Analog rotating black holes in a MHD inflow Based on ? Analog - - PowerPoint PPT Presentation

Analog rotating black holes in a MHD inflow

Sousuke Noda (Nagoya Univ. Japan)

Collaborators

Yasusada Nambu (Nagoya Univ.)

Masaaki Takahashi (Aichi Edu. Univ.)

NARUTO Strait in Tokushima, Japan

(Tidal whirlpool)

Based on

• Phys. Rev. D .95, 104055 (2017)

“Analog rotating black holes in a magnetohydrodynamic inflow” S.N., Yasusada Nambu, and Masaaki Takahashi

Analog geometry for acoustic waves

No flow With flow

FLOW

Ray’s trajectory = straight line Ray’s trajectory = curve Light rays in flat spacetime

Ray

Wave front

Ray

Wave front

Light rays in curved spacetime

ds2 = gµνdxµdxν = 0 ds2 = ηµνdxµdxν = 0 ds2 =

?

Acoustic Black Holes

Montecello dam (CA. USA)

sonic surface

“Acoustic” Black Hole

~ horizon

Source

• 4

2 2 4

horizon ergosurface

stream line

superradiance

http://amazinglist.net/2013/03/bottomless-pit-dam-monticello-drain-hole/

Acoustic Black Holes Black Holes

ds2 = gµνdxµdxν

metric Einstein eq.

Solutions of Einstein eq.

Physical Properties Phenomena

curved spacetime

Horizon Ergoregion (rotating BHs) Photon sphere No hair theorem . . . Superradiance Quasi Normal Modes Hawking radiation Black Hole Shadow . . .

ω

ergoregion Kerr BH

Gµν = κTµν

wave amplification

ω < mΩH

superradiant condition

Efgective geometry for acoustic waves

NO !

Acoustic horizon

Partially YES!

Acoustic ergoregion

sonic point

PhoNon sphere Beautiful theorems

Partially YES

• 4

2 2 4

horizon ergosurface

stream line

Superradiance Hawking radiation

theory: Unruh (1981) experiment: Steinhauer (2016) supersonic subsonic

“Hawking rad”

Fluid Dynamics

ρ ∂v ∂t + (v · r)v

• = rp + Fex

BH physics in Lab.

(Astrophysical) Jet QPO e.g.)

• 2. We “use” Black Hole physics to understand fluid phenomena.

Our work

Analog model

e.g.) Hawking rad, Superradiance etc.

Acoustic BH in labs

The motivation of our work

BH physics Fluid dynamics

sonic points

BH physics Fluid dynamical phenomena

Magnetic field Analog model

• 1. We discuss how to define the analog BH for MHD waves.

(This talk) (Future works)

(Astrophysical) Jet QPO e.g.)

• 2. We “use” Black Hole physics to understand fluid phenomena.

Our work

Analog model

e.g.) Hawking rad, Superradiance etc.

The motivation of our work

BH physics Fluid dynamics

sonic points

BH physics Fluid dynamical phenomena

Magnetic field Analog model

• 1. We discuss how to define the analog BH for MHD waves.

(This talk) (Future works)

Accretion disk

MHD wave instability

vertical magnetic field

Alfven wave in z-direction

• 1. Acoustic Black Holes
• 2. Magneto-acoustic BHs
• 3. Results

Contents

(perfect fluid) (MHD)

• 1. Acoustic Black Holes
• 2. Magneto-acoustic BHs
• 3. Results

Contents

(perfect fluid) (MHD)

Acoustic metric

Unruh (1981) Moncrief (1980)

velocity potential

, ,

Klein-Gordon eq. in a “ curved” spacetime

Perfect fluid (irrotational) Wave eq. for

Background Perturbation

p = p(ρ)

∂ρ ∂t + r · (ρv) = 0

ρ ∂v ∂t + (v · r)v

• = rp + Fex

1 √−s ∂ ∂xµ ✓√ −ssµν ∂δΦ ∂xν ◆ = ⇤sδΦ = 0

v = rΦ

Acoustic metric

ds2 = sµνdxµdxν

Acoustic line element

v → v + δv

Φ → Φ + δΦ ρ → ρ + δρ ∂ ∂t  c−2

s ρ

✓∂δΦ ∂t + v · rδΦ ◆ + r · ⇢ c−2

s ρ

✓∂δΦ ∂t + v · rδΦ ◆ v ρrδΦ

• = 0

Acoustic waves feel Riemannian geometry

δΦ

Acoustic Horizon & ergoregion

ds2 = sµνdxµdxν = ρ cs ⇢ − ⇥ c2

s −

• (vR)2 + (vφ)2⇤

dt2 − 2vφ R dφ dt + dR2 1 − (vR/cs)2 + R2 dφ2

• Acoustic horizon

Acoustic ergosurface

cs = vR

cs = q (vR)2 + (vφ)2

Acoustic line element

• 4

2 2 4

stream line

ergosurface horizon

For stationary & axisymmetric BG flow, there exist Killing vectors v = v(R) and .

η2 = 0

ξ2

(t) = 0

ξ(t)

ξ(φ)

η = ξ(t) + βξ(φ)

β = vφ R

• RH

const

Acoustic BH in laboratories

• 4

2 2 4

horizon ergosurface

stream line

Superradiance Hawking radiation

theory: Unruh (1981) experiment: Steinhauer (2016) supersonic subsonic

“Hawking rad”

Draining Bathtub Model

(R, φ) cylindrical coordinates

Wave scattering by the acoustic BH

Axisymmetric inflow with the sonic surface

The Superradiant condition

ω < mvφ

0 (RH)

RH

Angular velocity of the BG flow at the acoustic horizon

purely ingoing

ω

amplified wave

• M. Visser, Class.Quant.Grav. 15, 1767 (1998).

Acoustic superradiance

(2D)

⇤sµνδφ = 0

δφ = ψ(R)e−iωteimφ

Wave equation radial part

Veff

Rtort

• 4

2 2 4

stream line

ergo

vR ∝ − 1 √ R

vφ ∝ 1 R

,

d2ψ dR2

tort

+ (ω2 − Veff)ψ = 0 ω < mΩH

Kerr case

Ray

Wave front

δΦ ∝ eiS

Acoustic superradiance in the eikonal limit

V − ≤ ω ≤ V +

Eikonal approx.

Eikonal eq. (RAY) Forbidden region for RAYS

Radial eq. & efgective potential

✓dR dλ ◆2 = 1 c2

s

(ω − V +)(ω − V −) ≥ 0

sµνkµkν = 0

Klein-Gordon eq. (WAVE)

⇤sδΦ = 0

kµ = ∂S ∂xµ = sµν dxν dλ

V ± = mvφ ± p c2

s − (vR)2

R

S = −ω t + m φ + SR(R)

We can separate S as Efgective potential

stt ω2 − 2stφ ω m + sφφ m2 + sRR ✓dSR dR ◆2 = 0

RE

mvφ(RH) RH

transmitted wave

reflected wave incident wave negative phase velocity mode

RH

Forbidden for rays

level crossing

ergo Superradiant condition

ω < m ΩH = mvφ(RH) RH

tangent vector

Efgective potential & Acoustic superradiance

Superradiant condition

vφ = 0

case

vφ 6= 0

ω ω

R

V +

V −

RH Forbidden（for rays）

ω < m ΩH = mvφ(RH) RH

negative ω positive ω

Analog Schwarzshild BH Analog rotating BH

ω=const

mvφ(RH) RH

transmitted wave

reflected wave incident wave negative phase velocity mode

Forbidden for rays）

level crossing

ergo

R

V + V −

ω=const

V − ≤ ω ≤ V +

Forbidden region for RAYS

V ± = mvφ ± p c2

s − (vR)2

R

Efgective potential

horizon horizon

• 1. Acoustic Black Holes
• 2. Magneto-acoustic BHs

Contents

• 3. Summary & Future work

Difgerence between perfect fluid & MHD

ds2 =

Riemannian geometry

sµνdxµdxν

Multiple modes: Anisotropic propagation Perfect fluids single wave mode: Isotropic propagation BH-like structure : rotating BH

Superradiance

ds2 =

?

(2D case)

ω2 = c2

sk2

dispersion relation

r2 = ∂2

x + ∂2 y

⇤sδΦ = 0

Riemannian geometry ?????

fast & slow

dispersion relation

ω2 = f(k2, k · B)

B

k

θ

Veff

Rtort

Superradiant scat. of MHD waves ?? rotating BH ? MHD

Ray

Wave front

4 2 2 4

v

B

MHD wave equation (2D)

Ideal MHD

Lagrange derivative

MHD wave equation (2D)

∂ρ ∂t + r · (ρv) = 0

∂B ∂t = r ⇥ (v ⇥ B)

r · B = 0

,

,

∂v ∂t + (v · r)v + rp ρ + 1 4πρB ⇥ (r ⇥ B) + Fex = 0 Helmholtz theorem

Alfven velocity

v → v + δv

B → B + δB

ρ → ρ + δρ

Fex → Fex

D Dt = ∂ ∂t + v · r

VA = 1 √4πρB

δv = ✓∂xΦ ∂yΦ ◆ + ✓ ∂yΨ −∂xΨ ◆ = ✓∂x ∂y ∂y −∂x ◆ ✓Φ Ψ ◆ D2 Dt2 ✓Φ Ψ ◆ = ✓c2

s r2

◆ + ✓ ˆ L2

1

ˆ L1 ˆ L2 ˆ L1 ˆ L2 ˆ L2

2

◆ ✓Φ Ψ ◆

ˆ L1 = (VA)y ∂x − (VA)x ∂y ˆ L2 = (VA)x ∂x + (VA)y ∂y

How is the acoustic metric introduced ??

Wave Eq. for multiple wave modes

Background Perturbation

MHD wave & Acoustic wave

MHD wave eq (2D) B = 0 case

Acoustic metric = coeffjcients of the eikonal equation

case (MHD) B 6= 0 sµν ∂S ∂xµ ∂S ∂xν = 0

VA ∝ B

Eikonal equation (up to conformal factor)

ˆ L1 = (VA)y ∂x − (VA)x ∂y ˆ L2 = (VA)x ∂x + (VA)y ∂y D2 Dt2 ✓ Φ Ψ ◆ = ✓ c2

s r2

◆ + ✓ ˆ L2

1

ˆ L1 ˆ L2 ˆ L1 ˆ L2 ˆ L2

2

◆ ✓ Φ Ψ ◆

1 √−s ∂ ∂xµ ✓√ −ssµν ∂Φ ∂xν ◆ = 0

Φ = |Φ| eiS, Ψ = |Ψ| eiS

✓∂S ∂t + v · rS ◆4 (c2

s + V 2 A)|rS|2

✓∂S ∂t + v · rS ◆2 + (c2

s |rS|2)(VA · rS)2 = 0

Eikonal eq. for MHD waves (2D)

M µνλσ ∂S ∂xµ ∂S ∂xν ∂S ∂xλ ∂S ∂xσ = 0

Non quadratic form…..

eikonal limit

We introduce the acoustic metric through the eikonal equation.

Magneto-acoustic geometry

M µνλσ ∂S ∂xµ ∂S ∂xν ∂S ∂xλ ∂S ∂xσ = 0

Quartic metric

ds2 = F(x, y) =

• M µνλσyµyνyλyσ

1/4

.

yµ = ∂µS

F(x, σy) = σF(x, y)

If we define line element as

unit “circle” in a Finsler manifold

Anisotropic propagation of MHD

dispersion relation

ω2 = f(k2, k · B)

B

k

θ

Center 1 1 1

Finsler geometry

analogue geometry

Eikonal eq. for 2D MHD waves NON Riemannian !

satisfies the homogeneity condition: Finsler metric

σ = const

,

F(x, y)

• 1. Generalization of Riemannian geometry

Finsler geometry

• 2. Distance depends on the direction.

. . .

Magneto-acoustic metric

Unfortunately…..

I’m not familiar with calculations of Finsler geometry. (for now)

Almost isotropic propagation

magnetic pressure

The Finslerian magneto-acoustic metric M µνλσ Riemannian magneto-acoustic metric M µν Reduction of metric In strong (or weak) case

B

V 2

A c2 s

V 2

A ⌧ c2 s

• r

Alfven velocity

VA = 1 √4πρB

Center

1 1 1

gas pressure

∼ V 2

A

∼ c2

s

sound velocity

cs = p ∂p/∂ρ

• f MHD waves (fast mode)

η ⌘ ✓cs VA V 2

M

◆2 ⌧ 1

ω2 = V 2

M(k2 x + k2 y)

2 2 41 + s 1 − 4η ✓b · k k ◆2 3 5

• 0.5

0.0 0.5

• 0.5

0.0 0.5

Dispersion relation of fast mode wave front of fast mode

η = 0.29

η = 0.1

(v = 0)

B

for small η

fast mode slow mode

+

• Reduced Magneto-acoustic metric

✓∂S ∂t + v · rS ◆2 = V 2

M |rS|2

2 2 41 ± s 1 4 ✓cs VA V 2

M

◆2 ✓b · rS |rS| ◆2 3 5

η ⌘ ✓cs VA V 2

M

◆2 ⌧ 1

✓∂S ∂t + v · rS ◆2

We solve the eikonal eq. for

M µν

fast =

✓ −1 −vi −vi V 2

M δij −

• vi vj + η V 2

M bibj

◆ , M µν

slow =

✓ −1 −vi −vi −

• vi vj − η V 2

M bibj

◆ , i, j = 1, 2

Two eikonal eqs.

M µν

fast

∂S ∂xµ ∂S ∂xν = 0 M µν

slow

∂S ∂xµ ∂S ∂xν = 0 ,

Are these “metrics” ?

assign the coeffjcients

✓∂S ∂t + v · rS ◆2 ⇡ ( V 2

M

• |rS|2 η (b · rS)2

(fast mode) η V 2

M (b · rS)2

(slow mode)

fast mode slow mode b = B/B

V 2

M ≡ V 2 A + c2 s

√

can be expanded

Fast mode

has the inverse

Magneto-acoustic line element

ds2

fast ∝ −

⇥ (V 2

M − v2) − η (b · v)2⇤

dt2 − 2 [vi + η bi(b · v)] dt dxi + (δij + η bibj) dxidxj

ds2

fast ∝ −

⇥ (V 2

M − v2) − η (b · v)2⇤

dt2 − 2 ⇥ vφ + η bφ(b · v) ⇤ R dt dφ + dR2 1 − η (bR)2 − (vR/VM)2 + ⇥ 1 + η (bφ)2⇤ R2dφ2

Magneto-acoustic horizon & ergosurface

dt → dt + vR V 2

M − η V 2 M(bR)2 − (vR)2 dR

dφ → dφ + vR vφ + η bR bφ V 2

M

V 2

M − η V 2 M (bR)2 − (vR)2

dR R

M µν

fast

(Mfast)µν

We can define the inner product Distance

ds2

fast = (Mfast)µνdxµdxν

,

v2 = V 2

M − η(b · v)2

(vR)2 = V 2

M − ηV 2 M(bR)2

Magneto-acoustic horizon Magneto-acoustic ergosurface

Propagation speed in the radial direction

M µν

fast

Slow mode

∂S ∂t + v± · rS = 0

v± ≡ v ± (csVA/VM)b

M µν

slow

no inverse no inner pruduct The eikonal eq. for the slow mode is advective equation , BG MHD flow

v±

It’s impossible to introduce acoustic metric

v±

“distance”

Slow wave mode propagate along 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.

Equation of state

(stationary)

,

p ∝ ρΓ

r · (ρv) = 0 r ⇥ (v ⇥ B) = 0 r · B = 0

∂v ∂t + (v · r)v + rp ρ + 1 4πρB ⇥ (r ⇥ B) + Fex = 0

,

1 ≤ Γ ≤ 4

axisymmetric inflow

v = vR(R) e ˆ

R + vφ(R) e ˆ φ

B = BR(R) e ˆ

R + Bφ(R) e ˆ φ

R vφ − BR 4πρ vR RBφ = const. ≡ L

vRBφ − vφBR = const. ≡ −ΩF R BR

conserved quantities

ρ vRR = const. < 0

RBR = const. > 0

,

inflow

• utward

, angular velocity of the magnetic field lines angular momentum

η ⌘ ✓cs VA V 2

M

◆2 ⌧ 1

gas-pressure dominated mag-pressure dominated

Condition for reduction of the magneto-acoustic metric

✓∂S ∂t + v · rS ◆2 = V 2

M |rS|2

2 2 41 ± s 1 4 ✓cs VA V 2

M

◆2 ✓b · rS |rS| ◆2 3 5

Near R=0 and far region, V 2

A c2 s

For entire region, V 2

A c2 s

c2

s V 2 A

5 10 15 20 25 30 0.00 0.02 0.04 0.06 0.08 0.10

Rc/R∗

R/R∗

(cs/VA)2

(α, β) = (0.1, 0.04)

Can our BG flow satisfy η ≪ 1 ?

Only Mag-pressure dominated flow Which is allowed for our BG flow ?

ηc(α, β)

η ⌘ ✓cs VA V 2

M

◆2 ⌧ 1

ηc(α, β) ⌧ 1

can be realize for BG flow under η ⌧ 1

ds2

fast ∝ −

⇥ (V 2

M − v2) − η (b · v)2⇤

dt2 − 2 ⇥ vφ + η bφ(b · v) ⇤ R dt dφ + dR2 1 − η (bR)2 − (vR/VM)2 + ⇥ 1 + η (bφ)2⇤ R2dφ2 (coeffjcients of the eikonal eq.) η ⌘ ✓cs VA V 2

M

◆2 ⌧ 1

① ② ③

Magneto-acoustic metric

Analog horizon & ergoregion for the fast wave mode

v2 = V 2

M − η(b · v)2

(vR)2 = V 2

M − ηV 2 M(bR)2

Magneto-acoustic horizon Magneto-acoustic ergosurface

Background flow

Brief summary so far..

y/R∗

• 4
• 2

2 4

• 4
• 2

2 4

x/R∗

β > 0

si

v B

Flow & wave velocities are characterized by (α, β)

β ≡ ΩFR∗/vR

∗

α ≡ −cs(R∗)/vR

∗ > 0

~sound velocity @ ~angular velocity of B @

Magnetic-pressure dominated

R∗ R∗

(Riemannian) Magneto-acoustic metric for the fast mode

Equation of state

p ∝ ρΓ 1 ≤ Γ ≤ 4

Magneto-acoustic geometry

kµ ≡ dxµ dλ = M µν

fast

∂S ∂xν , (Mfast)µν kµkν = 0

How do we examine?

Magneto-acoustic horizon

Motion of magneto-acoustic ray

RE

ω

R

V + V −

RH

Forbidden

ω 一

Efgective potentials tangent vector separation of variables

S = −ωt + mφ + SR(R)

✓dR dλ ◆2 = 1 V 2

A

⇥ 1 − η (bR)2⇤ (ω − V +)(ω − V −)

Radial part V ± = m −(Mfast)tφ ± q (Mfast)2

tφ − (Mfast)φφ(Mfast)tt

(Mfast)φφ = m R vφ + ηbφ(b · v) ± p V 2

M − (vR)2 − η [(vR)2 − (bφ)2V 2 M]

1 + η (bφ)2 Depending on (α,β), these condition

√ =0 V − =0

Magneto-acoustic ergosurface

V − ≤ ω ≤ V +

Forbidden region for RAYS

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 (a)

(c)

β α

(b)

0.3

0.5 0.7

0.9

E(Re; α, β) = 0 H(Rh; α, β) = 0

ηc = 0.1

β ≡ ΩFR∗/vR

∗

α ≡ −cs(R∗)/vR

∗ > 0

~sound velocity @ ~Angular velocity of B @

5 10 15 20

• 1

1 2 3 4 5

R/R∗

(a)

V + V − 5 10 15 20

• 1

1 2 3 4 5

(b)

R/R∗

V − V + 5 10 15 20

• 1

1 2 3 4 5

(c)

R/R∗

V + V −

Classification of Magneto-acoustic geometry

V ± = m −(Mfast)tφ ± q (Mfast)2

tφ − (Mfast)φφ(Mfast)tt

(Mfast)φφ

3 types of geometry

Efgective potential for rays

NO ergo

Rotating BH Star R∗

R∗

Normal scatt

Star with the ergoregion

NO Horizon

ergo ergo ηc(α, β) ⌧ 1

horizon

Type (a) Analogue rotating black hole

5 10 15

• 1

1 2 3 4

ω (vR

∗ /R∗)

R/R∗

V + V −

Forbidden

The magnetic pressure

Superradiance for MHD waves

Amplified wave efgective potential for type (a) ergo region

positive energy mode negative energy mode

Magnetoacoustic horizon

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 (a)

(c)

β

α

(b) ηc = 0.1

0.3 0.5 0.7

0.9

V ± ≈ ±m R VM

R/R∗

Type (b)

Ergoregion instability by MHD waves scattering

5 10 15

• 1

1 2 3 4

V +

V −

Forbidden

ω (vR

∗ /R∗)

efgective potential for type (b) ergo region

positive energy mode negative energy mode

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 (a) (c)

β α

(b) ηc = 0.1 0.3 0.5 0.7

0.9

H(Rh; α, β) = 0

Ultraspinning star

NO Horizon

Applications (Future work)

Magneto-acoustic BH in astrophysical situations

Accretion disk

MHD wave instability

vertical magnetic field

Alfven wave in z-direction

MHD-analog superradance or ergoregion instability Central object does not have to be a BH MHD Wave around a star peculiar motion of accretion disk ?

POINT

v2 = V 2

M − η(b · v)2

(vR)2 = V 2

M − ηV 2 M(bR)2

Magneto-acoustic horizon Magneto-acoustic ergosurface

MHD Flow makes analog BH QNMs of a Magneto-acoustic BH

Radial Alfven point

Regularity condition

vφ = ΩF R M 2

A L R−2 Ω−1 F − 1

M 2

A − 1

L = ΩFR2

∗

M 2

A(R) ≡

✓ vR V R

A

◆2

When Radial Mach number

vφ = ΩF R∗ vR

∗

vR − (R/R∗)vR

∗

M 2

A − 1

Bφ = − BR ΩF vR

∗ R∗

R2 − R∗

2

M 2

A − 1

Bφ = ΩFRBR vR M 2

A(LR−2 − Ω−1 F )

M 2

A − 1

R vφ − BR 4πρ vR RBφ = const. ≡ L

vRBφ − vφBR = const. ≡ −ΩF R BR

becomes unit R = R∗ are singular.

Radial Alfven point Regularity condition @

RHS is written by vR

BR

and RBR = const. > 0

vφ

Bφ

and ,

R∗

We just solve vR .

ds2

fast ∝ −

⇥ (V 2

M − v2) − η (b · v)2⇤

dt2 − 2 ⇥ vφ + η bφ(b · v) ⇤ R dt dφ + dR2 1 − η (bR)2 − (vR/VM)2 + ⇥ 1 + η (bφ)2⇤ R2dφ2 η ⌘ ✓cs VA V 2

M

◆2 ⌧ 1

Magneto-acoustic metric

Analog horizon & ergoregion for the fast wave mode

v2 = V 2

M − η(b · v)2

(vR)2 = V 2

M − ηV 2 M(bR)2

Magneto-acoustic horizon Magneto-acoustic ergosurface

Magneto-acoustic metric for the fast mode

Summary

v±

“distance”

no metric for slow mode rotating BHs ultraspinning star ratating star MHD wave superradiance ergoregion instability normal scatteing

Analog wave phenomena

y/R∗

• 4
• 2

2 4

• 4
• 2

2 4

x/R∗

β > 0

si

v

B

Analog superradiance & ergoregion instability for MHD waves

(α, β)

BACK UP

Equation for

vR dvR dR − vφ R + 1 ρ dp dR + 1 4πρ Bφ R d dR(RBφ) + F R

ex = 0

Euler’s equation

vR

F R

ex

• Eq. for

draining bathtub type inflow

vR = vR

∗

✓ R R∗ ◆−1/2

vR

and

There exists one F R

ex for a given

Give F R

ex and solve this equation for vR

① ②

vR

✓

Background flow & propagation velocities

Alfven velocity

BR = BR

∗

R∗ R

vR = vR

∗

✓ R R∗ ◆−1/2

vφ = −β vR

∗

"✓ R R∗ ◆1/2 + ✓ R R∗ ◆−1/2 + 1 #

Bφ = −βBR

∗

" 1 + ✓ R R∗ ◆1/2# " 1 + ✓ R R∗ ◆−1#

, , β ≡ ΩFR∗/vR

∗

~ang velo of B @ R∗

y/R∗

• 4
• 2

2 4

• 4
• 2

2 4

x/R∗

β > 0

sink

v

B Background MHD flow Wave velocities

V 2

M = V 2 A + c2 s

c2

s = α2(vR ∗ )2

✓ R R∗ ◆−(Γ−1)/2

V 2

A = (vR ∗ )2

8 < : ✓ R R∗ ◆−3/2 + β2 ✓ R R∗ ◆1/2 " 1 + ✓ R R∗ ◆1/2#2 " 1 + ✓ R R∗ ◆−1#29 = ;

,

BG flow & Velocies are characterized by α ≡ −cs(R∗)/vR

∗ > 0

~sound velo @ R∗

α β

and

Equation of state

p ∝ ρΓ

1 ≤ Γ ≤ 4

Acoustic superradiance

1 √−s ∂ ∂xµ ✓√ −ssµν ∂δΦ ∂xν ◆ = 0 δΦ = ψ(R) R(7−Γ)/16 ei(−ω t+m φ)

Separation of variable

Wave equation for the acoustic disturbance

m = ±0, ± 1, · · ·

− d2ψ dR2

tort

+ Veff(R; ω, m, Γ)ψ = 0

radial eq.

Veff = − 1 c2

s

✓ ω − m vφ R ◆2 − g(R) (7 − Γ)(9 + Γ) 4096 g(R) R2 − 7 − Γ 16R dg(R) dR − m2 R2

• ∂/∂Rtort = g(R) · ∂/∂R

efgective potential ψ = 8 > > > > < > > > > : exp i Z Rtort 1 cs ✓ ω mvφ R ◆ dRtort ! for R ⇠ RH Cin exp i Z Rtort ω cs dRtort ! + Cout exp i Z Rtort ω cs dRtort ! for R ⇠ Rf RH

• Cout

Cin

• 2

+ cs(RH) cs(Rf) · ω − m ΩH ω

• 1

Cin

• 2

= 1

WKB sol Conservation of Wronskian

transmission rate reflection rate

ω < m ΩH = mvφ(RH) RH

Superradiant condition

g(R) = 1 − (vR/cs)2

scattering with amplification

• Cout

Cin

• > 1

Future work ①

MHD-superradanceやergoregion instabilityと

, Astrophysical situation The magnetoacoustic BHs in 3D

Accretion disk

MHD wave instability

vertical magnetic field

Alfven wave in z-direction

D2δv Dt2 c2

sr(r · δv) + VA ⇥ r ⇥ (r ⇥ (δv ⇥ VA)) = 0

MHD wave方程式

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。 JETなどの天体現象は？ ３次元にすると、Alfven waveも出てくる。

ωf,s = ω(k)f,s

ωAl = ω(k)Al

Future work ③

完全流体の場合 音波方程式 MHDの場合

磁気優勢 or 流体優勢

MHD wave方程式 KG型 ???? もしできたら。。。 Magneto-acoustic BH の Quasi Normal Modes や Bombの計算

速度ポテンシャル

か？？） 降着流 天体

MHD wave

音速面

輻射

1 √−s ∂ ∂xµ ✓√ −ssµν ∂δΦ ∂xν ◆ = 0

Clebsch ポテンシャルを使う？

ωQNM

ωBomb

や との輻射の関係は？ 円盤振動学？

B = rα ⇥ rβ

eikonalでは波の増幅率等は計算できない

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？

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

(vR)2 = V 2

M − ηV 2 M(bR)2

v2 = V 2

M − η(b · v)2

Magneto-acoustic horizon Magneto-acoustic ergosurface 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

(MHD waveが動径方向に出ようとしても出られない面)

Magneto-ergoregion instabiliry 疑問 質問 MRIなどの不安定性 不安定性の条件は？

Analogue BHの不安定性はBG flowの不安定性

Superradiant condition

Klein-Gordon eq. (WAVE)

⇤sδΦ = 0

dψ dRtort + (ω2 − Veff)ψ = 0

• Cout

Cin

• 2

+ cs(RH) cs(Rf) · ω − m ΩH ω

• 1

Cin

• 2

= 1

Asymptotic sol Conservation of Wronskian

transmission rate reflection rate

ω < m ΩH = mvφ(RH) RH

δφ = ψ(R)e−iωteimφ

,

ψ =

e−i

⇣ ω− mvφ

RH

⌘ Rtort

Cine−iωRtort + CouteiωRtort

• Cout

Cin

• > 1

Superradiant condition in the eikonal limit

Klein-Gordon eq. (WAVE)

⇤sδΦ = 0

dψ dRtort + (ω2 − Veff)ψ = 0

• Cout

Cin

• 2

+ cs(RH) cs(Rf) · ω − m ΩH ω

• 1

Cin

• 2

= 1

Asymptotic sol Conservation of Wronskian

transmission rate reflection rate

ω < m ΩH = mvφ(RH) RH

δφ = ψ(R)e−iωteimφ

,

Eikonal eq. (RAY)

S = −ω t + m φ + SR(R)

sµν ∂S ∂xµ ∂S ∂xν = 0 ∂S ∂xµ = sµν dxν dλ

tangent vector of RAYS

,

ψ =

e

−i ⇣ ω− mvφ

RH

⌘ Rtort

Cine−iωRtort + CouteiωRtort

• Cout

Cin

• > 1

✓dR dλ ◆2 = 1 c2

s

(ω − V +)(ω − V −) ≥ 0

V ± = mvφ ± p c2

s − (vR)2

R

RE

mvφ(RH) RH

反射波 入射波

RH 禁止領域（for rays）

level crossing

ergo 正と負の間の

V − ≤ ω ≤ V +

Forbidden region f

Acoustic Black Holes Black Holes

ds2 = gµνdxµdxν

metric Einstein eq.

Solutions of Einstein eq.

Physical Properties Phenomena

curved spacetime

Horizon Ergoregion (rotating BHs) Photon sphere No hair theorem . . . Superradiance Quasi Normal Modes Hawking radiation Black Hole Shadow . . .

ω

ergoregion Kerr BH

Gµν = κTµν

wave amplification

ω < mΩH

superradiant condition

Efgective geometry for acoustic waves

NO !

Acoustic horizon

Partially YES!

Acoustic ergoregion

sonic point

PhoNon sphere Beautiful theorems

Partially YES

• 4

2 2 4

horizon ergosurface

stream line

Superradiance Hawking radiation

theory: Unruh (1981) experiment: Steinhauer (2016) supersonic subsonic

“Hawking rad”

Fluid Dynamics

ρ ∂v ∂t + (v · r)v

• = rp + Fex

Slow mode

v± = v ± (csVA/VM)b ≈ v ± csb

∂S ∂t + v± · rS = 0 ,

BG MHD flow

に縛られて伝播する。

v±

v± vR

+ = vR + csbR = |vR ∗ |

✓ R R∗ ◆−1/2 2 6 6 4−1 + α r 1 + β2 ⇣ 1 + p R/R∗ ⌘2 (1 + R/R∗)2 ✓ R R∗ ◆(3−Γ)/4 3 7 7 5

eikonal方程式

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.00 0.01 0.02 0.03 0.04

β

α

0.3 0.5

0.7

0.9

ηc = 0.1

内向きか外向きかどちらに伝播するか？ 今考えている(α,β)の範囲内では vR

± < 0

R成分の符号 全領域で内向きでhorizonのような境界はない