Analog Rotating Black Holes in a MHD Inflow

NARUTO Strait in Tokushima, Japan (Tidal whirlpool)

Analog rotating black holes in a MHD inﬂow

Sousuke Noda (Nagoya Univ. Japan) Based on

“Analog rotating black holes in a magnetohydrodynamic inﬂow” S.N., Yasusada Nambu, and Masaaki Takahashi Collaborators Phys. Rev. D .95, 104055 (2017) Yasusada Nambu (Nagoya Univ.)

Masaaki Takahashi (Aichi Edu. Univ.) Analog geometry for acoustic waves

No ﬂow With ﬂow

FLOW Ray

Ray Wave front Wave front

Ray’s trajectory = straight line Ray’s trajectory = curve

ds2 = ?

Light rays in ﬂat spacetime Light rays in curved spacetime

2 µ ⌫ 2 µ ⌫ ds = ⌘µ⌫ dx dx =0 ds = gµ⌫ dx dx =0 Acoustic Black Holes

4 stream line

0 horizon superradiance -2 ergosurface -4 Source -4 -2 0 2 4

“Acoustic” Black Hole sonic surface ~ horizon

Montecello dam (CA. USA) http://amazinglist.net/2013/03/bottomless-pit-dam-monticello-drain-hole/ Black Holes Acoustic Black Holes

Solutions of Einstein eq. NO ! Einstein eq. metric Efective geometry for acoustic waves 2 µ ⌫ Gµ⌫ = Tµ⌫ ds = gµ⌫ dx dx @v Fluid Dynamics ⇢ +(v )v = p + Fex @t · r r curved spacetime  Physical Properties Partially YES! Horizon Acoustic horizon sonic point Ergoregion (rotating BHs) Acoustic ergoregion Photon sphere PhoNon sphere No hair theorem Beautiful theorems . . Phenomena Partially YES BH physics in Lab. wave ampliﬁcation Superradiance Hawking radiation Hawking radiation ergoregion 4 stream line Black Hole Shadow 2 Quasi Normal Modes “Hawking rad” horizon ! 0 Superradiance Kerr BH supersonic subsonic -2 . superradiant condition . ergosurface !

Acoustic BH in labs

sonic points BH physics Fluid dynamics Analog model e.g.) Hawking rad, Superradiance etc.

Our work (Astrophysical) BH physics Fluid dynamical phenomena Analog model e.g.) Jet QPO Magnetic ﬁeld

1. We discuss how to deﬁne the analog BH for MHD waves. (This talk) 2. We “use” Black Hole physics to understand ﬂuid phenomena. (Future works) The motivation of our work

sonic points BH physics Alfven0 Fluidwave in z-directiondynamics Analog model e.g.) Hawking rad, Accretion disk vertical magnetic ﬁeld Superradiance etc.

MHD wave instability

Our work (Astrophysical) BH physics Fluid dynamical phenomena Analog model e.g.) Jet QPO Magnetic ﬁeld

1. We discuss how to deﬁne the analog BH for MHD waves. (This talk) 2. We “use” Black Hole physics to understand ﬂuid phenomena. (Future works) Contents

1. Acoustic Black Holes (perfect ﬂuid)

2. Magneto-acoustic BHs (MHD)

3. Results Contents

1. Acoustic Black Holes (perfect ﬂuid)

2. Magneto-acoustic BHs (MHD)

3. Results Acoustic metric

Perfect ﬂuid (irrotational)

@⇢ Background Perturbation + (⇢v)=0 , v = @t r · rvelocity potential v v + v + ! ! @v ⇢ ⇢ + ⇢ ⇢ +(v )v = p + Fex , p = p(⇢) @t · r r ! 

Wave eq. for

@ 2 @ 2 @ c ⇢ + v + c ⇢ + v v ⇢ =0 @t s @t · r r · s @t · r r  ✓ ◆ ⇢ ✓ ◆

Klein-Gordon eq. in a “ curved” spacetime Acoustic line element 1 @ @ 2 µ ⌫ Moncrief (1980) µ⌫ ds = sµ⌫ dx dx p ss = ⇤s =0 Unruh (1981) p s @xµ @x⌫ ✓ ◆ Acoustic metric

Acoustic waves feel Riemannian geometry Acoustic Horizon & ergoregion

Acoustic line element ⇢ dR2 ds2 = s dxµdx⌫ = c2 (vR)2 +(v)2 dt2 2v Rd dt + + R2 d2 µ⌫ c s 1 (vR/c )2 s ⇢ s ⇥ ⇤ For stationary & axisymmetric BG ﬂow, there exist Killing vectors ⇠(t) and ⇠() . v = v(R)

2 R 2 2 ⇠(t) =0 cs = (v ) +(v ) Acoustic ergosurface ⌘ = ⇠(t) + ⇠() q const 2 v ⌘ =0 R = Acoustic horizon cs = v R RH stream line 4 horizon Acoustic BH in laboratories Hawking radiation Superradiance 2 4 stream line

0 2 “Hawking rad” 0 horizon -2 supersonic subsonic -2 ergosurface ergosurface -4 theory: Unruh (1981) -4

experiment: Steinhauer (2016) -4 -2 0 2 4 -4 -2 0 2 4 Draining Bathtub Model (2D) M. Visser, Class.Quant.Grav. 15, 1767 (1998). Axisymmetric inﬂow with the sonic surface

1 1 stream line vR v (R, ) 4 /pR , / R cylindrical coordinates 2 Wave equation 0 i!t im = (R)e e ⇤sµ⌫ =0 -2 ergo

radial part 2 d 2 -4 2 +(! Ve↵) =0 dRtort -4 -2 0 2 4 Acoustic superradiance

Wave scattering by the acoustic BH The Superradiant condition Kerr case ampliﬁed wave v(R ) Ve↵ 0 H !

Eikonal approx. Ray Klein-Gordon eq. (WAVE) Eikonal eq. (RAY) Wave front eiS @S dx⌫ / µ⌫ k = = s ⇤s =0 s kµk⌫ =0 µ @xµ µ⌫ d tangent vector We can separate S as 2 tt 2 t 2 RR dSR S = ! t + m + SR(R) s ! 2s ! m + s m + s =0 dR ✓ ◆ Radial eq. & efective potential Efective potential 2 dR 1 + v c2 (vR)2 = 2 (! V )(! V ) 0 s d cs V ± = m ± ✓ ◆ p R

Forbidden region for RAYS

level crossing + v (RH) V ! V m reﬂected wave   RH transmitted wave incident wave Superradiant condition Forbidden for rays v(R ) negative phase !

RH RE Efective potential & Acoustic superradiance

v =0 case !

Analog Schwarzshild BH positive ω ! =const V + Forbidden（for rays） R RH V Efective potential 2 R 2 negative ω v cs (v ) V ± = m ± p R horizon v =0 Forbidden region for RAYS 6 + Analog rotating BH ! V ! V   Superradiant condition

level crossing v (RH) m reﬂected wave v (RH) !

1. Acoustic Black Holes

2. Magneto-acoustic BHs

3. Summary & Future work Diference between perfect ﬂuid & MHD

Perfect ﬂuids MHD

-2

Ray -4 v Wave front -4 -2 0 2 4 B 2 µ ⌫ 2 ds = sµ⌫ dx dx ds = ? Riemannian geometry Riemannian geometry ?????

single wave mode: ⇤s =0 Multiple modes: fast & slow (2D case) Isotropic propagation Anisotropic propagation B dispersion relation dispersion relation k 2 2 ✓ !2 = c2k2 2 = @2 + @2 ! = f(k , k B) s r x y · BH-like structure : rotating BH rotating BH ?

Superradiance Ve↵ Superradiant scat. of MHD waves ??

Rtort MHD wave equation (2D) Ideal MHD Background Perturbation @⇢ @B + (⇢v)=0 , = (v B) , B =0 @t r · @t r⇥ ⇥ r · v v + v ⇢ ⇢ + ⇢ ! ! @v p 1 B B + B Fex Fex +(v )v + r + B ( B)+Fex =0 ! ! @t · r ⇢ 4⇡⇢ ⇥ r⇥

MHD wave equation (2D) Lagrange derivative D @ = + v Helmholtz theorem Dt @t · r

@ @ @ @ Alfven0 velocity v = x + y = x y @y @x @y @x 1 ✓ ◆ ✓ ◆ ✓ ◆✓ ◆ V = B A p4⇡⇢ Wave Eq. for multiple wave modes

Lˆ1 =(VA)y @x (VA)x @y D2 c2 2 0 Lˆ2 Lˆ Lˆ = s r + 1 1 2 Dt2 00 Lˆ Lˆ Lˆ2 ˆ ✓ ◆ ✓ ◆ ✓ 1 2 2 ◆ ✓ ◆ L2 =(VA)x @x +(VA)y @y

How is the acoustic metric introduced ?? MHD wave & Acoustic wave MHD wave eq (2D) Lˆ1 =(VA)y @x (VA)x @y 2 2 2 ˆ2 ˆ ˆ D cs 0 L1 L1L2 Lˆ2 =(VA)x @x +(VA)y @y 2 = r + 2 Dt 00 Lˆ1Lˆ2 Lˆ ✓ ◆ ✓ ◆ ✓ 2 ◆ ✓ ◆ V B A / B =0 case eikonal limit 1 @ @ @S @S p ssµ⌫ =0 sµ⌫ =0 Eikonal equation p s @xµ @x⌫ @xµ @x⌫ ✓ ◆

Acoustic metric = coefcients of the eikonal equation

B =0 case (MHD) We introduce the acoustic metric through the eikonal equation. 6 (up to conformal factor)

Eikonal eq. for MHD waves (2D) = eiS, = eiS | | | | @S 4 @S 2 + v S (c2 + V 2) S 2 + v S +(c2 S 2)(V S)2 =0 @t · r s A |r | @t · r s |r | A · r ✓ ◆ ✓ ◆

@S @S @S @S M µ⌫ =0 Non quadratic form….. @xµ @x⌫ @x @x Magneto-acoustic geometry Eikonal eq. for 2D MHD waves @S @S @S @S M µ⌫ =0 Quartic metric NON Riemannian ! @xµ @x⌫ @x @x Magneto-acoustic metric Finsler metric If we deﬁne line element as

2 µ⌫ 1/4 ds = F (x, y)= M yµy⌫ yy yµ = @µS , F (x, y) satisﬁes the homogeneity condition: F (x, y)=F (x, y). = const

Finsler geometry unit “circle” in a Finsler manifold 1. Generalization of Riemannian geometry 1 2. Distance depends on the direction. 1 . Center . 1 Anisotropic propagation of MHD analogue geometry dispersion relation B Finsler geometry !2 = f(k2, k B) k · ✓ Unfortunately…..

I’m not familiar with calculations of Finsler geometry. (for now) Reduction of metric 1 1 The Finslerian magneto-acoustic metric M µ⌫ Center 1

Riemannian magneto-acoustic metric M µ⌫

Alfven0 velocity V 2 c2 2 2 A s or VA cs 1 In strong (or weak) B case ⌧ V = B A p4⇡⇢ 2 magnetic pressure VA ⇠ sound velocity (v = 0) wave front of fast mode gas pressure c2 s c = @p/@⇢ B ⇠ s p 0.5 Dispersion relation of fast mode

2 2 2 2 2 2 VM(kx + ky) b k cs VA 0.0 ! = 1+ 1 4⌘ · ⌘ 1 2 2 s k 3 ⌘ V 2 ⌧ ✓ ◆ ✓ M ◆ 4 5

-0.5 Almost isotropic propagation of MHD waves (fast mode) for small ⌘ -0.5 0.0 0.5 ⌘ =0.29 ⌘ =0.1 Reduced Magneto-acoustic metric

@S 2 + v S We solve the eikonal eq. for @t · r b = B/B ✓ ◆

@S 2 V 2 S 2 c V 2 b S 2 + fast mode + v S = M |r | 1 1 4 s A · r @t · r 2 2 ± s V 2 S 3 ✓ ◆ ✓ M ◆ ✓ |r | ◆ - slow mode 4 V 2 V 2 + c2 5 M ⌘ A s 2 cs VA ⌘ 1 p can be expanded ⌘ V 2 ⌧ ✓ M ◆

Two eikonal eqs.

2 @S V 2 S 2 ⌘ (b S)2 (fast mode)fast mode + v S M |r | · r @t · r ⇡ ⌘ V 2 (b S)2 (slowslow mode) mode ✓ ◆ ( M · r assign the coefcients @S @S @S @S M µ⌫ =0 M µ⌫ =0 Are these “metrics” ? fast @xµ @x⌫ , slow @xµ @x⌫

i i µ⌫ 1 v µ⌫ 1 v M = ,M = ,i,j=1, 2 fast vi V 2 ij vi vj + ⌘ V 2 bibj slow vi vi vj ⌘ V 2 bibj ✓ M M ◆ ✓ M ◆ µ⌫ Fast mode Mfast

µ⌫ We can deﬁne the inner product has the inverse Mfast (Mfast)µ⌫ 2 µ ⌫ Distance dsfast =(Mfast)µ⌫ dx dx Magneto-acoustic line element

ds2 (V 2 v2) ⌘ (b v)2 dt2 2[v + ⌘ b (b v)] dt dxi +( + ⌘ b b ) dxidxj fast / M · i i · ij i j ⇥ ⇤ vR vR v + ⌘ bR b V 2 dR d d + M dt dt + 2 2 R 2 R 2 dR 2 2 R 2 R 2 ! V ⌘ V (b ) (v ) , ! VM ⌘ VM (b ) (v ) R M M

ds2 (V 2 v2) ⌘ (b v)2 dt2 2 v + ⌘ b(b v) Rdtd fast / M · · dR2 ⇥ + ⇤ ⇥ + 1+⌘ (b)2⇤ R2d2 1 ⌘ (bR)2 (vR/V )2 M ⇥ ⇤

Magneto-acoustic horizon & ergosurface

Magneto-acoustic horizon Magneto-acoustic ergosurface

(vR)2 = V 2 ⌘V 2 (bR)2 v2 = V 2 ⌘(b v)2 M M M · Propagation speed in the radial direction Slow mode

µ⌫ Mslow no inverse no inner pruduct

The eikonal eq. for the slow mode is advective equation

@S + v S =0 v v (csVA/VM)b @t ± · r , ± ⌘ ± BG MHD ﬂow Slow wave mode propagate along v ± v ± “distance” It’s impossible to introduce acoustic metric for the propagation of the slow mode. 2D Background MHD inﬂow

Analog rotating black holes in a MHD inﬂow 2D Axisymmetric MHD inﬂow Basic equations & Conserved quantities

Ideal MHD eq. (stationary)

(⇢v)=0 , (v B)=0 , B =0 Equation of state r · r⇥ ⇥ r · @v p 1 p ⇢ 1 4 +(v )v + r + B ( B)+Fex =0 /   @t · r ⇢ 4⇡⇢ ⇥ r⇥ axisymmetric inﬂow R R v = v (R) eRˆ + v (R) eˆ , B = B (R) eRˆ + B (R) eˆ conserved quantities

R R ⇢ v R = const. < 0 , RB = const. > 0 inﬂow outward

BR Rv RB = const. L 4⇡⇢ vR ⌘ angular momentum

vRB vBR = const. ⌦ RBR angular velocity of the magnetic ﬁeld lines ⌘ F Can our BG ﬂow satisfy η ≪ 1 ?

For entire region, Condition for reduction of the magneto-acoustic metric 2 c V 2 2 2 2 2 s A @S VM S cs VA b S ⌘ 2 1 + v S = |r | 1 1 4 · r ⌘ V ⌧ @t · r 2 2 ± s V 2 S 3 ✓ M ◆ ✓ ◆ ✓ M ◆ ✓ |r | ◆ 4 5

mag-pressure dominated gas-pressure dominated V 2 c2 c2 V 2 Which is allowed for our BG ﬂow ? A s s A

0.10 Near R=0 and far region, V 2 c2 (↵, )=(0.1, 0.04) A s 0.08

0.06 ⌘c(↵, ) Only Mag-pressure dominated ﬂow (c /V )2 s A can be realize for BG ﬂow under ⌘ 1 0.04 ⌧ 0.02 2 cs VA ⌘ 1 ⌘c(↵, ) 1 0.00 ⌘ V 2 ⌧ ⌧ 0 5 10 15 20 25 30 ✓ M ◆ Rc/R R/R ⇤ ⇤ Brief summary so far..

2 cs VA ① ⌘ 2 1 (Riemannian) Magneto-acoustic metric for the fast mode ⌘ VM ⌧ ✓ ◆ (coefcients of the eikonal eq.) Magneto-acoustic metric

ds2 (V 2 v2) ⌘ (b v)2 dt2 2 v + ⌘ b(b v) Rdtd fast / M · · dR2 ⇥ + ⇤ ⇥ + 1+⌘ (b)2⇤ R2d2 1 ⌘ (bR)2 (vR/V )2 M ⇥ ⇤ ② Analog horizon & ergoregion for the fast wave mode

Magneto-acoustic horizon Magneto-acoustic ergosurface

(vR)2 = V 2 ⌘V 2 (bR)2 v2 = V 2 ⌘(b v)2 M M M ·

③ Background ﬂow Magnetic-pressure dominated > 0

4 Equation of state Flow & wave velocities are characterized by (α, β) 2

y/R 0 ⇤ p ⇢ 1 4 R si   ↵ cs(R )/v > 0 ~sound velocity @R -2 / ⌘ ⇤ ⇤ ⇤ -4 R v ⌦FR /v ~angular velocity of B @ R -4 -2 0 2 4 x/R B ⌘ ⇤ ⇤ ⇤ ⇤ Magneto-acoustic geometry How do we examine?

Motion of magneto-acoustic ray separation of variables dxµ @S kµ = M µ⌫ , (M ) kµk⌫ =0 S = !t + m + S (R) ⌘ d fast @x⌫ fast µ⌫ R tangent vector

Radial part Forbidden region for RAYS 2 dR 1 R 2 + + = 1 ⌘ (b ) (! V )(! V ) V ! V d V 2   ✓ ◆ A ⇥ ⇤ Efective potentials !

(M ) (M )2 (M ) (M ) fast t ± fast t fast fast tt V ± = m ! q (M ) 一 fast V + Forbidden m v + ⌘b(b v) V 2 (vR)2 ⌘ [(vR)2 (b)2V 2 ] R = · ± M M 2 V R p 1+⌘ (b ) RH RE

Magneto-acoustic horizon √ =0 Depending on (α,β), these condition

V =0 Magneto-acoustic ergosurface Classiﬁcation of Magneto-acoustic geometry

0.14 (c) ↵ c (R )/vR > 0 ~sound velocity @ R ⌘c =0.1 0.3 s ⌘ ⇤ ⇤ ⇤ 0.12 0.5 E(Re; ↵, )=0 R (b) 0.7 ⌦FR /v ~Angular velocity of B @ R ⌘ ⇤ ⇤ 0.10 0.9 ⇤ 0.08 Efective potential for rays 0.06 (a) 2 H(Rh; ↵, )=0 (M ) (M ) (M ) (M ) 0.04 fast t ± fast t fast fast tt V ± = m q (M ) 0.02 ⌘ (↵, ) 1 fast c ⌧ 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ↵ 3 types of geometry

5 (a) 5 (b) 5 Normal scatt (c) 4 horizon 4 4 V + 3 3 3 + 2 V + 2 V 2 1 1 1 0 0 0

-1 V -1 -1 ergo ergo V V 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 R/R NO Horizon R/R NO ergo R/R ⇤ ⇤ ⇤ Rotating BH Star with the ergoregion Star 0.14 (c) ⌘c =0.1 0.3 Type (a) Analogue rotating black hole 0.12 0.5 E(Re; ↵, )=0 (b) 0.7 0.10 0.9 0.08 0.06 (a) H(Rh; ↵, )=0 Superradiance for MHD waves 0.04

0.02

0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 The magnetic pressure efective potential for type (a) ↵ m 4 V ± V ⇡ ± R M 3 Ampliﬁed wave

! 2 V + (vR/R ) ⇤ ⇤ 1 positive energy mode ergo Forbidden region 0 negative energy mode

-1 V 0 5 10 15 Magnetoacoustic horizon R/R ⇤ 0.14 (c) ⌘c =0.1 Type (b) Ultraspinning star 0.3 0.12 0.5 E(Re; ↵, )=0 (b) 0.7 0.10 0.9 0.08 Ergoregion instability by MHD waves scattering 0.06 (a) H(Rh; ↵, )=0 0.04

0.02

0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 efective potential for type (b) ↵ 4 NO Horizon 3

! 2 V + (vR/R ) ⇤ ⇤ positive energy mode 1 ergo Forbidden region negative energy mode 0

-1 V 0 5 10 15 R/R ⇤ Applications (Future work)

POINT Magneto-acoustic horizon Magneto-acoustic ergosurface

R 2 2 2 R 2 2 2 2 (v ) = V ⌘V (b ) v = VM ⌘(b v) M M ·

Magneto-acoustic BH in astrophysical situations

Alfven0 wave in z-direction

Accretion disk vertical magnetic ﬁeld

MHD wave instability

Central object does not have to be a BH MHD Flow makes analog BH

MHD-analog superradance or ergoregion instability MHD Wave around a star

QNMs of a Magneto-acoustic BH peculiar motion of accretion disk ? Radial Alfven0 point

Regularity condition

2 2 1 R R R M LR ⌦ 1 v B v B = const. ⌦F RB v = ⌦ R A F ⌘ F M 2 1 A R B R 2 2 1 Rv RB = const. L ⌦FRB MA(LR ⌦F ) R B = 4⇡⇢ v ⌘ vR M 2 1 A When Radial Mach number

R 2 2 v L = ⌦ R2 MA(R) R becomes unit R = R , v and B are singular. F ⌘ VA ⇤ ⇤ ✓ ◆ Radial Alfven0 point Regularity condition @ R ⇤

R R ⌦F R v (R/R )v ⇤ ⇤ R R v = R 2 ⇤ RHS is written by v and B v MA 1 ⇤

R 2 2 R B ⌦F R R RB = const. > 0 ⇤ B = R 2 v R MA 1 ⇤ ⇤ R We just solve v . Summary 2 cs VA Magneto-acoustic metric for the fast mode ⌘ 2 1 ⌘ VM ⌧ ✓ ◆ no metric for slow mode Magneto-acoustic metric v ± ds2 (V 2 v2) ⌘ (b v)2 dt2 2 v + ⌘ b(b v) Rdtd fast / M · · “distance” dR2 ⇥ + ⇤ ⇥ + 1+⌘ (b)2⇤ R2d2 1 ⌘ (bR)2 (vR/V )2 M ⇥ ⇤

Analog horizon & ergoregion for the fast wave mode

Magneto-acoustic horizon Magneto-acoustic ergosurface

(vR)2 = V 2 ⌘V 2 (bR)2 v2 = V 2 ⌘(b v)2 M M M ·

Analog superradiance & ergoregion instability for MHD waves

> 0 4 Analog wave phenomena 2 rotating BHs MHD wave superradiance y/R 0 ⇤ si (α, β) -2 ultraspinning star ergoregion instability

-4 ratating star normal scatteing v -4 -2 0 2 4 x/R B ⇤ BACK UP Equation for vR

Euler’s equation

dvR v 1 dp 1 B d Eq. for vR + + (RB)+F R =0 vR and F R dR R ⇢ dR 4⇡⇢ R dR ex ex

R R ① Give Fex and solve this equation for v draining bathtub type inﬂow

R R 1/2 ✓ ② There exists one Fex for a given v R vR = vR ⇤ R ✓ ⇤ ◆ Background ﬂow & propagation velocities

Background MHD ﬂow Equation of state 1/2 1/2 1/2 p ⇢ R R R R 1 4 vR = vR v = v + +1 /   ⇤ R , ⇤ " R R # ✓ ⇤ ◆ ✓ ⇤ ◆ ✓ ⇤ ◆ R ⌦FR /v 1/2 1 ⌘ ⇤ ⇤ R R R R R B = B ⇤ , B = BR 1+ 1+ ~ang velo of B @ R ⇤ R ⇤ R R ⇤ " ✓ ⇤ ◆ #" ✓ ⇤ ◆ #

Wave velocities

Alfven0 velocity > 0 3/2 1/2 1/2 2 1 2 2 R 2 R 2 R R R 4 VA =(v ) + 1+ 1+ ⇤ 8 R R R R 9 ✓ ⇤ ◆ ✓ ⇤ ◆ " ✓ ⇤ ◆ # " ✓ ⇤ ◆ # < = 2 : ( 1)/2 ; 2 2 R 2 R 2 2 2 cs = ↵ (v ) V = V + c y/R 0 ⇤ R M A s ⇤ sink ✓ ⇤ ◆ , -2 R ↵ cs(R )/v > 0 ~sound velo @ R ⌘ ⇤ ⇤ ⇤ -4

-4 -2 0 2 4 BG ﬂow & Velocies are characterized by and x/R v ↵ ⇤ B Acoustic superradiance

Wave equation for the acoustic disturbance Separation of variable (R) 1 @ µ⌫ @ i( ! t+m ) p ss =0 = (7 )/16 e p s @xµ @x⌫ R ✓ ◆ m = 0, 1, ± ± ··· radial eq. efective potential

2 2 d 1 mv (7 )(9 + ) g(R) 7 dg(R) m2 + Ve↵(R; !, m, ) =0 Ve↵ = ! g(R) 2 c2 R 4096 R2 16R dR R2 dRtort s ✓ ◆  g(R)=1 (vR/c )2 s @/@R = g(R) @/@R tort · WKB sol Rtort 1 mv exp i ! dR for R R c R tort ⇠ H 8 Z s ✓ ◆ ! = > R R > tort ! tort ! <>C exp i dR + C exp i dR for R R R in c tort out c tort ⇠ f H Z s ! Z s ! > > :> Cout scattering with ampliﬁcation > 1 Conservation of Wronskian C in C 2 c (R ) ! m ⌦ 1 2 Superradiant condition out + s H H =1 Cin cs(Rf) · ! Cin v (RH) !

The magnetoacoustic BHs in 3D , Astrophysical situation

Alfven0 wave in z-direction

Accretion disk vertical magnetic ﬁeld

MHD wave instability

Magneto-acoustic BH (or geometry)を考える上では、中心天体はBHでなくても良い。

MHD-superradanceやergoregion instabilityとJETなどの天体現象は？

３次元にすると、Alfven waveも出てくる。

MHD wave方程式 2 !f,s = !(k)f,s D v 2 2 cs ( v)+VA ( (v VA)) = 0 Dt r r · ⇥r⇥ r⇥ ⇥ !Al = !(k)Al か？？） Future work ③ eikonalでは波の増幅率等は計算できない完全流体の場合 1 @ @ 音波方程式 p ssµ⌫ =0 p s @xµ @x⌫ ✓ ◆ 速度ポテンシャル MHDの場合 MHD wave方程式 KG型 ???? B = ↵ r ⇥r 磁気優勢 or 流体優勢 Clebsch ポテンシャルを使う？もしできたら。。。 Magneto-acoustic BH の Quasi Normal Modes や Bombの計算

!QNM や !Bomb との輻射の関係は？円盤振動学？

天体輻射降着流

音速面

MHD wave 疑問点、質問

MHD ﬂowをAcoustic BHとして見てみると、不安定性が起こるか否か

R 2 2 2 R 2 (v ) = VM ⌘VM(b ) Magneto-acoustic horizon (MHD waveが動径方向に出ようとしても出られない面)

v2 = V 2 ⌘(b v)2 Magneto-acoustic ergosurface M · Background ﬂowがこれらを満たすかで決まった。（少なくとも今回のﬂowでは） horizonやergoの存否

質問流体の方程式をそのまま解いて、MHDの計算をする場合にも上記の条件は重要なものなのか？不安定性の条件は？

疑問 Magneto-ergoregion instabiliry MRIなどの不安定性 Analogue BHの不安定性はBG ﬂowの不安定性 Superradiant condition

Klein-Gordon eq. (WAVE)

i!t im ⇤s =0 , = (R)e e

d 2 +(! Ve↵) =0 dRtort

Asymptotic sol

mv i ! Rtort e RH = ⇣ ⌘ i!Rtort i!Rtort Cine + Coute

Conservation of Wronskian reﬂection rate transmission rate C 2 c (R ) ! m ⌦ 1 2 out + s H H =1 C c (R ) · ! C in s f in

Cout v (RH) > 1 !

Klein-Gordon eq. (WAVE) Eikonal eq. (RAY)

i!t im ⌫ =0 , = (R)e e µ⌫ @S @S @S dx ⇤s s =0 = sµ⌫ @xµ @x⌫ , @xµ d d 2 tangent vector of RAYS +(! Ve↵) =0 dRtort S = ! t + m + S (R) R Asymptotic sol 2 mv dR 1 + i ! R Rtort e H = 2 (! V )(! V ) 0 ⇣ ⌘ d c = ✓ ◆ s i!Rtort i!Rtort Cine + Coute Forbidden region for RAYS 2 R 2 Conservation of Wronskian v cs (v ) V ± = m ± + R V ! V p   reﬂection rate transmission rate 2 2 Cout cs(RH) ! m ⌦H 1 + =1 正と負の間の Cin cs(Rf) · ! Cin level crossing v (RH) m 反射波 RH 入射波

禁止領域（for rays） Cout v (RH) > 1 !

Solutions of Einstein eq. NO ! Einstein eq. metric Efective geometry for acoustic waves 2 µ ⌫ Gµ⌫ = Tµ⌫ ds = gµ⌫ dx dx @v Fluid Dynamics ⇢ +(v )v = p + Fex @t · r r curved spacetime  Physical Properties Partially YES! Horizon Acoustic horizon sonic point Ergoregion (rotating BHs) Acoustic ergoregion Photon sphere PhoNon sphere No hair theorem Beautiful theorems . . Phenomena Partially YES wave ampliﬁcation Superradiance Hawking radiation Hawking radiation ergoregion 4 stream line Black Hole Shadow 2 Quasi Normal Modes “Hawking rad” horizon ! 0 Superradiance Kerr BH supersonic subsonic -2 . superradiant condition . ergosurface !

1/2 (3 )/4 R R R R R ↵ R v+ = v + csb = v 2 1+ 3 | ⇤ | R 2 R ✓ ⇤ ◆ 2 2 ✓ ⇤ ◆ 6 1+ 1+ R/R (1 + R/R ) 7 6 r ⇤ ⇤ 7 4 ⇣ p ⌘ 5

0.04 0.9 0.7 内向きか外向きかどちらに伝播するか？ 0.5 0.03 0.3 ⌘c =0.1 R 0.02 今考えている(α,β)の範囲内では v < 0 ±

0.01 全領域で内向きでhorizonのような境界はない

0.00 0.0 0.1 0.2 0.3↵0.4 0.5 0.6 0.7