Acoustic Black Holes
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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)
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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
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1.Introduction:
“What is an Acoustic Black Hole?”
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Interest and Difficulty in Black Hole Physics
Black holes are the most fascinating objects in GR.
Hawking radiation (quantum)
: thermal emission from BHs
Numerous quantum / classical phenomena have been
predicted. 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!
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What is an Acoustic Black Hole?
“Acoustic BH” = Transonic Flow
down 1M1M 1M up
sonic point
effsc
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
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-- wave eq. for velocity potential perturbation
Sound Waves in Inhomogeneous Fluid Flow
Perturbation:
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This is precisely the eq. for a massless scalar fieldin a geometry with metric
jiij
iis
s
dxdxdtdxvdtcc
ds 2)( 2222 v
,22
dxvc
vdtdT
s
22212
22
2
222 )1()1( dzdy
cdx
c
vdT
c
vc
cds
ssss
s
,)0,0,(vv
Unruh, Phys. Rev. Lett. 46, 1351 (1981)
2~sd
“Acoustic Metric”: Metric for Sound Waves
Furthermore, setting
“Acoustic Metric”
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212
222
2
22 )1()1(~ dx
c
vdTc
c
vsd
ss
s
212
2ff22
2
2ff2 )1()1(~ dr
c
vdtc
c
vsd S
“Acoustic Metric” Schwarzschild Metric
sonic point horizon
“Acoustic Metric”: Metric for Sound Waves
Unruh, 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
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2. Acoustic BH Experiment Project:
Black Holes in Laval nozzles
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throat
“Laval Nozzle”:Convergent-Divergent Nozzle
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Two Types of Steady Flow in Laval Nozzles
flow flow
Pressure difference pu / pd determines the flow in the nozzle:
pupdthroa
tthroat
Subsonic flow : max M at throat, but M<1 everywhere.
Transonic flow : M=1 at throat; supersonic region exists. (may have a steady shock downstream)
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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.
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compressor
mass flow meter
settling chamber
Laval nozzle
flow
20cm
Configuration
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xb=8mm
throat
R=200mm
100mm 100mm
61.6mm 61.6mmx=0
Form of our Laval Nozzle
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Preliminary Experiment: Acoustic Black Hole Formation
subsonic
transonic
Acoustic BH is materialized in our experiments!!
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3. Application I:
Classical Analogue of Hawking Radiation
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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!
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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”
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Positive & Negative Frequency Mode Mixing
observer infinity
deformed
horizon
collapse
BH
positive freq. mode
(CLASSICAL)
surfacre gravity
exponential redshift
Nonstationary evolution of ST Change of vacuum state
star before collapse
negative freq. part appears! Particle Creation!!quantization
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Classical Counterpart of Hawking Radiation
Inner product (Fourier tr.):
Planck distribution!!
negative freq. mode from infinitypositive freq. mode for an observer
(Nouri-Zunoz & Padmanabhan, 1998)
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Experimental Setting
Step 1: subsonic background flow ( no horizon ).
Send sinusoidal sound wave against the flow.
Step 2: transonic background flow ( horizon present ).
Observe the waveform at upstream region.
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Redshift due to surface gravityincident freq:15kHz
horizon formed
Numerical Waveform (quasi-stationary flow, geometric acoustics
limit)
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Redshift due to surface gravityincident freq:15kHz
horizon formed
Numerical Waveform (quasi-stationary flow, geometric acoustics
limit)
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sinusoidal wave(t<0)
(next slide)
incident freq:15kHz
Numerical Spectrum(quasi-stationary flow, geometric acoustics
limit)
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Numerical Spectrum(quasi-stationary flow, geometric acoustics
limit)
penetrates into positive frequency range!
(next slide)
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Numerical Spectrum(quasi-stationary flow, geometric acoustics
limit)
500 1000 1500 2000 2500 3000fHz
1 107
2 107
3 107
4 107
5 107
Sf
500 1000 1500 2000 2500 3000f 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
Numerical
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Observation in a Laboratory
Signal is buried in noise.
However, output of LIA implies that redshift occurs.
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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?
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3. Application II:
Quasinormal Ringing
Okuzumi & Sakagami
“Quasionormal Ringing of Acoustic Black Holes in Laval Nozzles”
in preparation
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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 Modes, QNMs).
QNM frequencies contain the information on (M,J) of BHs.
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Quasinormal Ringing of a BH
NS-NS marger to a BH (Shibata & Taniguchi, 2006)
QN ringing
inspiral phase marger phase
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Mathematical Description of QNMs
Schrodinger-type Eq. outgoing B.C.
with..
In general, QNMs are defined as solutions of
V(): effective potential barrier
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Examples of Schroedinger-type Equation
(1) Schwarzschild Black Hole
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Examples of Schroedinger-type Equation
(1) Schwarzschild Black Hole
horizon
spatial infinity
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Examples of Schroedinger-type Equation
(2) Acoustic Black Hole in a Laval Nozzle
cs0: sound speed at stagnation points
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Potential Barrier for Different Laval Nozzles
Consider two-parameter family of Laval nozzle.
nozzle radius
: radius of the throatK : integer
1.0
tank 1 tank 2nozzleflow
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Potential Barrier for Different Laval Nozzles
1.04
3.92
11.4
1.19
flow
sonic horizon
flow
sonic horizon
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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)
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Numerical Simulation of Acoustic QN Ringing
We perform two types of simulations:
“Acoustic BH Formation”
initial state: no flow
set sufficiently large pressure 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
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Example of Transonic Flow
flow
sonic horizon
supersonic subsonic
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Result 1: Weak Shock Infall
steady shock
horizon
weak shock
QN ringing
gif
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QNM fit
numerical
nonlinear phase
ringdown phase
Result 2: Acoustic BH Formation
observed waveform
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QNM fit
numerical
nonlinear phase
ringdown phase
Result 2: Acoustic BH Formation
observed waveform
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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
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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 ?
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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.
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Appendix
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Standard Procedure for Calculating QNM Freq’s
Calculate the “S-matrix” for the potential barrier V():
Then, impose the outgoing B.C. ,
and obtain ’s that meet the boundary condition.
: “S-matrix”
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WKB Approach
0
Region (I) & (III): WKB solutions for truncated V() Around : exact solution for truncated V()
Expand V() in a Taylor series about the maximum point 0:
(I)
(II) (III)
1st order: Schutz & Will, 1985
3rd order: Iyer & Will, 1987
6th order: Konoplya, 2004
Matching
matching regions2312
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WKB Approach: S-Matrix
Here, is related to by
where
(1st WKB)
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QNM Solutions by WKB Approach
Conditions for QNMs:
i.e.
QNM frequency
(1st WKB value)
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Partially Reflected Quasinormal Modes (PRQNMs)
outgoing B.C. + “half mirror” B.C.
“half mirror”
c
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Example: Contact Surface in Perfect Fluid
Contact surface (contact discontinuity):• discontinuity of the density .
• 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.)
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Example: Contact Surface in Perfect Fluid
vcv v
1 2
If vc(= v) << cs ,
refl. coeff. R() for sound waves propagating from 1 to 2is given by [e.g. Landau & Lifshitz, Fluid Mechanics]
C.S.
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PRQNM Solutions by WKB ApproachIn region (III),
right-going WKB sol.
left-going WKB sol.
c
region (III) region (IV)
23
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PRQNM Solutions by WKB ApproachIn region (III),
right-going WKB sol.
left-going WKB sol.
Furthermore, if clies far away from the potential barrier,
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PRQNM Solutions by WKB Approach
Partially Reflecting B.C. :
PRQNM frequency:
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Example: Contact Surface in Perfect Fluid
Re ReIm Im
Table: the least damped PRQNM for an acoustic BH
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QNM fit
PRQNM fit
Numerical Simulation of PRQNMs
For t <15, an “ordinary” QNM (not PRQNM) dominates,
since the potential barrier is not yet “aware”
of the contact surface.