Polarized image of a synchrotron emitting ring around a static hairy black hole in Horndeski theory

In this paper, we investigate the polarization images of a synchrotron-emitting ﬂuid ring surrounding a static hairy black hole within the framework of Horndeski’s theory. Our ﬁndings indicate that the characteristics of these polarization images are predominantly inﬂuenced by the hairy parameter, alongside the magnetic ﬁeld near the black hole, the ﬂuid’s velocity, and the observer’s inclination angle. Speciﬁcally, the hairy parameter primarily affects the polarized intensity and the apparent radius of the ring in the images. Conversely, the impacts of the magnetic ﬁeld, ﬂuid velocity, and inclination angle on the polarization images are found to be independent of the hairy parameter and closely resemble those observed in the context of Schwarzschild black holes. Additionally, the polarization direction is sig-niﬁcantly inﬂuenced by the magnetic ﬁeld orientation, while the inclination angle crucially determines the apparent ﬂat-ness of the images. Variations in the ﬂuid velocity direction also markedly affect the trend in polarized intensity. Furthermore, we explore how these parameters inﬂuence the Stokes Q − U loops, revealing distinct behaviors in response to changes in the aforementioned variables. This comprehensive analysis enhances our understanding of the intricate dynamics and observational signatures of black holes within alternative gravitational theories.


Introduction
Gravitational phenomena observed at solar system scales are well-explained by general relativity (GR).However, to account for the observed acceleration in the expansion of the universe at cosmological scales, theorists must introduce a mysterious component known as dark energy.Additionally, a e-mail: has624@lehigh.edub e-mail: zhut05@zjut.edu.cn(corresponding author) the presence of dark matter is posited within the framework to address the discrepancies related to the missing mass evident at galactic scales.Yet, experimental validation of dark matter through the large hadron collider remains elusive.These challenges cast doubt on the sufficiency of GR at cosmological scales.Consequently, numerous alternative theories of gravity have been proposed, among which modified gravity theories stand out as particularly promising and effective, see Refs.[1][2][3][4], and references therein.
Among a broad class of modified gravity theories, Horndeski and beyond Horndeski theory [5][6][7] play a significant role in explaining acceleration of the expansion of the universe [8].It is interesting to mention that the Horndeski theory is the most general scalar-tensor theory in four dimensions that possess the second-order field equations and is free of Ostrogradski ghost [5,6].Note that an extension of the Horndeski theory has also been explored in Ref. [9].Recently, the black hole solutions in both the Horndeski and beyond Horndeski theories have attracted a great deal of attention in the scientific community [10][11][12][13][14][15][16][17].Specifically, by considering the quartic scalar field model, a static and spherical symmetric hairy black hole solution of Horndeski gravity has been obtained in Ref. [10], which has also been extended to the case of rotating hairy Horndeski black holes in Ref. [18].Implications of the astrophysical phenomena of this black hole solution then has been extensively studied in the literature, such as the Bondi-Hoyle-Lyttleton accretion process [19], thin accretion disk [20,21], scalar field perturbations [22,23], quasiperiodic oscillations [24], orbits of test particles [25,26], shadows and images [27][28][29][30], and strong gravitational lensing [18,31,32].
On the other hand, the Event Horizon Telescope collaboration has released the first images of the black hole in the central of M87* galaxy [33][34][35][36][37][38][39][40][41][42].This represents a great breakthrough and discovery in the field of astronomy.In the fifth paper of the above works [37], the asymmetric ring was confirmed and in Refs.[39][40][41], the magnetic field structure near the Event Horizon and the polarization of the ring were also discovered.These discoveries provide many subjects with research value.One of those is studying the polarization structure of black hole images to study and probe the magnetic field around the black holes, the motion of the accretion disks, and the geometric details of the black holes.Motivated by this, a great number of works were done on the polarization images of different types of black holes, see Refs.  and references therein.
In most studies of this kind, predicting the polarization images of specific types of black holes requires complex numerical simulations.These simulations are challenging and costly to implement in extensive and wide-ranging research projects.In 2021, a toy model was proposed to predict the polarization images of synchrotron-emitting fluid around Schwarzschild and Kerr black holes [69,70].Their results indicated that the properties of polarization, including polarized intensity, linear polarization angle, and the shape of Stokes Q − U loops, depend on the geometry and mass of the specific black hole, the velocity of the synchrotronemitting fluid, the inclination angle from the observer's sky, and the magnetic field around the asymmetric ring.Moreover, although this model could only compute results for a single radius value, the images of a fluid ring with finite width could be generated by summing contributions from results at different radii [71].
In this paper, we adopt the above mentioned toy model [69,70] to predict the polarization images of synchrotronemitting fluid around the static hairy black hole within the framework of Horndeski theory.Our results show that the characteristics of these polarization images are predominantly influenced by the hairy parameter, alongside the magnetic field near the black hole, the fluid's velocity, and the observer's inclination angle.This paper is organized as follows.In Sect.2, we provide a brief introduction of the hairy black hole in a specific Horndeski theory and the formulas for calculating the observed polarization vector in the image of an emitting ring in this spacetime, and then in Sect.3, we present our main results of the hairy effects on the polarization images of the synchrotron emitting ring.And finally, Sect. 4 succinctly summarizes our findings and initiates a discussion.

Observed polarization field of for the orbiting model in the static hairy black hole spacetime
We consider a subclass of the Horndeski theory with a scalar field φ, the dubbed quartic Horndeski gravity, whose action can be described as [11] where χ = −∂ μ φ∂ μ φ/2 is the canonical kinetic term of the scalar field, Q i (i = 2, 3, 4) are arbitrary functions of χ , Q i,χ is the derivative of Q i with respect to χ1 , R is the Ricci scalar, ∇ μ denotes the covariant derivative operator, while the operator is defined as ≡ ∇ μ ∇ μ .
For the static hairy black hole solutions that exist in the theory with the above action, one sets [10] (2.2) ) where κ = √ 8π and the coefficients α 21 , α 31 , w 2 , and w 4 take the values of ) to ensure the 4-current vanishes at infinity and the finiteness of energy.With these conditions, one can obtain the static and spherically symmetric solution as [10] with the metric where Here M is the black hole mass and Q is the hairy parameter with the dimension of length which is related to the coefficients α 42 , α 22 , and κ via (2.9) It is obvious one has to set α 42 /α 22 < 0. The scalar profile associated with the above black hole solution read (2.10) The above metric reduces to the Schwarzschild metric when Q → 0, and it is asymptotically flat.With the above background metric and scalar field in hand, we are in a position to consider a synchrotron emitting ring around a static hairy black hole.The ring is assumed to lie on the equatorial plane of the black hole as in [70].Following the setup in [70], we consider the photons emitted from the fluid element in the ring plane.These photons follow null Fig. 1 Geometry of the G-frame, or geodesic frame.Photons emitted from point P on the fluid ring sweep along the null geodesic line to the observer.x − z plane give the geodesic plane.We define ψ as the angle between the axis x and the direction of the observer and φ as the angle between the direction of the observer and the normal of the accretion disk.Significantly, in 3-dimensions, as shown in Fig. 2, axis z does not coinside with the normal direction n geodesics and travel from the fluid element located at point P to the observer, as shown in Figs. 1 and 2 in the geodesic frame or G-frame2 .In Figs. 1 and 2, the ring is on the x − ỹ plane and z is the normal direction which is perpendicular to the x − ỹ plane with the black hole located at the center o.There is an angle θ between the normal direction and the orientation of the distant observer as shown in Figs. 1 and  2. The fluid element at point P located at azimuthal angle φ measured from the datum line, which is perpendicular to the plane including normal direction and observer direction, to the direction of x in the ring plane.The photon was radiated from the point P with an emitting angle ω and it moves along the null geodesic which is located at the geodesic plane to the observer at infinity and sweeps the angle ψ between the direction of x and the orientation of the distant observer.
From the Fig. 2, one can find that angle ψ and azimuth angle φ satisfy cos ψ = − sin θ sin φ. (2.11) We consider a null geodesic with conserved energy k t = −1 traveling from P to the observer.Thus, at P in the G-frame, the corresponding orthogonal time component and orthogonal spatial component can be expressed as Fig. 2 Geometry of the whole geometric model.Axis x and axis x share the same radical direction lying on the accretion disk.The photon is emitted from point P with an angle ω which is given by Eq. (2.16) between axis x. ξ is the angle defined between x-z plain of G-frame and the x-ẑ plain of P-frame Fig. 3 The details of the P-frame.In the P-frame, axis x is radical, and axis ŷ lies in the azimuthal direction.Axis ẑ is parallel with the normal of the accretion disk.We define the velocity of the fluid is β, which has an angle χ from axis x. η is defined as an angle between the equatorial magnetic field vector B eq and the axis x where the subscript "G" means the quantity measured in the G-frame and ω is the emission angle of the photon, which can be calculated from the relation given by [54] cos At point P, one can then establish an additional local Cartesian coordinate system, the P-frame, that shares its x axis with the G-frame.The ŷ axis aligns with the azimuthal direction, while the ẑ axis is orthogonal to the ring plane, as depicted in Fig. 3.A rotation of angle ξ around the x axis allows for the transformation between these two coordinate systems, from which one obtains cos ξ = cos θ sin φ , sin ξ = sin θ cos φ sin ψ . (2.17) Through the application of the rotation to k (G) , it is possible to derive the corresponding orthogonal components in P-frame [70], ) ) The fluid moves on the x − ŷ plain of P-frame and it has a velocity β at the point P, which has an angle χ from x axis in Fig. 3 and could be described as Applying a Lorentz boost [70] with velocity β, one obtains the orthogonal components k (F) in the fluid frame (F-frame) (2.26) The radiation emitted along k μ (F) in F-frame is Doppler shifted by the time it reaches the observer.We set k t (O) in the observer frame is equal to unity, the Doppler factor δ which includes both Doppler shift and gravitational redshift from velocity will be . (2.27) In F-frame, we assume the magnetic field has the form [70] which describes the field components in the disk plane in terms of the B eq and B z and an orientation η shown in Fig. 3.The angle ζ between the magnetic field B and the 3-vector k (F) which affects the intensity of synchrotron radiation obeys the relation written in [70] sin In the fluid frame, it is well-known that the E-vector of the radiation is going along the direction of k (F) × B. Thus, the orthonormal components of the polarization 4-vector f μ could be written as and then the normalized polarization vector satisfies ) We use the inverse Lorentz transform to obtain the polarization vector of photon f μ at point P in P-frame from those in F-frame [70], (2.36) (2.39) The Cartesian unit vectors ( x, ŷ, ẑ) in P-frame are oriented along the spherical polar unit vectors (r , φ, − θ) of the static hairy Black hole spacetime, so the orthonormal components in static hairy black hole spacetime are and (2.41) In the static hairy Black hole spacetime, we can describe the coordinates x and y of the photon radiated from point P along the null geodesic to the observer at infinity [72] x = − Rk φ sin θ , (2.42) We compute the polarization vector by the Walker-Penrose constant κ [73].First of all, in static spherically symmetric spacetime, we could obtain the second order Killing-Yano tensors Y μν which have the form of [63,74] where θ o is the angle between the disk of the black hole and its normal.It further generates a Ỹμν given by Hodge dual, i.e.Ỹμν = μναβ Y αβ .These symmetries give rise to two additional constants of motion κ 1 and κ 2 along the null geodesics, ) At position P, we have and

.50)
Since κ 1 and κ 2 are conserved along the geodesic, we could obtain the normalized polarization electric field vector E along the x and y directions in the observer's sky [75] The intensity of the linearly polarized light of the synchrotron radiation that reaches the observer from the point P can be approximated as [70] |I | = δ 3+α ν l p | B| 1+α ν sin 1+α ν ζ. (2.54) where the power α ν depends on the ratio of the emitted photon energy hν to the electron temperature kT , l p is the path length through the emitting region and could be expressed with the height of the disk H as H. (2.55) We set α ν = 1 as in [70], so the intensity |I | of the linearly polarized synchrotron radiation that reaches the observer from the location P is Therefore, the polarized vector components observed along the x and y direction and the total polarization intensity are ) and the electric vector position angle E V P A can be expressed as

.60)
Here the Stokes parameters Q and U are given by

Effect of the hairy parameter on polarization images in the static hairy black hole spacetime
Now we are in a position to explore the effects of the hairy parameter on the polarization of the equatorial emitting ring around the static hairy black hole spacetime.

Effects of hairy parameter on the polarization images
Let's examine how hairy parameters impact polarization images.Figures 4, 5  hole with varying hairy parameter, magnetic field surrounding the black hole, fluid velocity, and the observer inclination angle.Figure 4 specifically demonstrates the influence of black hole radius (R) on the polarization image with a purely vertical magnetic field.And Figs. 5, 6, 7, 8, 9 and 10 showcase the effect of hairy parameter (Q) on polarization images, polarized intensity, electric vector position angle (E V P A), and its variation ratio ( E V P A/E V P A Q=0 , where 3 ) for photons emitted from various points on the equatorial emitting ring.Specifically, Fig. 4 presents the polarization images and their enlarged counterparts for emitting rings with varying radii, considering a purely vertical magnetic field (B z = 1).It is shown that when the fluid velocity is zero (β = 0), the photon polarization direction remains horizontal.At β = 0.3, the angle of fluid velocity (χ ) noticeably impacts the polarization, similar to its influence on the ring with R = 6.Generally, while the hairy parameter (Q) affects image size, this influence diminishes with increasing radius (R). 3

Here we set E V P A = E V P A α − E V P A α=0
Furthermore, polarized intensity decreases as the value of Q increases.
In Fig. 5, we illustrate the case with a purely vertical magnetic field (B z = 1) and fixed parameters: R = 6, M = 1, and θ = 20 • .The shapes of polarization images for static hairy black holes with different hairy parameter values resemble the Schwarzschild case [10], with the hairy parameter (Q) primarily influencing the radius of the polarization images.Notably, polarized intensity decreases as Q increases, a trend also evident in the |i| − φ plots of Fig. 5.The minimum photon polarized intensity shifts to a smaller φ value when fluid velocity increases from β = 0 to β = 0.3.Additionally, the E V P A curves for β = 0.3 appear smoother than those for β = 0, and the E V P A/E V P A Q=0 plots become more intricate with this change in fluid velocity.Changing the angle of fluid velocity from χ = −90 • to χ = −180 • brings the minimum and maximum values of photon polarized intensity closer, indicating reduced fluctuation in the curves.The E V P A curves shift towards increasing φ values.Generally, the extreme values of E V P A/E V P A Q=0 consistently occur between the null points and the point where The effects of hairy parameter Q on the polarized vector with different inclination angles θ are shown in Fig. 6.Notably, the effect of the observed inclination angle on the overall shape of the polarization images for emitting rings of the static hairy black hole mirrors that observed in Schwarzschild black holes [10].Specifically, the polarization images become increasingly flattened with an increase in the observed inclination angle.Furthermore, as the observed inclination angle increases, the curves representing polarized intensity exhibit more pronounced fluctuations.The difference in extreme values of polarized intensity becomes more evident with varying hairy parameters (Q), particularly at larger inclination angles.Additionally, the absolute values of these curves increase with increasing inclination angle.Despite these variations, the general shape of the polarized intensity images remains similar to the image observed at θ = 20 • .The E V P A plots, however, become more complex with increasing θ .As the observed inclination angle increases, the first three E V P A curves tend to merge, and the monotonic increase of the last curve becomes less steep.Simul-taneously, the peak observed in the E V P A/E V P A Q=0 image around φ = π 3 gradually diminishes.In contrast, the other peak located to the right of the image becomes more prominent and pronounced.
In Figs. 7 and 8, we focus on the effects of magnetic field components when the emitting rings lie on pure equatorial magnetic field with fixed parameters R = 6, M = 1, χ = −90 • and β = 0.3.In Fig. 7, we show the case with only angular magnetic field.The polarized vectors plotted in the polarization image are all radical.In this case, the values of polarized intensity with different hairy parameter Q at the same angular position of emitting rings differ much in height from the others.The minimum value of the polarized intensity moves to near π and the maximum value moves to near 2π or 0.Moreover, the curves of E V P A show a consistent monotone increment at different angular position of the emitting ring.On the contrary, the polarized vectors in the case with only radical magnetic are more tangential.the effects of θ are similar to those in other cases, which means the shape of the polarization images were flattened during the observed inclination angle increases.Nevertheless, When the observed inclination angle θ increases to a certain extent, the polarized direction will experience a reversal at the bottom of the polarization images.This phenomenon could be observed in the graphics of curves of values of polarized intensity.The reversal appears at the angular position of around 5π 6 .This property becomes more obvious with θ increasing, and meanwhile the minimum value of the polarized intensity moves to the position of reversal.Moreover, the monotonically increasing property of E V P A − φ function becomes more complex, which is mainly reflected in the central curve.
In Fig. 8, we present polarization images in the case where the equatorial magnetic field has both angular and radial components with fixed parameters R = 6, M = 1, θ = 20 • , χ = −90 • and β = 0.3.As B r < B φ , the relative images are similar to those images in the case of pure angular magnetic field.As the B r increases and B φ decreases, the images finally become so similar to those in the case of pure radical magnetic field.
In Fig. 9, we focus on the effects of the angle of the fluid velocity χ for pure equatorial magnetic field including B r = 0.87 and B φ = 0.5 with fixed parameters R = 6, M = 1, θ = 20 • and β = 0.3.The effects of χ on the polarization image of the emitting rings are insignificant, but it is easy to discover that the curves of the value of the polarized intensity become more flattered and the extreme values from different curves with different Q are much closer.Moreover, as χ increases, the peak of the E V P A/E V P A Q=0 increases.
In Fig. 10, we plot the polarization images of emitting rings for different observed inclination angles with fixed parameters R = 6, M = 1, χ = −90 • , β = 0.3 and pure equatorial magnetic field including B r = 0.87 and B φ = 0.5.Since the components of magnetic field are close to the pure radical magnetic field, the changes of the shapes of polarization images during observed inclination angle increases are similar to the cases of pure radical magnetic field except the reversal position angle decreased.Besides, the changes of E V P A curves are also similar to the cases of pure radical magnetic field.Figures 11 and 12 depict the Q − U loop diagrams for varying fluid direction angles (χ ) and observed inclination angles (θ ), respectively.These plots assume a pure vertical magnetic field and fixed parameters: R = 6, M = 1, and β = 0.3.Notably, the radius of the inner loops increases as χ changes from −90 • to −180 • , while the loops extend significantly into the third quadrant with increasing θ .
Figure 13 illustrates the influence of a pure equatorial magnetic field, composed of varying components, on the Q − U loop diagrams.The parameters are fixed at R = 6, M = 1, β = 0.3, χ = −90 • , and θ = 20 • .As the magnetic field components shift from B r = 0.97 and B φ = 0.26 to B r = 0.5 and B φ = 0.87, the loops exhibit a counterclockwise rotation.However, the overall shapes of the loops remain largely unaffected by these changes in magnetic field components.
Then in Figs. 14 and 15, we explore the distinct effects of fluid direction angles (χ ) and observed inclination angles (θ ) on Q − U loop diagrams for a pure equatorial magnetic field (B r = 0.87, B φ = 0.5).As χ shifts from −90 • to −180 • , the loops undergo a counterclockwise rotation of approximately 90 • with a slight decrease in radius.Conversely, as θ rises from 20 • to 87 • , the outer loops expand significantly into the first quadrant, while the inner loops remain largely unchanged.

Conclusions
This paper examines the polarization images of a synchrotronemitting fluid ring around a static hairy black hole, analyzing the influence of several parameters.Polarization intensity and properties like EVPA are significantly affected by the magnetic field surrounding the emitting ring, the fluid velocity, and the observer's inclination angle.We find that the hairy parameter Q influences the size of polarization images, primarily reflecting the intensity and direction of polarization for photons emitted from various locations on the ring.Furthermore, polarized intensity increases as Q decreases.The effects of χ (fluid direction angle) and β (velocity) are only apparent when β is nonzero.
With a purely vertical magnetic field, χ significantly impacts the direction of polarization vectors in the images.As the inclination angle (θ ) increases, the images flatten, and polarized intensity fluctuates strongly.E V P A curves are also substantially affected by θ .
Conversely, with a purely equatorial magnetic field, the polarization vector directions are more sensitive to the ratio of magnetic field components (B r and B φ ) than to χ .The influence of θ is similar to the vertical field case, but a minimum polarized intensity emerges around φ = π at larger θ values.E V P A curves exhibit similar changes as before but with greater smoothness.
Finally, we examine the influence of the hairy parameter Q and other parameters on Q − U loops.In each case, two loops, whose radii increase with Q, enclose the coordinate origin.Variations in the equatorial magnetic field components induce a counterclockwise rotation of the loops.As the  observer's inclination angle increases, the outer loops expand significantly in different directions depending on whether the magnetic field surrounding the emitting ring is equatorial or vertical.
Specifically, with a purely vertical magnetic field, the radius of the inner loops increases as the fluid velocity angle (χ ) transitions from −90 • to −180 • .Conversely, for the same χ variation but with a purely equatorial magnetic field, the loops undergo a counterclockwise rotation.
While the effects of magnetic field parameters, fluid velocity, and observer inclination angle on polarization images are evidently more complex than those of the hairy parameter Q, studying these images for a synchrotron-emitting fluid ring around a static hairy black hole remains valuable for researching Horndeski theory and testing the validity of GR.
, 6, 7, 8, 9 and 10 illustrate polarization images of equatorial emitting rings for a static hairy black

Fig. 4
Fig.4 Polarization patterns corresponding to cases with the pure vertical magnetic field B z = 1.In the pictures in the left column, the directions of the ticks indicate the orientation of the E-vector of each photon emitted from the different locations around the ring as viewed in the sky.The length of the ticks is the value of the polarized intensity

Fig. 5 Fig. 6
Fig.5 Effects of the hairy parameter Q on the polarized vector in the static hairy black hole spacetime for the different value of fluid velocity β in the cases of the pure vertical magnetic field B z = 1.In these cases, we set R = 6, M = 1 and the observer inclination angle θ = 20 •

3. 2
Effects of hairy parameter on the Q − U loop diagram Now let us turn to present our results of the effects of hairy parameter on the Q −U loop diagram in the static hairy black hole.In Figs.11, 12, 13, 14 and 15, we present the effects of the hairy parameter Q on Stokes Q − U loop diagrams in static hairy black hole spacetime.The loops of linearly varying Stokes parameters Q and U illustrate the continuous variability of the polarization of emitting photons around the black hole.Generally, each diagram contains two loops enclosing the origin, and the radius of these loops increases with the value of the hairy parameter Q.

Fig. 7 3 (Fig. 8 3 Fig. 9
Fig.7 Effects of the hairy parameter Q on the polarized vector in the static hairy black hole spacetime for the different observer inclination angles θ in the cases of the pure equatorial magnetic field.The top row

Fig. 10 Fig. 11
Fig.10 Effects of hairy parameter Q on the polarized vector in the static hairy black hole spacetime for the different observer inclination angles θ.In these cases, we set R = 6, M = 1, the fluid direction angle χ = −90 • , the value of fluid velocity β = 0.3, B r = 0.87, B φ = 0.5 and B z = 0

Fig. 12 1 Fig. 13 Fig. 14 Fig. 15
Fig. 12 The Q − U diagram for different observed inclination angles θ with pure vertical magnetic field in static hairy black hole spacetime.Here we set R = 6, M = 1,β = 0.3, χ = −90 • , B r = 0, B φ = 0 and B z = 1 Funding This work is supported by the National Key Research and Development Program of China under Grant No.2020Y FC2201503, the National Natural Science Foundation of China under Grants No.12275238 and No. 11675143, the Zhejiang Provincial Natural Science Foundation of China under Grants No.LR21A050001 and No. LY20A050002, and the Fundamental Research Funds for the Provincial Universities of Zhejiang in China under Grant No. RF-A2019015.