Dynamics of null particles and shadow for general rotating black hole

Black hole may be the most thoroughly-studied object before its discovery in the history of science. Recently, the Event Horizon Telescope (EHT) collaboration announced the images of the supermassive BHs M87* and SgrA*[1, 2], these discoveries together with the detections of gravitational waves (GWs)[3, 4] confirm the existence of black holes, and open a new era of testing gravity in strong-field regime and far beyond the scale of solar system.

Shadow is the image formed by photons which are sent off by the central BH and reach the observer finally. All the unstable orbits of photons form light rings around BH, the projection of the light rings on the observing screen is what called shadow. The unstable orbits are the critical ones, on which small perturbations would cause the photons to be either captured by the central compact object or sent off to infinity. The light rays forming shadow pass very close to the event horizon and are bent a lot due to the strong gravitational lensing effect, therefore observations of shadow make it possible to test the strong-field properties of gravity.

As we know, according to no-hair theorem[5, 6, 7], Kerr metric is the unique neutral, stationary, and asymptotically flat BH solution of vacuum Einstein equations in four dimensions. While, it is possible BHs are not isolated and are surrounded and influenced by other fields, as well as, the underlying gravity theory is possible to be some modified gravity theory, which means Kerr hypothesis may be be deviated. Recently, non-Kerr BHs have been studied extensively by considering coupling matter fields to gravity[8, 9, 10, 11], or modifying theories of gravity [12, 13, 14, 15, 16, 17], or immersing the BHs in astrophysical environment [18, 19, 20, 21, 22, 23]. Testing no-hair theorem, and thus testing general relativity, is one of the most important topics in black hole physics and astrophysics.

Observationally, it is quite sensible to treat the deviations of Kerr BH in a model-independent way, so that the effects of the deviations can be studies universally. The model-independent strong-field tests require a modified spacetime which deviates from the Kerr metric in a parametrized form. In Ref.[24], Johannsen constructed a general asymptotically flat rotating BH that depends nonlinearly on the deviation functions, and the spacetime symmetries are held meanwhile. The Johannsen BH admits three constants of motion, i.e., the energy, angular momentum and Carter constant, which is regular outside the event horizon and free of closed timelike curves. The BH possesses correct Newtonian limit in the non-relativistic regime, and is consistent with all current weak-field tests. Up to leading truncation order, the BH possesses four deviation parameters that measure potential deviations from the Kerr metric in the strong-field regime. Apparently, the BH is not solution of any particular theories of gravity, but can be embedded into several known modified theories of gravity invoked in astrophysics and cosmology by suitably choosing the deviation parameters. Significantly Johannsen BH does not suffer from pathologies in the exterior domain, it hence fairly serves as a phenomenological framework for strong-field tests of the no-hair theorem in general classes of gravity theories.

The shadow of Schwarzschild BH was first studied by Synge[25], and Bardeen first studied the shadow of Kerr[26], the result showed that, compared to the spherical BH, the shadow of Kerr is no longer circular, the distortion of shadow is understood as the effect of frame dragging. Aside from the distortion effects of rotating parameter, studies show that the deviation parameters lead to deformation of the shadows significantly as well, thus shadow can be used to extract information of BHs so as to detect the deviations from Kerr and constrain the BH parameters in the coming observations[27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. The announcement of BH shadow inspire many attempts to test no-hair theorem, alternative theories of gravity, the candidates of dark matter, and quantum effects of gravity utilizing the data of shadow [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100]. With the improvements of detection accuracy, BH shadow is expected to serve as independent measurement or complementary to GW detections to provide a powerful way of constraining the BH parameters and distinguishing different BHs.

Aside from the deviation effects, the luminous source is another factor that affect the shadow. We know BHs accrete matters and form accretion disk surrounding them, the matters in accretion disk would emit photons and serve as luminous source of the shadow, which is expected to be affected significantly by accretion disk. Compared to the ideal treatment of BH shadow, taking into account accretion disk is more realistic in astrophysics.

In this paper, we study the shadow of Johannsen BH, our results show that the shadow are affected significantly by rotating and deviation parameters, including the shape of shadow boundaries, and the luminosity, the central dark region and the region with high luminosity of the accretion disk images. With the observational results of M87* and SgrA* we constrain the deviation parameters further based on the theoretical constraints. Due to the universal features of Johannsen BH, this study is helpful to test no-hair theorem and modified gravities in a model-independent way.

The paper is organised as, in section 2 we review the general rotating BH proposed by Johannsen. In section 3, we plot the shadow boundaries on celestial plane, calculate the shadow radius, and constrain the deviation parameters with observational data. In section 4, we give the accretion disk images of Johannsen BHs by ray-tracing method. We summarize our results in the last section 5. In this work, the geometrized units are used with $G=c=1$.

In this section we review Johannsen BH proposed in Ref.[24]. Johannsen BH is a general neutral, stationary and asymptoticall flat BH, which is a generalization of Kerr. Let’s start with Kerr, in Boyer-Lindquist coordinates Kerr BH is given by

	$\displaystyle ds^{2}$	$\displaystyle=-\left(1-\frac{2Mr}{\Sigma}\right)dt^{2}-\frac{2Mar\sin^{2}\theta}{\Sigma}dtd\phi+\frac{\Sigma}{\Delta}dr^{2}$
		$\displaystyle+\Sigma d\theta^{2}+\left(r^{2}+a^{2}+\frac{2Ma^{2}r\sin^{2}\theta}{\Sigma}\right)\sin^{2}\theta d\phi^{2},$		(1)

with

$\displaystyle\Delta\equiv r^{2}-2Mr+a^{2},\quad\quad\quad\Sigma\equiv r^{2}+a^{2}\cos^{2}\theta.$

(2)

We know general stationary and axisymmetric BHs are of Petrov type I which admit two constants of motion, the energy $E$ and angular momentum $L_{z}$. While Kerr BH is of Petrov type D, since Carter found there exist a third constant of motion, the Carter constant $Q$[101], by separating variables of the Hamilton-Jacobi equations of test particles

$\displaystyle-\frac{\partial S}{\partial\tau}=\frac{1}{2}g^{\alpha\beta}\frac{\partial S}{\partial x^{\alpha}}\frac{\partial S}{\partial x^{\beta}}.$

(3)

The successful separation of variables enable the geodesic motions of test particles in Kerr spacetime to be integrable. If the Kerr metric is required to be modified in the manner by holding the Carter symmetry, i.e., the motions of test particle in this novel spacetime is expected to be integrable as well, a metric constructed by Johannsen satisfies this requirement, the metric can be expressed in contravariant form as

		$\displaystyle g^{\alpha\beta}\frac{\partial}{\partial x^{\alpha}}\frac{\partial}{\partial x^{\beta}}$		(4)
	$\displaystyle=$	$\displaystyle-\frac{1}{\Delta\tilde{\Sigma}}\left[(r^{2}+a^{2})A_{1}(r)\frac{\partial}{\partial t}+aA_{2}(r)\frac{\partial}{\partial\phi}\right]^{2}$
		$\displaystyle+\frac{\Delta}{\tilde{\Sigma}}A_{5}(r)\left(\frac{\partial}{\partial r}\right)^{2}+\frac{1}{\tilde{\Sigma}}A_{6}(\theta)\left(\frac{\partial}{\partial\theta}\right)^{2}$
		$\displaystyle+\frac{1}{\tilde{\Sigma}\sin^{2}\theta}\left[A_{3}(\theta)\frac{\partial}{\partial\phi}+a\sin^{2}\theta A_{4}(\theta)\frac{\partial}{\partial t}\right]^{2},$

where

$\displaystyle\tilde{\Sigma}\equiv\Sigma+f(r)+g(\theta).$

(5)

The explicit form of the deviation functions $A_{i}(r)$ and $A_{j}(\theta)$ will be discussed below.

To see the motions of test particle in the spacetime described by the metric (4) are integrable, we suppose the Hamilton-Jacobi function of a particle with mass $m_{0}$ to be of the form

$\displaystyle S=\frac{1}{2}m_{0}^{2}\tau-Et+L_{z}\phi+S_{r}(r)+S_{\theta}(\theta),$

(6)

which indicates $p_{r}=\textmd{d}S_{r}/\textmd{d}r$ and $p_{\theta}=\textmd{d}S_{\theta}/\textmd{d}\theta$. Inserting the metric (4) into the Hamilton-Jacobi equation (3), we have

	$\displaystyle-m_{0}^{2}=$	$\displaystyle-\frac{1}{\Delta\tilde{\Sigma}}\left[-(r^{2}+a^{2})A_{1}(r)E+aA_{2}(r)L_{z}\right]^{2}$		(7)
		$\displaystyle+\frac{1}{\tilde{\Sigma}\sin^{2}\theta}\left[L_{z}-aE\sin^{2}\theta\right]^{2}$
		$\displaystyle+\frac{\Delta}{\tilde{\Sigma}}A_{5}(r)\left(\frac{\partial S_{r}}{\partial r}\right)^{2}+\frac{1}{\tilde{\Sigma}}\left(\frac{\partial S_{\theta}}{\partial\theta}\right)^{2}.$

After separating variables of Eq. (7) we have

	$\displaystyle C=$	$\displaystyle\frac{1}{\Delta}\left[-(r^{2}+a^{2})A_{1}(r)E+aA_{2}(r)L_{z}\right]^{2}$
		$\displaystyle-m_{0}^{2}\left[r^{2}+f(r)\right]-\Delta A_{5}(r)\left(\frac{\partial S_{r}}{\partial r}\right)^{2},$		(8)
	$\displaystyle C=$	$\displaystyle\frac{1}{\sin^{2}\theta}\left[L_{z}-aE\sin^{2}\theta\right]^{2}+m_{0}^{2}(a^{2}\cos^{2}\theta+g(\theta))+\left(\frac{\partial S_{\theta}}{\partial\theta}\right)^{2},$		(9)

where $C$ is the separation constant, through which we could define the Carter constant

$\displaystyle Q=C-(L_{z}-aE)^{2}.$

(10)

Since Carter symmetry is a critical property of Johannsen metric, it is quite sensible to discuss the underlying physics of Carter symmetry/Carter constant. Carter constant reveals a deep and complicated symmetry in spacetime, it corresponds to a Killing tensor. For Johannsen BH, we find the Killing tensor is given by

$$K^{\mu\nu}=a^{2}\cos^{2}\theta g^{\mu\nu}+\frac{1}{\sin^{2}\theta}\delta^{\mu}_{\phi}\delta^{\nu}_{\phi}+a^{2}\sin^{2}\theta\delta^{\mu}_{t}\delta^{\nu}_{t}-2a\delta^{\mu}_{t}\delta^{\nu}_{\phi}+\delta^{\mu}_{\theta}\delta^{\nu}_{\theta}.$$

(11)

One checks that the Killing equation $K_{(\mu\nu;\lambda)}=0$ is satisfied, where the bracket denotes symmetrization of the indices. As Killing vectors $\partial_{t}$ and $\partial_{\phi}$ correspond to energy and angular momentum of the spacetime, Killing tensor describes a nontrivial symmetry of the spacetime and makes the general geodesic motion of probe particle to be integrable, for more motivations and applications of Carter constant see Cater’s original paper and related works [102, 103, 104, 105, 106].

Generally it relates to energy and angular momentum at $z$-direction. Here we concentrate on the physical implications of Cater constant in this article. The separation constant $C$ and Carter constant $Q$ in our paper are related through Eq.(10). We take new separation constant following [105]

$$\Lambda=C+2aEL_{z}-m_{0}^{2}a^{2},$$

(12)

then the separation equation (9) can be rewritten as

$$p_{\theta}^{2}+\frac{p_{\phi}^{2}}{\sin^{2}\theta}+m_{0}^{2}a^{2}\left[\left(\frac{E}{m_{0}}\right)^{2}-1\right]\sin^{2}\theta=\Lambda,$$

(13)

where $p_{\phi}=L_{z}$. Since we will see later $g(\theta)$ is set to be zero by observation requirement, here we ignore this term to clarify the physical meaning of Carter constant. Note that eq.(13) does not depend on the mass $M$ of gravitational source. Now we try to explain the terms on the left side of above equation. In the $a\rightarrow 0$ limit, i.e., the spherical symmetry case, the physical meaning of the first two terms are clear. They are the square of angular momentum $L^{2}=p_{\theta}^{2}+\frac{p_{\phi}^{2}}{\sin^{2}\theta}$, in which the separation constant $\Lambda$ equals to $L^{2}$. When $a\neq 0$, we consider the weak-field approximation, which implies that the observer locates at far field region, the spatial metric can be rewritten as

$$d\sigma^{2}=\frac{\Sigma}{r^{2}+a^{2}}dr^{2}+\Sigma d\theta^{2}+(r^{2}+a^{2})\sin^{2}\theta d\phi^{2},$$

(14)

with oblate spheroidal coordinates

	$\displaystyle x$	$\displaystyle=$	$\displaystyle(r^{2}+a^{2})^{1/2}\sin\theta\cos\phi,$		(15)
	$\displaystyle y$	$\displaystyle=$	$\displaystyle(r^{2}+a^{2})^{1/2}\sin\theta\sin\phi,$		(16)
	$\displaystyle z$	$\displaystyle=$	$\displaystyle r\cos\theta.$		(17)

With the asymptotic metric (14) the square modulus of particle’s linear momentum observed at infinity is given by

$$p^{2}=g^{rr}_{\infty}p_{r}^{2}+g^{\theta\theta}_{\infty}p_{\theta}^{2}+g^{\phi\phi}_{\infty}p_{\phi}^{2}.$$

(18)

In the limit $r\rightarrow\infty$, we have

$$g^{\theta\theta}_{\infty}p_{\theta}^{2}\longrightarrow 0,\quad\quad\quad\quad g^{\phi\phi}_{\infty}p_{\phi}^{2}\longrightarrow 0.$$

(19)

Thus

$$g^{rr}_{\infty}p_{r}^{2}\longrightarrow p^{2}=m_{0}^{2}\left[\left(\frac{E}{m_{0}}\right)^{2}-1\right].$$

(20)

Now the separation equation (13) can be rewritten as

$$p_{\theta}^{2}+\frac{p_{\phi}^{2}}{\sin^{2}\theta}+|p_{r}|^{2}a^{2}\sin^{2}\theta=\Lambda.$$

(21)

This equation implies that, if a particle at infinity possess nonvanishing momentum $p_{r}$ it also possesses nonvanishing angular momentum with respect to the center $O$ of $r=0$ disk. In the Boyer-Lindquist coordinates, the $\theta=constant$ surfaces are not cones with vertices at the origin, but are hyperboloids of rotation, which cross the $r=0$ disk into a circle with radius $a\sin\theta$. At infinity the vector $\vec{p}_{r}$ coincide with the asymptotes of the hyperboloid, which is to say $\vec{p}_{r}$ is tangent to $\theta=constant$ surfaces, thus parallel transport along $\theta=constant$ surface maintains $\vec{p}_{r}$ to be tangent to the hyperboloid. When crossing the $r=0$ disk, the point of maximum approach to the center $O$ is reached, now $\vec{p}_{r}$ is orthogonal to the $r=0$ disk. So its momentum with respect to the center $O$ has a square modulus given by $|p_{r}|^{2}a^{2}\sin^{2}\theta$, which is just the term appearing in eq.(21). It should be mentioned, for bounded orbits of particles the sign before the third term on the left side of eq.(21) should be changed, for more details please refer to Ref.[105].

From eq.(21) we see that, the separation constant has the meaning of ‘extended’ angular momentum which come not only from the angular motions but also from the radial motion. The radial angular momentum is not obvious, since for the spacetime of black hole with spherical symmetry, angular momentum comes only from angular motions. While, for the spacetime of rotating black hole, the radial angular momentum appears. It arises from the particular choice of coordinates which in turn ensures the separability of the Hamilton-Jacobi equation.

Solving the variables-separated Eqs. (8) and (9) we have

$\displaystyle S_{r}(r)=\pm\int dr\frac{1}{\Delta}\sqrt{\frac{R(r)}{A_{5}(r)}},\;S_{\theta}(\theta)=\pm\int d\theta\sqrt{\Theta(\theta)},$

(22)

with

	$\displaystyle R(r)$	$\displaystyle\equiv P^{2}-\Delta\left\{m_{0}^{2}\left[r^{2}+f(r)\right]+(L_{z}-aE)^{2}+Q\right\},$		(23)
	$\displaystyle\Theta(\theta)$	$\displaystyle\equiv Q+(L_{z}-aE)^{2}-m_{0}^{2}(a^{2}\cos^{2}\theta+g(\theta))-\frac{1}{\sin^{2}\theta}\left[L_{z}-aE\sin^{2}\theta\right]^{2},$		(24)
	$\displaystyle P$	$\displaystyle\equiv(r^{2}+a^{2})A_{1}(r)E-aA_{2}(r)L_{z}.$		(25)

The explicit integration expression of coordinates and proper time can be obtained by setting the partial derivative of Hamilton-Jacobi function with respect to the constants of motion to be zero, that’s to say the motions of particles in spacetime (4) are integrable.

In order to write the metric in explicit form, the derivation functions $A_{i}(r)$, $i=1,2,5$, can be expanded as a power series of $M/r$[24]:

$\displaystyle A_{i}(r)\equiv\sum_{n=0}^{\infty}\alpha_{in}\left(\frac{M}{r}\right)^{n},~{}~{}~{}~{}i=1,2,5,$

(26)

as well as

$\displaystyle f(r)\equiv\sum_{n=0}\epsilon_{n}\frac{M^{n}}{r^{n-2}},\;\;g(\theta)\equiv M^{2}\sum_{k,l=0}^{\infty}\gamma_{kl}\sin^{k}\theta\cos^{l}\theta,$

(27)

Expanding the metric in power of $1/r$ and requiring the metric to be asymptotically flat, the undetermined parameters and functions are fixed to be $\alpha_{10}=\alpha_{20}=\alpha_{50}=1$, $\epsilon_{0}=\epsilon_{1}=0$, and $A_{3}(\theta)=A_{4}(\theta)=A_{6}(\theta)=1$. If the parameter $M$ is required to be the mass of the central object, the parameters can be further constrained to be $\alpha_{11}=\alpha_{21}=\alpha_{51}=0$. The metric should be consistent with the the weak-field tests, in the parameterized post-Newtonian (PPN) formulism, the general metric can be written in the form [107, 108]

$$ds^{2}=-\left[1-\frac{2M}{r}+2(\beta_{PPN}-\gamma_{PPN})\frac{M^{2}}{r^{2}}\right]dt^{2}+(1+2\gamma_{PPN}\frac{M}{r})dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}).$$

(28)

For general relativity, $\beta_{PPN}=\gamma_{PPN}=1$. Values of $\beta_{PPN},\gamma_{PPN}$ corresponding to some other metric theories can be found in Chapter 5 of Ref.[108]. We perform the large $r$ expansion of Johannsen metric and find

$$ds^{2}=-\left[1-\frac{2M}{r}-\frac{M^{2}(2\alpha_{12}-\epsilon_{2})-g(\theta)}{r^{2}}\right]dt^{2}+(1+\frac{2M}{r})dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}).$$

(29)

Note that the spin parameter $a$ appear in the higher order expansion of $\frac{1}{r}$. Comparing metric (29) with metric (28) we have

	$\displaystyle 2(\beta_{PPN}-\gamma_{PPN})$	$\displaystyle=$	$\displaystyle\epsilon_{2}-2\alpha_{12}+\frac{g(\theta)}{M^{2}},$		(30)
	$\displaystyle\gamma_{PPN}$	$\displaystyle=$	$\displaystyle 1,$		(31)

which implies

$$\beta_{PPN}-1=\frac{1}{2}\left[\epsilon_{2}-2\alpha_{12}+\frac{g(\theta)}{M^{2}}\right]$$

(32)

This quantity is constrained by observations to be [108]

$$\beta_{PPN}-1=(0.2\pm 2.5)\times 10^{-5}.$$

(33)

In order to avoid any fine-tuning between the parameters $\epsilon_{2}$, $\alpha_{12}$ and the function $g(\theta)$, one can set $\epsilon_{2}=\alpha_{12}=g(\theta)=0$ for simplicity.

Now the metric can be expressed in covariant form as

	$\displaystyle g_{tt}$	$\displaystyle=-\frac{\tilde{\Sigma}[\Delta-a^{2}A_{2}(r)^{2}\sin^{2}\theta]}{[(r^{2}+a^{2})A_{1}(r)-a^{2}A_{2}(r)\sin^{2}\theta]^{2}},$
	$\displaystyle g_{t\phi}$	$\displaystyle=-\frac{a[(r^{2}+a^{2})A_{1}(r)A_{2}(r)-\Delta]\tilde{\Sigma}\sin^{2}\theta}{[(r^{2}+a^{2})A_{1}(r)-a^{2}A_{2}(r)\sin^{2}\theta]^{2}},$
	$\displaystyle g_{rr}$	$\displaystyle=\frac{\tilde{\Sigma}}{\Delta A_{5}(r)},$
	$\displaystyle g_{\theta\theta}$	$\displaystyle=\tilde{\Sigma},$
	$\displaystyle g_{\phi\phi}$	$\displaystyle=\frac{\tilde{\Sigma}\sin^{2}\theta\left[(r^{2}+a^{2})^{2}A_{1}(r)^{2}-a^{2}\Delta\sin^{2}\theta\right]}{[(r^{2}+a^{2})A_{1}(r)-a^{2}A_{2}(r)\sin^{2}\theta]^{2}}.$

with

	$\displaystyle A_{1}(r)$	$\displaystyle=1+\sum_{n=3}^{\infty}\alpha_{1n}\left(\frac{M}{r}\right)^{n},$		(34)
	$\displaystyle A_{2}(r)$	$\displaystyle=1+\sum_{n=2}^{\infty}\alpha_{2n}\left(\frac{M}{r}\right)^{n},$
	$\displaystyle A_{5}(r)$	$\displaystyle=1+\sum_{n=2}^{\infty}\alpha_{5n}\left(\frac{M}{r}\right)^{n},$
	$\displaystyle\tilde{\Sigma}$	$\displaystyle=r^{2}+a^{2}\cos^{2}\theta+f(r),$
	$\displaystyle f(r)$	$\displaystyle=\sum_{n=3}^{\infty}\epsilon_{n}\frac{M^{n}}{r^{n-2}}.$

Thus to leading order, the metric is characterized by mass $M$, spin $a$ and four deviation parameters $\alpha_{13}$, $\alpha_{22}$, $\alpha_{52}$, and $\epsilon_{3}$. By requiring the metric to be regular outside the event horizon, the determinant of the metric to be non-negative, and the absence of closed timelike curves, the four deviation parameters can be further constrained to be

	$\displaystyle\alpha_{52}$	$\displaystyle>-\frac{\left(M+\sqrt{M^{2}-a^{2}}\right)^{2}}{M^{2}},$		(35)
	$\displaystyle\epsilon_{3}$	$\displaystyle>-\frac{\left(M+\sqrt{M^{2}-a^{2}}\right)^{3}}{M^{3}},$
	$\displaystyle\alpha_{13}$	$\displaystyle>-\frac{\left(M+\sqrt{M^{2}-a^{2}}\right)^{3}}{M^{3}},$
	$\displaystyle\alpha_{22}$	$\displaystyle>-\frac{\left(M+\sqrt{M^{2}-a^{2}}\right)^{2}}{M^{2}}.$

For more details on constraining the deviation parameters from theoretical aspect please refer to Ref.[24]. It should be mentioned that, generally the Johannsen metric (2) is not a solution of any particular gravity theory, but can be mapped to known four-dimensional BH solution of theories of modified gravity by suitably choosing the deviation parameters, e.g., the metric can be mapped to modified gravity bumpy Kerr metric, rotating braneworld BHs, slowly rotating BHs in dynamical Chern-Simons gravity, and static BHs in Einstein-Dilaton-Gauss-Bonnet gravity[106, 110, 111, 112].

In this section, we study the effects of the BI parameters on the shadow boundary. Shadow boundary is expected to be formed by all the unstable orbits of photons known as light rings. The unstable orbits are the critical ones which separate the photons to escape to infinity or to be captured by the central BH.

BHs accrete surrounding matter to form the accretion disk, which is expected to be an important luminous source of shadow, thus the information of accretion disk can be extracted through observing photons on the image plane of observers. We simulate the photons emitted from accretion disk by tracing backward along the trajectory of photon from the image plane to the accretion disk plane. The radiation flux from accretion disk corresponds to the luminosity on the image plane. Since accretion disks are quite complex physical systems which are difficult to simulate, for the purpose of this work, we use the simple Page-Thorne thin disk model to describe the radiation flux of accretion disk[114].

The Lagrangian of a free particle is given by

$${\cal L}=\frac{1}{2}g_{\alpha\beta}\dot{q}^{\alpha}\dot{q}^{\beta},$$

(45)

where $\dot{q}^{\alpha}\equiv\frac{dq^{\alpha}}{d\tau}$. The Hamiltonian could be defined in the standard way

$$H=p_{\sigma}\dot{q}^{\sigma}-{\cal L}=\frac{1}{2}g^{\alpha\beta}p_{\alpha}p_{\beta},$$

(46)

where $p_{\alpha}\equiv\frac{\partial{\cal L}}{\partial\dot{q}^{\alpha}}=g_{\alpha\beta}\dot{q}^{\beta}$ is the momentum conjugate to $q^{\alpha}$. For the Johannsen metric we obtain

$\displaystyle H=\frac{\Delta A_{5}(r)}{2\tilde{\Sigma}}p_{r}^{2}+\frac{p_{\theta}^{2}}{2\tilde{\Sigma}}-\frac{R(r)}{2\Delta\tilde{\Sigma}}-\frac{\Theta}{2\tilde{\Sigma}},$

(47)

from which we obtain the equations of motion of photon

	$\displaystyle\dot{r}$	$\displaystyle=\frac{\Delta A_{5}(r)}{\tilde{\Sigma}}p_{r},$		(48)
	$\displaystyle\dot{p}_{r}$	$\displaystyle=-\left(\frac{\Delta A_{5}(r)}{2\tilde{\Sigma}}\right)^{\prime}p_{r}^{2}-\left(\frac{1}{2\tilde{\Sigma}}\right)^{\prime}p_{\theta}^{2}+\left(\frac{R+\Delta\Theta}{2\Delta\tilde{\Sigma}}\right)^{\prime},$
	$\displaystyle\dot{\theta}$	$\displaystyle=\frac{p_{\theta}}{\tilde{\Sigma}},$
	$\displaystyle\dot{p}_{\theta}$	$\displaystyle=-\left(\frac{\Delta A_{5}(r)}{2\tilde{\Sigma}}\right)_{,\theta}p_{r}^{2}-\left(\frac{1}{2\tilde{\Sigma}}\right)_{,\theta}p_{\theta}^{2}+\left(\frac{R+\Delta\Theta}{2\Delta\tilde{\Sigma}}\right)_{,\theta},$
	$\displaystyle\dot{t}$	$\displaystyle=\frac{1}{2\Delta\tilde{\Sigma}}\frac{\partial}{\partial E}\left(R+\Delta\Theta\right),$
	$\displaystyle\dot{p_{t}}$	$\displaystyle=0,$
	$\displaystyle\dot{\phi}$	$\displaystyle=-\frac{1}{2\Delta\widetilde{\Sigma}}\frac{\partial}{\partial L}\left(R+\Delta\Theta\right),$
	$\displaystyle\dot{p_{\phi}}$	$\displaystyle=0,$

where ^′ denotes derivation with respect to $r$, and the subscript $,\theta$ denotes derivation with respect to $\theta$. We consider an observer viewing the central BH from a large distance $d$ (see Fig.5). According to the equations of motion (48), the photons with initial positions and momenta on the observer’s image plane can be traced backwards to the accretion disk so as to extract information of the accretion disk. For convenience, we convert the coordinates of the photon on the image plane at $(\alpha_{0},\beta_{0})$ to Boyer-Lindquist coordinates[115, 116]

	$\displaystyle r_{i}$	$\displaystyle=\sqrt{\alpha_{0}^{2}+\beta_{0}^{2}+d^{2}},$		(49)
	$\displaystyle\theta_{i}$	$\displaystyle=\arccos{\frac{\beta_{0}\sin\theta_{0}+d\cos\theta_{0}}{r_{i}}},$
	$\displaystyle\phi_{i}$	$\displaystyle=\arctan{\frac{\alpha_{0}}{d\sin\theta_{0}-\beta_{0}\cos\theta_{0}}},$

where $\theta_{0}$ is the inclination angle of the photon as shown in Fig.5. The photons that can be observed on the image plane are those with momentum perpendicular to the image plane and against the line of sight, the components of the photon momentum $\vec{k}_{0}$ are given by

	$\displaystyle k_{r}$	$\displaystyle=\frac{d}{r_{i}}k_{0},$
	$\displaystyle k_{\rm\theta}$	$\displaystyle=\frac{-\cos\theta_{0}+(\beta_{0}\sin\theta_{0}+d\cos\theta_{0})\frac{d}{r_{i}^{2}}}{\sqrt{\alpha_{0}^{2}+\left(d\sin\theta_{0}-y^{\prime}\cos\theta_{0}\right)^{2}}}k_{0},$
	$\displaystyle k_{\rm\phi}$	$\displaystyle=\frac{-\alpha_{0}\sin\theta_{0}}{\alpha_{0}^{2}+(d\sin\theta_{0}-\beta_{0}\cos\theta_{0})^{2}}k_{0}.$

The radiation flux of accretion disk is closely related to circular geodesic motion of baryons on the equatorial plane. Obviously there are two conserved quantities for the baryons on circular orbits, energy and angular momentum. With the external spacetime metric, energy and angular momentum can be expressed as $p_{t}=-E=g_{tt}\dot{t}+g_{t\phi}\dot{\phi}$, $p_{\phi}=L=g_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi}$, with which the geodesic motion on the equatorial plane can be expressed as

	$\displaystyle\frac{dt}{d\tau}$	$\displaystyle=\frac{Eg_{\phi\phi}+Lg_{t\phi}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\,,$		(50)
	$\displaystyle\frac{d\phi}{d\tau}$	$\displaystyle=-\frac{Eg_{t\phi}+Lg_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\,,$		(51)
	$\displaystyle g_{rr}\left(\frac{dr}{d\tau}\right)^{2}$	$\displaystyle=-1+\frac{E^{2}g_{\phi\phi}+2ELg_{t\phi}+L^{2}g_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\,.$		(52)

In the above derivation we use the normalization condition $p_{\alpha}p^{\alpha}=m_{0}^{2}$ and set the mass of the massive particle $m_{0}$ to be 1. Note that $\dot{\theta}=0$, there is no particle motion in the direction of $\theta$. Eqs.(50) and (51) allow us to re-express energy and angular momentum as

	$\displaystyle E$	$\displaystyle=$	$\displaystyle\frac{g_{tt}-g_{t\phi}\Omega}{\sqrt{g_{tt}-2g_{t\phi}\Omega-g_{\phi\phi}\Omega^{2}}}\,,$		(53)
	$\displaystyle L$	$\displaystyle=$	$\displaystyle\frac{g_{t\phi}+g_{\phi\phi}\Omega}{\sqrt{g_{tt}-2g_{t\phi}\Omega-g_{\phi\phi}\Omega^{2}}}\,,$		(54)

where $\Omega=d\phi/dt$ is the angular velocity of the massive particle on the equatorial plane. The circular equatorial orbits require $\dot{r}=\ddot{r}=0$, this condition along with Eq.(52) give rise to the circular orbits condition $g_{tt,r}+2g_{t\phi,r}\Omega+g_{\phi\phi,r}\Omega^{2}=0$, from which one obtains the angular velocity

$\displaystyle\Omega=\frac{d\phi}{dt}=\frac{-g_{t\phi,r}\pm\sqrt{(g_{t\phi,r})^{2}-g_{tt,r}g_{\phi\phi,r}}}{g_{\phi\phi,r}}\,.$

(55)

From Eq.(52) one may introduce an effective potential

$$V_{\textmd{\scriptsize{eff}}}(r)=-1+\frac{E^{2}g_{\phi\phi}+2ELg_{t\phi}+L^{2}g_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\,.$$

(56)

The position of the innermost stable circular orbit (ISCO) is determined by the effective potential through the conditions

$\displaystyle V_{\textmd{\scriptsize{eff}}}(r)=V_{\textmd{\scriptsize{eff}}}^{\prime}(r)=V_{\textmd{\scriptsize{eff}}}^{\prime\prime}(r)=0\,.$

(57)