Acoustic Black Holesーブラックホール物理を実験室で検証するー
京都大学大学院 人間・環境学研究科
宇宙論・重力グループ M2
奥住 聡
共同研究者:阪上雅昭(京大 人・環), 吉田英生(京大 工)
Outline
1. Introduction: “What is an Acoustic Black Hole”?
2. “Acoustic BH Experiment Project”
3. Application I: Hawking Radiation (classical analogue)
4. Application II: Quasinormal Ringing
Interest and Difficulty in Black Hole Physics
Black holes are the most fascinating objects in GR.Black holes are the most fascinating objects in GR.Black holes are the most fascinating objects in GR.Black holes are the most fascinating objects in GR.
Hawking radiation (quantum)
: thermal emission from BHs
Numerous quantum / classical phenomena have beenpredicted. For example,
Quasinormal Ringing (classical)
: characteristic oscillation of BHs
However, many of them are difficult to observe.
To examine them, an alternative way is nessesary!
What is an Acoustic Black Hole?
“Acoustic BH” = Transonic Flow
down 1−>M1−<M 1−=M up
sonic point
eff
sc
velocityfluid:
velocitysound:
v
cs)1(eff ±=±= Mccvc sss
“effective” sound velocity in the lab
Acoustic BH region
In the supersonic region,
sound waves cannot propagate against the flow
= sonic horizon
→ “Acoustic Black Hole”
0>+ scv0=+ scv0<+ scv
-- wave eq. for velocity potential perturbation
Sound Waves in Inhomogeneous Fluid Flow
Perturbation:
This is preciselypreciselypreciselyprecisely the eq. for a massless scalar fieldin a geometry with metric
[ ]ji
ij
ii
s
s
dxdxdtdxvdtcc
ds δρ
+−−−= 2)( 2222 v
,22dx
vc
vdtdT
s −+≡
( )2221
2
22
2
222 )1()1( dzdy
cdx
c
vdT
c
vc
cds
sss
s
s
++
−+−−= − ρρ
,)0,0,(v=v
Unruh, Phys. Rev. Lett. 46, 1351 (1981)
2~sd≡
“Acoustic Metric”: Metric for Sound Waves
Furthermore, setting
“Acoustic Metric”
21
2
222
2
22 )1()1(~ dx
c
vdTc
c
vsd
s
s
s
−−+−−= 21
2
2
ff22
2
2
ff2 )1()1(~ drc
vdtc
c
vsd S
−−+−−=
“Acoustic Metric” Schwarzschild Metric
sonic point horizon
“Acoustic Metric”: Metric for Sound WavesUnruh, Phys. Rev. Lett. 46, 1351 (1981)
coordinate axial:
velocityfluid:
sound of speed:
22
x
dxvc
vtT
v
c
s
s
∫ −+=
coordinate radial:
timeildSchwarzsch:
velocityfall-free:)/(
light of speed:
2/1
ff
r
t
rrcv
c
S
g−=
Two Types of Steady Flow in Laval Nozzles
flow flow
Pressure difference pu / pd determines the flow in the nozzle:
pupdthroat throat
Subsonic flow : max M at throat, but M<1 everywhere.
Transonic flow Transonic flow Transonic flow Transonic flow : M=1 at throat; supersonic region exists.(may have a steady shock downstream)
THEORY
Graduate School of H&E Studies
EXPERIMENT
Graduate School of Engineering
TARGETS
• Hawking Radiation
• Quasinormal Ringing
numerical
Planckian fit
Acoustic BH Experiment Project at Kyoto Univ.
Preliminary Experiment:Acoustic Black Hole Formation
subsonicsubsonicsubsonicsubsonic transonictransonictransonictransonic
Acoustic BH is materialized in our experiments!!
Thermal emission from BHs.
Quantum phenomenon; derived from QFT in curved ST.( mixing of positive & negative freq. modes)
: “surface gravity”
Properties of Hawking Radiation
Too weak to observe in the case of astrophysical BHs!
How can we study Hawking Radiation?
Hawking radiation of phonon in airflow: impossible!!
(possible for BEC transonic flow ? [Garay et al., 2000] )
Nevertheless, some classical phenomena in acoustic BHs
will shed light on quantum aspects of Hawking radiation.
“classical counterpert of Hawking radiation”
Positive & Negative Frequency Mode Mixing
observer infinity
deformed
horizon
collapse
BH
positivepositivepositivepositive freq. mode(CLASSICAL)
surfacre gravity
exponential exponential exponential exponential redshiftredshiftredshiftredshift
Nonstationary evolution of ST � Change of vacuum state
star before collapse
negativenegativenegativenegative freq. part appears! Particle Creation!!quantization
Classical Counterpart of Hawking Radiation
Inner product (Fourier tr.):
Planck distribution!!
negativenegativenegativenegative freq. mode from infinity
positivepositivepositivepositive freq. mode for an observer
(Nouri-Zunoz & Padmanabhan, 1998)
Experimental Setting
Step 1: subsonic background flow ( no horizonno horizonno horizonno horizon ).Send sinusoidal sound wave against the flow.
Step 2: transonic background flow ( horizon presenthorizon presenthorizon presenthorizon present ).Observe the waveform at upstream region.
Redshift due to surface gravityincident freq:15kHz
horizon formed
Numerical Waveform (quasi-stationary flow, geometric acoustics limit)
Redshift due to surface gravityincident freq:15kHz
horizon formed
Numerical Waveform (quasi-stationary flow, geometric acoustics limit)
sinusoidal wave(t<0)
(next slide)
incident freq:15kHz
Numerical Spectrum(quasi-stationary flow, geometric acoustics limit)
Numerical Spectrum(quasi-stationary flow, geometric acoustics limit)
penetrates into positive frequency range!
(next slide)
Numerical Spectrum(quasi-stationary flow, geometric acoustics limit)
500 1000 1500 2000 2500 3000
f@HzD
1 ´ 10- 7
2 ´ 10- 7
3 ´ 10- 7
4 ´ 10- 7
5 ´ 10- 7
SHfL
500 1000 1500 2000 2500 3000
f Hz
1エ10- 7
2エ10- 7
3エ10- 7
4エ10- 7
5エ10- 7
S
Planckian fit
1)exp(2 −∝
κπω
ω
Numerical
Observation in a Laboratory
Signal is buried in noise.
However, output of LIA implies that redshift occurs.
Classical Counterpart of HR: Discussion
Recently, full order calculation has been performed.
Furuhashi, Nambu and Saida, CQG 23, 5417 (2006)
Their results agree with our calculation.
Planckian distr. seems to be robust.
Does the thermal emission of phonon really occur
in quantum fluids (BEC / superfluid) ?
How about the effect of high frequency dispersion?
3. Application II:
Quasinormal Ringing
Okuzumi & Sakagami
“Quasionormal Ringing of Acoustic Black Holes in Laval Nozzles”
in preparation
Quasinormal Ringing
“Characteristic ‘sound’of BHs (and NSs)”
Arises when the geometry around a BH is perturbed and settles down into its stationary state.
e.g. after BH formation / test particle infall
Described as a superposition of a countably infinite number
of damped sinusoids (QuasiNormal QuasiNormal QuasiNormal QuasiNormal Modes, Modes, Modes, Modes, QNMsQNMsQNMsQNMs).
QNM frequencies contain the information on (M,J) of BHs.
Quasinormal Ringing of a BH
NS-NS marger to a BH (Shibata & Taniguchi, 2006)
QN ringing
inspiral phase marger phase
Mathematical Description of QNMs
Schrodinger-type Eq. outgoing B.C.with..
In general, QNMs are defined as solutions of
V(ξ): effective potential barrier
Examples of Schroedinger-type Equation
(2) Acoustic Black Hole in a Laval Nozzle
cs0: sound speed at stagnation points
Potential Barrier for Different Laval Nozzles
Consider two-parameter family of Laval nozzle.
nozzle radius
: radius of the throat
K : integer
1.0
tank 1 tank 2nozzleflow
Potential Barrier for Different Laval Nozzles
1.04
3.92
11.4
1.19
flow
sonic horizon
flow
sonic horizon
QNM Frequencies of Different Laval Nozzle(the least-damped (n=0) mode; 3rd WKB value)
easier to observe
Re/Im ~ 4
(WKB approx.is not good)
Numerical Simulation of Acoustic QN Ringing
We perform two types of simulations:
“Acoustic BH Formation”
initial state: no flowset sufficiently largepressure difference
final state: transonic flow
“Weak Shock Infall”
initial state: transonic flow‘shoot’ a weak shock into the flow
final state: transonic flow
~ BH formation ~ test particle infall
Numerical Simulation: Summary
In both types of simulations, QNMs are actually excited.
The results agree with WKB analysis well ( for K >1 ).
cf. Schwarzschild, l = 2 , least-damped mode
Typical values in laboratories:
� similar to values for astrophysical BHs
Numerical Simulation: Discussion
For future experiments, larger Q-value is wanted.
However, Q is at most ~ 2 for planar wave modes.
QNMs of an Acoustic BH surrounded by a “half-mirror”(contact surface)
QNMs for non-planar waves
Can matched filtering be used in our experiments ?
Summary
“Acoustic BH” = Transonic Flow
wave eq. for sound in perfect fluid
� wawe eq. for a massless scalar field in curved ST
sonic point � event horizon of a BH
Results of numerical simulations strongly suggest
that classical counterpart of HR and QN ringing
can be realized in a laboratory.
Standard Procedure for Calculating QNM Freq’s
Calculate the “S-matrix” for the potential barrier V(ξ):
Then, impose the outgoing B.C. , and obtain κ ’s that meet the boundary condition.
: “S-matrix”
WKB Approach
ξ0
κ2
Region (I) & (III): WKB solutions for truncated V(ξ)Around ξ0 : exact solution for truncated V(ξ)
Expand V(ξ) in a Taylor series about the maximum point ξ0:
(I) (II) (III)
1st order: Schutz & Will, 1985
3rd order: Iyer & Will, 1987
6th order: Konoplya, 2004
MatchingMatchingMatchingMatching
matching regionsmatching regionsmatching regionsmatching regions
ξ23ξ12
WKB Approach: S-Matrix
Here, Here, Here, Here, νννν is related to is related to is related to is related to κκκκ by by by by
where
(1st WKB)
QNM Solutions by WKB Approach
Conditions for QNMs:
i.e.
QNM frequencyQNM frequencyQNM frequencyQNM frequency(1st WKB value)
Partially Reflected Quasinormal Modes(PRQNMs)
outgoing B.C. + ““““half mirrorhalf mirrorhalf mirrorhalf mirror”””” B.C.B.C.B.C.B.C.
“half mirrorhalf mirrorhalf mirrorhalf mirror”
ξc
Example: Contact Surface in Perfect Fluid
Contact surfaceContact surfaceContact surfaceContact surface (contact discontinuity):• discontinuity of the density ρ .• the pressure p and the fluid velocity v are continuous.• moves with the surrounding fluid, i.e., vc= v .• partially reflects sound waves.
vcv v
1 2
Contact Surface
(C.S.)
Example: Contact Surface in Perfect Fluid
vcv v
1 2
If vc(= v) << cs ,refl. coeff. R(κ) for sound waves propagating from 1 to 2is given by [e.g. Landau & Lifshitz, Fluid Mechanics]
C.S.
PRQNM Solutions by WKB ApproachIn region (III),
right-going WKB sol. left-going WKB sol.
ξc
region (III) region (IV)
ξ23
PRQNM Solutions by WKB ApproachIn region (III),
right-going WKB sol. left-going WKB sol.
Furthermore, if ξc lies far away from the potential barrier,
Example: Contact Surface in Perfect Fluid
Re ReIm Im
TableTableTableTable: the least damped PRQNM for an acoustic BH