Testing Einstein Maxwell Power-Yang-Mills Hair via Black Hole Photon Rings

Although general relativity (GR) has achieved great success in classical scenarios, it still faces significant challenges under extreme conditions Penrose (1965); Hawking and Penrose (1970). The singularity theorems of Penrose and Hawking indicate that GR loses its predictive ability in certain regions, which prompts physicists to propose many alternative theories of gravity or consider quantum effects to eliminate singularities.Rovelli (2004); Brunnemann and Thiemann (2006); Modesto (2007); Bodendorfer et al. (2013); Long et al. (2019); Long and Ma (2020); Bojowald (2001). The Einstein-Cartan theory avoids the appearance of singularities by introducing torsionHehl et al. (1976); quantum gravity frameworks such as string theory and loop quantum gravity provide a completely new explanatory perspective for avoiding the problem of spacetime singularities Green and Schwarz (1983); Han et al. (2007); Ashtekar and Singh (2011); Zhang et al. (2023); Gambini and Pullin (2008); Ashtekar and Singh (2011). In addition, the Einstein-Yang-Mills theory, by unifying non-Abelian gauge fields and gravity within the geometrized framework of curved spacetime, provides a new tool for studying the interactions between gauge fields and gravitational fields in complex backgroundsJackiw and Rebbi (1976); Brink et al. (1977); Dorigoni et al. (2024); Koutrolikos (2024). This theory not only expands the scope of application of the GR but also has great significance in describing asymptotic symmetries, black hole physics and the research on quantum gravityBarnich and Lambert (2013). The proposal of alternative theories of gravity has provided new ideas for GR to overcome its limitations under extreme conditions.

black hole, as one of the classic predictions of GR, is not only an important tool for verifying existing gravitational theories but also a crucial experimental object for testing the validity of alternative theories and distinguishing differences among theories. In recent years, the exploration of black holes has always been at the forefront of theoretical physics research. With the release of the images of the supermassive black hole M87* by the Event Horizon Telescope (EHT) in 2019 Akiyama et al. (2019a, b, c, d, e, f) and the images of the supermassive black hole SgrA* at the center of the Milky Way in 2022 Akiyama et al. (2022), the existence of black holes, the mysterious celestial bodies predicted by the GR, has also been proven. It can be clearly seen from the pictures released by the EHT that there is an obviously dark region in the central part of the black hole. Usually, this specific dark region is named the shadow of the black hole. The shadow is the region inside the critical curve, which is closely related to the radially unstable spherical photon orbit Guo and Gao (2021). Meanwhile, the shadow is also a prominent feature of the image of the accretion flow around supermassive black holes Johannsen (2013). The shape and size of the shadow depend on the spacetime metric. The research on black hole shadows provides us with an important method to extract effective information about black hole spacetime and helps us test various alternative theories of gravity. After Synge’s pioneering work Synge (1966), a large number of studies on black hole shadows have been carried out in various gravitational theories Johannsen and Psaltis (2010); Li and He (2021); Sui et al. (2024); Abdujabbarov et al. (2016); Wei and Liu (2013); Mizuno et al. (2018); Konoplya (2019); Zare et al. (2024).

There is a bright ring surrounding the black hole shadow. This bright ring is called the photon ringWang et al. (2023). It is an important concept in the research on black hole imaging. Through the strong gravitational lensing effect, the light around the black hole is repeatedly deflected around it, and finally forms photon trajectories one after another Wang et al. (2022). These photon trajectories together present a ”ring-shaped” image around the black hole. This phenomenon not only provides crucial clues for black hole imaging but also serves as an important tool for studying physical phenomena in the extreme environment near black holes. Through observations of the photon ring, people have gained a deeper understanding of the properties of black holes such as the structure of the accretion disk, the gravitational lensing effect, and the rotation speed of black holes Sui et al. (2024); Gan et al. (2021a); Meng et al. (2024); Yang et al. (2023). The photon ring provides a method for testing the no-hair theorem of black holes Yang et al. (2024) and studying the gravitational effects of black holes Gralla et al. (2019), as well as a way to extract information about black hole parameters such as mass, spin, and event horizon size Gogoi and Ponglertsakul (2024). In addition, people’s research on the higher-order ring images of black hole accretion disks has also made it possible to use them to distinguish different accretion models Bisnovatyi-Kogan and Tsupko (2022). Therefore, the observational images of the photon rings and shadows of black holes are an effective tool Rincon and Gómez (2024), which can help us better understand the properties of black holes.

The no-hair theorem indicates that usually a black hole can be described by three physical parameters, namely mass, angular momentum, and electric charge. This implies that although the internal conditions before the formation of a black hole may be extremely complicated, it becomes a relatively simple celestial body after its formation. However, for modified theories of gravity, the introduction of modification terms adds extra degrees of freedom, which can explain phenomena that are difficult to be fully explained by the GR. In fact, in the process of exploring modified theories of gravity, people are committed to introducing additional matter fields into the theoretical system. The introduction of these additional matter fields has brought about a significant result, that is, a black hole can no longer be characterized solely by the traditional three parameters but requires more ”hairy” parameters to fully describe its properties. It is precisely based on such a background that hairy black holes, as a new research object, have been widely constructed and deeply analyzed in related fields Sui et al. (2024); Hawking et al. (2016); Herdeiro and Radu (2014); Antoniou et al. (2018a, b); Hod (2011).

Recently, Gabriel Gómez et al. obtained the spherically symmetric solution of the Einstein Maxwell theory in the presence of the Yang-Mills field with a power-termGómez et al. (2023). This spherically symmetric black hole solution includes the interaction between the Yang-Mills field and the mass-charge of the black hole. We hope to use the black hole photon ring to explore how the Yang-Mills field really has an impact on the black hole spacetime. Specifically, we focus on the model with power terms $p=1/2$ and $Q=0.6$. This special case represents a black hole possessing both Abelian charges $Q$ and non-Abelian charges $q_{YM}$ and is asymptotically non-flat. When our Yang-Mills hair parameter $Q_{YM}$ is set to zero, it reverts to the standard Reissner-Nordström (RN) black hole solution. Therefore, this paper aims to carry out research by analyzing this model and leveraging the structural characteristics of the photon ring and shadow mentioned earlier. Since there are differences in the structural characteristics of the photon ring and shadow among different black hole solutions, it is expected that the changes in these characteristics can be used to distinguish different black hole solutions, thereby testing the Yang-Mills hair.

The organizational structure of this paper is as follows: In Section II, the Einstein Maxwell power-Yang-Mills black hole solution is presented. Meanwhile, the value situation of the event horizon of this black hole under the action of the hairy parameter $Q_{YM}$ is explored. In Section III, the geodesic equation of the hairy black hole is first derived. Then the effective potential and the photon sphere radius are discussed. In Section IV, the light bending around the black hole is analyzed. Assuming the existence of an optically and geometrically thin accretion disk around the black hole, the geodesic trajectories under the influence of different hairy parameters are given. Moreover, the total emission and total observed intensities as well as the optical appearance under three different toy models with different values of the hairy parameter $Q_{YM}$ are studied and compared. Section V is the summary. In addition, throughout this paper, the natural unit system is adopted, i.e., $c=G=1$, and $M=1$ is set. The metric signature $(-,+,+,+)$ is used in this paper.

In this section, first briefly introduce the static and spherically symmetric hairy black hole under the standard Einstein-Maxwell theory with the p-power Yang-Mills term. We consider the spherically symmetric spacetime (in Schwarzschild coordinates), and the line element of this black hole can be written asGómez et al. (2023)

$$\begin{split}ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right)\end{split},$$

(1)

where

$$\begin{split}f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\frac{Q_{YM}}{r^{4p-1}}\end{split}.$$

(2)

This solution can be understood as a linear combination of the Yang-Mills term and the standard RN solution,and the Yang-Mills charge $q_{YM}$ is related to its normalized version asMazharimousavi and Halilsoy (2009)

$$\begin{split}Q_{YM}\equiv\frac{2p-1}{4p-3}q_{YM}^{2p}\end{split}.$$

(3)

It can be seen from Eq. (2) that, compared with the standardized RN black hole solution, the hole metric in this paper only introduces a nonlinear parameter $p$ and a hairy parameter $Q_{YM}$. Their introduction indicates the geometric deformation on the radial and time metric components. In this paper, the cases of $p=1/2$ and the Abelian charge $Q=0.6$ are taken. Therefore, when the hairy parameter $Q_{YM}=0$, it can revert to the standard RN black hole solution. Since this metric is static and spherically symmetric, and the event horizon, as a hypersurface with spacetime symmetry, should also be static and spherically symmetric. Therefore, when solving for the event horizon, it should only be a function of $r$, independent of $t$, $\theta$ and $\varphi$. That is to say, the event horizon should satisfy

$$\begin{split}g^{rr}=0\end{split},$$

(4)

we have

$$\begin{split}f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\frac{Q_{YM}}{r^{4p-1}}=0\end{split}.$$

(5)

When we determine the values of the power term $p=1/2$ and the Abelian charge $Q=0.6$, solving Eq. (5) can yield the horizon situations under different $Q_{YM}$, and they are shown in Fig. 1. The black dashed line in the figure represents the position of the event horizon radius of the RN black hole. We find that when $Q_{YM}=1.77778$, there is only one root corresponding to the Einstein Maxwell power-Yang-Mills black hole having only one horizon. When $Q_{YM}$is less than this value, the black hole will have two horizons; conversely, when $Q_{YM}$ is greater than this value, there is no horizon.

In order to study the influence of the hairy parameter $Q_{YM}$ of the Einstein Maxwell power-Yang-Mills black hole on the photon ring and the shadow, we need to first study the equation of motion of photons in the spacetime of this black hole and discuss the effective potential of photons. In the spherically symmetric spacetime, we consider the photon motion constrained to a fixed plane, $\theta=\pi/2$, that is

$$\begin{split}\sin\theta=1,\frac{d\theta}{d\tau}=0\end{split}.$$

(6)

For the spacetime of the line element (1), the geodesic equation can be derived according to the Lagrangian

$$\begin{split}2\mathcal{L}=g_{ij}\frac{dx^{i}}{d\tau}\frac{dx^{j}}{d\tau}\end{split},$$

(7)

where $\tau$ is the affine parameter along the geodesic. From the spacetime metric, the Lagrangian is

$$\begin{split}\mathcal{L}=\frac{1}{2}\left[-f(r)\dot{t}^{2}+\frac{1}{f(r)}\dot{r}^{2}+r^{2}\dot{\theta}^{2}+(r^{2}\sin^{2}\theta)\dot{\varphi}^{2}\right]\end{split}.$$

(8)

Where the dot indicates the derivative with respect to $\tau$, and the corresponding canonical momenta are

$$\begin{split}p_{t}&=\frac{\partial\mathcal{L}}{\partial\dot{t}}=-f(r)\dot{t},\\ p_{r}&=-\frac{\partial\mathcal{L}}{\partial\dot{r}}=-\frac{1}{f(r)}\dot{r},\\ p_{\theta}&=-\frac{\partial\mathcal{L}}{\partial\dot{\theta}}=-r^{2}\dot{\theta},\\ p_{\varphi}&=-\frac{\partial\mathcal{L}}{\partial\dot{\varphi}}=-r^{2}\sin^{2}\theta\dot{\varphi}.\end{split}$$

(9)

In the Schwarzschild spacetime, there are two Killing vector fields, namely the timelike Killing vector field $\left(\frac{\partial}{\partial t}\right)^{a}$ and the axial Killing vector field $\left(\frac{\partial}{\partial\varphi}\right)^{a}$. These two Killing vector fields will lead to two conserved quantities, that is, the conservation of energy $E$ and the conservation of angular momentum $L$Gralla et al. (2019).

$$\begin{split}E=-g_{00}\frac{dt}{d\tau}=f(r)\frac{dt}{d\tau}\end{split},$$

(10)

$$\begin{split}L=g_{33}\frac{d\varphi}{d\tau}=r^{2}\frac{d\varphi}{d\tau}\end{split}.$$

(11)

Substituting Eq. (10) and (11) into (8), we get:

$$\begin{split}2\mathcal{L}=-\frac{E^{2}}{f(r)}+\frac{1}{f(r)}\dot{r}^{2}+\frac{L^{2}}{r^{2}}\end{split}.$$

(12)

Also, because the Lagrangian of the null geodesic is zero:

$$\begin{split}\frac{1}{f(r)}\left(\frac{dr}{d\tau}\right)^{2}=\frac{E^{2}}{f(r)}-\frac{L^{2}}{r^{2}}\end{split}.$$

(13)

From Eq. (13), an ordinary differential equation for the radius $r$ with respect to the azimuthal angle $\varphi$ in the orbital plane can be further obtainedGogoi and Ponglertsakul (2024)

$$\begin{split}\left(\frac{dr}{d\varphi}\right)^{2}=\frac{1}{b^{2}}r^{4}-f(r)r^{2}\equiv V(r)\end{split}.$$

(14)

Here, $b=L/E$ is the impact parameter. It can be seen that the photon trajectory is determined by the impact parameter $b$. In order to study the hole shadow, we need to determine the radius of the unstable photon sphere, which is determined by the following equationYang et al. (2023); Gogoi et al. (2023)

$$\begin{split}\left.V(r)\right|_{r=r_{ph}}=0\end{split},$$

(15)

$$\begin{split}\left.\frac{dV(r)}{dr}\right|_{r=r_{ph}}=0.\end{split}$$

(16)

Due to spherical symmetry, Eq. (16) can be simplified to

$$\begin{split}\left.\frac{d}{dr}\frac{f(r)}{r^{2}}\right|_{r=r_{ph}}=0\end{split}.$$

(17)

After obtaining the radius of the photon sphere from Eq. (17), substituting it into Eq. (15) will yield the corresponding critical impact parameter as

$$\begin{split}b_{ph}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}\end{split}.$$

(18)

The effective potential for the radial motion of photons can be used to distinguish different photon orbits, and the effective potential at the radial distance $r$ can be expressed asGogoi and Ponglertsakul (2024)

$$\begin{split}V_{\text{eff}}(r)\equiv f(r)\frac{1}{r^{2}}\end{split}.$$

(19)

Therefore, the effective potential at the photon sphere is

$$\begin{split}V_{\text{eff}}(r_{ph})=\frac{1}{b_{ph}^{2}}\end{split}.$$

(20)

Above, $r_{ph}$ is the radius of the photon sphere, and $b_{ph}$ is the corresponding critical impact parameter. Their relationship is shown in Fig. 2. Here, the case corresponding to a RN black hole is taken as an example.

In the figure, for an RN black hole with $Q=0.6$, the event horizon radius is $r_{h}=1.8$, the photon sphere radius is $r_{ph}=2.73693$ and the critical impact parameter is $b_{ph}=1.85869$. The maximum value in the figure represents the radius of the unstable photon sphere. Further, we show the photon effective potential images for different hairy parameters $Q_{YM}$ in Fig. 3 . It is not difficult to see from Fig. 3 that as $Q_{YM}$ increases, the photon effective potential will gradually increase, and the event horizon radius of the black hole and the photon sphere radius of the photons will also decrease.

The value of the effective potential is of crucial significance in the study of photon orbits. When the effective potential reaches its maximum value, the corresponding orbit is an unstable photon sphere. Once photons on such an orbit are slightly disturbed, they may either fall into the black hole or escape to infinity and be captured by distant observers. Conversely, when the effective potential takes on a minimum value, the orbit at this time is stable, allowing photons to move continuously and stably on the corresponding orbit. And the unstable photon sphere that can be transmitted to distant observers has an important influence on observing the characteristics of the accretion imageGan et al. (2021b). Therefore, in the following discussion, we will focus on the unstable photon sphere orbits.

The fine structure of the photon ring involves the complex behaviors of photon orbits near black holes, especially the distribution of photons on different orbits. Since photons on different orbits appear as layers with different brightnesses when imaged, it helps observers identify the characteristics of black holes. In this part, we will adopt the method of backward ray tracing to analyze the behaviors of null geodesics near the Einstein Maxwell power-Yang-Mills hairy black hole and further explore how the optical appearance of the black hole presents when a very thin accretion disk serves as a light source. Here, the accretion disk is idealized as an optically and geometrically extremely thin plane, that is, the disk surface has no absorption or scattering effects on photons. During the research process, we assume that the background radiation has no impact on the results and that the accretion disk is the only light source. To simplify the analysis, we fix the accretion disk on the equatorial plane and assume that the observer is located in the direction of the North Pole at the same time. Due to the strong gravitational pull of the black hole, the null geodesics of photons may intersect with the accretion disk multiple times before falling into the black hole or escaping to infinity. The difference in the number of such intersections will have an important impact on the total light intensity received by the observer. Therefore, we need to classify these light rays according to the number of intersections between the null geodesics and the accretion disk and calculate the imaging effect of the Einstein Maxwell power-Yang-Mills hairy black hole based on this.

In this paper, the null geodesics near the Einstein Maxwell power-Yang-Mills black hole are studied. Meanwhile, based on the distribution of light rays, the optical images of the black hole surrounded by an optically and geometrically thin accretion disk are calculated, and the influence of the hairy parameter on both is discussed.

Firstly, the Einstein Maxwell power-Yang-Mills black hole solution is briefly reviewed. In this solution, a p-power Yang-Mills term and a hairy parameter related to non-Abelian charge are introduced, and the influence of the hairy parameter on the radius of the event horizon is analyzed. By analyzing the change of the hairy parameter $Q_{YM}$ when $Q=0.6$, it is found that a critical situation occurs for the black hole when $Q_{YM}=1.77778$; when $Q_{YM}<1.77778$, the Einstein Maxwell power-Yang-Mills black hole has two horizons, corresponding to the Cauchy horizon and the event horizon respectively; when $Q_{YM}>1.77778$, there is no event horizon and it is a naked singularity situation. Due to the cosmic censorship hypothesis proposed by Penrose, which prohibits the appearance of naked singularities, in this paper, we mainly consider the case where $Q_{YM}<1.77778$.

Secondly, the equations of motion trajectories and the effective potential of photons around this black hole were studied. Starting from the Lagrangian of the black hole, the geodesic equations and the effective potential of photons were obtained. Based on the effective potential, the radius of the unstable photon sphere orbit could be determined. Subsequently, the null geodesics around the hairy black hole were explored through the backward ray-tracing method. It was found that the hairy parameter $Q_{YM}$ had an impact on both the distribution and classification of geodesics. Moreover, when the hairy parameter increased, the event horizon radius $r_{h}$, photon sphere radius $r_{ph}$, the radius of the innermost stable circular orbit $r_{isco}$ and critical impact parameter $b_{ph}$ of the black hole would all decrease. The interval of the impact parameters corresponding to the photon ring and the lensing ring would also become narrower, which would further affect the image of the null geodesics of the black hole. The data and images could be seen in Table 1 and Fig. 6 respectively.

Finally, the total emission intensity, the total observed intensity, and the optical appearance corresponding to each of the three simplified models with different values of the hairy parameter $Q_{YM}$ under the irradiation of optically and geometrically thin accretion disks were studied. It was found that as the hairy parameter increased, the total observed intensity of each model would become stronger, and all the peaks of the total observed intensity were shifting in the direction of decreasing $b$, leading to a reduction in the ring radius. In addition, all three emission models showed that the direct image dominated the total flux, the lensing ring was a bright yet extremely narrow ring that was difficult to observe, and the contribution of the photon ring was negligible and almost invisible. In conclusion, different hairy parameters would result in different optical appearances, which also differed from those of the standard RN black hole. Further analysis led to the conclusion that when the hairy parameters were different, there was no degeneracy in their optical appearances and geodesics. Therefore, it could be considered that the geodesics and optical appearance images of photons could be used to distinguish different spacetime properties. Theoretically speaking, it was possible to distinguish the spacetime metrics of Einstein Maxwell power-Yang-Mills black holes with different hairy parameters. This characteristic creates a potential opportunity for characterizing various black hole solutions by observing the photon rings of black holes in the future, thus providing a valuable research approach and direction for delving deeper into the mysteries of black holes.

We acknowledge the anonymous referee for a constructive report that has significantly improved this paper. This work was supported by Guizhou Provincial Basic Research Program (Natural Science) (Grant No. QianKeHeJiChu-[2024]Young166), the Special Natural Science Fund of Guizhou University (Grant No.X2022133), the National Natural Science Foundation of China (Grant No. 12365008) and the Guizhou Provincial Basic Research Program (Natural Science) (Grant No.QianKeHeJiChu-ZK[2024]YiBan027 and QianKeHeJiChu-ZK[2025]General Program680.).