1. Introduction
Since the Event Horizon Telescope (EHT) collaboration released the images of the supermassive black holes M87* [1–9] and Sgr A* [10–15], it suggests that the theoretical results such as polarized image [16] and the black hole shadow [17–19], also referred as to the photon ring [20, 21] could be found on the black hole images in the future [22]. And it is promising to be used for testing the black hole physics and gravity theories [23, 24], or as a probe of the electromagnetism [25], accretion [26], dark matter [27] and exotic matter [28]. Henceforth, it is conceivable that black hole imagery holds importance for both theoreticians and astrophysicists.
Derived from the recent theoretical interest of the black hole shadow, it is also motivated to study the escape probability of a photon emitted near the a black hole. The escape probability formulates the probability of the emission from a point emitter that can escape from the black holes. Synge was the first to study the photon escape probability in Schwarzschild spacetime [29]. The author also estimated the critical cone and presented its relation to the distance of emitters to the black hole. Recently, the critical cone in Kerr spacetime was studied by Semerak [30]. And it was also extended to the Kerrde Sitter [31], and KerrSen [32]. From these studies, it is expected that the escape probability are closely related to the black hole parameters.
In recent studies, the escape probability were obtained for given velocities of emitters case by case. Reference [33] revisited the isotropic emitter at rest in a locally non-rotating frame on the equatorial plane. Later on, there have been related studies on other equatorial Kerr emitters, which extended previous studies to emitters on stable circular orbits (ISCO) [34, 35], emitters falling from ISCO [36], and geodesic emitters following various trajectories [37]. References [38, 39] discussed necessary and sufficient conditions for photons emitted from an arbitrary spacetime position of the extreme Kerr black hole to escape to infinity. Besides, the analytical approaches for equatorial emitters were also explored by pioneers [40, 41].
However, it is noted that, current studies are all limited to the emitters on the equatorial plane of a rotating black holes. Although no ambiguity on the notion of the escape probability, there seems no general approach that can provide a comprehensive study on this topic. To confront with this situation, this paper introduces a new and general approach determining escape probability for the emitters in arbitrary velocity and locations. It is based on the previous study on Kerr black hole with astrometric [42, 43], and extended the approaches to calculate the escape probability. It will be shown that escape probability for emitters far from the equatorial plane can be well calculated.
The rest of the paper is organized as follows. In section 2 , we review the astrometric approach for calculating the black hole shadow, and present the extension to the escape probability of the emitters. In section 3 , we calculated the escape probability of a photon near Kerr black hole and studies its relation with r, inclination θ and relative speed. Finally, conclusions and discussions are summarized in section 4 .
2. Algorithm for the escape probability
The escape probability of emitters can be determined by the size of the shadow with respect to finite-distant reference locations [33]. Therefore, to investigate the escape probability in a general situation, we adopt the approach previously employed for calculating shadows for finite-distant observers with astrometric [42–45]. This approach allows us to obtain the escape probability of emitters outside of the equatorial plane.
For illustration, we will show the approach with Kerr black hole. Kerr metric with mass M and spin a can be given by
$\begin{eqnarray}\begin{array}{l}{\rm{d}}{s}^{2}\,=-\displaystyle \frac{{\rm{\Delta }}}{{\rm{\Sigma }}}{\left({\rm{d}}t-a{\sin }^{2}\theta {\rm{d}}\phi \right)}^{2}\\ \,+\,\displaystyle \frac{{\sin }^{2}\theta }{{\rm{\Sigma }}}{\left(a{\rm{d}}t-({r}^{2}+{a}^{2}){\rm{d}}\phi \right)}^{2}\\ \,+\,\displaystyle \frac{{\rm{\Sigma }}}{{\rm{\Delta }}}{\rm{d}}{r}^{2}+{\rm{\Sigma }}{\rm{d}}{\theta }^{2}\,,\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\rm{\Delta }}={r}^{2}-2{Mr}+{a}^{2},\end{eqnarray}$
$\begin{eqnarray}{\rm{\Sigma }}={r}^{2}+{a}^{2}{\cos }^{2}\theta .\end{eqnarray}$
By making use of Hamilton–Jacobi method for the null geodesic, one can obtain the 4-momentum of emissions from an emitter in the form of $\begin{eqnarray}\displaystyle \frac{1}{E}{p}_{t}=-1,\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{1}{{E}^{2}}{\left({p}_{r}\right)}^{2}=\displaystyle \frac{{\left(({r}^{2}+{a}^{2})-a\lambda \right)}^{2}-{\rm{\Delta }}\kappa }{{{\rm{\Delta }}}^{2}},\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{1}{{E}^{2}}{\left({p}_{\theta }\right)}^{2}=\kappa -\displaystyle \frac{{\left(\lambda -a{\sin }^{2}\theta \right)}^{2}}{{\sin }^{2}\theta },\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{1}{E}{p}_{\phi }=\lambda .\end{eqnarray}$
The critical condition to determine whether the emission will escape is provided by the unstable circle orbit rc of null geodesics, specifically, ${p}^{r}{| }_{{r}_{{\rm{c}}}}=({p}^{r})^{\prime} {| }_{{r}_{{\rm{c}}}}=0$. The 4-momentum of the emissions satisfying the above critical condition have integral constants as follows, $\begin{eqnarray}\kappa ({r}_{{\rm{c}}})=-\displaystyle \frac{12{r}_{{\rm{c}}}^{2}({r}_{{\rm{c}}}(2M-{r}_{{\rm{c}}})-{a}^{2})}{{\left(M-{r}_{{\rm{c}}}\right)}^{2}},\end{eqnarray}$
$\begin{eqnarray}\lambda ({r}_{{\rm{c}}})=\displaystyle \frac{{r}_{{\rm{c}}}^{2}({r}_{{\rm{c}}}-3M)+{a}^{2}({r}_{{\rm{c}}}+M)}{a(3M-{r}_{{\rm{c}}})}.\end{eqnarray}$
The assemble of the unstable circle orbits is also referred to as the photon sphere.For the black hole shadow, the keys to the astrometric approaches lies in determining the location on the image plane with the reference geodesic [42]. For our study focusing on the escape probability, the notion of location on image plane is changed into emission direction with respect to the emitters. For simplicity, we choose the reference 4-velocities of the emissions on the photon sphere, denoted as k and w, which can be derived from the pμ in equations (3 ), namely3c ) larger than zero with given κ and λ in equations (4a ).
$\begin{eqnarray}k=p{| }_{{r}_{{\rm{c}}}={r}_{{\rm{c}},\min }},\quad w=p{| }_{{r}_{{\rm{c}}}={r}_{{\rm{c}},\max }},\end{eqnarray}$
where the range of rc is determined by the right hand side of equation (For emitters with given velocity u, the emitted direction of an emission, formulated by the 4-momentum l ≡ p, with respect to the reference directions k and w can be expressed as
$\begin{eqnarray}\cos \alpha =\displaystyle \frac{l\cdot k}{(u\cdot k)(u\cdot l)}+1,\end{eqnarray}$
$\begin{eqnarray}\cos \beta =\displaystyle \frac{w\cdot l}{(u\cdot l)(u\cdot w)}+1,\end{eqnarray}$
$\begin{eqnarray}\cos \gamma =\displaystyle \frac{w\cdot k}{(u\cdot k)(u\cdot w)}+1.\end{eqnarray}$
By making use of spherical trigonometric, we can express spherical coordinate Φ, $\Psi$ in terms of α, β and γ, namely $\begin{eqnarray}{\rm{\Psi }}=\arccos \pm \sin \beta \sqrt{1-{\left(\tfrac{\cos \alpha -\cos \beta \cos \gamma }{\sin \beta \sin \gamma }\right)}^{2}},\end{eqnarray}$
$\begin{eqnarray}{\rm{\Phi }}=\arccos \tfrac{\cos \beta }{\sin {\rm{\Psi }}}.\end{eqnarray}$
To obtain the escape probability, we simulate uniformly distributed points on a sphere labeled by ($\Psi$, Φ) to represent isotropic emissions from the emitters. The schematic diagram is shown in the left panel of figure 1. The black solid curve represents emissions that satisfy the critical condition for escaping, and it also corresponds to the boundary of the shadow. In this sense, the escape probability is determined by
$\begin{eqnarray}P=1-\displaystyle \frac{{\rm{Area}}\,{\rm{of}}\,{\rm{the}}\,{\rm{shadow}}}{4\pi }\,.\end{eqnarray}$
To calculate the area of the shadow, we apply Dan Sunday’s winding number algorithm on the plane of ($\Psi$, Φ), as illustrated in the right panel of figure 1. This method is implemented by calculating how many times the curve circles the point P. A point is outside only if the curve does not surround the point, that is, if the surround number equals zero. The encircling number of a point P on the plane with respect to any continuous closed curve is wn(P, C). For a horizontal ray R to the right, if the edge goes across R from the bottom up, wn+1; otherwise, wn−1. We go through all the edges, and we end up with a total wn(P, C).
Figure 1. Left panel: uniform-distributed points on the sphere. Right panel: Dan Sunday’s winding number algorithm for determining the points within the critical curves. The critical curve is sketched with the black curve. |
3. Escape probability of emitters near a Kerr black hole
Because emitters at different locations and velocities can influence the escape probability, we thus introduce the representative 4-velocities as follows,
$\begin{eqnarray}{u}_{r{\rm{p}}({\rm{m}})}={ \mathcal E }\left(\displaystyle \frac{1}{-{g}_{00}}{\partial }_{t}+(-)\tfrac{1}{\sqrt{{\rm{\Sigma }}}}\sqrt{{{\rm{\Delta }}}_{r}\left(\tfrac{1}{-{g}_{00}}-\tfrac{1}{{{ \mathcal E }}^{2}}\right)}{\partial }_{r}\right),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{u}_{\theta p(m)}={ \mathcal E }\left(\displaystyle \frac{1}{-{g}_{00}}{\partial }_{t}+(-)\displaystyle \frac{1}{\sqrt{{\rm{\Sigma }}}}\sqrt{\displaystyle \frac{1}{-{g}_{00}}-\displaystyle \frac{1}{{{ \mathcal E }}^{2}}}{\partial }_{\theta }\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{u}_{\phi {\rm{p}}({\rm{m}})}={ \mathcal E }\left(\displaystyle \frac{1}{-{g}_{00}}+(-)\displaystyle \frac{{g}_{03}}{{g}_{00}}\sqrt{{g}^{33}\left(\displaystyle \frac{1}{-{g}_{00}}-\displaystyle \frac{1}{{{ \mathcal E }}^{2}}\right)}{\partial }_{t}\right.\\ \,\left.-(+)\sqrt{{g}^{33}\left(\displaystyle \frac{1}{-{g}_{00}}-\displaystyle \frac{1}{{{ \mathcal E }}^{2}}\right)}{\partial }_{\phi }\right),\end{array}\end{eqnarray}$
which is derived from the 4-velocities of timelike geodesic used in [43].To quantify the speed of the emitters, we introduce the relative speed with respect to the static frame,
$\begin{eqnarray}v=\displaystyle \frac{| {\gamma }^{* }u| }{{u}_{\mathrm{st}}\cdot u}=\sqrt{1-{\left(N/{ \mathcal E }\right)}^{2}}.\end{eqnarray}$
It will be demonstrated that the speed also has an impact on the probability of escape.In the following parts, we will calculate the escape probabilities for the emitters with velocities given in equations 12 (a)–(c). For illustrations, we will refer to emitters with velocities urp(m), uθp(m), and uφp(m) as r+(−)-emitter, θ+(−)-emitter, and φ+(−)-emitter, respectively. And we conventionally name the emitters with subscripts + and—as ‘positive direction’ and ‘negative direction’, respectively.
Figure 2 shows the escape probability as a function of the distance of r+-emitters to the black hole. As expected, the escape probability of an emitter increases monotonically with the distance r. Moreover, the escape probabilities of emitters are larger for a highly rotating black hole. The approach presented in section 2 allows us to study the emitters outside of the equatorial plane with arbitrary velocities. Figure 3 shows the escape probability as function of inclination θ for r+-emitters. It is found that the escape probability increases with θ for a highly rotating black hole, while it decreases with θ for a slowly rotating black hole. Figure 4 shows the escape probability as a function of speed in positive directions. It is found that the escape probability for r+-emitters is significantly suppressed compared to that for θ+-emitters and φ+-emitters. Additionally, the escape probability increases with the speed for r-emitters, while it is shown to decrease with the speed for θ+-emitters and φ+-emitters. The conclusions hold for Kerr black hole with different spins. We further present the escape probability as a function of θ for different velocities along positive direction in figure 5. For r+-emitters and φ+-emitters at low speeds, the escape probability decreases monotonically with inclination θ. In contrast, for these emitters at high speeds, the escape probability increases with θ. The speed of r+-emitters can reduce the escape probability. The escape probability for θ+-emitters as a function of θ shows a different behavior compared to that for r+-emitters and φ+-emitters. It is observed that the escape probability decreases with θ for both low-speed and high speed emitters. Figure 6 shows the escape probability as a function of speed in negative direction. It is found that the spin of the black hole does not influence the tendency of the relationship between the escape probability P and speed v. The escape probability for r−-emitters is shown to be much larger and more sensitive to v than that for r+-emitters.
Figure 2. Escape probability P as function of r-emitters’ distance r/M to the black hole with different spins. We consider the r+-emitters at inclination angle θ = π/2 and speed v = 0.5c. |
Figure 3. Escape probability as function of inclination θ for different spin of Kerr black hole. We consider r+-emitters located at 10M with a fixed speed v = 0.9c |
Figure 4. Escape probability P as function of speed v/c for different velocities of emitters located at r = 10M, θ = π/2. These velocities are along the positive direction. The spin Kerr black hole is set to be a = 0.1M (left panel), a = 0.5M (medium panel), and a = 0.9M (right panel), respectively. |
Figure 5. Escape probability is plotted as a function of θ for different emitters located at r = 6M along the positive direction, both in low and high speeds. The spin of Kerr black hole is set to be a = 0.99M. |
Figure 6. Escape probability P as function of speed v/c of emitters, but for the different 4-velocities in negative direction. We consider the emitters are located at r = 10M, θ = π/2, and the black hole spin a = 0.1M (left panel), a = 0.5M (medium panel), and a = 0.9M (right panel), respectively. |
Figure 7 shows the escape probability as function of the θ for different emitters along negative direction in low and high speed. Differed from the results shown in figure 5, the escape probability decreases with the inclination θ for emitters along negative direction. It is also found that the tendency of the relationship between P and θ does no affect by the speed of emitters. Figure 8 shows the escape probability as function of r−-emitters’ distance to black hole. As expected, the escape probability increases with the distance.
Figure 7. Escape probability as function of the θ for different emitters along negative direction in low and high speed. The spin of Kerr black hole is set to be a = 0.99M. |
Figure 8. Escape probability P as function of emitters’ distance r/M for different black hole spin. We consider the r−-emitters are located in θ = π/2 with speed v = 0.5c in negative direction. |
4. Discussions
In this paper, we proposed a novel approach for calculating the escape probability for a point emitter in arbitrary velocities near a Kerr black hole. For illustration, we showed the escape probability as function of inclination angle θ, emitter’s distance to the black hole r, and emitter’s velocities (including speed and motion direction).
It was found that the velocities of the emitters can significantly affect the escape probability. The escape probability decrease significantly with speed of the out-going emitters, and vice versa. It suggests that the in-falling emitters might have more possible to be observed by distant observers than expected. As expected, it is shown that the escape probability increases with the distance of the emitters. It is consistent with our intuitive that the emission from farther emitters can more likely escape.