Black Hole Physics in BEC

Transcription

1 1 Black Hole Physics in BEC Acoustic black hole realized by transonic flow Grad. Sch. Human and Enironmental Studies, Kyoto Uni. Masa-aki SAKAGAMI

2 Black Hole : Most interesting Object in General Relatiity A spacetime haing horizon where een light cannot escape from Interesting Phenomena of Black Hole Hawking Radiation (Quantum) Non-stational Quantum emittion of radiation from BH Classical analogue of Hawking radiation Quasi-Normal Mode (Classical) Graitational wae from perturbed BH with ringdown wae form

3 Sketch to form laal nozzle in low temp. atomic gas 3

4 BlackHole spacetime (1) M r = RH = = c GM c Free falling Local Inertial frame r = = GM r = 0 Horizon Local inertial frame falls down with light elocity Light cannot escape from the horizon Stretching effect: tidal force (falling el. Is dependent on r)

5 BH spacetime () Stretching effect: tidal force Free falling Local inertial frame 5 Horizon Local inertial frame falls at light elocity

6 Hawking Radiation : particle creation 6 Energy Emittion from BH Photon (positie energy) Photon (negatie energy) Tidal force breaks up irtual photon pair Stretching of wae function due to Tidal force occurs Created photon number κ Surface graity el. gradient at horizon 1 < Nω > = πω κ T e = κ π k B 1

7 Hawking Radiation: wae dynamics(1) 7 space-time diagram of Black Hole formation time horizon out Aφ + out* Bφ a world line of light (or massless scalar) BH It has a narrow escape from a formation of eent horizon Non stationary eolution of space-time is essential in φ past infinity r=0 space surface of collapsing star This wae suffers stretching effects due to tidal force near the horizon in out φ ω = Aω, σφσ + σ B ω, σ φ out* ω

8 Hawking Radiation () wae dynamics 8 General Relatiity + Quantum Theory Vacuum in past infinity Stretching Tidal Force Collapse BH Negatie freq. Ω < 0 e iω ( t+ r / c) Obserer Positie freq. part appears particle creation horizon Star before graitational Collapse Infinity Vacuum

9 Acoustic Black Hole Correspondence between BH and its sonic analogue Black Hole spacetime light flow with sonic point sound wae Horizon Sonic Horizon Horizon plays essential role in Hawking radiation We can easily realize acoustic black hole In laboratories. Obseration of HR is quite difficult In case of BEC T 7 nk

10 Sound Waes in Perfect Fluid 10 Euler eq. continuity ρ + = p t ρ + ( ρ ) = 0 t p c s ρ γ = const. dp γp = dρ ρ = 0 = Φ adiabatic irrotational ( t, x) = Φ ( x) + φ ( t, x Φ >> Φ 0 ) 0 ρ ( t, x) ρ ( x) + δρ( t, x ρ >> = 0 ) 0 φ δρ t ρ (, ) + φ x t 0 = c t s ( ρ ) φ( x, t) 0 wae equation for perturbed part of elocity potential This simulates wae propagation in cured spacetime: e.g. black hole.

11 Acoustic Metric : Metric for Sound Waes 11 Unruh, Phys. Re. Lett. 46, 1351 (1981) This is precisely the eq. for a massless scalar field in a geometry with metric ds ds Furthermore, setting = ρ = s c s ρ c cs [ i j ] i i ( c ) dt dtdx + δ dx dx dt dt + c dx s 1 s ( 1 ) dt + (1 ) dx dz + cs cs cs d ~ s, ij = (, ρ 0, 0) ( dy + ) Acoustic Metric,

12 1 Metric of Perfect Fluid ~ 1 = (1 ) c dt (1 ) dx s + cs cs ds metric of perfect fluid ds Schwartzschild metric = c c 1 ( 1 ) c dt + (1 ) dr c T s : speed of sound : fluid elocity = t + c s x : spatial coordinate sonic point dx ( r) = c c : speed of light r g r : freefall elocity T : time coordinate r : radial coordinate horizon Unruh, Phys. Re. Lett. 46, 1351 (1981)

13 Form of our Laal Nozzle (Classical wae in air) 13 R=00mm κ eff dcs ( x) dx x= 0 = rad/s cross section: ( ) A ( x) R x + R + = π throat 61.6mm 61.6r3 x=0 mm flow b 100mm b=8mm 100mm x

14 14 Two Types of Flow in Laal Nozzle p 0 M = 0 d p 0 > p 1 throat flow da < 0 da = 0 da > 0 > 0 d = 0 M =1 d d > 0 < 0 p 1 M =1 M throat transonic flow subsonic flow :subsonic flow :transonic flow flow p 0 p 1 determines which type of flow is realized in the Laal nozzle.

15 Laal Nozzle: Flow with Horizon 15 ρ density of fluid A area continuity elocity of fluid ρ A = const (1) Bernoulli's princ. pressure func. (enthlpy) 1 + h(ρ) = const () sound el. p( ρ ) dp ρ c dρ dp h ρ = = s ( ) c s = ρ ρ dρ deriatie of(1)()

16 Sonic Horizon 16 Mach num. c s flaw in Laal Nozzle d M = M = ( 1) da A M < 1 da < 0 d > 0 M > 1 da > 0 d > 0 M < 1 accel. M < 1 M < 1 decel. M < 1 accel. M = 1 M > 1 accel.

17 17 Acoustic Hawking Radiation propagation of downstream-region modes in subsonic-transonic flow Stretching Positie freq component creation! graitational collapse negatie freq. mode iω( t+ x cs ) ( Ω < 0) sonic point(x=0) c s κ x e Acoustic BH subsonic flow transonic flow x surface graity κ d dx ( c s ) x=0

18 18 Results (waeform obsered upstream) incident freq:15khz affected portion by surface graity horizon appeared at the throat Re[ δp( t)] Re[ φ( t)]

19 Spectrum of Acoustic Hawking Radiation (WKB analysis) 19 Fourier component ~ dz Ω z c φ ( ω) = e z 0 z downstream region negatie frequency iω κ i ω κ c sh i sh πω κ = Γ ~ φ ( ω) = Furthermore, δp( x, t) δp ~ ( ω) Ω πκ ω exp( φ ( x, t) t upstream region positie frequency 1 πω κ ω πω exp( ) 1 e ) 1 ( iω κ ) in the upstream region, κ

20 Results (power spectrum of δp(t) ) 0 Negatie freq. Positie freq. incident freq:15khz sinusoidal portion(t<0) zoom(next)

21 1 Numerical Results (power spectrum of δp(t) ) Negatie freq. Positie freq. penetrates into positie frequency range! zoom(next)

22 Numerical Results (power spectrum of δp(t) ) Planckian fit κ =11700 rad/s S f exp( ω πω κ ) f Hz

23 Summary 3 Hawking radiation Non-stationary eol. of space-time (or fluid): without Horizon with horizon Stretching effect near horizon spectrum of radiation Planck distribution Acoustic Black Hole by Laal Nozzle (transonic flow ) erification of Hawking radiation in laboratories This system is similar as point contact in mesoscopic systems Quasi-normal Mode outgoing BC emittion of Graitation wae by BH

24 Sketch to form laal nozzle in low temp. atomic gas 4

25 nd law for sonic horizon? upstream p, flow downstream u, Tu Hu p d, Td, H d 貯気漕貯気漕 エントロピー生成は下流の貯気漕で起こる Horizon 内部 :chaotic or ergodic 5 Sonic horizon ではエントロピー生成は起こっていない BH の場合 : S = A ( 3 l c ) 1/ p G BH 4l p = Quantum Graity? 上流の static obserer から見ると, 下流で生成されたエントロピーが sonic horizon に投影される? Holography? 面積則は無いが, 第 法則は存在する? BH エントロピーの微視的な理解に示唆を与える

26 Numerical Simulation of Acoustic QN Ringing Acoustic BH Formation initial state: no flow fluid pumped out from the outlet final state: transonic flow Two types of 1D simulation : Weak Shock Infall initial state: transonic flow a weak shock injected from the inlet final state: transonic flow ~ BH formation ~ test particle infall flow

27 Acoustic BH Formation の例 ( 流れの時空図 ) sonic horizon 下流から順に流れ始める 上流では 流れが形成されると x~1.0l から遠方に向かって減衰振動が放射される QN ringing に違いない flow

28 Weak Shock Infall の例 Numerical QNM fitting