Quasinormal Ringing ofAcoustic Black Holes

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）

What is an Acoustic Black Hole?

“Acoustic BH” = Transonic Flow

down 1|| M1|| M 1|| M 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

throat

“Laval Nozzle”:Convergent-Divergent Nozzle

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.

compressor

mass flow meter

settling chamber

Laval nozzle

flow

20cm

Configuration

Quasinormal Ringing

“Characteristic ‘sound’ of BHs (and NSs)”

occurs 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.

Quasinormal Ringing of a BH

NS-NS merger to a BH (Shibata & Taniguchi, 2006)

QN ringing

inspiral phase merger phase

Definition of QNMs

Schroedinger-type wave eq. outgoing BCswith

QNMs are defined as solutions of

horizon

infinity

solutions:

n: complex

Example: Schwarzschild Black Hole

horizon

spatial infinity

Example: Schwarzschild Black Hole

~ rg-2

Example: Schwarzschild Black Hole

fundamental (n=0) mode

Goal of This Study

What form of nozzle yield the QNM easiest to observe?A measure of the detectability: “Q-value”

Are QNMs excited in experimentally feasible situations?

Numerical simulation (full order calculation)

Find the form of the nozzle which yield large Q

Experimental testing of gravitational-wave analysis

(such as matched filtering)

Does acoustic BH really have QNMs?

Questions

-- wave eq. for velocity potential perturbation

Sound Waves in Inhomogeneous Fluid Flow

Perturbation:

1D flow approx.:

Schroedinger-type Wave Equation

cs0: sound speed at stagnation points

, v, A : independent of t

M : Mach number

where

“effective potential”

Potential Barrier for Different Laval Nozzles

Consider a family of Laval nozzle:

nozzle radius

K : integer

r

tank 1 tank 2nozzleflow

Potential Barrier for Different Laval Nozzles

1.04L

3.92L-2

11.4L-2

1.19L

flow

sonic horizon

flow

sonic horizon

Procedure for Calculating QN Freq’s

Calculate the “S-matrix” for the potential barrier V():

Then, impose the outgoing B.C. ,

and find ’s that satisfy the boundary condition.

: “S-matrix”

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

WKB Approach: S-Matrix

Here, is related to by

where

(1st WKB)

QNM Solutions by WKB Approach

Conditions for QNMs:

i.e.

QNM frequency

(1st WKB value)

QNM Frequencies of Different Laval Nozzles

(WKB approx.breaks down)

Q = 2

Q = 1

WKB solution:Schutz & Will (1985)Iyer & Will (1987)

: peak point of V

n=0 mode freq.

(3rd WKB value)

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

let a weak shock “fall” into the horizon

final state: transonic flow

~ BH formation ~ test particle infall

flow

Example of Transonic Flow

flow

sonic horizonsupersonic subsonic

Result 1: Weak Shock Infall

steady shock

horizon

gif

weak shock

Result 1: Weak Shock Infall

steady shock

horizon

weak shock

QN ringing

gif

QNM fit(3rd WKB)

numerical

ringdown phase

observed waveform

Result 1: Weak Shock Infall

QNM fit(3rd WKB)

numerical

nonlinear phase

ringdown phase

Result 2: Acoustic BH Formation

observed waveform

QNM fit (3rd WKB)

numerical

nonlinear phase

ringdown phase

Result 2: Acoustic BH Formation

observed waveform

Numerical waveform agrees with the least damped QNM very well!

Numerical Simulation: Discussion

In both types of simulations, QNMs are actually excited.

The results agree with WKB analysis well ( for K

>1 ).Typical values in laboratories:

cf. real BH:(l=m=2, least-

damped)>2.0

LN

0~L

cs

QNMs of acoustic BHs decay too quickly. Difficult to detect in experiments…?

Partially Reflected Quasinormal Modes (PRQNMs)

outgoing B.C. + “half mirror” B.C.

“half mirror”

c

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.)

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.

PRQNM Solutions by WKB Approach

Partially Reflecting B.C. :

Example: Contact Surface in Perfect Fluid

Re ReIm Im

Table: the least damped PRQNM frequency (3rd WKB value)

contact surface enhances Q-value!!

observed waveform

PRQNM fit (3rd WKB)

numerical

Numerical Simulation of PRQNMs

Numerical Simulation of PRQNMs

no contact surface contact surface present

SummaryFor future experiments, we have studied QN ringing of acoustic BHs in Laval nozzles.

A contact surface elongates the damping times of QNMs.

Acoustic BHs (transonic fluid flow) do have QNMs.

QNMs are excited in experimentally feasible situations.

A wider range of Q becomes accessible!

Experimental testing of gravitational-wave analysis(such as matched filtering) .Astrophysical BH surrounded by a “half mirror” ??

Future Works

PRQNM Solutions by WKB ApproachIn region (III),

right-going WKB sol.

left-going WKB sol.

c

region (III) region (IV)

23

PRQNM Solutions by WKB ApproachIn region (III),

right-going WKB sol.

left-going WKB sol.

Furthermore, if clies far away from the potential barrier,

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.