Observational Signature of a Near-extremal Kerr-like Black Hole at - - PowerPoint PPT Presentation

▶
observational signature of a near extremal kerr like
SMART_READER_LITE
LIVE PREVIEW

Observational Signature of a Near-extremal Kerr-like Black Hole at - - PowerPoint PPT Presentation

Oct 06, 2023 169 likes •385 views

Observational Signature of a Near-extremal Kerr-like Black Hole at the Event Horizon Telescope Haopeng Yan Niels Bohr Institute, University of Copenhagen based on PRD 98, 084063 with Minyong Guo and Niels A. Obers January 4, 2019 Nordic Winter

slide-1
SLIDE 1

Observational Signature of a Near-extremal Kerr-like Black Hole at the Event Horizon Telescope

Haopeng Yan

Niels Bohr Institute, University of Copenhagen based on PRD 98, 084063 with Minyong Guo and Niels A. Obers

January 4, 2019

Nordic Winter School on Particle Physics and Cosmology

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

1 / 18

slide-2
SLIDE 2

Outline

1

Introduction

2

Orbiting emitter Kerr-like black hole in the MOG theory Equations of photon trajectory Near-extremal solutions

3

Observational appearance

4

Summary

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

2 / 18

slide-3
SLIDE 3

Introduction

The Event Horizon Telescope (EHT) will announce the first image of a black hole in 2019. EHT website. The observational signature of a near-extremal Kerr black hole (predicted by General Relativity) has been theoretically studied in

Gralla, Lupsasca and Strominger, 2017

GR is supposed to be modified, what about the signatures in alternative gravitational theories? I will introduce a generalization to one of these alternative theories: the MOG theory Moffat, 2005 -a modified gravity theory without invoking dark matter.

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

3 / 18

slide-4
SLIDE 4

The Kerr-MOG spacetime

The rotating solution is given by the Kerr-MOG metric ds2 = −∆Σ Ξ dt2 + Σ ∆dr 2 + Σdθ2 + Ξ sin2 θ Σ (dφ − ωdt)2, (1) where ∆ = r 2 − 2GMr + a2 + αGNGM2, Σ = r 2 + a2 cos2 θ, (2) Ξ = (r 2 + a2)2 − ∆a2 sin2 θ, ω = a(a2 + r 2 − ∆) Ξ , (3) where α is the modified parameter and G = (1 + α)GN is called as an enhanced gravitational constant.

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

4 / 18

slide-5
SLIDE 5

The Kerr-MOG spacetime

The rotating solution is given by the Kerr-MOG metric ds2 = −∆Σ Ξ dt2 + Σ ∆dr 2 + Σdθ2 + Ξ sin2 θ Σ (dφ − ωdt)2, (1) where ∆ = r 2 − 2GMr + a2 + αGNGM2, Σ = r 2 + a2 cos2 θ, (2) Ξ = (r 2 + a2)2 − ∆a2 sin2 θ, ω = a(a2 + r 2 − ∆) Ξ , (3) where α is the modified parameter and G = (1 + α)GN is called as an enhanced gravitational constant. For simplicity, we set GN = 1, Mα ≡ MADM = (1 + α)M, β2 = α 1 + αM2

α,

(4) thus ∆ = r 2 − 2Mαr + a2 + β2. (5)

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

4 / 18

slide-6
SLIDE 6

The Kerr-MOG spacetime

Mass dependent gravitational charge K =

  • αGNM.

(6) The event horizon is obtained for ∆ = 0, r± = Mα ±

  • M2

α − (a2 + β2).

(7) The extremal condition a2 + β2 = M2

α.

(8)

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

5 / 18

slide-7
SLIDE 7

The Kerr-MOG spacetime

Mass dependent gravitational charge K =

  • αGNM.

(6) The event horizon is obtained for ∆ = 0, r± = Mα ±

  • M2

α − (a2 + β2).

(7) The extremal condition a2 + β2 = M2

α.

(8) Note that the quantities under the square roots of (6) and (7) should be nonnegative, thus we obtain physical bounds on α as 0 ≤ α ≤ M2

α

a2 − 1. (9)

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

5 / 18

slide-8
SLIDE 8

Orbiting emitter

We assume the emitter (“hot spot”) is on a circular orbit with radius rs at the equatorial plane. The angular velocity is Ωs = dφ dt = ± Γ(rs) r 2

s ± aΓ(rs),

(10) where Γ2(r) = Mαr − β2. (11)

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

6 / 18

slide-9
SLIDE 9

Orbiting emitter

Photons originate from the emitter

There are four conserved quantities along each photon trajectory: the invariant mass µ2 = 0, the total energy E, the angular momentum L and the Carter constant Q. Introducing two rescaled quantities, ˆ λ = L E , ˆ q = √Q E . (12)

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

7 / 18

slide-10
SLIDE 10

Orbiting emitter

Equations of Photon trajectory

Using Hamilton-Jacobi method, we can obtain the equations, − ˆ ro

rs

dr ±

  • R(r) = −

ˆ θo

θs

dθ ±

  • Θ(θ),

(13) ∆φ = φo − φs = − ˆ ro

rs

a(2Mαr − β2 − aˆ λ) ±∆

  • R(r)

dr + − ˆ θo

θs

ˆ λ csc2 θ ±

  • Θ(θ)dθ,

(14) ∆t = to − ts = − ˆ ro

rs

  • r 4 + a2r 2 + 2Mαr − β2 − a

2Mαr − β2ˆ

λ

  • ±∆
  • R(r)

dr + − ˆ θo

θs

a2 cos2 θ ±

  • Θ(θ)dθ,

(15) where R(r) =

r 2 + a2 − aˆ

λ

2 − ∆

  • ˆ

q2 +

a − ˆ

λ

2

, (16) Θ(θ) = ˆ q2 + a2 cos2 θ − ˆ λ2 cot2 θ. (17)

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

8 / 18

slide-11
SLIDE 11

Orbiting emitter

Photon trajectory

Each photon trajectory can be labeled by a pair of conserved quantities (ˆ λ, ˆ q), which connects the star (ts, rs, θs, φs) to an observer (to, ro, θo, φo). We decouple the coordinates ts and φs by using φs = Ωsts, and made the following choice, θs = π/2, ro → ∞ and φo = 2πN. For given choice of rs and θo, solving these equations gives the time-dependent trajectories [ˆ λ(to), ˆ q(to)] corresponding to a track of the emitter’s image. Plugging ˆ λ and ˆ q into the functions of observables: position [x(ˆ λ, ˆ q], y(ˆ λ, ˆ q)), redshift g(ˆ λ, ˆ q) and flux Fo(ˆ λ, ˆ q).

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

9 / 18

slide-12
SLIDE 12

Near-extremal solutions

Near-extremal expansion

Set Mα = 1 and introduce dimensionless coordinate R = r − 1, The near-extremal condition (ǫ ≪ 1) a2 + β2 = 1 − ǫ3, β2 = α/(1 + α). (18) We use a as modified parameter instead of α (to avoid square root), α = 1/a2 − 1 + O(ǫ3). (19) The emitter is located on (or near) ISCO Rs = ǫ¯ R + O(ǫ2), where ¯ R ≥ ¯ RISCO =

  • 2a2/(2a2 − 1)

1/3

. (20) Introducing new quantities λ and q to track the small corrections ˆ λ = 1 + a2 a (1 − ǫλ), ˆ q =

  • 4 − 1

a2 − q2. (21)

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

10 / 18

slide-13
SLIDE 13

Near-extremal solutions

r − θ equation: − ´ ro

rs dr ±√ R(r) = −

´ θo

θs dθ ±√ Θ(θ),

Introducing a separation scaling of ǫp (ǫ ≪ ǫp ≪ 1) with p ∈ (0, 1) and split the radial integral into two pieces (set Mα = 1), Ir = ˆ ǫpC

ǫ¯ R

dR √ R + ˆ Ro

ǫpC

dR √ R . (22) Performing the radial integral by using matched asymptotic expansion method. The angular integral is given by elliptic integral. From the r − θ equation, we can write λ in terms of q, i.e., we get a function λ(q).

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

11 / 18

slide-14
SLIDE 14

Near-extremal solutions

∆t and ∆φ equation: ∆φ − Ωs∆t = −Ωsto + φo (set φo = 2πN),

We introduce a dimensionless time coordinate ˆ to, ˆ to = to Ts = toΩs 2π = ato 2(1 + a2)πMα + O(ǫ). Plugging λ(q) into this equation gives functions of q(ˆ to) and λ(ˆ to).

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

12 / 18

slide-15
SLIDE 15

Near-extremal solutions

Observational quantities

Observational quantities in terms of λ and q: The image position x = −1 + a2 a 1 sin θo , (23) y = s

  • 4 − 1

a2 − q2 + a2 cos2 θo − (1 + a2)2 a2 cot2 θo. (24) The image redshift g = 1

√ 4a2−1 a

+ 2a(1+a2)

√ 4a2−1 λ ¯ R

. (25)

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

13 / 18

slide-16
SLIDE 16

Near-extremal solutions

Observational quantities

Observational quantities in terms of λ and q: The image flux (relative to the comparable “Newtonian flux”)

Cunningham & Bardeen, 1972. is given by

Fo FN = √ 4a2 − 1ǫ¯ R 2a2Ds qg3

  • 4 − 1

a2 − q2

Θ0(θo) sin θo

  • det ∂(B, A)

∂(λ, q)

  • −1

, where Ds =

  • q2 ¯

R2 + 4(1 + a2)λ¯ R + (1 + a2)2λ2, (26) and A and B are functions associated with the trajectory equations.

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

14 / 18

slide-17
SLIDE 17

Observational appearance

Making a choice of the modified parameter α and parameters ǫ, ¯ R, Ro, θo.

0.0 0.2 0.4 0.6 0.8 1.0

  • 1.0
  • 0.5

0.0 0.5 1.0 y/ymax a=1 (\alpha=0) 0.0 0.2 0.4 0.6 0.8 1.0

  • 1.0
  • 0.5

0.0 0.5 1.0 a=0.8 (\alpha=0.563) 0.0 0.2 0.4 0.6 0.8 1.0

  • 1.0
  • 0.5

0.0 0.5 1.0 a = 0.717 (\alpha=0.945) 0.0 0.2 0.4 0.6 0.8 1.0 20 40 60 80 100 Fo / FN

  • ϵ / log ϵ

0.0 0.2 0.4 0.6 0.8 1.0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 t

  • g

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 t

  • (Phase)

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 t

  • (Phase)

(Phase)

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

15 / 18

slide-18
SLIDE 18

Observational appearance

The entire image at EHT

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

16 / 18

slide-19
SLIDE 19

Summary

We study the observaitonal signature of a near-extremal Kerr-like black hole in the modified gravity theory (MOG), in particular, we study the optical appearance of an emitter orbiting near the BH. There are typical signatures away from the Kerr case which may be tested by the Event Horizon Telescope (EHT). Outlook: black hole surroundings, such as plasma and accretion disk. ...

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

17 / 18

slide-20
SLIDE 20

Thank you for your attention!

Haopeng Yan (NBI, UCPH) Observational Signature at EHT

  • Jan. 4, 2019, NWSPPC

18 / 18