Wave optics in the Kerr BH Sousuke Noda (Nagoya Univ.) Wave optics - PowerPoint PPT Presentation

Wave optics in the Kerr BH Sousuke Noda (Nagoya Univ.) Wave optics in black hole spacetimes: Schwarzschild case Y. Nambu and S.N Wave optics in black hole spacetimes: Kerr case (in prep.) S.N and Y. Nambu HTGRG3 2017 1 August @ Quy Nhon, Vietnam

Sousuke Noda (Nagoya Univ. Japan) Collaborators Yasusada Nambu (Nagoya Univ.) Masaaki Takahashi (Aichi Edu. Univ.) Based on Physical Rev. D .95, 104055 (2017) S.N., Y.Nambu, and M.Takahashi On August 4 (Fri) Analog rotating black holes in a MHD inflow

Wave optics in the Kerr BH Sousuke Noda (Nagoya Univ.) Wave optics in black hole spacetimes: Schwarzschild case Y. Nambu and S.N Wave optics in black hole spacetimes: Kerr case (in prep.) S.N and Y. Nambu HTGRG3 2017 1 August @ Quy Nhon, Vietnam

1. Introduction & Motivation 2. Wave scattering by a Kerr BH 3. Interference patterns & Images

1. Introduction & Motivation 2. Wave scattering by a Kerr BH 3. Interference patterns & Images

Violation of geometrical optics approximation Why wave optics ? Wave optics Geometrical optics eikonal approx. short wavelength lens lens Supernumerary rainbow Brocken spectra Caustics alexander's dark band w o b n Primary rainbow i a r y r a d n o c e S z Envelope of rays Brightness = ∞ Airy (1836) Mie scattering Primary in the geometrical optics 0.30 alexander's dark band 0.25 0.20 supernumerary rainbow 0.15 These phenomena cannot be understood in geometrical optics. 0.10 0.05 - 5 5 10

Violation of geometrical optics approximation Why wave optics ? Wave optics Geometrical optics eikonal approx. short wavelength lens lens Supernumerary rainbow Brocken spectra Caustics alexander's dark band w o b n Primary rainbow i a r y r a d n o c e S z Envelope of rays Brightness = ∞ Airy (1836) Mie scattering Primary in the geometrical optics 0.30 alexander's dark band 0.25 0.20 supernumerary rainbow 0.15 These phenomena cannot be understood in geometrical optics. 0.10 0.05 - 5 5 10

Scattering in geometrical optics Violation of geometrical approximation 2. Rainbow scattering 1. Orbiting (glory) Why wave optics ? Di ff erential cross section ◆ − 1 ✓ d Θ d σ b Ray Θ ( b ) scattering angle d Ω = sin Θ db b scatterer BH case Unstable Circular orbit Θ ( b ) b c sin Θ = 0 b scatterer − π Θ = − π , − 2 π , ... − 2 π − 3 π Caustics Θ ( b ) b c � d Θ b � scatterer = 0 � db � b = b c same angle

MODEL of a gravitational lensing Lens Deflection angle ∝ 1 /b b : impact parameter Small aperture Source e b g a m I Screen Pinhole camera

screen Spherical symmetric case image lens

Non-spherical case Small aperture Source Einstein cross e g a m I by Hubble Space Telescope cross image Screen pinhole caustics Pinhole camera

Caustics & # of image screen image plane Small aperture Source e g a m I Pinhole camera

Non-spherical case pinhole caustics

unstable circular orbits Kerr BH = Non-spherical lens + unstable circular orbit No unstable circular orbits in this model Lens (deflection angle ∝ 1/b) mode caustics Small aperture Source e g a m I mode caustics ?

unstable circular orbits Kerr BH = Non-spherical lens + unstable circular orbit No unstable circular orbits in this model Lens (deflection angle ∝ 1/b) mode caustics Small aperture Source e g Kerr a m I mode caustics ?

Our goal Wave scattering by a Kerr BH Kerr BH Φ interference pattern Optical caustics? screen image plane unstable circular orbit Imaging (Wave optics) 2D Fourier transform = imaging Unstable Circular Orbit Black Hole Shadow (image) E ff ect of the unstable circular orbit in the scattered wave.

1. Introduction & Motivation 2. Wave scattering by a Kerr BH 3. Interference patterns & Images

BH obs source Setup ( θ , φ ) ( θ s , φ s ) r Equatorial plane r s Kerr spacetime (Boyer-Lindquist) dt 2 − 4 Mar sin 2 θ ∆ dr 2 + Σ d θ 2 + A sin 2 θ ✓ ◆ 1 − 2 Mr dtd φ + Σ ds 2 = g µ ν dx µ dx ν = − d φ 2 Σ Σ Σ Σ = r 2 + a 2 cos 2 θ A = ( r 2 + a 2 ) 2 − ∆ a 2 sin 2 θ , , ∆ = r 2 − 2 Mr + a 2 short wavelength ( M � λ ) M ω � 1 interference scalar wave , monochromatic S = − e − i ω t ⇤ Φ ( x, x s ) = S √− g δ 3 ( x − x s ) stationary point source Φ ( x, x s ) = G ( x , x s ) e − i ω t 1 1 � √− gg jk ∂ k G − ω 2 g tt G − 2 ω g t φ ∂ φ G + √− g δ 3 ( x − x s ) � = − √− g ∂ j

Partial wave expansion of G ( x , x s ) ` ∞ ψ ` m ( r, r s ) X X ` m ( θ s ) e im � e − im � s G ( x , x s ) = s + a 2 S ` m ( θ ) S ∗ r 2 + a 2 p √ r 2 Spheroidal harmonics ` =0 m = − ` The radial part d 2 ` m + Q ( r ; ` , m ) ` m = − � ( r − r s ) dr 2 source term ∗ Q ( r ; ` , m ) = [ ! ( r 2 + a 2 ) − ma ] 2 − ∆ ( A ` m + a 2 ! 2 − 2 am ! ) for M ω � 1 ( r 2 + a 2 ) 2 To obtain ψ ` m ( r, r s ) , we use a property of the Green function: W :Wronskian ψ ` m ( r, r s ) = − u 1 ( r s ) u 2 ( r ) θ ( r − r s ) − u 1 ( r ) u 2 ( r s ) θ ( r s − r ) , W W where and are independent solutions of the homogeneous eq. u 1 u 2 d 2 u ` m ✓ ◆ ! r ∗ − ⇡` + Q ( r ; ` , m ) u ` m = 0 2 + � ` m u ` m ∼ sin r ∗ → ∞ dr 2 ∗ phase shift

Radial equation and the phase shift A in e − i ω r ∗ 1 d 2 u ` m e + i ω r ∗ + Q ( r ; ` , m ) u ` m = 0 e − i $ r ∗ dr 2 A out ∗ IN mode ( e − i $ r ∗ ( r ∗ → −∞ ) purely ingoing @ horizon u IN = A out e i ! r ∗ + A in e − i ! r ∗ ( r ∗ → + ∞ ) UP mode ( B out e i $ r ∗ + B in e − i $ r ∗ ( r ∗ → −∞ ) purely outgoing @ infinity u UP = e i ! r ∗ ( r ∗ → + ∞ ) , e 2 i � ` m ≡ ( − ) ` +1 A out S matrix ( ) Green function A in with r > r s ψ ` m ( r, r s ) = − u IN ( r s ) u UP ( r ) i r, r s � M = u IN ( r s ) u UP ( r ) 2 ω A in W = e i ! r ∗ = e i ! r ∗ ⇢ i � h i Sum over the partial waves h �` m �` m 2 i ω ( − ) ` n o i ! r s ∗ + r +2 � ` m − i ! r s ∗ − 2 ! (1 /r s − 1 /r ) e i [ ! r s ∗ +2 � ` m ] − ( − ) ` e − i ! r s ∗ 2 i ω ( − ) ` − ( − ) ` e e 2 ! ˜ does not converge. λ ` m = A ` m + a 2 ω 2 Fresnel di ff raction r = (1 /r + 1 /r s ) − 1 ˜

Wave scattering by a Kerr BH Green function ` G ( x , x s ) ≡ e i ! ( r ∗ + r s ∗ ) ∞ �` m X X r e 2 i � ` m Z ` m ( θ , φ ) Z ∗ ( − ) ` e i ` m ( θ s , φ s ) 2 ! ˜ 2 i ω rr s Numerical cal. ` =0 m = − ` S matrix V e ff A out S = e 2 i � ` m ≡ ( − ) ` +1 A out A in A in horizon r ∗ unstable circular orbit phase shift (Schwarzschild case) ω M = 5 Deflection angle Reflection rate 1.0 2 impact parameter Θ = 2Re d � ` | e 2 i δ ` | d ` 1 0.8 b = ` / ! M 0 0.6 - 1 - 2 0.4 b c - 3 0.2 - 4 - 5 Unstable Circular Orbit 0.0 20 40 60 80 100 0 10 20 30 40 50 60 √ ` c = 3 3 M ! b c

Prüfer method Prüfer WKB no turning point turning point WKB method in two different forms Asymptotic form ① ② Phase shift ( d 2 u ` m e − i $ r ∗ ( r ∗ → −∞ ) + Q ( r ; ` , m ) u ` m = 0 u ` m = A sin [ ω r ∗ + ζ ] ( r ∗ → + ∞ ) , dr 2 ∗ � ` m − ⇡` / 2 u ` m ingoing @ horizon 0 0 0 = P ( r ∗ )) R dP dr ⇤ P ( r ⇤ ) + P 2 + Q = 0 ( u/u u ` m = e P = − i $ ( r ∗ → −∞ ) dr ∗ ˜ P = f ( P ) d ˜ ✓ ◆ P ω − Q 0 = ω cot [ ω r ∗ + ˜ sin 2 ( ω r ∗ + ˜ ˜ + P ) = 0 P = ζ ( r ∗ → + ∞ ) u/u P ( r ∗ )] dr ∗ ω Z ∞ ✓ ◆ ` + 1 Q − ! ) + ⇡` p � WKB = dr ∗ ( − ! r 0 ∗ ` m 2 2 r 0 ∗ turning point ˜ P ` < ` c P − i $ ζ ` > ` c V e ff critical value √ ` c = 3 3 M ! r ∗ r 0 ∗

Phase shift (Kerr case) m = ± ` a = 0 . 6 M ω M = 30 rays on the equatorial plane prograde orbit 1.0 | e 2 i δ `` | 1.0 Re[ e 2 i δ `` ] 0.8 m = ` 0.5 0.6 0.0 0.4 - 0.5 0.2 - 1.0 0.0 0 100 200 300 400 0 100 200 300 400 ` ` retrograde orbit 1.0 Re[ e 2 i δ ` , − ` ] 1.0 | e 2 i δ ` , − ` | 0.8 m = − ` 0.5 0.6 0.0 0.4 - 0.5 0.2 - 1.0 0.0 0 100 200 300 400 0 100 200 300 400 ` `

observer’s sky source plane Sum over the partial waves ` G ( x , x s ) ≡ e i ! ( r ∗ + r s ∗ ) ∞ �` m X X ( − ) ` e i r e 2 i � ` m Z ` m ( θ , φ ) Z ∗ ` m ( θ s , φ s ) 2 ! ˜ 2 i ω rr s cv ` =0 m = − ` For , ` max = 420 ω M = 30 G ( x , x s ) ∼ 160 , 000 terms equatorial plane ϑ s 10 , 000 points on the obs. plane ϕ s r s 160 , 000 × 10 , 000 = 1 , 600 , 000 , 000 ϑ terms φ r It takes 2~4 days with a PC (8 cores)

observer’s sky source plane 1. Introduction & Motivation 2. Wave scattering by a Kerr BH 3. Interference patterns & Images ϑ s ϕ s r s ϑ φ r

Interference patterns ω M = 30 a = 0 . 5 M a = 0 a = 0 . 3 M concentric -0.5 0 0.5 -0.5 0 0.5 +0.1 0.4 0.7 Diamond-shaped caustic a = 0 . 9 M a = 0 . 7 M a/M = 0 . 7 ϑ +0.2 1.0 φ +0.3 1.2 0.7

Caustics by the winding mode G = G direct + G wind | G wind | ∼ 10 − 1 | G direct | ` ≤ ` c ` > ` c a=0 a=0.7 Dragging e ff ects 0.015 on and G wind 0.025 G direct E ff ect of unstable circular orbit are di ff erent. 0.020 0.010 0.015 0.005 0.010 0.005 - 0.3 - 0.2 - 0.1 0.1 0.2 0.3 0.4 0.6 0.8 1.0