Shadows and observational appearance of a new family of Ayón-Beato-García black holes

Ping-Hui Mou ,Yun-Xian Chen ,Ke-Jian He and Guo-Ping Li,?

1 School of Physics and Astronomy,China West Normal University,Nanchong 637000,China

2 Department of Physics and Chongqing Key Laboratory for Strongly Coupled Physics,Chongqing University,Chongqing 401331,China

Abstract In this paper,we study the shadows and observational appearance of the Ayón-Beato-García(ABG) black hole,which is surrounded by a thin disk accretion.For a four-dimensional ABG black hole,as the charge q increases,the event horizon r+,radius rp and impact parameter bp of the photon sphere decrease,while the maximum value of effective potential Veff increases.However,as the term γ associated with nonlinear electrodynamics increases,the related physical quantities r+,rp and bp increase,but the maximum value of Veff decreases.When the light ray reaches the vicinity of the ABG black hole,one can observe the trajectories of the light rays appearing as three types,i.e.,direct emission,lens ring and photon ring.Furthermore,the shadows and the observational appearance of the ABG black hole are studied.The results show that the charge q and the term γ have a great influence on the shape and intensity of the black hole shadow.When the relevant state parameters change or the emission model changes,it can be argued that it is the direct emission that plays a decisive role in the total observed intensity of the ABG black hole,while the contribution of the lens ring to the total observed intensity is small and the photon ring makes almost no contribution.

Keywords: black hole shadow,thin disk accretion,rings

1.Introduction

As one of the predictions of general relativity,black holes exist in our universe as extremely dense objects.Moreover,with the improvement of observation technology,people have been able to observe this mysterious object in the Universe.In 2015,gravitational waves from the merger of two black holes were monitored by the Laser Interferometer Gravitational Wave Observatory (LIGO) [1].In 2019,the Event Horizon Telescope (EHT) collaboration released an image of a supermassive black hole located at the center of galaxy M87[2–7].And recently,the EHT Collaboration released the first image of a supermassive black hole at the center of the Milky Way[8].These works strongly proved the existence of black holes and successfully confirmed the prediction of general relativity.In the two images released by EHT,similar bright rings and dark areas appeared.This bright ring is called a photon ring,and the dark central region is the shadow of black holes.Due to the gravitational lens effect,the shadows of black holes are formed,and some of the light rays near the black hole deflect and rotate around the black hole,which leads to the formation of a photon ring [9–12].The shadows of black holes are widely studied,which will further reveal the characteristics of black holes [13–41].

In the process of observing black holes,the accretion matter around black holes plays an indispensable role.In the early 1970s,it was thought that the outward transfer of angular momentum of the accretion matter would cause an accretion disk to form around the black hole [42–44].This type of accretion disk mentioned in the [42–44] is geometrically thin and is called the thin disk.The gas of the thin disk is optically thick and has a very low temperature compared to the Verri temperature [45].In 1976,a geometrically thin and optically thin accretion model was proposed by Shapiro[46].Composed of high-temperature gas in the inner region,this model is able to explain the x-ray emission seen in some black hole sources,but it is unstable [47].Immediately after,the thick disk was also studied[48–51].In 1988,the slim disk was proposed considering the case where the accretion gas is optically too thick so that the thin disk is no longer applicable[45,52].The Lorentz violation parameterlwas shown to have a large effect on the energy flux,temperature distribution and emission spectrum of the thin disk [53].Farajiet alin their recent paper confirmed that the quadrupole parameter plays a non-negligible role in the physical properties of the thin disk solution around twisted Schwarzschild spacetime[54].Recently,efforts have been devoted to studying the shadows and the observational appearance of black holes which are surrounded by spherically symmetric accretion and thin disk accretion [21–36].Considering both the static and infalling spherically symmetric accretion models,the inner region of the infalling one is darker than that of the static one due to the Doppler effect [23,24].Also,under the background that black holes are surrounded by the thin disk accretion,the light rays around black holes can be divided into three types,i.e.,direct emission,lens ring and photon ring [23,24,27,30,32–36].Moreover,there are differences in the contributions of direct emission,lens rings and photon rings to the observed intensity of black holes.It can be seen that studying the observational appearance of a black hole surrounded by accretion can reflect some characteristics of the black hole,and these studies can provide theoretical support for further observations of black holes in the Universe.

It is well known that the singularity problem is an important issue in general relativity.Among them,Ayón Beato and García coupled nonlinear electrodynamics to the Einstein field equation to obtain some regular black hole solutions which have no singularities and their gravitational field is regularized[55–58].When the chargeq=0,this class of black hole solutions transfers to singular Schwarzschild solutions.When the chargeq≠0,the chargeqappears as a nonlinear electromagnetic fieldFμν,while this class of black holes is obtained without singularities,and the associated gravitational field is regularized.In [59],Caiet alstudied a class of ABG-related black hole solutions containing massm,chargeq,and three terms α,β,and γ associated with nonlinear electrodynamics.The chargeqand the term γ are shown to have a strong influence on the Hawking temperature and the radius of an event horizon.Furthermore,Caiet alobtained an upper limit on the chargeqbased on the analysis of data from the shadow of M87*[59].That is,in the 1 σ confidence region,the upper limit ofqis 0.7.While in the 2 σ confidence region,the upper limit ofqis always below 1.Recently,the thermodynamic phase transition of this generalized ABG black hole was studied by Ghasemiet al.The results suggest that the thermodynamic system of this ABG black hole is involved in the Hawking-Page phase transition,in which the ABG black hole of unbalanced evaporation eventually reaches the AdS vacuum space [60].Obviously,this ABG black hole has many properties that deserve to be studied.

Based on the previous work,it is an interesting practice to explore the shadow and observational appearance of the ABG black hole surrounded by the thin disk accretion.In this paper,when a thin disk exists around the ABG black hole,we have studied its shadow by using a ray-tracing method.First,by analyzing the behavior of photons around the ABG black hole,we obtained the trajectories of light rays in the vicinity of the ABG black hole surrounded by the thin disk accretion.Then,the trajectories of light rays emitted from the north pole direction of the ABG black hole are analyzed.The results show that the light rays are distinguished into three types,i.e.,direct emission,lens rings and photon rings.Further,the characteristics of the three rings and differences in their contribution to the observed intensity are studied.Especially,with the emission function at different positions,we also study whether changes in the chargeqand the term γ have an impact on the shadow and observational appearance of the ABG black hole.This could help us to understand the importance ofqand γ for the ABG black hole.

The paper is structured as follows.In section 2,the effective potential and photon orbits of the ABG black hole are discussed.In section 3,when the ABG black hole is surrounded by a thin disk,we studied its shadows and rings.Section 4 concerns summary and discussion.

2.The effective potential and photon orbits of the ABG black hole

In this section,we will discuss the effective potential and photon orbits of the ABG black hole.The premise of studying the deflection of a light beam near the ABG black hole is to understand the motion of photons around the ABG black hole.The nonlinear Einstein Maxwell action function is obtained by coupling the nonlinear electrodynamics to the Einstein field equations by Ayón Beato and García [55],which is

Table 1.The event horizon r+,the radius rp and the impact parameter bp of the photon sphere when q takes different values,where m=1 and γ=1.

Table 2.The event horizon r+,the radius rp and the impact parameter bp of the photon sphere when γ takes different values,where m=1 and q=0.04.

It is a class of Ayón-Beato-García (ABG) related black hole solutions with massm,chargeqand three terms α,β and γ associated with nonlinear electrodynamics[59].This class of black hole solutions is regular under the condition,that is,αγ ≥6,βγ ≥8 and γ>0.According to the saturation casesthe metric function appears in the following form

With the help of the Euler–Lagrange equation,we can study the null geodesic of the ABG black hole,which is

where λ is the affine parameter and ˙xμis the four-dimensional velocity of the photon.In this spacetime,the Lagrangian quantity of the particle is

Under the condition of settingthe motion of the photon is fixed on the equatorial plane.In addition,the metric coefficients in equation (2) do not depend on timetand azimuth φ.Therefore,there are two conserved quantities,energyEand angular momentumL,which are denoted as

As can be seen from table 1,the event horizonr+,the radiusrpand the impact parameterbpof the photon sphere decrease as the chargeqincreases.In table 2,the event horizonr+,the radiusrpand the impact parameterbpof the photon sphere increase as the term γ associated with nonlinear electrodynamics increases.However,after γ increases to 4,the values ofr+,rpandbpno longer increase.In addition,the effective potentialVeffis plotted whenqand γ take different values,as shown in figure 1.

The relationship between effective potentialVeffandris shown in figure 1.It should be noted thatVeff=0 corresponds to the position of the event horizon.Asrincreases,the value ofVeffbegins to increase,and reaches the maximum at the photon sphere wherer～rp.In subfigure(a),when the term γ associated with nonlinear electrodynamics is fixed,the maximum value ofVeffincreases with the increase ofq,while the corresponding event horizon decreases with the increase ofq.In subfigure (b),whenqis fixed,the maximum value ofVeffdecreases with the increase of γ.Here,an interesting phenomenon emerges,when γ=2,γ=3,γ=8 and γ=10,the lines ofVeffhave almost the same trend and are difficult to distinguish.In other words,when γ increases to a certain value,the change ofVeffis very small,and the corresponding event horizon almost does not change.Comparing the two subfigures in figure 1,it can be found that the chargeqhas a greater effect onVeff,while the term γ associated with nonlinear electrodynamics has a smaller effect onVeff.

Figure 1.The profiles of the effective potential Veff and r,in which m=1.The left figure corresponds to the case where γ is fixed and q varies,and the right figure corresponds to the case where q is fixed and γ varies.

Figure 2.The trajectories of light rays in the polar coordinates(r,φ).Here,the black disk represents the ABG black hole and the gray dotted circle represents the photon sphere.

Now,let us focus on the trajectory of the light ray.Based on equation (8),the resulting equation is

By redefining a parameteru=1/rin equation (10),it can be rewritten as

With the help of the ray-tracing method,the trajectories of light rays are obtained for different values of the chargeqand the term γ,as shown in figure 2.

In figure 2,these different colored lines have different physical meanings.It is true that whenbc=bp,the light ray will always surround the black hole infinite times,and neither be deflected nor fall into the black hole.Due to the numerical accuracy in the mathematical program,it is difficult for us to accurately show the light rays for the casebc=bp.So,we selected the case thatbcis very close to but smaller thanbpto show the light ray that is very close to the photon sphere,which corresponds to the red line in figure 2.Whenbc>bp,the light rays correspond to the green line,and the light rays are deflected to form the shadow of the ABG black hole.Whenbc

Table 3.The regions of direct emission,lens rings and photon rings relative to the impact parameter bc.

3.Shadows and rings of the ABG black hole

3.1.Direct emission,lens ring and photon ring

In this section,we will study the observational appearance of the ABG black hole surrounded by the thin disk accretion.In this model,we consider the thin disks are horizontally placed on the equatorial plane of the ABG black hole3Obviously,it is also very interesting to discuss a tipped disk,which we leave for future work..The trajectory of the light ray near a black hole is an important basis for studying the shadow and rings of a black hole.Grallaet alsuggest that light rays will have different types when light rays reach the vicinity of a black hole [32].Therefore,following the method in [32],we use the total number of orbitsn=to distinguish the trajectories of the light rays as direct emission,lens ring and photon ring.The total number of orbitscorresponds to the direct emission,where the trajectories of the light rays will intersect the equatorial plane once.When the total number of orbitscorresponds to the lens ring,the trajectories of the light rays will intersect the equatorial plane at least twice.Whencorresponds to the photon ring,the trajectories of the light rays will intersect the equatorial plane at least three times.For the ABG black hole,the regions of direct emission,lens ring and photon ring with respect to the impact parameterbc,are given in table 3 and figure 3 whenqand γ take different values.

As can be seen from table 3,the regions of direct emission,lens ring and photon ring change as the chargeqand the term γ associated with nonlinear electrodynamics increase.In figure 3,the relationship between the total number of orbitsnand the impact parameterbcis shown in the coordinate system and the regions of direct emission,lens ring and photon ring are also shown.In figure 3,the value ofbcdecreases when γ is fixed andqincreases,and increases whenqis fixed and γ increases.Thus,changes inqand γ have opposite effects on the position of these three regions.To visualize the effect of a change inqand γ on these three regions,we give the trajectories of photons near the ABG black hole in figure 4.

Figure 3.The relationship between the total number of orbits n and the impact parameter bc.

In figure 4,the trajectories of photons near the ABG black hole are shown in the polar coordinate system (bc,φ).From figure 4,we find that γ is fixed whileqincreases and the area of the lens ring and photon ring becomes wider.The opposite is the case,whereqis fixed and γ increases,and the area of the lens ring and photon ring becomes narrower.Obviously,the chargeqand the term γ associated with nonlinear electrodynamics have a great influence on the trajectories of light rays.

3.2.Observed specific intensity and transfer function

We can study the observed specific intensity under the model of a thin disk surrounding the ABG black hole.It is assumed that the thin disk accretion locates at the stationary frame of the static world line and that the photons emitted from the disk are isotropic.When the static observer locates at the north pole of the ABG black hole,the observed specific intensity can be expressed as [23]

whereIobs(r)is the observed specific intensity,and frequency ν andIemi(r) is the emitted specific intensity with frequency νe.The total specific intensityI(r) is obtained by integrating over all frequencies ofIobs(r),which is denoted as

Figure 4.Trajectories of photons near the ABG black hole.Here,the gray dashed line represents the photon sphere,the black disk represents the ABG black hole,the red line represents the direct emission,the blue line represents the lens ring and the green line represents the photon ring.

Figure 5.Relationship between the transfer function rn(bc) and the impact parameter bc.Here,the red,blue and green lines represent the radial coordinates of the first,second and third transfer functions intersecting the thin disk.

As the light ray reaches the vicinity of the ABG black hole,it will intersect with the thin disk located in the equatorial plane of the ABG black hole.The light ray intersects the thin disk once,which corresponds to the case of direct emission.The light ray intersects the thin disk twice,which forms the lens ring.The light ray intersects the thin disk three times,which forms the photon ring.The additional luminosity gained by the photon ring increases as the number of times the light ray intersects the thin disk increases.Therefore,the observed intensity is the sum of the intensities at these intersections,which is

Here,rn(bc) is defined as the transfer function,which is used to represent the radial position of the thin disk plane with thenth intersection point.As discussed above,different light rays will intersect with the thin disk different times.Each intersection will make the light rays obtain the additional luminosity and this luminosity is dependent on the radial positions,which will give rise to the different appearances of ABG black holes.Therefore,it is only when the transfer function is considered that the lens ring and the photon ring can be formed,which will be shown in a later section.It is worth noting that we do not consider the absorption of a light ray by the thin disk,as this would lead to a reduction in the observed intensity.In addition,the slope of the transfer function determines the amplification ratio.For differentqand γ,the relationship between the transfer functionrn(bc) and the impact parameterbcis plotted in figure 5.

As can be seen from figure 5,the slope of the red line is small,the slope of the blue line is large and the slope of the green line tends to infinity.Here,the red line represents the radial coordinate of the first transfer functionr1(bc) intersecting the thin disk,which corresponds to the direct emission and is related to the redshift source.The blue line represents the radial coordinates of the second transfer functionr2(bc)intersecting the thin disk,which represents the lens ring.The green line represents the radial coordinates of the third transfer functionr3(bc) intersecting the thin disk,which represents the photon ring.

3.3.Observational characteristics of direct emission and ring

Once the form and position of a typical emission function have been determined,we can study the observed specific intensity.Considering the fact that universality of exponential decay,we assume that the shadow luminosity intensity decreases exponentially,which is

Figure 6.The observed specific intensity and its optical appearance,γ=1 and q=0.04.The top row is the relation about I(r)versus bc.The bottom row is the two-dimensional of the observed specific intensity.

Figure 7.The observed specific intensity and its optical appearance,γ=1 and q=0.08.The top row is the relation regarding I(r)versus bc.The bottom row is the two-dimensional of the observed specific intensity.

Here,rinis the innermost position of the thin disk accretion,which is assumed to have three cases.For the first case,the thin disk accretion is located at the innermost stable circular orbit withrin=risco,whereriscois the innermost stable circular orbit of the ABG black hole.For the second case,the thin disk accretion is located at the photon sphere withrin=rp.For the third case,the thin disk accretion is located at the event horizon withrin=r+.Whenqand γ take different values,the observed specific intensity and its optical appearance are plotted in figures 6,7 and 8 with the help of equations (14) and (15).

Figure 8.The observed specific intensity and its optical appearance,γ=3 and q=0.04.The top row is the relation regarding I(r)versus bc.The bottom row is the two-dimensional of the observed specific intensity.

The first,second and third columns correspond to the observed intensity and the optical appearance of the observed intensity when the thin disk accretion located atrisco,rpandr+,respectively.In the first column of figures 6,7 and 8,the thin disk accretion is located at the innermost stable circular orbit.The top row shows that for γ=1 andq=0.04,the first peak of the observed intensity is 0.38 atbc?5m(0.1 atbc?4.8mfor γ=1 andq=0.08) (0.41 atbc?5.2mfor γ=3 andq=0.04),which corresponds to the photon ring.Then,the second peak of the observed intensity is 0.46 atbc?5.4m(0.44 atbc?5.2mfor γ=1 andq=0.08)(0.47 atbc?5.6mfor γ=3 andq=0.04),which corresponds to the lens ring.Finally,the third peak is 0.45 atbc?7.3m(0.44 atbc?7mfor γ=1 andq=0.08) (0.46 atbc?7.5mfor γ=3 andq=0.04),which corresponds to direct emission.It can be seen from the bottom row of the first column of figures 6,7 and 8,that the lens ring is a thin ring and the photon ring is a very narrow ring.In the second columns of figures 6,7 and 8,the thin disk accretion is located at the photon sphere.For the cases of γ=1 andq=0.04 and γ=3 andq=0.04,the observed intensity has two peaks.As can be seen from the bottom rows of figures 6 and 8,the lens ring and the photon ring can be distinguished.However,for the case of γ=1 andq=0.08,there is only one peak of the observed intensity.The lens ring and photon ring are superimposed in the bottom row of figure 7.In the third columns of figures 6,7 and 8,the thin disk accretion is located at the event horizon.For the cases of γ=1 andq=0.04 and γ=3 andq=0.04,the lens ring and the photon ring can be distinguished.However,for the case of γ=1 andq=0.08,the lens ring and photon ring are superimposed.In summary,the direct emission determines the total observed intensity,the lens ring makes a small contribution to the observation,and the photon ring makes almost no contribution.

Comparing these three sets of figures,some meaningful conclusions can be drawn.Firstly,by comparing figure 6 and figure 7,it can be seen that all the peaks and positions of the peaks of the observed intensity decrease as the chargeqincreases.In particular,for the case ofq=0.08,the corresponding lens ring and photon ring are overlapped when the thin disk accretion is located atrpandr+,which is a very different result fromq=0.04.This suggests that the chargeqhas a significant effect on the observed appearance of the ABG black hole.Secondly,all the peaks of the observed intensity and the positions of the peaks increase with the increase of the term γ associated with nonlinear electrodynamics.Meanwhile,the brightness of the ring with γ=3 is higher than that of γ=1.This indicates that the brightness of the ring is closely related to γ.Thirdly,the observed appearances of the ABG black hole are different when the thin disk accretion is located at different positions.In particular,shadows and photon rings are more easily distinguished when the thin disk accretion is located atrisco.Obviously,the chargeqand the term γ associated with nonlinear electrodynamics have a great influence on the observation of the ABG black hole,which may help us to distinguish the ABG black hole from other black holes.

4.Conclusion and discussion

In this paper,we have studied the shadows and observational appearance of the ABG black hole surrounded by the thin disk accretion using a ray-tracing method.For the ABG black hole,the effective potential and the motion of photons in the vicinity of the ABG black hole are discussed.When the ABG black hole is surrounded by the thin disk accretion,we find that the trajectories of light rays are redefined as direct emission,lens ring and photon ring.After introducing the transfer and emission functions,we further study the observational appearance of the ABG black hole surrounded by the thin disk accretion.

After completing the above works,we have obtained some interesting conclusions.Firstly,the event horizonr+,the radiusrpand the impact parameterbpof the photon sphere decrease as the chargeqincreases,while the maximum value of the effective potentialVeffincreases.However,the event horizonr+,the radiusrp,and the impact parameterbpof the photon sphere increase with the increases of the term γ associated with nonlinear electrodynamics,and the maximum value of the effective potentialVeffdecreases.Secondly,the effects of the chargeqand the term γ associated with nonlinear electrodynamics on the trajectories of light rays are significant.The regions of direct emission,lens ring and photon ring all change withqand γ.This means thatqand γ also have an effect on the observational appearance.In particular,as the chargeqincreases,the peak and the position of the peak decreases for all observed intensities.Also,we note that the lens ring and the photon ring for γ=1 andq=0.08 are overlapped when the thin disk accretion located at the photon sphere and the event horizon,while the lens ring and the photon ring for γ=1 andq=0.04 are not overlapped.When the chargeqis fixed,the peak and the position of the peak of all observed intensities corresponding to γ=3 are always greater than that of γ=1.In particular,the brightness of the photon ring corresponding to γ=3 andq=0.04 is higher than that of γ=1 andq=0.04.Finally,the position of the thin disk accretion also has a great influence on the observation of the ABG black hole.The size of the shadows and the observed appearance of the ABG black hole are different when the thin disk accretion is located at different positions.

In addition,the direct emission plays a decisive role in the total intensity of the observation,while the lens ring makes a small contribution to the observation and the photon ring makes almost no contribution.It is worth noting that the chargeqand the term γ associated with nonlinear electrodynamics have a strong influence on the shadows of the ABG black hole,which will probably help us to distinguish the ABG black hole from other black holes in different gravitational contexts.As a brief outlook,it will be interesting to study the shadow of the ABG black hole when it is surrounded by the spherically symmetric accretion,which may further reveal more characteristics of the ABG black hole.

Acknowledgments

The authors would like to thank the anonymous reviewers for their helpful comments and suggestions,which helped to improve the quality of this paper.This work is supported by the National Natural Science Foundation of China (Grant Nos.11 875 095 and 11 903 025),and by the starting fund of China West Normal University (Grant No.18Q062),and by the Sichuan Youth Science and Technology Innovation Research Team (21CXTD0038),and by the Chongqing Science and Technology Bureau (cstc2022ycjh-bgzxm0161),and by the Natural Science Foundation of SiChuan Province(2022NSFSC1833).