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

Wave optics in the Kerr BH

Wave optics in black hole spacetimes: Kerr case (in prep.)

HTGRG3 2017 1 August @ Quy Nhon, Vietnam S.N and Y. Nambu

Sousuke Noda (Nagoya Univ.)

Wave optics in black hole spacetimes: Schwarzschild case

• Y. Nambu and S.N

Analog rotating black holes in a MHD inflow

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)

Wave optics in the Kerr BH

Wave optics in black hole spacetimes: Kerr case (in prep.)

HTGRG3 2017 1 August @ Quy Nhon, Vietnam S.N and Y. Nambu

Sousuke Noda (Nagoya Univ.)

Wave optics in black hole spacetimes: Schwarzschild case

• Y. Nambu and S.N

• 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

Why wave optics ?

Violation of geometrical optics approximation

Brocken spectra Supernumerary rainbow

Airy (1836)

Caustics

Mie scattering Brightness = ∞ Envelope of rays in the geometrical optics

These phenomena cannot be understood in geometrical optics. lens Wave optics Geometrical optics lens

short wavelength eikonal approx.

alexander's dark band Primary rainbow S e c

• n

d a r y r a i n b

• w

z

supernumerary rainbow

• 5

5 10 0.05 0.10 0.15 0.20 0.25 0.30

Primary alexander's dark band

Why wave optics ?

Violation of geometrical optics approximation

Brocken spectra Supernumerary rainbow

Airy (1836)

Caustics

Mie scattering Brightness = ∞ Envelope of rays in the geometrical optics

These phenomena cannot be understood in geometrical optics. lens Wave optics Geometrical optics lens

short wavelength eikonal approx.

alexander's dark band Primary rainbow S e c

• n

d a r y r a i n b

• w

z

supernumerary rainbow

• 5

5 10 0.05 0.10 0.15 0.20 0.25 0.30

Primary alexander's dark band

Θ(b)

b

scatterer scattering angle Ray

Why wave optics ?

Scattering in geometrical optics

• 2. Rainbow scattering
• 1. Orbiting (glory)

Violation of geometrical approximation

dσ dΩ = b sin Θ ✓dΘ db ◆−1

Differential cross section Θ(b)

b

bc

Θ(b)

b

bc

−π

−2π

−3π

scatterer same angle

Caustics

dΘ db

• b=bc

= 0

Θ = −π, −2π, ...

scatterer

sin Θ = 0

BH case

Unstable Circular orbit

MODEL of a gravitational lensing

Source

Lens

Screen Small aperture

I m a g e

Pinhole camera

Deflection angle ∝ 1/b

: impact parameter

b

b

Spherical symmetric case

lens screen

image

Source Screen Small aperture

I m a g e

Pinhole camera

Non-spherical case

caustics pinhole cross image

Einstein cross

by Hubble Space Telescope

Source Small aperture

I m a g e

Pinhole camera

Caustics & # of image

image plane screen

caustics pinhole

Non-spherical case

unstable circular orbits

Source Lens (deflection angle ∝1/b) Small aperture

I m a g e

No unstable circular orbits in this model Kerr BH = Non-spherical lens + unstable circular orbit

caustics mode caustics mode

?

unstable circular orbits

Source Lens (deflection angle ∝1/b) Small aperture

I m a g e

No unstable circular orbits in this model Kerr BH = Non-spherical lens + unstable circular orbit

caustics mode caustics mode

?

Kerr

Wave scattering by a Kerr BH Imaging (Wave optics)

screen

Φ

image plane

Kerr BH interference pattern unstable circular orbit 2D Fourier transform = imaging

Unstable Circular Orbit

Black Hole Shadow (image) Effect of the unstable circular orbit

Our goal

Optical caustics? in the scattered wave.

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

Kerr spacetime (Boyer-Lindquist) scalar wave stationary point source , monochromatic

Setup

Equatorial plane

BH source

(θ, φ) (θs, φs)

• bs

r

rs

ds2 = gµνdxµdxν = − ✓ 1 − 2Mr Σ ◆ dt2 − 4Mar sin2 θ Σ dtdφ + Σ ∆dr2 + Σdθ2 + A sin2 θ Σ dφ2

A = (r2 + a2)2 − ∆a2 sin2 θ

∆ = r2 − 2Mr + a2 Σ = r2 + a2 cos2 θ

, ,

⇤Φ(x, xs) = S

Φ(x, xs) = G(x, xs)e−iωt

S = −e−iωt √−g δ3(x − xs)

−ω2gttG − 2ωgtφ∂φG + 1 √−g ∂j √−ggjk∂kG

• = −

1 √−g δ3(x − xs)

short wavelength

Mω 1

(M λ)

interference

Partial wave expansion of G(x, xs)

G(x, xs) =

∞

X

`=0 `

X

m=−`

ψ`m(r, rs) √ r2 + a2p r2

s + a2 S`m(θ)S∗ `m(θs)eime−ims

d2 `m dr2

∗

+ Q(r; `, m) `m = −(r − rs)

Q(r; `, m) = [!(r2 + a2) − ma]2 − ∆(A`m + a2!2 − 2am!) (r2 + a2)2

The radial part for Mω 1 To obtain ψ`m(r, rs), we use a property of the Green function:

ψ`m(r, rs) = −u1 (rs) u2 (r) W θ(r − rs) − u1 (r) u2 (rs) W θ(rs − r)

u1 u2

,

where and are independent solutions of the homogeneous eq.

d2u`m dr2

∗

+ Q(r; `, m)u`m = 0

Spheroidal harmonics source term

u`m ∼ sin ✓ !r∗ − ⇡` 2 + `m ◆

phase shift

r∗ → ∞

W :Wronskian

uIN = ( e−i$r∗ (r∗ → −∞) Aoutei!r∗ + Aine−i!r∗ (r∗ → +∞) uUP = ( Boutei$r∗ + Bine−i$r∗ (r∗ → −∞) ei!r∗ (r∗ → +∞),

e2i`m ≡ (−)`+1 Aout Ain

Radial equation and the phase shift

d2u`m dr2

∗

+ Q(r; `, m)u`m = 0

IN mode UP mode purely ingoing @ horizon purely outgoing @ infinity

ψ`m(r, rs) = −uIN(rs)uUP(r) W = i 2ωAin uIN(rs)uUP(r)

= ei!r∗ 2iω (−)` n ei[!rs∗+2`m] − (−)`e−i!rs∗

• e−iωr∗

e+iωr∗

Ain

Aout

1

Green function Sum over the partial waves does not converge.

Fresnel diffraction

= ei!r∗ 2iω (−)` ⇢ e

i h !rs∗+

`m 2!˜ r +2`m

i

− (−)`e

−i h !rs∗−

`m 2! (1/rs−1/r)

i

λ`m = A`m + a2ω2

˜ r = (1/r + 1/rs)−1

e−i$r∗

with r > rs

S matrix ( )

r, rs M

Wave scattering by a Kerr BH

ωM = 5

20 40 60 80 100

• 5
• 4
• 3
• 2
• 1

1 2

`c = 3 √ 3M!

Θ = 2Red` d`

Reflection rate Deflection angle Numerical cal.

Green function

S = e2i`m ≡ (−)`+1 Aout Ain

|e2iδ`|

10 20 30 40 50 60 0.0 0.2 0.4 0.6 0.8 1.0

b = `/!M

impact parameter

Unstable Circular Orbit

bc

phase shift (Schwarzschild case)

S matrix

horizon

unstable circular orbit

r∗

Ain

Aout

Veff

bc

G(x, xs) ≡ ei!(r∗+rs∗) 2iωrrs

∞

X

`=0 `

X

m=−`

(−)`ei

`m 2!˜ r e2i`mZ`m(θ, φ)Z∗

`m(θs, φs)

Prüfer method

`m − ⇡`/2

in two different forms Asymptotic form

① ②

˜ P = f(P)

dP dr∗ + P 2 + Q = 0 d ˜ P dr∗ + ✓ ω − Q ω ◆ sin2 (ωr∗ + ˜ P) = 0

˜ P = ζ (r∗ → +∞)

P = −i$ (r∗ → −∞)

`c = 3 √ 3M!

critical value

WKB

`m

= Z ∞

r0∗

dr∗( p Q − !) + ⇡` 2 ✓ ` + 1 2 ◆ − !r0∗

turning point

Phase shift

ingoing @ horizon

d2u`m dr2

∗

+ Q(r; `, m)u`m = 0

u`m

r∗

P

˜ P

ζ −i$

Veff

` < `c ` > `c

Prüfer turning point

r0∗

no turning point WKB

u`m = e

R dr

⇤P (r ⇤)

u/u

0 = ω cot [ωr∗ + ˜

P(r∗)]

(u/u

0 = P(r∗))

u`m = ( e−i$r∗ (r∗ → −∞) A sin [ωr∗ + ζ] (r∗ → +∞), WKB method

100 200 300 400

• 1.0
• 0.5

0.0 0.5 1.0 100 200 300 400 0.0 0.2 0.4 0.6 0.8 1.0 |e2iδ``|

Re[e2iδ``]

` `

100 200 300 400 0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 400

• 1.0
• 0.5

0.0 0.5 1.0 Re[e2iδ`,−`]

|e2iδ`,−`|

` `

a = 0.6M

Phase shift (Kerr case)

ωM = 30 m = ±`

rays on the equatorial plane

m = ` m = −`

prograde orbit retrograde orbit

Sum over the partial waves

G(x, xs) ≡ ei!(r∗+rs∗) 2iωrrs

∞

X

`=0 `

X

m=−`

(−)`ei

`m 2!˜ r e2i`mZ`m(θ, φ)Z∗

`m(θs, φs)

It takes 2~4 days with a PC (8 cores)

ωM = 30

For , `max = 420

G(x, xs) ∼ 160, 000 terms

cv

• bserver’s sky

source plane

ϑ

r

rs

ϕs

ϑs

φ

equatorial plane

10, 000 points on the obs. plane

160, 000 × 10, 000 = 1, 600, 000, 000

terms

• 1. Introduction & Motivation
• 2. Wave scattering by a Kerr BH
• 3. Interference patterns & Images
• bserver’s sky

source plane ϑ

r

rs

ϕs

ϑs

φ

+0.1 0.7

• 0.5

0.5

• 0.5

0.5

a = 0 a = 0.3M a = 0.5M

a = 0.9M

φ

ϑ

a/M = 0.7

a = 0.7M

0.7 +0.3 1.2

Interference patterns

0.4 +0.2 1.0

Diamond-shaped caustic

ωM = 30

concentric

Caustics by the winding mode

a=0 a=0.7

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.005 0.010 0.015 0.4 0.6 0.8 1.0 0.005 0.010 0.015 0.020 0.025

Effect of unstable circular orbit

Dragging effects

• n

and

G = Gdirect + Gwind

` ≤ `c

` > `c

Gdirect

Gwind

are different. |Gwind| ∼ 10−1|Gdirect|

Detection of waves from unstable circular orbit

0.4 0.6 0.8 1.0 0.005 0.010 0.015 0.020 0.025

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.005 0.010 0.015

Schwarzschild Kerr Dragging efgects on direct mode winding mode

difgerence

NO peak shift peak shift

position of obs position of obs

Imaging in wave optics

aperture an imaging system (telescope)

W Ham(φ, ϑ) = 0.54 − 0.46 cos(2π p φ2 + ϑ2)

Hamming window function

˜ G = F[G(φ, ϑ) × W Ham(φ, ϑ)]

| ˜ G| : image

Imaging = 2D Fourier transform

• bserver’s sky

source plane

ϑ

r

rs

ϕs ϑs

φ

φ ϑ

• 6
• 4
• 2

• 4
• 2

Mω = 30 a = 0.7M

The number of image & observer’s position

aperture Rim of Black Hole Shadow

• 6
• 4
• 2

• 4
• 2

• 6
• 4
• 2

• 4
• 2

Is the interference patterns observable?

Interference in the Power spectrum may be observable ・Too short coherence time

|Φ|2(ω)

mass

∆ω

Supermassive Stellar size Intermediate

106M 103M 10M 40kHz 40Hz 40MHz Typical freq. These are sensitive to the posisions of source and obs (Schwarzschild)

・Too much large to catch Modulation (effect of USCO)

What about pulser?

Φ

θ

We can use the motion of a source

short coherence time interference in Fourier space

Summary

screen image plane

2D Fourier transform (imaging)

• 6
• 4
• 2

• 4
• 2

aperture

G(x, xs) ≡ ei!(r∗+rs∗) 2iωrrs

∞

X

`=0 `

X

m=−`

(−)`ei

`m 2!˜ r e2i`mZ`m(θ, φ)Z∗

`m(θs, φs)

G

G = Gdirect + Gwind

|G|

|F[G]|

0.4 0.6 0.8 1.0 0.005 0.010 0.015 0.020 0.025

|Gwind|

|Gdirect|

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.005 0.010 0.015

position of obs Kerr Schwarzschild |Gdirect|

|Gwind|