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
Outline Introduction 1 Orbiting emitter 2 Kerr-like black hole in the MOG theory Equations of photon trajectory Near-extremal solutions Observational appearance 3 Summary 4 Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 2 / 18
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
The Kerr-MOG spacetime The rotating solution is given by the Kerr-MOG metric ∆ dr 2 + Σ d θ 2 + Ξ sin 2 θ ds 2 = − ∆Σ Ξ dt 2 + Σ ( d φ − ω dt ) 2 , (1) Σ where ∆ = r 2 − 2 GMr + a 2 + α G N GM 2 , Σ = r 2 + a 2 cos 2 θ, (2) ω = a ( a 2 + r 2 − ∆) Ξ = ( r 2 + a 2 ) 2 − ∆ a 2 sin 2 θ, , (3) Ξ where α is the modified parameter and G = ( 1 + α ) G N is called as an enhanced gravitational constant. Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 4 / 18
The Kerr-MOG spacetime The rotating solution is given by the Kerr-MOG metric ∆ dr 2 + Σ d θ 2 + Ξ sin 2 θ ds 2 = − ∆Σ Ξ dt 2 + Σ ( d φ − ω dt ) 2 , (1) Σ where ∆ = r 2 − 2 GMr + a 2 + α G N GM 2 , Σ = r 2 + a 2 cos 2 θ, (2) ω = a ( a 2 + r 2 − ∆) Ξ = ( r 2 + a 2 ) 2 − ∆ a 2 sin 2 θ, , (3) Ξ where α is the modified parameter and G = ( 1 + α ) G N is called as an enhanced gravitational constant. For simplicity, we set α β 2 = 1 + α M 2 G N = 1 , M α ≡ M ADM = ( 1 + α ) M , α , (4) thus ∆ = r 2 − 2 M α r + a 2 + β 2 . (5) Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 4 / 18
The Kerr-MOG spacetime Mass dependent gravitational charge � K = α G N M . (6) The event horizon is obtained for ∆ = 0, � α − ( a 2 + β 2 ) . M 2 r ± = M α ± (7) The extremal condition a 2 + β 2 = M 2 α . (8) Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 5 / 18
The Kerr-MOG spacetime Mass dependent gravitational charge � K = α G N M . (6) The event horizon is obtained for ∆ = 0, � α − ( a 2 + β 2 ) . M 2 r ± = M α ± (7) The extremal condition a 2 + β 2 = M 2 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 ≤ M 2 α a 2 − 1 . (9) Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 5 / 18
Orbiting emitter We assume the emitter (“hot spot”) is on a circular orbit with radius r s at the equatorial plane. The angular velocity is Ω s = d φ Γ( r s ) dt = ± s ± a Γ( r s ) , (10) r 2 where Γ 2 ( r ) = M α r − β 2 . (11) Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 6 / 18
Orbiting emitter Photons originate from the emitter There are four conserved quantities along each photon trajectory: the invariant mass µ 2 = 0, the total energy E , the angular momentum L and the Carter constant Q . Introducing two rescaled quantities, √ Q λ = L ˆ E , ˆ q = E . (12) Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 7 / 18
Orbiting emitter Equations of Photon trajectory Using Hamilton-Jacobi method, we can obtain the equations, ˆ θ o ˆ r o dr d θ − R ( r ) = − Θ( θ ) , (13) � � ± ± r s θ s ˆ r o ˆ θ o λ csc 2 θ a ( 2 M α r − β 2 − a ˆ ˆ λ ) ∆ φ = φ o − φ s = − dr + − Θ( θ ) d θ, (14) � � ± ∆ R ( r ) ± r s θ s r 4 + a 2 � r 2 + 2 M α r − β 2 � − a � � 2 M α r − β 2 � ˆ � ˆ r o λ ∆ t = t o − t s = − dr � ± ∆ R ( r ) r s ˆ θ o a 2 cos 2 θ + − Θ( θ ) d θ, (15) � ± θ s where � 2 − ∆ � r 2 + a 2 − a ˆ q 2 + � � 2 � � a − ˆ R ( r ) = λ ˆ λ , (16) q 2 + a 2 cos 2 θ − ˆ λ 2 cot 2 θ. Θ( θ ) = ˆ (17) Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 8 / 18
Orbiting emitter Photon trajectory Each photon trajectory can be labeled by a pair of conserved quantities (ˆ λ, ˆ q ) , which connects the star ( t s , r s , θ s , φ s ) to an observer ( t o , r o , θ o , φ o ) . We decouple the coordinates t s and φ s by using φ s = Ω s t s , and made the following choice, θ s = π/ 2, r o → ∞ and φ o = 2 π N . For given choice of r s and θ o , solving these equations gives the time-dependent trajectories [ˆ λ ( t o ) , ˆ q ( t o )] corresponding to a track of the emitter’s image. Plugging ˆ λ and ˆ q into the functions of observables: position [ x (ˆ q ] , y (ˆ q )) , redshift g (ˆ q ) and flux F o (ˆ λ, ˆ λ, ˆ λ, ˆ λ, ˆ q ) . Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 9 / 18
Near-extremal solutions Near-extremal expansion Set M α = 1 and introduce dimensionless coordinate R = r − 1, The near-extremal condition ( ǫ ≪ 1) a 2 + β 2 = 1 − ǫ 3 , β 2 = α/ ( 1 + α ) . (18) We use a as modified parameter instead of α (to avoid square root), α = 1 / a 2 − 1 + O ( ǫ 3 ) . (19) The emitter is located on (or near) ISCO R s = ǫ ¯ R + O ( ǫ 2 ) , where � 1 / 3 2 a 2 / ( 2 a 2 − 1 ) � R ≥ ¯ ¯ R ISCO = . (20) Introducing new quantities λ and q to track the small corrections � λ = 1 + a 2 4 − 1 ˆ a 2 − q 2 . ( 1 − ǫλ ) , q = ˆ (21) a Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 10 / 18
Near-extremal solutions ´ r o ´ θ o dr d θ ± √ ± √ r − θ equation: − R ( r ) = − Θ( θ ) , θ s r s Introducing a separation scaling of ǫ p ( ǫ ≪ ǫ p ≪ 1) with p ∈ ( 0 , 1 ) and split the radial integral into two pieces (set M α = 1), ˆ R o ˆ ǫ p C dR dR I r = √ + √ . (22) R R ǫ ¯ ǫ p C R 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
Near-extremal solutions ∆ t and ∆ φ equation: ∆ φ − Ω s ∆ t = − Ω s t o + φ o (set φ o = 2 π N ), We introduce a dimensionless time coordinate ˆ t o , = t o Ω s t o = t o at o ˆ = + O ( ǫ ) . 2 ( 1 + a 2 ) π M α T s 2 π Plugging λ ( q ) into this equation gives functions of q (ˆ t o ) and λ (ˆ t o ) . Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 12 / 18
Near-extremal solutions Observational quantities Observational quantities in terms of λ and q : The image position x = − 1 + a 2 1 , (23) a sin θ o � a 2 − q 2 + a 2 cos 2 θ o − ( 1 + a 2 ) 2 4 − 1 cot 2 θ o . y = s (24) a 2 The image redshift 1 g = . (25) √ + 2 a ( 1 + a 2 ) 4 a 2 − 1 λ √ ¯ a 4 a 2 − 1 R Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 13 / 18
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 √ 4 a 2 − 1 ǫ ¯ − 1 qg 3 � det ∂ ( B , A ) F o R � � � � = , � � 2 a 2 D s � ∂ ( λ, q ) F N 4 − 1 a 2 − q 2 � Θ 0 ( θ o ) sin θ o � where � q 2 ¯ R 2 + 4 ( 1 + a 2 ) λ ¯ R + ( 1 + a 2 ) 2 λ 2 , D s = (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
Observational appearance Making a choice of the modified parameter α and parameters ǫ, ¯ R , R o , θ o . a = 1 (\alpha=0) a = 0.8 (\alpha=0.563) a = 0.717 (\alpha=0.945) 1.0 1.0 1.0 0.5 0.5 0.5 y / y max 0.0 0.0 0.0 - 0.5 - 0.5 - 0.5 - 1.0 - 1.0 - 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 100 100 100 80 80 80 60 60 60 F o / F N - ϵ / log ϵ 40 40 40 20 20 20 0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.5 1.5 1.5 1.0 1.0 1.0 g 0.5 0.5 0.5 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t t t ( Phase ) o ( Phase ) o ( Phase ) o Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 15 / 18
Observational appearance The entire image at EHT Haopeng Yan (NBI, UCPH) Observational Signature at EHT Jan. 4, 2019, NWSPPC 16 / 18
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