ーー物理実験室検證ー.pdf

Acoustic Black Holes 物理 実験室 検証 京都大學(xué)大學(xué)院 人間 環(huán)境學(xué)研究科 宇宙論 重力 M2 奧住 聡 共同研究者 阪上雅昭 京大 人 環(huán) 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 1 Introduction What is an Acoustic Black Hole 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 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 What is an Acoustic Black Hole Acoustic BH Transonic Flow down 1 M 1 s cv0 s cv0 s cv wave eq for velocity potential perturbation Sound Waves in Inhomogeneous Fluid Flow Perturbation This is preciselypreciselypreciselyprecisely the eq for a massless scalar field in a geometry with metric ji ij ii s s dxdxdtdxvdtc c ds 2 2222 v 22 dx vc v dtdT s 2221 2 2 2 2 2 22 1 1 dzdy c dx c v dT c v c c ds sss s s 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 2 22 2 2 2 1 1 dx c v dTc c v sd s s s 21 2 2 ff 22 2 2 ff 2 1 1 dr c v dtc c v sd S Acoustic Metric Schwarzschild Metric sonic pointhorizon Acoustic Metric Metric for Sound Waves Unruh Phys Rev Lett 46 1351 1981 coordinate axial velocityfluid sound of speed 22 x dx vc v tT v c s s coordinate radial timeildSchwarzsch velocityfall free light of speed 2 1 ff r t rrcv c S g 2 Acoustic BH Experiment Project Black Holes in Laval nozzles throat Laval Nozzle Convergent Divergent Nozzle Two Types of Steady Flow in Laval Nozzles flowflow Pressure difference pu pd determines the flow in the nozzle pu pd throatthroat 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 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 observerinfinity 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 gravity incident freq 15kHz horizon formed Numerical Waveform quasi stationary flow geometric acoustics limit Redshift due to surface gravity incident freq 15kHz horizon formed Numerical Waveform quasi stationary flow geometric acoustics limit sinusoidal wave t1 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 Appendix 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 WKB Approach 0 2 Region I III WKB solutions for truncated V Around 0 exact solution for truncated V Expand V in a Taylor series about the maximum point 0 I II III 1storder Schutz Will 1985 3rdorder Iyer Will 1987 6thorder Konoplya 2004 MatchingMatchingMatchingMatching matching regionsmatching regionsmatching regionsmatching regions 23 12 WKB Approach S Matrix Here Here Here Here is related to is related to is related to is related to by by by by where 1stWKB QNM Solutions by WKB Approach Conditions for QNMs i e QNM frequencyQNM frequencyQNM frequencyQNM frequency 1stWKB 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 c Example Contact Surface in Perfect Fluid Contact surfaceContact surfaceContact surfaceContact 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 vcvv 12 Contact Surface C S Example Contact Surface in Perfect Fluid vcvv 12 If vc v cs refl coeff R for sound waves propagating from 1 to 2 is given