Shadow revisiting and weak gravitational lensing with Chern-Simons modification

Einstein’s general relativity (GR) is a successful theory in modern physics and passes lots of tests in astrophysics as well as astronomy, however, it is still facing some challenges, such as the explanation of the Universe expansion history, the large scale structure and the understanding of the quantum gravity and so on. Thus, physicists are very interested in finding alternative or higher dimensional theories of gravity to explain our Universe or to further understand the gravity. In particular, recent achievements on astronomical observations, such as the detection of the gravitational waves released from binary black hole mergers Abbott et al. (2016) and shadows of supermassive black holes Akiyama et al. (2019, 2022), make it possible to test alternative theories of gravity. These developments dredge a powerful venue to distinguish or constrain black holes deviating from those in GR.

Since the EHT observation of the M87* black hole was published, the shadow of black holes, as an important astronomical observation of strong gravitational lensing, has received considerable attention. In history, the work of shadow was started by Synge Synge (1966) and Luminet Luminet (1979), who first gave the photon capture region of the Schwarzschild black hole. After that, Bardeen considered the shadow of the Kerr black hole, and found that the shadow cast of the rotating Kerr black hole is not a standard perfect circle Bardeen (1973). Since the shadow of black hole can effectively reflect the information of strong field regime, it is extensively investigated in GR and modified theories of gravity (MoG), see Cunha and Herdeiro (2018); Perlick and Tsupko (2022); Chen et al. (2022) for reviews. Later, it was proposed in Hioki and Maeda (2009); Kramer et al. (2004) that the deformation and size of shadow could determine the black hole parameters, such that it can be further used to test different theories of gravity, see for examples Wei and Liu (2013); Tsupko (2017); Allahyari et al. (2020); Hou et al. (2021); Gan et al. (2021); Khodadi et al. (2020); Badía and Eiroa (2021); Meng et al. (2022); Kuang and Övgün (2022) and references therein. Considerable studies indicate that the shadow related observables can diagnose and even constrain black hole parameters in alternative gravity theories, though some constraints are still rough. On the other hand, during the black hole mergence, the final ringdown phase describes a perturbed black hole radiating gravitational wave with quasi-normal mode (QNM), which is one of the characteristic properties of the black hole and could provide near horizon information of the black holes. The QNM frequencies should typically be extracted from the perturbation theory. But Cardoso et al. proposed that in the eikonal limit, the real part of QNM frequency is connected with the angular velocity of the circular null geodesics while the imaginary part is connected with the Lyapunov exponent Cardoso et al. (2009), which is also valid in rotating black holes. Then the real part of the QNM in the eikonal limit was further related to the shadow radius of static black hole as $\omega_{Re}=\lim\limits_{\ell\gg 1}\frac{\ell}{R_{sh}}$Jusufi (2020a); Liu et al. (2020), and more recently this connection was extended into the rotating black holes Jusufi (2020b). This connection may stem from the fact that the gravitational waves could be treated as photons or other massless particles propagating along the last timelike unstable orbit out to infinity, but more research is needed to fully comprehend this correspondence.

One particular modified theory of gravity is the Chern-Simons (CS) gravity which is frequently used to discuss quantum gravity and black holes in an exclusively theoretical setup. The CS gravity was characterized by the coupling between a scalar field $\varphi$ and the first-class Pontryagin density $\ast R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ with the dual Riemann tensor $\ast R_{\alpha\beta\rho\sigma}=\frac{1}{2}\varepsilon_{\rho\sigma}^{~{}~{}~{}\mu\nu}R_{\alpha\beta\mu\nu}$ Alexander and Yunes (2009). This term is dubbed CS term which is parity-violating. There exist two types of CS modified gravity. The first is the non-dynamical CS (ndCS) gravity, in which the scalar field is a priori prescribed function so that $\ast R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ vanishes to ensure the diffeomorphism invariance of the theory, but the CS term have non-trivial contribution on the solution Alexander et al. (2008). The second is dynamical CS (dCS) gravity, in which the scalar field is treated as a dynamical field such that the CS term evolves in terms of the field equations Yunes and Pretorius (2009). It should be emphasized that despite the fact that the ndCS gravity action can be produced from a specific limit of the dCS action, they are typically viewed as two nonequivalent theories.

The dCS gravity theory is widely considered in a phenomenological context in cosmology and relativistic astrophysics, so we shall concentrate on this theory with the action

where $\kappa=(16\pi G)^{-1}$ and $\gamma$ is the CS coupling constant. Regarding to the black hole solution in dCS theory, the Schwarzschild black hole of GR is also a solution to the dCS theory because this theory is parity-violating in the sense that modification from GR due to the dCS term only appears in systems with broken parity symmetry Cardoso and Gualtieri (2009), but the rotating black hole would be modified and the rotating black hole with scalar hair emerges. The analytical spinning black hole solution with first order in small rotation was proposed in Yunes and Pretorius (2009); Konno et al. (2009) , which has been extended to arbitrary order in small rotation Cano and Ruipérez (2019). A non-perturbative numerical solution for spinning black hole in dCS gravity was constructed by directly solving the fields’ equations of motions Delsate et al. (2018).

The leading order correction to the slowly rotating solution reads as Yunes and Pretorius (2009); Konno et al. (2009)

where $ds^{2}_{SK}$ is the metric for slowly rotating Kerr black hole,

with $B=1-2M/r$. Meanwhile, the scalar field $\varphi$ has the configuration $\varphi=\left(\frac{5}{2}+\frac{5M}{r}+\frac{9M^{2}}{r^{2}}\right)\frac{\gamma a\cos\theta}{4Mr^{2}}$. It is obvious that as $\gamma\to 0$, the modified term in (3) vanishes and the solution goes back to the slowly rotating Kerr black hole. The superradiant spectrum in dCS gravity in the context of the slowly rotating black hole has been analyzed in Alexander et al. (2022). The optical phenomena in strong field regime of this slowly rotating black hole, such as the flux and the emission spectrum of the accretion disks Harko et al. (2010), the shadow cast Amarilla et al. (2010), geodetic precession and strong gravitational lensing Chen and Jing (2010), have partly disclosed how the CS correction affects the observational signals.

The aim of this paper to further study the shadow observables and the gravitational lensing effect of the slowly rotating black hole in dCS gravity. Our complementary research to these issues could help to improve our understanding on the effect of this theory and provide theoretical prediction of the test of the dCS theory with future more precise EHT observations. The remaining of this paper is organized as follows. In section II, by revisiting the shadow cast, we will check the possible constraint on the CS coupling from the EHT observation of the supermassive black holes, and then analyze the QNM frequencies in eikonal limit. In section III, we calculate the gravitational deflection angle in weak field limit with the use of Gauss-Bonnet theorem. Section IV contributes to our conclusion and discussion. We shall use the unit $G=c=1$ and denote the dimensionless quantities $a/M\to a$ and $\gamma/M^{2}\to\gamma$ for simplification unless otherwise noted.

The achievement of the EHT is remarkable since it opens a new window to explore the strong gravity regime through the direct observation. It is known that the core physics in such observation is the light deflection by the gravitational field. In the strong gravity regime, there is a photon region of black hole where the light rays from the light source get captured and it provides key properties in the black hole shadow. The existence of unstable photon regions outside the event horizon provides the possibility to observe the black hole directly. The photons that escape from the spherical orbits form the boundary of the dark silhouette of the black hole. Then according to the outside communicators, this dark silhouette is known as black hole shadow. In particular, the information of the first black hole image of M87* published by EHT collaborations in 2019 Akiyama et al. (2019) gives constraints on some shadow observables. Furthermore, from the second black hole image of SgrA*, we know that the angular shadow diameter is evaluated as $d_{sh}=48.7\pm 7\mu as$ Akiyama et al. (2022). The shadow related physics from EHT observations is consistent with the prediction from the Kerr black hole geometry in GR, but they cannot exclude other black holes in GR or some exotic black holes in MoG. Alternatively, the EHT observations of shadow can be applied as a tool to constrain the black hole parameters in various theories of gravity.

A preliminary image of the shadow cast of slowly rotating black hole (3) in dCS gravity was studied in Amarilla et al. (2010), in which how the dCS coupling parameter affect the shadow size $R_{s}$ and distortion $\delta_{s}$ were also explored. Since in general $R_{s}$ and $\delta_{s}$ may not adequately reflect the black hole shadow. Here we shall broaden our analysis to include the shadow area and oblateness. Then we will connect the theoretical study on the angular shadow diameter by the slowly rotating black hole with the EHT observations of the supermassive black holes, and discuss the potential constraint on the dCS coupling parameter.

It is well known that the light rays will deflect when they propagate in gravity field, due to the gravitational lensing effect. This effect mainly depends on the nature of the central source, so it is a useful tool to diagnose theories of gravity or to test exotic object in our Universe. The strong gravitational lensing effect by the slowly rotating black hole in dCS gravity has been well studied in Chen and Jing (2010). In this section, we extend the study into the weak gravitational lensing and focus on the dCS correction on the gravitational deflection angle of light. To this end, we shall use the Gibbons-Werner method Gibbons and Werner (2008), in which the Gauss-Bonnet theorem in the optical geometry is resorted. Specifically, Gibbons and Werner proposed that for a static and spherically symmetric black hole, the deflection angle of light can be calculated by integrating the Gaussian curvature of the optical metric outwards from the light ray, as a consequence of the focusing of light rays emerges as a global topological effect. Their method was then extended to compute the light deflection angle of a static spherical black hole with cosmological constants for the finite distances from the black hole to the light source and the observer Ishihara et al. (2016). Moreover, Gibbons-Werner method was soon used in stationary and axisymmetric black holes Werner (2012). Calculating the light deflection angle with this method has been widely investigated in various spherically symmetric or axisymmetric black holes. For a comprehensive review on the applications of the Gauss-Bonnet theorem to gravitational deflection angle of light in weak field limit, one refers to the review work Ono and Asada (2019), by following the steps of which we will proceed.

We firstly study the orbit equation of photon on the equatorial plane ($\theta=\pi/2$). For the stationary and axisymmetric spacetime (3), we mention in the appendix that there exists two constants of motion, $E$ and $L_{z}$, so we can define the impact parameter $b$ as

The null condition $ds^{2}=0$ leads to the following orbit equation of photon

where we have inserted the impact parameter (16). For convenience, we introduce $r\equiv 1/u$ and then the orbital equation of photon can be rewritten as

In principle, we can get the orbit trajectory by inserting metric functions (3) and (4) into the (18). However, due to the complexity, we cannot solve the differential equation analytically. Instead we use the approximations for the weak field, slow rotation and small dCS coupling $\gamma$ contribution to get the analytical approximation solution. The orbital function of the photon is thus obtained as ¹¹1Hereafter we will omit the term $\mathscr{O}(M^{2},a^{2},\gamma^{3})$ for simplicity.

Meanwhile we solve (19) for $\phi$ obtained as

where we can choose the domain of $\phi$ to be $-\pi<\phi\leq\pi$ without loss of generality. As shown in Fig. 9, the range of the angular coordinate value $\phi_{S}$ at the source ($S$) point is $-\pi/2\leq\phi_{S}<\pi/2$. But for the receiver ($R$) point , the range of $\phi_{R}$ is $|\phi_{R}|>\pi/2$. We assume the infinite distance limit for the source and receiver, namely $u_{S},u_{R}\rightarrow 0$, which leads to the angles $\phi_{S}\rightarrow 0$ and $\phi_{R}\rightarrow\pi$.

For the source and receiver in the equatorial plane in the axisymmetric spacetime, the definition of deflection angle is

Here $\Psi_{S}$ and $\Psi_{R}$ are the included angles of the connecting line between the source and the lens, and the connecting line between the observer and the lens and the radial direction of the light rays respectively. $\phi_{RS}$ is the longitude separation angle between source and observer (cf. Fig.9). In addition, we denote the integral region of the quadrilateral ($R_{\infty},R,S,S_{\infty}$) as $\mathscr{D}$.

According to Gauss-Bonnet theorem, the deflection angle can also be written by

where $K_{o}$ and $\kappa_{g}$ are the Gaussian curvature of the surface $\mathscr{D}$ and the geodesic curvature of light rays, respectively, $dS$ is the area element of surface, and $d\ell$ is infinitesimal line element along the boundary of surface. Now let us compute these two terms in (21). To this end, we rewrite $dt$ from the null condition $ds^{2}=0$ for the metric (3) as

where $i$, $j$ run from 1 to 3, and $\rho_{ij}$ and $\beta_{i}$ are

Firstly, for the light propagation in the equatorial plane, by considering the 2-dimensional Riemann tensor ${}^{(2)}R_{r\phi r\phi}$, the Gaussian curvature can be defined as

where ${}^{(2)}R_{r\phi r\phi}$, ${}^{(2)}\Gamma_{rr}^{\phi}$ and $\text{det}\rho_{ij}^{(2)}$ are defined by the optical metric $\rho_{ij}$ on the equatorial plane. Therefore, the integral of Gaussian curvature over the closed surface is Ono and Asada (2019)

where $r(\phi)$ is the orbit equation of photon and we have inserted $dS=\sqrt{\text{det}\rho^{(2)}}drd\phi$. Taking into account the above mentioned approximations, the Gaussian curvature $K_{o}$ and area element $dS$ can be easily obtained respectively

Thus the surface integral of Gaussian curvature is obtained as

Secondly, for the photon in the equatorial plane, the geodesic curvature is

and recalling (24), we have

In addition, the line element in the integral is given by

Thus, the path integral of geodesic curvature $\kappa_{g}$ is

Finally in terms of (21), (28) and (32), the deflection angle in weak field limit of the slowly rotating black hole in dCS gravity is given as

We can easily find that when $\gamma=0$, the deflection angle of the slowly rotating black hole in dCS gravity will return to the result of the Kerr black hole in GR. Note that the result (33) is for the light in the prograde motion. However for the retrograde case, the spin parameter $a$ will be changed into $-a$.

The dCS gravity is an appealing modified gravity theory and it is one of the most widely investigated parity violating gravity theory. In this theory, the static solution is still described by Schwarzschild metric of general relativity, but Kerr black hole is not a solution to this theory and the rotating black holes will be modified by the CS coupling term. Up to date, the complete form of fast rotating solutions is still missing, and the only axisymmetric solution was perturbatively obtained from Schwarzschild metric in terms of small spinning parameter. In this paper, we focused on the slowly spinning solution in the leading order correction and analyzed the effect of CS coupling term on shadow related quantities and gravitational deflection effect in weak field limit.

We analyzed the null geodesic using the Hamilton-Jacobi method, and revisited the shadow observables of the slowly rotating black hole in dCS modified gravity which are characterized by the shape and distortion of the shadow appearance. We found that the slowly rotating black hole with CS correction corresponds to larger black hole shadow since both the shadow radius and area are larger comparing to that in GR. As the CS coupling increases, the distortion of shadow decreases but the oblateness increases, meaning that the CS coupling suppresses the shadow deformation brought by the spinning of black hole. Then, using the EHT observation of supermassive black hole M87* and SgrA*, we attempted to constrain the CS parameter. The spinning parameter for M87* is not lower than 0.4 from EHT observation which could exclude all slowly rotating black hole in GR or alternative gravity theories, so it is not a good probe to detect the parameters of slowly rotating black hole. While the spinning parameter for SgrA* is inconclusive, and our result showed that the shadow diameter in slowing rotating black hole of dCS theory is consistent with the EHT observation $d_{sh}=48.7\pm 7\mu as$ for SgrA*, indicating that it could be a candidate for supermassive black hole in SgrA*. In addition, we also found that the CS term enhances the real part of QNM frequency in eikonal limit using its connection with the circular null geodesic on equatorial plane. Finally, we investigated the gravitational deflection angle of light in weak field limit of slowly rotating black hole spacetime with CS correction. It was shown that the CS could enhance the deflection for prograde photon while suppresses the deflection for retrograde photon.

It is noticed that here we partly studied the shadow observables, QNM in eikonal limit and weak gravitational lensing effect, which are observations related with the null geodesic, in a slowly rotating solution which is analytically perturbative in dCS theory. Our findings can be considered as a complementary to the earlier research on shadow cast Amarilla et al. (2010) and strong gravitational lensing effect Chen and Jing (2010). Since the non-perturbative spinning solutions to dCS gravity have been numerically constructed, it would be interesting to explore those optical observations for general rotating black hole with CS correction. In particular, we expect future generations of EHT observations may help to detect the CS modification from GR, and even give possible constraints on the CS coupling.