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

Sousuke Noda (Nagoya Univ. Japan) Collaborators Yasusada Nambu (Nagoya Univ.) Masaaki Takahashi (Aichi Edu. Univ.) NARUTO Strait in Tokushima, Japan (Tidal whirlpool) Phys. Rev. D .95, 104055 (2017) “Analog rotating black holes in a magnetohydrodynamic inflow” S.N., Yasusada Nambu, and Masaaki Takahashi Analog rotating black holes in a MHD inflow Based on

? Analog geometry for acoustic waves With flow No flow FLOW Ray Ray Wave front Wave front Ray’s trajectory = curve Ray’s trajectory = straight line ds 2 = Light rays in flat spacetime Light rays in curved spacetime ds 2 = η µ ν dx µ dx ν = 0 ds 2 = g µ ν dx µ dx ν = 0

Acoustic Black Holes stream line 4 2 horizon 0 superradiance 2 ergosurface 4 Source - - “Acoustic” Black Hole sonic surface ~ horizon Montecello dam (CA. USA) http://amazinglist.net/2013/03/bottomless-pit-dam-monticello-drain-hole/

Acoustic Black Holes Black Holes Solutions of Einstein eq. NO ! Einstein eq. metric E fg ective geometry for acoustic waves ds 2 = g µ ν dx µ dx ν G µ ν = κ T µ ν  ∂ v � Fluid Dynamics ρ ∂ t + ( v · r ) v = �r p + F ex curved spacetime Partially YES! Physical Properties Horizon Acoustic horizon sonic point Acoustic ergoregion Ergoregion (rotating BHs) PhoNon sphere Photon sphere Beautiful theorems No hair theorem . . . BH physics in Lab. Partially YES Phenomena wave amplification Superradiance Hawking radiation Hawking radiation ergoregion stream line 4 Black Hole Shadow 2 Quasi Normal Modes “Hawking rad” horizon 0 Kerr BH ω Superradiance supersonic subsonic . superradiant condition . 2 ergosurface . ω < m Ω H theory: Unruh (1981) 4 experiment: Steinhauer (2016) - -

The motivation of our work Acoustic BH in labs sonic points Fluid dynamics BH physics Analog model e.g.) Hawking rad, Superradiance etc. Our work (Astrophysical) BH physics Fluid dynamical phenomena Analog model Magnetic field e.g.) Jet QPO 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 Fluid dynamics Alfven wave in z-direction BH physics 0 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 Magnetic field e.g.) Jet QPO 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

velocity potential , , Acoustic metric Perfect fluid (irrotational) Background Perturbation ∂ρ ∂ t + r · ( ρ v ) = 0 v = �r Φ v → v + δ v Φ → Φ + δ Φ  ∂ v � ρ → ρ + δρ ∂ t + ( v · r ) v = �r p + F ex p = p ( ρ ) ρ Wave eq. for δ Φ  ✓ ∂δ Φ ◆� ⇢ ✓ ∂δ Φ ◆� � ∂ c − 2 c − 2 s ρ ∂ t + v · r δ Φ s ρ ∂ t + v · r δ Φ v � ρ r δ Φ + r · = 0 ∂ t Klein-Gordon eq. in a “ curved” spacetime Acoustic line element Moncrief (1980) ✓ √ ◆ ∂ − ss µ ν ∂δ Φ 1 ds 2 = s µ ν dx µ dx ν = ⇤ s δ Φ = 0 Unruh (1981) √− s ∂ x µ ∂ x ν Acoustic metric Acoustic waves feel Riemannian geometry

ergosurface horizon Acoustic Horizon & ergoregion Acoustic line element dR 2 ⇢ � ds 2 = s µ ν dx µ dx ν = ρ ( v R ) 2 + ( v φ ) 2 �⇤ dt 2 − 2 v φ R d φ dt + 1 − ( v R /c s ) 2 + R 2 d φ 2 c 2 ⇥ � − s − c s ξ ( t ) and ξ ( φ ) For stationary & axisymmetric BG flow, there exist Killing vectors . v = v ( R ) q ( v R ) 2 + ( v φ ) 2 ξ 2 Acoustic ergosurface c s = ( t ) = 0 η = ξ ( t ) + β ξ ( φ ) const β = v φ η 2 = 0 � Acoustic horizon c s = v R � � R � R H stream line Acoustic BH in laboratories 4 Superradiance Hawking radiation 2 stream line 4 2 0 “Hawking rad” horizon 0 2 supersonic subsonic 2 ergosurface theory: Unruh (1981) 4 4 experiment: Steinhauer (2016) - - - -

ergo Draining Bathtub Model M. Visser, Class.Quant.Grav. 15, 1767 (1998). (2D) Axisymmetric inflow with the sonic surface v φ ∝ 1 v R ∝ − 1 stream line ( R, φ ) , √ 4 cylindrical coordinates R R 2 Wave equation 0 δφ = ψ ( R ) e − i ω t e im φ ⇤ s µ ν δφ = 0 2 radial part d 2 ψ 4 + ( ω 2 − V e ff ) ψ = 0 dR 2 tort - - Acoustic superradiance Wave scattering by the acoustic BH The Superradiant condition Kerr case ω < mv φ amplified wave 0 ( R H ) V e ff ω < m Ω H R H purely ingoing ω Angular velocity of the BG flow at the acoustic horizon R tort

Acoustic superradiance in the eikonal limit Eikonal approx. Ray Klein-Gordon eq. (WAVE) Eikonal eq. (RAY) Wave front δ Φ ∝ e iS k µ = ∂ S dx ν ⇤ s δ Φ = 0 s µ ν k µ k ν = 0 ∂ x µ = s µ ν d λ tangent vector We can separate S as ◆ 2 ✓ dS R s tt ω 2 − 2 s t φ ω m + s φφ m 2 + s RR S = − ω t + m φ + S R ( R ) = 0 dR Radial eq. & e fg ective potential E fg ective potential ◆ 2 ✓ dR = 1 V ± = mv φ ± p c 2 s − ( v R ) 2 ( ω − V + )( ω − V − ) ≥ 0 c 2 d λ R s Forbidden region for RAYS level crossing V − ≤ ω ≤ V + mv φ ( R H ) reflected wave R H transmitted wave Superradiant condition incident wave Forbidden for rays ω < m Ω H = mv φ ( R H ) negative phase ergo velocity mode R H R H R E

E fg ective potential & Acoustic superradiance v φ = 0 ω case Analog Schwarzshild BH positive ω ω =const V + Forbidden （ for rays ） R R H E fg ective potential V − V ± = mv φ ± p c 2 s − ( v R ) 2 negative ω R horizon v φ 6 = 0 Forbidden region for RAYS ω V − ≤ ω ≤ V + Analog rotating BH Superradiant condition level crossing mv φ ( R H ) ω < m Ω H = mv φ ( R H ) reflected wave R H R H ω =const transmitted wave incident wave negative phase velocity mode V + Forbidden for rays ） ergo R V − horizon

Contents 1. Acoustic Black Holes 2. Magneto-acoustic BHs 3. Summary & Future work

? Di fg erence between perfect fluid & MHD Perfect fluids MHD 4 2 0 2 Ray 4 v Wave front B ds 2 = ds 2 = s µ ν dx µ dx ν Riemannian geometry ????? Riemannian geometry single wave mode: Multiple modes: fast & slow ⇤ s δ Φ = 0 (2D case) Isotropic propagation Anisotropic propagation B dispersion relation dispersion relation k ω 2 = f ( k 2 , k · B ) θ ω 2 = c 2 s k 2 r 2 = ∂ 2 x + ∂ 2 y BH-like structure : rotating BH rotating BH ? Superradiance V e ff Superradiant scat. of MHD waves ?? R tort

MHD wave equation (2D) Ideal MHD Background Perturbation ∂ρ ∂ B , , ∂ t + r · ( ρ v ) = 0 ∂ t = r ⇥ ( v ⇥ B ) r · B = 0 ρ → ρ + δρ v → v + δ v 1 ∂ v ∂ t + ( v · r ) v + r p F ex → F ex B → B + δ B ρ + 4 πρ B ⇥ ( r ⇥ B ) + F ex = 0 Lagrange derivative MHD wave equation (2D) Dt = ∂ D ∂ t + v · r Helmholtz theorem ✓ ∂ y Ψ ✓ ∂ x Φ ◆ ◆ ✓ ∂ x ◆ ✓ Φ ◆ 0 Alfven velocity ∂ y δ v = + = ∂ y Φ − ∂ x Ψ ∂ y − ∂ x Ψ 1 V A = √ 4 πρ B Wave Eq. for multiple wave modes ˆ ✓ ˆ L 1 = ( V A ) y ∂ x − ( V A ) x ∂ y D 2 L 1 ˆ ˆ ✓ Φ ◆ ✓ c 2 ◆ ◆� ✓ Φ ◆ s r 2 L 2 L 2 0 1 = + L 1 ˆ ˆ ˆ L 2 Dt 2 Ψ 0 0 Ψ ˆ L 2 L 2 = ( V A ) x ∂ x + ( V A ) y ∂ y 2 How is the acoustic metric introduced ??

MHD wave & Acoustic wave MHD wave eq (2D) ˆ L 1 = ( V A ) y ∂ x − ( V A ) x ∂ y ✓ ˆ L 1 ˆ ˆ D 2 ✓ ◆ ✓ ◆ ◆� ✓ ◆ c 2 s r 2 L 2 Φ 0 L 2 Φ ˆ L 2 = ( V A ) x ∂ x + ( V A ) y ∂ y 1 = + L 1 ˆ ˆ ˆ L 2 Ψ Ψ Dt 2 0 0 L 2 2 V A ∝ B B = 0 case eikonal limit ✓ √ ◆ s µ ν ∂ S ∂ S ∂ − ss µ ν ∂ Φ 1 Eikonal equation ∂ x ν = 0 = 0 √− s ∂ x µ ∂ x µ ∂ x ν Acoustic metric = coe ffj cients of the eikonal equation case (MHD) We introduce the acoustic metric through the eikonal equation. B 6 = 0 (up to conformal factor) Eikonal eq. for MHD waves (2D) Φ = | Φ | e iS , Ψ = | Ψ | e iS ◆ 4 ◆ 2 ✓ ∂ S ✓ ∂ S s | r S | 2 )( V A · r S ) 2 = 0 � ( c 2 s + V 2 A ) | r S | 2 + ( c 2 ∂ t + v · r S ∂ t + v · r S M µ νλσ ∂ S ∂ S ∂ S ∂ S Non quadratic form….. ∂ x σ = 0 ∂ x ν ∂ x λ ∂ x µ

Magneto-acoustic geometry Eikonal eq. for 2D MHD waves M µ νλσ ∂ S ∂ S ∂ S ∂ S Quartic metric NON Riemannian ! ∂ x σ = 0 ∂ x ν ∂ x λ ∂ x µ Magneto-acoustic metric Finsler metric If we define line element as ds 2 = F ( x, y ) = � 1 / 4 M µ νλσ y µ y ν y λ y σ y µ = ∂ µ S � , satisfies the homogeneity condition: . F ( x, σ y ) = σ F ( x, y ) F ( x, y ) σ = 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 B dispersion relation Finsler geometry ω 2 = f ( k 2 , k · B ) k θ

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