinuestoredshift(for- wavepacket and decomposing it into is Fourier compo-
ward in time) as the wavepacket climbs away from the nents, each of which would pick up a negative frequency
horizon,the redshifting ofthe Killing frequency is essen- part appropriate for the corresponding frequency. This
tially complete by the time the wavepacket reaches the is how Unruh checked the spectrum in his model [6].
turningpoint. Thus,pasttheturningpoint,eachKilling SincetheKillingfrequencyisnotconservedonourlattice
frequency componentpropagatesautonomously,moreor we cannot avail ourselves of this option.3 Thus instead
less as it would in the continuum. As described in the we compute the occupation numbers for a sequence of
Introduction, this, together with the fact that the evolu- wavepacketsϕ (k) with different profiles:
s
tion of ω at the lattice scale is adiabatic, is the reason
why the usual Hawking effect is reproduced. ϕ (k)=k eexp[ (k 0.05s)2], s=1,...,9 (4.1)
s
− −
(ineunits with κ = 1, as usual). The form of these
IV. LATTICE WAVE EQUATION ANALYSIS wavepacketswas dictated by the need to have them well
enough localized to be contained in the flat region of
In this section we discuss the results of our numerical the spacetime (so we could construct the correspond-
calculations. The basic method is to take an outgoing ing positive frequency initial data) while at the same
positive frequency wavepacket and evolve it backwards time containing enough power at low frequencies of or-
in time. After it has “bounced” from the horizon and der κ to have a measurable Hawking occupation num-
propagatedback into the flat regionof the spacetime we ber. To shift the wavepacket to the desired starting po-
decompose it into positive and negative frequency parts. sition the Fourier transform (4.1) was multiplied by a
Since it is a purely left moving wavepacket at this stage phase factor exp( ikzinitial). The positive frequency ini-
−
the negative frequency part is just the part composed tial data was constructed by evolving each Fourier com-
of positive wavevectors. The occupation number (2.20) ponent one time step ∆t by multiplying with the phase
of the original outgoing packet is then given by minus factor exp( iω(k)∆t) and Fourier transforming the re-
−
the norm of this negative frequency part divided by the sult back into position space.
total norm. We typically computed the total norm in The time dependence of the lattice (illustrated in Fig.
positionspaceusing(2.17). Sincethe negativefrequency 1) means that the results will not be independent of
part ϕ(−) was projected out in the wavevector space we the “launch time” of the wavepacket from a given lo-
used an alternate expressionfor its norm, (ϕ(−),ϕ(−))= cation. If we start our backwards evolution too far in
2 ω(k) ϕ(−)(k)2, where the frequency ω(k) is the future then when the wavepacket nears the horizon
g−ivePnkb>y0(|3.1).|| | the lattice points will be pathologically sparse. On the
e other hand if we launch backwards at too early a time
thenthewavepacketcanbecomepartiallytrappedinthe
A. Numerical issues region where the lattice spacing gets very small. This
happens because a sufficiently high frequency wave is
turned around when it tries to propagate into a region
Our system of equations (2.16) is a coupled set of or-
of larger lattice spacing. We encountered this “chan-
dinarydifferentialequations,whichis tobe solvedinthe
neling” phenomenon in our simulations, and avoided it
limit that time is continuous but the spatial lattice is
just by launching later. The runs reported here were all
fixed. In order to satisfy the Courant stability criterion
done with wavepackets that reached the horizon around
the time step ∆t must satisfy ∆t < a(z,t)δ. (In this
t = 27, and were launched from far enough away to be
1+1dimensionalsetting theCourantconditioncoincides
contained in the flat region of the spacetime. Typically,
withtheconditionthatcausalityberespectedbythedif-
ferencing scheme.) In the asymptotic region a(z,t) = 1
so the stability condition there is ∆t < δ. For a fixed
∆t/δ the general condition would be violated wherever
a(z,t)<∆t/δ. Since this only happens deep behind the 3We studied the possibility that the WKB evolution could
horizon (for t > 0) we just modified a(z,t) inside the be used to establish a mapping between in and out frequen-
black hole so that it is never less than ∆t/δ. We used a cies,thusallowing ustochecktheentirespectrumofparticle
timediscretizationscheme(writtenoutintheAppendix) creation just by propagating a single wavepacket. This turns
witherrorsoforder(∆t)2,andfoundthat∆t=0.4δ was out to be a flawed idea however since the modes are mixed
by the evolution and, moreover, such a map misses the role
adequately small to obtain very good accuracy.
of the negativefrequency piece.
6
the launchtime wasaroundt=56 when the wavepacket A sample of the comparisons for κδ = 0.002 is listed
was centered at x=28. in Table I. The typical relative difference in this case is
All of the calculations reported here were done using of order half a percent.
the line element (2.1), with v (2.4) ranging from √2
max
to 30.
3. Lattice dependence
C. Results To study the dependence on the lattice we ran the
s=1 wavepacketbackward at several different values of
1. Generalities the lattice spacing and for several values of v (2.4)
max
from √2 to 30. The change of v produces only a
max
The behavior of a typical wavepacket throughout the rather small change in the shape of the function v(x)
process of bouncing off the horizon is illustrated in Fig. outside the horizon, however it produces a shift of the
6. The real part of the wavepacket is plotted vs. the position of the horizon (2.5). For large vmax the horizon
static coordinate x at several different times. Following is given approximately by xH ln(2vmax). The horizon
≃
backwardsintime, the wavestartsto squeezeupagainst positionrangesfromabout0.6for v =√2 to about4
max
the horizon and then a trailing dip freezes and develops for v =30. Thus a wavepacketlaunched backwardin
max
oscillationsthatgrowuntiltheyballoonout,forminginto time from a givenposition at a giventime will reachthe
a compact high frequency wavepacket that propagates horizon later in time for larger v , so that the proper
max
neatly away from the horizon backwards in time. lattice spacing at the horizon will be larger, though not
This ingoing wavepacket contains both positive and by a very large amount for the wavepackets considered
negative frequency components, in just the right combi- here.
nation to produce only an outgoing wave when sent in In Fig. 5 we plot the occupation numbers for lattice
towardsthe black hole, with no wave propagatingacross spacings δ ranging from 0.002 to 0.128, and for several
thehorizon. Toillustratethiswehavetakenoneofthese values of v (2.4), v = √2,2,2.5,3,4,5,10,20,30.
max max
ingoing wavepackets,decomposedit into its positive and In all cases the late time wavepacket was launched from
negative frequency parts, and sent them separately back the same position and time. All wavepackets had the
toward the horizon forward in time (Fig. 7). The nega- same envelope for their Fourier transform, however for
tive frequencypart(whichis the smallerofthe twosince different lattice spacings the wavepacket is necessarily
the surface gravity is fairly low compared to the typical different.
frequencies in the outgoing wavepacket), mostly crosses
0.025
the horizon,with justa bit bouncing backout. The pos-
itivefrequencypartmostlybounces,withjustalittle bit
0.02
going into the black hole. #
n Thermal Value
o 0.015
ti
a
2. Spectrum p
u 0.01
c
c
The“observed”occupationnumbersforthewavepack- O 0.005
ets (4.1) all agree well with the integrated thermal pre-
dictions (2.21)ofthe Hawkingeffect providedthe lattice 0
spacing is not too large.
0 0.05 0.1 0.15
Delta
TABLE I. Comparison of the black hole radiation on the
latticetothethermalHawkingoccupationnumbers(2.21)for FIG.5. Occupation numbersfor s=1wavepacketfor var-
thewavepackets(4.1)forthecasevmax =√2andκδ=0.002. ious values of δ and vmax. Higher occupation numbers corre-
spond to larger values of vmax.
s Thermal Lattice Rel. Diff.
1 0.01563 0.01557 0.0038
For all values of v , the occupation number con-
2 0.01409 0.01404 0.0035 max
vergesto the Hawkingpredictionasthe lattice spacingδ
3 0.01266 0.01262 0.0034
decreases.
4 0.01135 0.01130 0.0041
The pattern of deviations from the thermal prediction
5 0.01014 0.01009 0.0045
appears to depend quite strongly on the value of v , a
6 0.00903 0.00899 0.0039 max
result which we do not understand at present. The best
7 0.00801 0.00797 0.0051
wecandoistolistvariouseffectswhichwouldcontribute
8 0.00709 0.00705 0.0052
to the deviations: (i) the WKB turning point is mov-
9 0.00625 0.00622 0.0046
ing away from the horizon as δ grows, so one might ex-
7
pect thatthe effective surfacegravityv′(x ) andhence in others, which requires at least two dimensions to be
t.p.
theeffectiveHawkingtemperaturefeltbythewavepacket possible. At the atomic level such a volume-preserving
woulddecrease;(ii)atlargerv theproperlatticespac- flow involves erratic motions of individual atoms. One
max
ing at the horizon is larger at the time this particular possible improvement of the lattice model is to make a
wavepacket reaches the horizon; (iii) the lattice points latticethatmimicsthissortofvolumepreservingflow. It
are rather sparse near the horizon for the larger values isnotclearwhether the motionsofthe lattice points can
of δ, and their precise positions will be significantly dif- be slow enough to be adiabatic on the timescale of the
ferent for different values of v . Effect (i) would lead high frequency lattice modes. If they cannot, then the
max
to a decreaseinthe occupationnumber, while onemight time dependence of this erratic lattice background will
expect that effect (ii) would lead to a relative increase excite the quantum vacuum.
due to the extra particle creation associated with time In a fluid the lattice is a part of the system, not just
dependence ofthe lattice. Perhaps what we are seeing is a fixedbackground. The averageflow couldbe adiabatic
just a delicate balance between these two effects. for the fully coupled ground state of the system but not
for the field theory of the perturbations on the “back-
ground lattice”. Similar comments apply in quantum
V. DISCUSSION gravity: surely if in-out mode conversion is at play the
incoming high frequency modes are strongly coupled to
Wehaveshownthatlinearfieldtheoryonafallinglat- thequantumgravitationalvacuum. Ideally,therefore,we
tice reproduces the continuum Hawking effect with high shouldtrytofindamodelinwhichthebackgroundisnot
fidelity, thus verifying earlier expections [12]. For the decoupled from the perturbations.
smallest lattice spacing we studied, δ =0.002/κ,the de- As a first step in this direction one could study a one
viations from the thermal particle creation in wavepack- dimensional model in which the lattice points are non-
ets composed of frequencies ω < O(κ) were of order relativistic point masses, coupled to each other by near-
half a percent. This is remarkable∼,particularly since the estneighborinteractions,and“falling”orpropagatingin
proper lattice spacing encountered by the wavepacket at abackgroundpotential(withorwithoutperiodicbound-
the horizon was 0.08/κ. When κδ is pushed to higher ary conditions). The perturbations of such a lattice are
values eventually∼we see significant deviations from the the phonon field, and the back reaction to the Hawking
Hawking prediction. radiationisincluded(althoughthe backgroundpotential
There are several directions in which this work could isfixed). Inamodellikethisonecouldpresumablyfollow
be developed. One idea is to include self-interactions for in detail the nonlinear origin of the outgoing modes and
the quantum field theory. There are general arguments the transferofenergyfromthe meanflow tothe thermal
[21,22]that the thermalHawkingeffectholds alsoforin- radiation.
teracting fields. There are also perturbative calculations
[23] and calculations exploiting exact conformal invari-
ance of certain 1+1 dimensional field theories [24]. Also ACKNOWLEDGEMENTS
the “hadronization” of Hawking radiation from primor-
dial black holes has been analyzed by matching to high WearegratefultoMattChoptuikandDavidGarfinkle
energycollisiondata[25],butthephysicsofhowthether- for advice on numerical matters and to Steve Corley for
mal Hawking radiation is “dressed” by interactions as it many helpful discussions. This work was supported in
redshiftsawayfromthe horizonhasnotbeenstudied. In part by the National Science Foundation under grants
a 1+1 dimensional model it should not be prohibitively No. PHY98-00967 at the University of Maryland and
difficulttonumericallystudyHawkingradiationofinter- PHY94-07194at the Institute for Theoretical Physics.
actingfields, nowthatwe seeit canbe doneonalattice.
The falling lattice model has provided a satisfactory
mechanism—the Bloch oscillation—for how to get an
outgoing mode from an ingoing mode in a stationary
background, but there is a serious flaw in the picture:
thelatticeisconstantlyexpanding. Inthefluidmodelby
contrastthelatticeofatomsmaintainsauniformaverage
density. In a fundamental theory we might also expect
the scale of graininess of spacetime to remain fixed at,
say, the Planck scale or the string scale (since presum-
ably the graininess would define this scale) rather than
expand. Can the falling lattice model be improved to
share this feature?
Afluidmaintainsuniformdensityinaninhomogeneous
flow by compressing in some directions and expanding
8
APPENDIX: FINITE DIFFERENCING SCHEME
The equation of motion (2.16) for ϕ (t) is
m
2Dϕ (t)
m−1
∂ a (t)∂ ϕ (t) =D . (A1)
t m t m
(cid:16) (cid:17) (cid:18)am+1(t)+am(t)(cid:19)
We used a time discretization scheme for the left hand side arising from the replacement of f˙ by
f(t+∆t/2) f(t ∆t/2) /∆t=f˙+(1/24)f(3)(∆t)2+O((∆t)4), (A2)
h − − i
which yields
∂ (a (t)∂ ϕ (t)) a(n+1/2) ϕ(n+1) ϕ(n) a(n 1/2) ϕ(n) ϕ(n 1) /(∆t)2, (A3)
t m t m
→h (cid:16) − (cid:17)− − (cid:16) − − (cid:17)i
where the spatial index is omitted and the argument n stands for n∆t. We could have used this replacement, since
a(z,t) is a prescribed function that can be calculated at any t, but for some reasonwe substituted a(n 1/2) by the
±
average a(n)+a(n 1) /2. This yields the approximation
(cid:16) ± (cid:17)
a(n)+a(n+1) ϕ(n+1) ϕ(n) a(n)+a(n 1) ϕ(n) ϕ(n 1) /2(∆t)2 (A4)
h(cid:16) (cid:17)(cid:16) − (cid:17)−(cid:16) − (cid:17)(cid:16) − − (cid:17)i
1 1 1 1
=∂ (a∂ ϕ)(n)+ aϕ(4)+ a˙ϕ(3)+ a¨ϕ¨+ a(3)ϕ˙ (n)(∆t)2+O((∆t)4). (A5)
t t
12 6 4 4
(cid:16) (cid:17)
With (A4) in place of the left hand side of (A1) we solve for ϕ(n 1) in terms of ϕ(n) and ϕ(n+1) to obtain an
−
equation for evolving ϕ backwards in time by one time step given the following two steps.
[1] T. Jacobson, Phys. Rev. D44, 1731 (1991).
[2] Fora review see A.W. Peet, Class. Quant. Grav. 15, 3291 (1998).
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[4] V.Moncrief, Astrophys.J, 235, 1038 (1980).
[5] M. Visser, Class. Quantum Grav. 15, 1767 (1998).
[6] W.G. Unruh,Phys.Rev.D 51, 2827 (1995).
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[8] S.Corley and T. Jacobson, Phys.Rev. D 54,1568 (1996).
[9] S.Corley, Phys.Rev.D 57, 6280 (1998).
[10] Y.HimemotoandT.Tanaka,“AgeneralizationofthemodelofHawkingradiationwithmodifiedhighfrequencydispersion
relation”, gr-qc/9904076.
[11] H.Saida and M. Sakagami, “Black hole radiation with high frequency dispersion”, gr-qc/9905034.
[12] S.Corley and T. Jacobson, Phys.Rev. D 57, 6269 (1998).
[13] T.A. Jacobson and G.E. Volovik, Phys.Rev.D 58, 064021 (1998); JETP Lett. 68, 874 (1998).
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[16] S.Corley, Phys.Rev.D 57, 6280 (1998).
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[18] Seee.g. AppendixA of ref. [12].
[19] S.W. Hawking, Nature(London) 248 (1974) 30; Commun. Math. Phys. 43 (1975) 199.
[20] The Hamiltonian techniquewas first applied todispersive models of theHawking effect in ref. [7].
[21] G.W. Gibbons and M.J. Perry,Proc. Roy.Soc. Lond., Series A 358, 467-94 (1977).
[22] W.G. Unruhand N.Weiss, Phys.Rev.D 29, 1656 (1984).
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9
FIG.6. A typical wavepacket evolution. Here vmax =√2, κδ =0.004, and the wavepacket (4.1) has s=1. The ocsillations
of the incoming wavepacket are too dense toresolve in the plots.
10