Extracting information about the gravitational background from black hole images is both important and challenging. In this study we use a physically motivated plasma model typically applied to stationary axisymmetric spacetimes to demonstrate that in a rotating black hole spacetime the polarizations of emitted light near the event horizon depend solely on the spacetime geometry independent of the plasma flow geometry. We confirm that the framedragging effect of a rotating black hole governs the observed polarization structure in the nearhorizon image. This finding indicates a unique imprint of the black hole spin on the polarization of the nearhorizon image. We anticipate that refined observations of nearhorizon emissions by the nextgeneration Event Horizon Telescope will enable us to determine the black hole spin in a straightforward manner.
['General Relativity and Quantum Cosmology', 'Astrophysics - High Energy Astrophysical Phenomena']
Unique Imprint of Black Hole Spin on the Polarization of NearHorizon Images
{'Unique Imprint of Black Hole Spin on the Polarization of Near-Horizon Images': "Yehui Hou ¶ , 1 Jiewei Huang ¶ , 2 Yosuke Mizuno, 3, 4, ∗ Minyong Guo, 2, 5, † and Bin Chen 6, 7, 1, 8, ‡  \n1 School of Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871, P.R. China  \n2 School of physics and astronomy, Beijing Normal University, Beijing 100875, P. R. China  \n3  \nTsung-Dao Lee Institute, Shanghai Jiao-Tong University, Shanghai, 201210, P. R. China  \n4 School of Physics & Astronomy, Shanghai Jiao-Tong University, Shanghai, 200240, P. R. China  \n- 5 Key Laboratory of Multiscale Spin Physics, Ministry of Education, Beijing 100875, P. R. China 6 Institute of Fundamental Physics and Quantum Technology,  \nNingbo University, Ningbo, Zhejiang 315211, China  \n7 School of Physical Science and Technology, Ningbo University, Ningbo, Zhejiang 315211, China 8 Center for High Energy Physics, Peking University,  \nNo.5 Yiheyuan Rd, Beijing 100871, P. R. China  \nExtracting information about the gravitational background from black hole images is both important and challenging. In this study, we use a physically motivated plasma model, typically applied to stationary, axisymmetric spacetimes, to demonstrate that in a rotating black hole spacetime, the polarizations of emitted light near the event horizon depend solely on the spacetime geometry, independent of the plasma flow geometry. We confirm that the frame-dragging effect of a rotating black hole governs the observed polarization structure in the near-horizon image. This finding indicates a unique imprint of the black hole spin on the polarization of the near-horizon image. We anticipate that refined observations of near-horizon emissions by the next-generation Event Horizon Telescope will enable us to determine the black hole spin in a straightforward manner.  \nIntroduction -The images of the supermassive black holes, recently released by the Event Horizon Telescope (EHT) Collaboration [1-9], provide a new window to study various strong-field problems in astrophysics. One of the most crucial tasks is to determine the parameters of the central black hole embedded in the surrounding spacetime. The mass and spin of the black hole play a central role in this determination. Regarding spin measurements, in addition to the well-known indicators such as the photon ring [2, 10] and sub-images that could be resolved by the next-generation EHT [11-13] and space very-long baseline interferometry (VLBI) projects [14, 15], there are other methods available. These include brightness asymmetry [16], photon ring auto-correlation [17], inner shadow measurement [18, 19], and the polarization structure [4, 5]. Among these methods, the timeaveraged polarization structure of the image is particularly regarded as one of the most reliable for estimating black hole spin [20-23].  \nA rotating black hole theoretically exhibits the framedragging effect [24], which causes the accretion flow, threaded with magnetic fields, to adopt a spiral shape. This spiral shape imparts a distinctive signature in linear polarizations [25]. Consequently, measuring the helical behavior of the polarization structure could offer valuable insights into the black hole spin [23]. The method for determining the spin of black holes using polarized  \nimages typically involves comparing observational data with the polarized images generated from general relativistic magnetohydrodynamics (GRMHD) simulations and general relativistic radiative transfer (GRRT) calculations [26-29]. However, these simulations rely on several assumptions. In particular, the uncertainty of electron temperature significantly affects the resultant polarized images. Moreover, the GRMHD simulations are computationally intensive, making it challenging to explore larger parameter ranges. Additionally, concerns have been raised about the validity of fluid treatment at the horizon scale [30], underscoring the need for a more comprehensive approach to model the accretion flows. Alternative methods, such as Vlasov dynamics or particle-in-cell simulations, have been suggested [31, 32], although performing large-scale simulations remains difficult due to the spatial scale being determined by the ion Larmor radius. Despite these advancements, it remains a significant challenge to simulate the plasma environment around black holes accurately.  \nIndeed, the primary challenge arises from the unknown properties of the accreting plasma [33, 34]. It is particularly difficult to distinguish whether the observed polarization pattern is due to gravitational effects or the intrinsic electromagnetic behavior of the accreting plasma. This ambiguity complicates the process of inferring the black hole's spin from the overall polarization image. To simplify the measurement of black hole spin and reduce the reliance on complex physical models, we must ask: Is it possible to identify a polarization quantity that depends solely on gravitational effects, independent  \nof plasma flow geometry? Such a polarization quantity could provide more direct insights into black hole characteristics, thereby enhancing our understanding of their properties.  \nIt is hypothesized that near the event horizon, the magnetic field becomes highly toroidal due to the corotating plasma having an angular velocity that matches the event horizon's rotation. Furthermore, the emission from this near-horizon corotating plasma is believed to be detectable. Current research supports the magnetically arrested disk (MAD) model for both M87* and SgrA* [2, 5, 9, 35-37]. The simulation results suggest an increasing emission as the event horizon is approached [2, 38, 39]. Consequently, the polarization of the near-horizon image (NHI), which results from the synchrotron radiation of the near-horizon plasma, can be detected. This polarization has the potential to encode precisely the information about the black hole spin, primarily due to the extreme frame-dragging effects that overwhelm other influences. This enables us to read black hole spin from the polarizations of the NHI of black holes.  \nIn this letter, we confirm that there is a distinctive signature of spin on the polarization pattern of the NHI for a rotating black hole, independent of the plasma flow geometry. We begin by discussing the general properties of horizon-scale plasma within the context of a stationary, axisymmetric black hole spacetime. The plasma is highly conductive and exhibits a geometrically thin emission profile at the equatorial plane. The synchrotron radiations emitted by the electrons within the plasma are predominantly perpendicular to the magnetic field. For computational convenience, we then narrow our focus to Petrov Type D black holes. We reveal that the ratio of the real part to the imaginary part of the complex Penrose-Walker constant for the polarized light emitted from the equatorial near-horizon source depends solely on the black hole parameters and the conserved impact parameters. Most importantly, we show the unique imprint of Kerr spin on the polarization electric vector position angle (EVPA) of the NHI. Throughout the study, we use the units G = c = 1.  \nSetup -Plasma, a state of matter composed of charged particles such as electrons and ions, is attracted to black holes. In this study, we examine a general stationary axisymmetric spacetime that contains a rotating black hole. This spacetime is characterized by a line element, which can be expressed as d s 2 = g tt dt 2 +2 g tϕ dtdϕ + g ϕϕ dϕ 2 + g rr dr 2 + g θθ dθ 2 in the Boyer-Lindquist coordinates. The magnetic field is coupled with the plasma flow, which is required to obey the high conductivity condition so that any possible electric field will be eliminated by the fast rearrangement of the electrons, E µ = F µν U ν = 0 where U µ is the bulk velocity of the plasma, F µν is the electromagnetic tensor. Using the Bianchi identity ∇ µ ∗ F µν = 0, the magnetic field B µ = ∗ F µν U ν has a  \nsimple form[40, 41]:  \nB µ = Ψ √ -g U r ( U t U µ + δ µ t ) , (1)  \nwhere g is the determinant of the spacetime metric; Ψ denotes the overall strength, which is conserved along the bulk velocity, U µ ∂ µ Ψ ≈ U r ∂ r Ψ = 0. Note that we do not consider the region with U r = 0, where the accretion towards the black hole stops and Eq. (1) cannot be applied. This magnetic field matches the simulational results in [42] for the regions close to a Kerr black hole. The millimeter-wavelength emission is dominated by the collective synchrotron radiation of the thermal or nonthermal electrons within the plasma, whose polarization vector f µ is largely perpendicular to the global magnetic field [43]. Without imposing any specific distribution of electrons within the plasma, the covariant expression for f µ must take the following form  \nf µ = ϵ µναβ U ν p α ( B ⊥ ) β ω √ B 2 ⊥ , (2)  \nwhere ϵ µναβ is the Levi-Civita tensor; p µ denotes the four-momentum of photon; ω = -U µ p µ is the photon frequency in the plasma frame; B µ ⊥ = B µ -ω -2 ( B · p ) p µ ⊥ denotes the spatial magnetic field in the plasma frame with p µ ⊥ = p µ -ωU µ being the spatial four-momentum of photon. The expression Eq. (2) naturally satisfies the gauge condition f t = 0.  \nThe photon emitted from the synchrotron radiation of the plasma, as observed on the screen, is described by the coordinates ( α, β ) ∝ ( -sin θ o p ϕ , p θ ) [44] in asymptotic flat spacetime, where θ o represents the observational angle. Given the position ( α, β ), the electric vector polarization angle (EVPA) is computed from the components of f µ projected along the α and β directions on the observer's screen  \nχ = tan -1 f β f α , (3)  \nwhich describes the angle between the polarization vector and the α -axis.  \nPolarizations of NHI -In the following discussion, for computational simplicity (to make an analytical study), we restrict our attention to Petrov Type D spacetime. Neglecting in-medium effects, the polarization evolution can be determined analytically using the Penrose-Walker constant [45, 46], which is a complex number conserved along null geodesics. By choosing a null basis ( l, n, m, ¯ m ) with l and n as the principal null directions, the complex Penrose-Walker constant is given by:  \nκ = κ 1 + iκ 2 = 2Ψ -1 3 2 p µ f ν ( l [ µ n ν ] -m [ µ ¯ m ν ] ) , (4)  \nwhere p µ denotes the photon four-momentum, Ψ 2 is the only non-vanishing Weyl scalar in this framework [47],  \nand κ 1 , κ 2 represent the real and imagine part of κ , respectively.  \nThe Kerr spacetime, classified as a Petrov Type D spacetime, is of significant astrophysical relevance. In this context, we aim to use the Kerr geometry as an example to demonstrate the unique imprint of black hole rotation on the polarizations of photons emitted very near the event horizon. Specifically, Eq. (4) can be rewritten as  \nκ ≡ κ 1 + iκ 2 = ( Ai B )( r -ia cos θ ) , A = 2 p [ t f r ] +2 a sin 2 θp [ r f ϕ ] , B = 2sin θ [ ( r 2 + a 2 ) p [ ϕ f θ ] -ap [ t f θ ] ] , (5)  \nwhere a is the Kerr spin. For a Kerr black hole, the event horizon is determined by the larger root r + of the equation ∆ = ( r -r + )( r -r -) = 0. For simplicity, we set the ADM mass M = 1. To analyze the near-horizon behavior of the polarizations of emissions from the equatorial plasma, we need to take the near-horizon limit, r → r + of Eq. (2). Considering the fact that the accretion is concentrated on the equatorial plane and the near-horizon plasma flow is dominated by gravitational attraction, the bulk velocity satisfies | rU θ | ≪ | U r | and U t , U ϕ should be finite and vary slowly outside the horizon. This assumption is consistent with the simulated results of MADs presented in [39, 42]. Without loss of generality, we will set U θ = 0 and keep U t and U ϕ invariant when performing the near-horizon expansion. Consequently, for the bulk velocity of the equatorial plasma near r = r + , we obtain the following relations:  \nU t = -4 Y ∆ -( 1 + 2 r + ) U t + aU ϕ r 2 + ( r + -1) + o (∆ 0 ) , U r = -2 | Y | r + + o (∆ 0 ) , U θ = 0 , U ϕ = -2 aY r + ∆ + aU t + U ϕ r 2 + ( r + -1) + o (∆ 0 ) , (6)  \nwhere Y = U t +Ω h U ϕ , Ω h = a/ (2 r + ) is the angular velocity of the Kerr black hole. We have chosen U r < 0 for the accreting plasma. To the lowest order approximation, all components of U µ are proportional to Y . Specifically, we have:  \nU r U t = Ω h ∆ a | Y | Y , U ϕ U t = Ω h , (7)  \nwhich is a general relationship for the accretion velocity near the horizon. The near-horizon expansion of the magnetic field, described by Eq. (1), is given as follows:  \nB t = 2Ψ U t r + ∆ Y | Y | + o (∆ -1 ) , B r = Ψ r 2 + U t , B θ = 0 , B ϕ = Ψ U t a r 2 ∆ Y | Y | + o (∆ -1 ) . (8)  \n+  \nHere, B θ = 0 arises from U θ = 0; the plunging matter must have Y < 0, consistent with the second law of black hole thermodynamics [48]. Specifically, if a probe particle has Y > 0, it cannot classically reach the event horizon. Furthermore, the ratio B ϕ /B r ≈ -a ∆ -1 aligns with the results from the simulation presented in [42]. Expanding f µ near r = r + in the Kerr metric, we obtain:  \nX E f t = -2 aσ θ √ η r + ∆ + aσ θ √ η r 2 + ( r + -1) + o (∆ 0 ) , X E f r = aσ θ √ η r 2 + + R 1 ∆+ o (∆) , X E f θ = a ( aλ -2 r + ) -σ r a | aλ -2 r + | r 2 + ∆ +Θ 1 + o (∆ 0 ) , X E f ϕ = -a 2 σ θ √ η r 2 + ∆ + σ θ √ η r 2 + ( r + -1) + o (∆ 0 ) , (9)  \nwhere X = -Ψ -1 √ -g ( U µ p µ ) √ B 2 ⊥ , E = -p t , λ and η are conserved impact parameters that indicate the direction of photon emission [49]. The term σ θ denotes the sign of p θ . The terms R 1 and Θ 1 depend on the bulk velocity of the plasma, and their forms are rather complicated and thus are omitted here.  \nNote that the overall factor X/ E involves the bulk velocity, meaning that κ in Eq. (5) depends on the plasma motion. However, by combining Eqs. (5) and (9) and expanding the photon four-momentum, we find that the ratio  \nz ≡ κ 1 κ 2 = z 0 + z 1 ∆+ O (∆ 2 ) (10)  \nis independent of the plasma motion near the horizon. Here, we have introduced the following expressions  \nz 0 = σ θ √ η λ -a , z 1 = σ θ ( 1 + z 2 0 ) √ η Ω h 2 a 2 (Ω h λ -1) . (11)  \nThe ratio z is directly related to the EVPA on the observer's screen, as described by the equation  \nχ = tan -1 ( βz + µ β -µz ) = tan -1 ( µ β ) +tan -1 z , (12)  \nbased on Eq. (3), where µ = -( α + a sin θ o ). From this, we can derive the following equation:  \nχ = tan -1 ( µ β ) +tan -1 z 0 + z 1 ∆ 1 + z 2 0 + o (∆ 1 ) , (13)  \nwhich represents a fundamental and universal relationship. This equation indicates that the EVPA of the NHI is independent of the plasma flow and is solely influenced by the spacetime geometry. Specifically, it highlights a unique imprint of black hole spin on the polarization of the NHI.  \nVerification -In Fig. 1, we compare the results of the near-horizon expansion given by Eq. (10) with the exact value of z for the plasma with three different flow  \ntypes: the plasma infalling from infinity (Type I), the plasma plunging from the prograde innermost stable circular orbit (ISCO) (Type II), and the fitting result from the simulation of MAD introduced in [39] (Type III). We have chosen a typical direction for the emitted photons with ( λ, η ) = (2 , 8) for a = 0 . 94 and a = 0 . 2.  \nFIG. 1. Compare the universal relationship of z with the exact z for the plasma under three different flow conditions.  \n<!-- image -->  \nIt is evident that the expansion up to the next-toleading order of ∆ works quite well for 0 ≤ ∆ ≲ 10 -2 (corresponding to r = 1 . 01 r + for a = 0 . 94 and r = 1 . 003 r + for a = 0 . 2) for all three types of plasma flow. The inclusion of the next-next-to-leading term only improves the expansion in a very small interval around ∆ ∼ 5 × 10 -2 (corresponding to r = 1 . 05 r + for a = 0 . 94 and r = 1 . 01 r + for a = 0 . 2). Therefore, the validity of the near-horizon expansion is manifested.  \nIn Fig. 2, we present the polarization vectors at the event horizon for a Kerr black hole with spin parameters a = 0 . 2 and a = 0 . 94, observed at inclination angles θ o = 17 · and θ o = 80 · . In the plots, the projection of the black hole's rotation direction onto the screen aligns with the β axis. The EVPAs in the NHIs exhibit significant variations at χ = π/ 4 and χ = -π/ 4 when either the spin parameter a or the observation angle θ o is altered, indicating a mutual dependence on both the black hole's spin and the observation angle. By integrating these observations, we can infer the values of the black hole's spin and the observation angle.  \nConclusion and discussion -In this letter, we show that the EVPA in the NHI of a rotating black hole pro-  \nFIG. 2. The EVPAs in the NHIs of Kerr black holes are represented by different symbols: · indicates χ = π/ 4, and △ indicates χ = -π/ 4. The dashed lines represent the NHIs.  \n<!-- image -->  \nvides a novel measure, which is independent of the photon ring and inner shadow and can serve as an alternative means of detecting supermassive black hole spins. Our result shows a unique imprint of black hole spin on the polarization of NHI. The discussions can be readily extended to any Petrov Type D spacetime. In fact, in a companion paper [50], we have demonstrated that Eq. (10) remains invariant and have derived a similar expression for Eq. (13) in a Kerr-Newman-Taub-NUT spacetime. The universal relationship illustrates that the EVPA in the NHI is governed exclusively by spacetime geometry, irrespective of the plasma flow geometry. This phenomenon results from the frame-dragging effect near the event horizon of rotating black holes. Therefore, this relationship is expected to be valid in all rotating spacetimes.  \nOur study did not account for the Faraday rotation effects caused by plasma, whereas current polarization results from the EHT are depolarized by the Faraday rotation [22]. This raises the question whether our results can be useful when taking into account of the Faraday rotation. The influence of the Faraday rotation could be quantified through upcoming multi-frequency observations, which would allow it to be separated from the polarization signal. Specifically, the measurements at higher frequencies can distinguish the Faraday rotation from the underlying magnetic field structure [20, 30], thereby isolating the gravitational effects from the nearhorizon polarization. This issue could also be addressed by incorporating the frequency dependence of the emitted photons in our calculations, which was not considered in our current work. Additionally, it would be beneficial  \nto use GRMHD simulations and GRRT calculations to test realistic scenarios, including the presence of a Faraday screen.  \nThe work is partly supported by NSFC Grant Nos. 12205013 and 12275004. YM is supported by the National Key R&D Program of China (Grant No. 2023YFE0101200), the National Natural Science Foundation of China (Grant No. 12273022), and the Shanghai Municipality Orientation Program of Basic Research for International Scientists (Grant No. 22JC1410600).  \n- ∗ Corresponding author: mizuno@sjtu.edu.cn\n- † Corresponding author: minyongguo@bnu.edu.cn\n- ‡ Corresponding author: bchen01@pku.edu.cn\n- [1] Event Horizon Telescope Collaboration, K. Akiyama et al. , 'First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,' Astrophys. J. Lett. 875 (2019) L1, arXiv:1906.11238 [astro-ph.GA] .\n- [2] Event Horizon Telescope Collaboration, K. Akiyama et al. , 'First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring,' Astrophys. J. Lett. 875 no. 1, (2019) L5, arXiv:1906.11242 [astro-ph.GA] .\n- [3] Event Horizon Telescope Collaboration, K. Akiyama et al. , 'First M87 Event Horizon Telescope Results. VI. 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