Presentation on theme: "Quasinormal Ringing of Acoustic 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）"— Presentation transcript:
Quasinormal Ringing of Acoustic 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 downup sonic point “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”
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 phasemerger phase
Definition of QNMs Schroedinger-type wave eq. outgoing BCswith QNMs are defined as solutions of horizoninfinity solutions: n : complex
Example: Schwarzschild Black Hole
horizonspatial infinity Example: Schwarzschild Black Hole ~ r g -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 c s0 : 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 rr tank 1tank 2nozzle flow
Potential Barrier for Different Laval Nozzles 1.04L 3.92L L L 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 00 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 ) 1 st order: Schutz & Will, rd order: Iyer & Will, th order: Konoplya, 2004 Matching matching regions 23 12
WKB Approach: S-Matrix Here, is related to by where (1 st WKB)
QNM Solutions by WKB Approach Conditions for QNMs: i.e. QNM frequency (1 st 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. (3 rd 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 horizon supersonicsubsonic
QNM fit (3 rd WKB) numerical ringdown phase observed waveform Result 1: Weak Shock Infall
QNM fit (3 rd WKB) numerical nonlinear phaseringdown phase Result 2: Acoustic BH Formation observed waveform
QNM fit (3 rd WKB) numerical nonlinear phaseringdown 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 QNMs of acoustic BHs decay too quickly. Difficult to detect in experiments…?
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., v c = v. partially reflects sound waves. vcvc vv 12 Contact Surface (C.S.)
Example: Contact Surface in Perfect Fluid vcvc vv 12 If v c (= v) << c s, refl. coeff. R( ) for sound waves propagating from 1 to 2 is 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 Im Table: the least damped PRQNM frequency (3 rd WKB value) contact surface enhances Q-value!!
observed waveform PRQNM fit (3 rd WKB) numerical Numerical Simulation of PRQNMs
no contact surfacecontact surface present
Summary For 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 Approach In region (III), right-going WKB sol.left-going WKB sol. cc region (III)region (IV) 23
PRQNM Solutions by WKB Approach In region (III), right-going WKB sol.left-going WKB sol. Furthermore, if c lies 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.