Images of hairy Reissner-Nordström black hole illuminated by static accretions

Recently, the Event Horizon Telescope (EHT) has made significant achievements, providing us with a new window to test the essence of gravity in the strong field regime. One of the most impressive achievement among them is the release of images of the supermassive black hole M87* Akiyama et al. (2019a, b, c, d, e, f) and the central black hole of our Milky Way Akiyama et al. (2022a, b, c, d, e, f). The dark regions in the center of the images are the shadow of black hole, which are related to the strong gravitational lensing of photons around the black hole Virbhadra and Ellis (2000, 2002); Virbhadra and Keeton (2008); Synge (1966); Bardeen et al. (1972); Bozza (2010). The unstable photon orbit (photon sphere) in the innermost layer around the black hole define the radius of shadow, and the bright rings in the images are related to the outer photon orbits. Moreover, the bright ring-shaped construction is the product of various light rays as the black hole is always illuminated by complex accretion flows.

In fact, early on, the black hole shadow was known as the escaping cone of photon Synge (1966), which is defined as the impact parameter corresponding to the critical curve. Furthermore, the expression of the angular radius of the photon capture region for the Schwarzschild black hole was proposed Luminet (1979), and later Bardeen investigated the D-type shadow of the Kerr black hole Bardeen et al. (1972), which is caused by the dragging effect of the rotating black hole on the light rays. This has inspired a wave of research on the shadow of black hole, and numerous simulations of various black hole shadows have been widely discussed, see for examples Shen et al. (2005); Yumoto et al. (2012); Atamurotov et al. (2013); Papnoi et al. (2014); Abdujabbarov et al. (2015); Kumar and Ghosh (2020) and references therein. Those studies indicate that the properties of black hole shadows closely depend on the spacetime background. The shadows of black holes in different spacetime dimensions and modified theories of gravity have also been attracted extensive discussion, such as conformal gravity Meng et al. (2022), Gauss-Bonnet theory Ma and Lu (2020); Guo and Li (2020), the Chern-Simons type theory Meng et al. (2023a); Ayzenberg and Yunes (2018); Amarilla et al. (2010), f(R) gravity Addazi et al. (2021); Dastan et al. (2022), and so on Amarilla and Eiroa (2012, 2013); Amir et al. (2018); Mizuno et al. (2018); Eiroa and Sendra (2018); Vagnozzi and Visinelli (2019); Banerjee et al. (2020); Chowdhuri and Bhattacharyya (2021); Konoplya and Zhidenko (2019); Younsi et al. (2016); Olmo et al. (2023). In addition, the shadows of naked singularities have been studied in Shaikh et al. (2019a); Joshi et al. (2020); Dey et al. (2021), and the wormholes shadows were also explored in Rahaman et al. (2021); Kasuya and Kobayashi (2021); Shaikh et al. (2019b); Wielgus et al. (2020); Peng et al. (2021a); Neto et al. (2023); Tsukamoto et al. (2012) and references therein. On the other hand, images from EHT collaboration are simulated via Kerr black holes in general relativity (GR). However, due to the finite resolution of EHT, this also provides some space for alternative theories beyond GR. Therefore, plenty of works have been carried out to use the shadow observations of the EHT to test or constrain modified theories of gravity Vagnozzi et al. (2023); Afrin et al. (2021); Kuang et al. (2022); Tang et al. (2022, 2023); Kuang and Övgün (2022); Kumar et al. (2019); Shaikh et al. (2021); Wu et al. (2023a); Capozziello et al. (2023); Sui et al. (2023); Pantig et al. (2023); Ghosh et al. (2023); Tsukamoto et al. (2014).

The bright ring-shaped constructions in the EHT images are closely related to the accretion flow radiation around the black hole, and the geometric features of the accretion flow determine the optical appearance of the black hole. However, in the realistic astrophysical environments, the accretion flows around black holes are always very complex such that one need numerical simulation using general relativistic magnetohydrodynamic (GRMHD) to extract the black hole images Porth et al. (2019). It is worth noting that usually in order to investigate the main characteristics of black hole images and the gravity in strong field regime, giving some toy models of accretion structures is enough to put forward. Thus, Wald et al. first proposed an optically and geometrically thin toy accretion disk around Schwarzschild black holes, and defined direct $(n=1)$, lensed ring $(n=2)$ and photon ring $(n=3)$ emissions by the number of times ($n$) light intersecting with the accretion disk Gralla et al. (2019). They found that the total observed brightness of black hole images mainly comes from the direct emissions, while the contribution of lensed ring emissions is small and photon ring emissions can even be ignored. Subsequently, toy accretion disk models around various black holes have been extensively studied Dokuchaev and Nazarova (2019); Peng et al. (2021b); He et al. (2022); Eichhorn and Held (2021); Li and He (2021); Wang et al. (2023). The spherical accretions, as another toy accretion model, have also been attracted great interest Zeng et al. (2020); Saurabh and Jusufi (2021); Zeng and Zhang (2020); Qin et al. (2021); Narayan et al. (2019), in which the size and shape of shadows are mainly determined by the spacetime geometry of the black hole, rather than the details of the accretions. Moreover, the accretion constructions around black holes with two unstable photon spheres were also analyzed in Gan et al. (2021a, b); Meng et al. (2023b). Compared to black hole with single photon sphere, the black hole with double photon spheres could have additional photon rings in the image, and the photon ring and lensed ring emissions will contribute significantly to the total observed brightness. In general, the optical appearance is a potential probe to distinguish GR black holes from other alternative compact objects Boshkayev et al. (2022); Xavier et al. (2023); Sakai et al. (2014); Bacchini et al. (2021); Destounis et al. (2023) or black holes in alternative theories beyond GR Guo et al. (2023a, b); Archer-Smith and Zhang (2021); Okyay and Övgün (2022); Uniyal et al. (2023a); Hou et al. (2022); Uniyal et al. (2023b); Akbarieh et al. (2023); Gao et al. (2023); Theodosopoulos et al. (2023); Zeng et al. (2023); Meng et al. (2023c), but this does not always work, as even in GR, the images of black holes may exhibit degeneracies. For example, sufficiently dense boson stars Liebling and Palenzuela (2023); Rosa et al. (2023) may have shadows similar to those of classical black holes Rosa and Rubiera-Garcia (2022) and images of very relativistic rotating boson stars may resemble Kerr black holes Vincent et al. (2016).

On the other hand, when there are other additional sources around the black hole, the interaction between spacetime and additional sources can cause the black hole to carry global charge. This charge is also known as the hair of black hole, and in this case the spacetime usually will deviate from that in GR. In this scenario, Ovalle and Casadio first proposed the gravitational decoupling (GD) approach Ovalle et al. (2021); Contreras et al. (2021) to introduce additional sources, and obtain hairy black holes which describes deformations of the black holes in GR. Due to the fact that in the GD approach, the additional sources can be dark matter, scalar fields, tensor fields etc., so the GD approach can be used to study the general existence problem of solutions for hairy black holes. In particular, they take the Schwarzschild black hole as the seed black hole, and use the GD approach to construct the hairy Schwarzschild black hole which obtain extensive attention, such as the thermodynamics of rotating hairy black holes Mahapatra and Banerjee (2023), scalar perturbations and quasinormal modes of hairy black holes Cavalcanti et al. (2022); Yang et al. (2023a); Li (2023), strong gravitational lensing, black hole shadow and image Afrin et al. (2021); Islam and Ghosh (2021); Meng et al. (2023b), precession and Lense-Thirring effect Wu et al. (2023b) and gravitational waves from extreme mass ratio inspirals Zi and Li (2023).

In this work, we will focus on the optical features of the hairy RN black hole, which was also constructed via the GD approach by taking the Schwarzschild black hole in GR as the seed black hole. Since the hairy RN black hole includes electric charge and additional arbitrary hairy charge, thus, the aim of this work to investigate if the black hole shadow and image can probe the different charges in this scenario. In details, we will firstly explore the effects of characterized parameters of the hairy RN black hole on the light rays, and on the black hole rings and images when it is surrounded by the optically and geometrically thin accretion disks and spherical accretion. Then we shall discuss the possible distinguishable effects of the charges on the hairy RN black hole shadows and images, and potentially differentiate the hairy RN black hole from RN black hole.

This paper is organized as follows. In Sec. II, we will study the radius of photon sphere and impact parameters by analyzing the effective potential of photons around the hairy RN black hole. In Sec. III, we use the ray-tracing method to discuss the distribution of the light rays around the hairy RN black hole, and then investigate the optical appearance of hairy RN black hole illuminated by the optically and geometrically thin accretion disk. In Sec. IV, we present the black hole shadows and images illuminated by static spherical accretions. Sec. V gives our conclusions.

Black holes in classical GR are only described by mass, electric charge and spin because of the no-hair theorem Ruffini and Wheeler (1971). But the interaction between black hole spacetime and matters may introduce other charges, so the black hole could carry additional hairy charges and the physical effects of these hairy charges can modify the original background spacetime, namely, hairy black holes may form. Recently, Ovalle et.al used the GD approach to obtain a spherically symmetric black hole with additional hairy charges Ovalle et al. (2021); Contreras et al. (2021), which has the metric

with

This metric describes the deformation of the Schwarzschild black hole solution due to the introduction of additional sources, which can be dark matter, scalar fields, tensor fields, etc. Here $Q$ and $l_{0}$ represent potential charges, and both of them are proportional to $\alpha$, such that the above solution reduces to Schwarzschild case as $\alpha\to 0$. In appendix A, we review how the hairy RN black hole (1) is solved out from the seed metric via the GD approach. The technics of this approach to obtain the deformed solutions sound good, but the specific background theory, which can give the additional required tensor field in the approach, remains open. The authors of Ovalle et al. (2021) briefly discussed the potential background theories which still call for further investigations. Nevertheless, the uncertainty of the background theories did not hinder physicists’ attention on this type of deformed black hole, and lots of theoretical and observable properties of the deformed black hole have been investigated Mahapatra and Banerjee (2023); Cavalcanti et al. (2022); Yang et al. (2023a); Li (2023); Afrin et al. (2021); Islam and Ghosh (2021); Meng et al. (2023b); Wu et al. (2023b); Zi and Li (2023). The reasons may stem from two aspects: (i) Due to the uncertainty of the ingredients in our Universe and the dispute on no-hair theorem of black hole, people can temporarily suspend the unclear background theory and go ahead to study its possible outputs. (ii) More complete understanding of the hairy black hole could give more information to build its background theories, at least from bottom-up perspective. In particular, as addressed in Ovalle et al. (2021), $Q$ can be an electric charge, or a tidal charge of extradimensional origin or any other source depending on the specific background theory. When $Q$ represents an electric charge, the electrovacuum of the RN geometry also contains a tensor-vacuum whose components are those explicitly proportional to $\alpha$. In this paper, taking our motivations mentioned in the Introduction into account, we will treat $Q$ as the electric charge differentiating from the additional hairy charge $l_{o}$, and instead of clarifying its corresponding background theory, we shall discuss their possible probe from black hole shadow and images. So we denote the above geometry (1) as the hairy RN spacetime. Thus, in the following studies, $M$, $Q$, and $l_{o}$ are the parameters related with the mass, electric charge and hairy charge of the black hole, respectively. In addition, $l_{o}$ should satisfy $l_{o}\leq 2M$ to ensure the asymptotic flatness of spacetime.

Here we focus on the properties of motions of photon in the vicinity of the hairy RN black hole. Before proceeding, we briefly discuss the event horizon of hairy RN black hole, which can be solved out from

After numerical calculation, we show how the event horizon $r_{h}$ is affected by the parameters $\alpha$, $Q$, and $l_{o}$ with the others fixed in FIG.1. Obviously, with fixed $\alpha$, both the charge parameters $l_{o}$ and $Q$ will suppress the event horizon $r_{h}$, but as the deformation parameter $\alpha$ increases, the event horizon of the hairy RN black hole becomes larger.

Next, we will investigate the null geodesic motion in the hairy RN black holes. The motions of photons are described by the Euler-Lagrange equation,

where $\dot{x}^{\mu}=\frac{dx^{\mu}}{d\tau}$ represents the four-velocity of the photon and $\tau$ is the affine parameter. In the static spherically symmetric hairy RN black hole, the Lagrangian of the photon takes the form

Since $\partial_{t}$ and $\partial_{\phi}$ are Killing vector fields of the hairy RN spacetime, we can define the conserved energy, $E$, and the $z$ component of the angular momentum, $L_{z}$, of the photon as

Furthermore, due to the spherical symmetry of the spacetime, we can focus on the motions of photons on the equatorial plane $(\theta=\pi/2)$ without losing generality. Moreover, by considering $\mathscr{L}=0$ for photon and defining the impact parameter $b=L_{z}/E$, we can obtain three first-order differential equations of motion for the photons

where the effective potential is given as

The $\pm$ sign in (7) represent the clockwise and counterclockwise direction of photons’ trajectories, respectively. Then the fate of a photon is determined by the radial geodesic equation. In particular, the circular orbit should satisfy $\dot{r}=0$ and $\ddot{r}=0$, which will be translated to

where $r_{ph}$ is the radius of the photon sphere satisfying $V^{\prime\prime}_{\text{eff}}(r_{ph})<0$ , and $b_{ph}$ is the critical impact parameter which is related with the shadow radius of the black hole. As shown in the leftmost plot of FIG. 2, the light rays with $b>b_{ph}$, will be deflected by the black hole and escape to infinity; on the contrary, the light rays with $b<b_{ph}$ will be captured into the event horizon of the black hole. The light rays with $b=b_{ph}$ will circle around the black hole all the time with the radius $r_{ph}$. The effects of the two charges on the $r_{ph}$ and $b_{ph}$ are shown in the middle and rightmost plot of FIG. 2. It shows that as both charge parameters increase, both $r_{ph}$ and $b_{ph}$ become smaller, similar to what occurs for the event horizon, as shown in FIG. 1, as expected.

Based on the preliminary analysis, we shall study the optical appearance of the hairy RN black hole in various illumination conditions. It is known that the hot, optically thin accretion flows are surrounding M87*, Sgr A* and many other supermassive black holes in our Universe Yuan and Narayan (2014). Here we will consider the hairy RN black hole illuminated by statically thin disk accretions and spherical accretions, respectively, which are simple toy accretion models but enough for the purpose of this paper.

In this section, we shall use the ray-tracing method to analyze the distribution of light rays in the vicinity of the hairy RN black hole and further investigate the optical appearance of the black hole illuminated by an optically and geometrically thin accretion disk. For simplicity, we consider that the accretion disk is fixed at the equatorial plane and the observer is located at the North pole. Due to the strong gravitational lensing effect, the light rays around the black hole may intersect with the accretion disk arbitrary times before falling into the black hole or escaping to infinity, which brings different contributions to the total intensity received by the observer. For this purpose, we should first classify the light rays in terms of the number of times they intersect with the thin accretion disk, and then figure out the images of the hairy RN black hole.

In this section, we will investigate the images of hairy RN black holes surrounded by a static spherical accretion. When the materials in the Universe are trapped by a black hole, the disk-shaped accretion flow usually forms around the black hole and rotates with a large angular momentum, but the matter will flow radially to the black hole and form a spherically symmetric accretion when the angular momentum is extremely small Yuan and Narayan (2014). For an optically and geometrically thin static accretion with spherical symmetry, the observed specific intensity $I(\nu_{o})$ detected by an observer at infinity $r=\infty$ (measured in erg s^-1 cm^-2 str^-1 Hz^-1) is evaluated by integrating the specific emissivity along the photon path $\gamma$ Bambi (2013)

Here, $\nu_{e}$ and $\nu_{o}$ are the emitted photon frequency and the observed photon frequency, respectively. $g=\nu_{o}/\nu_{e}=f(r)^{1/2}$ is the redshift factor. $j(\nu_{e})$ is the emissivity per unit volume in the rest frame and usually taken the form $j_{e}(\nu_{e})\propto\delta(\nu_{r}-\nu_{e})/r^{2}$ with $\nu_{r}$ the emitter’s rest-frame frequency Bambi (2013). And $dl_{prop}$ is the infinitesimal proper length given by

where the $d\phi/dr$ is expressed in (10). Subsequently, by further integrating (16) over all the observed frequencies, we obtain the total observed intensity as

Then we calculate the total observed intensities, and depict them into a two dimensional plane as the images of hairy RN black hole with the same selected parameters as previous sections. The results are presented in FIG. 6. For the total observed intensities in each case, there exists a peak which corresponds to the bright ring in the images of the black hole. As the increasing of both $Q$ and $l_{o}$ (FIG.6a and 6e), the peak is enhanced and shifts to smaller impact parameters, indicating that the ring becomes brighter but the faint illuminating region in the center is smaller for larger $Q$ (Fig. 6b-6d) or $l_{o}$ (Fig. 6f-6h). In addition, FIG. 6i-6l show that the effect of increasing of $\alpha$ obeys a completely different rule from that of the charges. Namely, increasing $\alpha$ suppresses the peak but enlarges the corresponding impact parameters, thus, we see larger faint illuminating region in the images of black hole with larger $\alpha$. Thus, comparing to the RN black hole, the introduction of additional field can enlarge/suppress or keep the brightness as well as the size of photon ring, depending on the competition between the parameters $l_{o}$ and $\alpha$. This implies the possible degeneracies in the optical appearances of the hairy RN black hole and RN black hole surrounded by the static spherical accretion.

The black hole images released by the EHT collaborators open a new window for testing gravity in strong field regime and provide new inspiration for testing possible hairs around the black hole. In this paper, taking the hairy RN black hole constructed by GD proposal Ovalle et al. (2021), which could contain electric charge and additional arbitrary hairy charge, as the central object, we investigated its optical appearance illuminated by some toy accretion models to discuss the possible probes of the two charges.

As the preparation, we first analyzed the effects of electric charge, additional hairy charge and the deviation parameter on the photon sphere and critical impact parameters. We found that as both charges increase, the photon sphere and critical impact parameters of the hairy RN black hole both decrease, but the deviation parameter enlarges them. Then, using the back ray-tracing method, we studied the photon trajectories around the hairy RN black hole and found that the charges have a significant influence on the distributions and classification of light rays. In particular, for the hairy RN black hole with larger electric charge or additional hairy charge, the widths of the lensed ring and photon ring emissions of light rays is larger, on the contrary, the one with larger deviation parameter $\alpha$ corresponds to narrower widths of the lensed ring and photon ring emissions. This phenomena would indeed have been printed on the black hole images.

We then investigated the optical appearance of the hairy RN black holes illuminated by an optically and geometrically thin accretion disk, which is radiated in the form (14). There always exist two explicit peaks in the total observed intensity, which correspond to two bright rings in the image of the hairy RN black hole as we project the total observed intensity into a two dimensional plane. We found that hairy RN black hole with larger charge parameter $Q$ ($l_{o}$) gives rings with weaker brightness, while that with larger $\alpha$ has brighter rings. In addition, for larger charge parameter $Q$ ($l_{o}$), the central black hole shadow is smaller, but the larger deviation parameter $\alpha$ tends to enlarge the central shadow size. Finally, we figured out the optical appearances of hairy RN black holes illuminated by a static spherical accretion. The observed intensity shows a peak at critical impact parameter, indicating that in the optical appearance of the hairy RN black hole, a bright ring enclosing a shaded region appears and the brightness becomes weaker as the impact parameter increase due to the redshift effect. Moreover, we found that as the charge parameter $Q$ ($l_{o}$) increases, the ring becomes brighter but the enclosed shaded region in the center is smaller. While the stronger deviation parameter $\alpha$ corresponds to a darker ring but larger faint illuminated region in the central of black hole images, which are completely opposite to the influence of charge parameter $Q$ or $l_{o}$.

In conclusion, we investigated the effects of electric charge $Q$, additional hairy charge $l_{o}$ and the deviation parameter $\alpha$ on the accretion images of the hairy RN black holes in the scenario of GD proposal. Our results show that the electric charge and additional hairy charge have similar effects on the rings and images of the hairy RN black hole, while the deviation parameters show completely different influences. Thus, we discussed that one may not be able to distinguish the electric charge and additional hairy charge via the rings and images of the black hole in this scenario. Besides, there could be degeneracies in the optical appearances of the hairy RN black hole, caused by the competing effects of the charge parameters and the deviation parameter. This could also produce the undistinguished images between the hairy RN black holes and RN black holes.

The significance of the current study could exist in two aspects. From observable point, our study chooses the hairy RN black hole as a first attempt to utilize ring and image of black holes as potential probes to seek the possible distinguishing of electric charge and other charges. We hope that our study can inspire more related researches in this direction. From theoretical point, as previously mentioned, the complete background theory that generates the hairy RN black hole is unclear now, so our results may give some beneficial hints for building the background theory of hairy RN black hole from the phenomena besides for the existed thermodynamics Mahapatra and Banerjee (2023), quasinormal modes Cavalcanti et al. (2022); Yang et al. (2023a); Li (2023), strong gravitational lensing Afrin et al. (2021); Islam and Ghosh (2021); Meng et al. (2023b), precession and Lense-Thirring effect Wu et al. (2023b) and gravitational waves Zi and Li (2023).