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The metric of the Schwarzschild BH in the string cloud context is given by [37]
$ {\rm d}s^{2}=-f(r){\rm d}t^{2}+\frac{1}{f(r)}{\rm d}r^{2}+r^{2}{\rm d}\theta^{2}+r^{2}\sin^{2}\theta {\rm d}\phi^{2}, $
(1) where
$ f(r) $ is the metric potential, which is given as$ f(r)=1-a-\frac{2M}{r}, $
(2) where M represents the BH mass, and a is a parameter describing the string cloud. It is worth noting that as a approaches zero, the BH degenerates into a Schwarzschild BH. Therefore, the analysis also includes the scenario of a Schwarzschild BH. Using Eq. (2), we can express the radius of the BH horizon as follows:
$ r_{\rm H}=\frac{2M}{1-a}. $
(3) The string cloud model can provide an explanation for the field theory resulting from the distance interaction between particles, which is associated with a unique behavior of the gravitational field. Assuming that the gravitational field can be created by the elements of the string, we concentrate on the scenario where
$ a<1 $ , as it behaves as an attractive gravitational charge [47].The motion of photons satisfies the Euler-Lagrangian equation
$ \frac{{\rm d}}{{\rm d}\lambda}\Bigg(\frac{\partial {\rm \mathcal{L}}}{\partial \dot{x}^{\rm \alpha}}\Bigg) = \frac{\partial {\rm \mathcal{L}}}{\partial x^{\rm \alpha}}, $
(4) where λ is the affine parameter, and
$ \dot{x}^{\alpha} $ is the four-velocity of the photon. We only consider the photons that move on the equatorial plane; hence, the Lagrangian equation is$ \mathcal{L}=-\frac{1}{2}g_{\rm \alpha \beta}\frac{{\rm d} x^{\rm \alpha}}{{\rm d} \lambda}\frac{{\rm d} x^{\beta}}{{\rm d} \lambda}=0. $
(5) The energy and the angular momentum of the photons are conserved quantities, i.e.,
$ E=-g_{t t}\frac{{\rm d} t}{{\rm d} \lambda} = f(r)\frac{{\rm d} t}{{\rm d} \lambda},\; \; \; \; L=g_{\rm \phi \rm \phi}\frac{{\rm d} \phi}{{\rm d} \lambda}=r^{2} \frac{{\rm d} \phi}{{\rm d} \lambda}. $
(6) The four-velocity of the time, the azimuthal angle, and the radial components can be obtained by using Eqs. (2)–(5); we have
$ \begin{eqnarray} &&\frac{{\rm d} t}{{\rm d} \lambda}=\frac{1}{b}\Bigg(1- a - \frac{2M}{r}\Bigg)^{-1},\; \; \; \frac{{\rm d} \phi}{{\rm d} \lambda}=\frac{1}{r^{2}}, \end{eqnarray} $
(7) $ \begin{eqnarray} &&\frac{{\rm d} r}{{\rm d} \lambda}=\pm\sqrt{\frac{1}{b^{2}}-\frac{1}{r^{2}}\Bigg(1- a - \frac{2M}{r}\Bigg)}. \end{eqnarray} $
(8) Here, the symbol "
$ \pm $ " indicates that the photons move radially outward ($ - $ ) or inward ($ + $ ), and b is the impact parameter, which is defined as$ b\equiv L/E $ . Therefore, the effective potential of the Schwarzschild string cloud BH is$ {\rm V}_{\rm eff}=\frac{1}{r^{2}}\Bigg(1- a - \frac{2M}{r}\Bigg). $
(9) The radius of the photon ring can be obtained from the relations
$ {\rm V}_{\rm eff}=\dfrac{1}{b_{\rm c}} $ and$ {\rm V}_{\rm eff}'=0 $ . Thus, the radius of the photon sphere is$ r_{\rm s}=\dfrac{3M}{1-a} $ , and the critical impact parameter is$ b_{\rm c}=\dfrac{3\sqrt{3}M}{\sqrt{1-3a+3a^{2}-a^{3}}} $ . According to Eqs. (8) and (9), we have$ \Omega(u) \equiv \Bigg(\frac{{\rm d} u}{{\rm d} \phi}\Bigg)^{2} = 2 M u^{3} + au^{2} - u^{2} + \frac{1}{b^{2}}, $
(10) where u is defined as
$ u \equiv 1/r $ . If$ r>r_{s} $ , both photons with$ b>b_{c} $ and$ b<b_{c} $ can reach the distant observer at infinity. Therefore, in the case where$ b>b_{c} $ , we only consider the light that can reach the observer's plane. This is because when$ b<b_{c} $ , the light enters the BH and ultimately falls into the singularity. According the Cardano formula, Eq. (10) can be rewritten as$ \begin{aligned}[b]& 2Mu^{3}+au^{2}-u^{2}+\frac{1}{b^{2}} \equiv 2 M G(u)\\=& 2M (u-u_{1})(u-u_{2})(u-u_{3}). \end{aligned} $
(11) Here, the cubic polynomial
$ G(u) $ has two positive roots and one negative root, satisfying$ u_{1} \leq 0 < u_{2} < u_{3} $ . Following Ref. [20], these three roots can be given by the periastron distance P. By introducing a parameter$Q^{2} \equiv (P-2M) (P+6M)$ , one can obtain$ u_{1}=\frac{P-2M-Q}{4MP},\; \; \; u_{2}=\frac{1}{P},\; \; \; u_{3}=\frac{P-2M+Q}{4MP}, $
(12) and
$ b^{2}=\frac{1}{\dfrac{P-2M}{P^{3}} - a u^{2}}. $
(13) Hence, we can obtain the value of parameter b at infinity for a given periastron distance P, and vice versa. Furthermore, the bending angle of the light ray is
$ \psi (u) = \sqrt{\frac{2}{M}} \int_{0}^{u_{2}} \frac{{\rm d}u}{\sqrt{(u-u_{1})(u-u_{2})(u-u_{3})}} - \pi. $
(14) By converting the above equation into an elliptic integral, one obtains
$ \psi (u) = \sqrt{\frac{2}{M}} \Bigg(\frac{2 F(\Psi_{1},k)}{\sqrt{u_{3}-u_{1}}} - \frac{2 F(\Psi_{2},k)}{\sqrt{u_{3}-u_{1}}}\Bigg) - \pi, $
(15) where
$ \Psi_{1}=\dfrac{\pi}{2} $ ,$ \Psi_{2}=\sin^{-1}\sqrt{\dfrac{-u_{1}}{u_{2}-u_{1}}} $ , and$ k=\sqrt{\dfrac{u_{2}-u_{1}}{u_{3}-u_{1}}} $ . According to Eq. (12), the total change of the bending angle is$ \psi (u) = 2 \sqrt{\frac{P}{Q}} \Big(K(k)- F(\Psi_{2},k)\Big) - \pi, $
(16) where
$ K(k) $ is the complete elliptic integral of the first kind. The above calculation is based on Luminet's semi-analytic method for calculating the total bending angle of the light ray [20]. However, there are other methods for calculating the bending angle in strong fields, particularly the strong deflection limit proposed by Bozza [66]. In this limit, the deflection angle can be determined as follows:$ \phi (x) = - \overline a \log \Big(\frac{{\theta {D_{\rm OL}}}}{{{b_{\rm c}}}} - 1 \Big) + \overline b. $
(17) Here,
$ \overline a $ and$ \overline b $ are the strong deflection limit coefficients, which satisfy$ \overline a = \frac{{R(0,{x_{m}})}}{{2\sqrt {{\beta x_{m}}} }},\; \; \; \; \; \; \overline b = - \pi + {b_{\rm R}} + \overline a \log \frac{{2{\beta_{m}}}}{{{y_{m}}}},$
(18) where the subscript m denotes the value at
$x = x_{m}$ , and$ {b_{\rm R}} = \int_{0}^{1} g(z,{x_{ m}}){\rm d}z, $
(19) $ R(z,{x_{m}}) = \frac{2 \sqrt{B y}}{Cd(A)} (1-y_{m}) \sqrt{C_{ m}} $
(20) $ g(z,x_{m}) = R(z,x_{m})f(z,x_{m}) - R(0,x_{ m})f_{m}(z,x_{m}), $
(21) $ f(z,x_{m}) = \frac{1}{\sqrt{y_{ m}}-\big((1-y_{m})z + y_{m}\big)\dfrac{C_{0}}{C}}, $
(22) in which
$ A(x) $ ,$ B(x) $ , the$ C(x) $ are obtained from$ f(x) $ deformation; we have$ \begin{aligned}[b]& A(x) = B(x)^{-1} = 1 - \frac{1}{r} - a,\; \; \; C(x) = x^2,\; \; \;\\& y = A(x),\; \; \; z =\frac{y-y_{m}}{1 - y_{ m}}. \end{aligned} $
(23) The researchers extensively investigated the bending angle of light rays in strong gravitational fields, by using various modified gravity contexts as a means of obtaining the angle. This is the second method they employed [67–84].
By using a ray-tracing code, we can observe the trajectory of a light ray under different string cloud parameters, as illustrated in Fig. 1 through the bending angle function. It is evident from the figure that a larger value of a corresponds to a larger radius of the black disk. Light rays in proximity to the BH with a large a may be extremely curved, resulting in an increase in the light ray density for a distant observer.
Figure 1. (color online) The trajectory of the light ray for different a values with
$M=1$ in polar coordinates$(r,\phi)$ . Left Panel– string cloud parameter$a=0$ (Schwarzschild BH), Middle Panel– string cloud parameter$a=0.2$ , and Right Panel– magnetic charge$a=0.5$ . The green, blue, and red lines correspond to$b>b_{\rm c}$ ,$b=b_{\rm c}$ , and$b<b_{\rm c}$ , respectively. The BH is shown as a black disk. -
In this section, we employ the Luminet method to examine the direct and secondary images of the Schwarzschild string cloud BH. Our assumption is that the black hole is surrounded by an optically thick and geometrically thin accretion disk. The radiation is emitted from point M (the emission plane) with coordinates
$ (r,\varphi) $ and travels to point m (the observation plane) with coordinates$ (b,\alpha) $ . The coordinate system of the accretion disk can be seen in Fig. 2.Figure 2. (color online) The coordinate system. The design of this figure is taken from Ref. [20].
Two images of the accretion disk are derived on the observation plane, i.e., the direct image with coordinates
$ (b^{(d)},\alpha) $ and the secondary image with coordinates$ (b^{(s)},\alpha+\pi) $ , as reported in [20]. Assuming that the deflection angle from M to the observer is γ and the observer's inclination angle is$ \theta_{0} $ , we can derive the following equation:$ \begin{array}{l} \cos \alpha = \cos \gamma \sqrt{\cos^{2} \alpha + \cot^{2} \theta_{0}}. \end{array} $
(24) For the direct image of the accretion disk, Eq. (11) can be rewritten as
$ \gamma = \frac{1}{\sqrt{2M}} \int_{0}^{1/r}\frac{1}{\sqrt{G(u)}}{\rm d}u = 2 \sqrt{\frac{P}{Q}} \Big(F(\zeta_{\rm r},k) - F(\zeta_{\rm \infty}, k)\Big), $
(25) where
$F(\zeta_{r},k)$ and$ F(\zeta_{\rm \infty}) $ are the elliptical integrals, and$ k^{2}=\dfrac{Q-P+6M}{2Q} $ ,$\sin^{2}\zeta_{r}=\dfrac{Q-P+2M+4MP/r}{Q-P+6M}$ , and$ \sin^{2}\zeta_{\rm \infty}=\dfrac{Q-P+2M}{Q-P+6M} $ [20]. Thus, the radius r as a function of α and P is obtained; we have$ \frac{1}{r} = \frac{P-2M-Q}{4MP} + \frac{Q-P+6M}{4MP} sn^{2}\Bigg(\frac{\gamma}{2}\sqrt{\frac{Q}{P}} + F(\zeta_{\rm \infty}, k)\Bigg). $
(26) According to this equation, the iso-radial curves for a given angle
$ \theta_{0} $ are presented. For the$ (n+1)th $ order image of the accretion disk, Eq. (25) satisfies$ 2n \pi - \gamma = 2 \sqrt{\frac{P}{Q}} \Big(2K(k)- F(\zeta_{r},k) - F(\zeta_{\rm \infty}, k)\Big), $
(27) where
$ K(k) $ is the complete elliptic integral.Figure 3 displays the direct and secondary images of circular rings orbiting a Schwarzschild string cloud BH, located at various distances from the center (
$ r=16M $ ,$ r=18M $ ,$ r=20M $ , and$ r=22M $ , from inside to outside) and at different inclination angles ($ \theta_{0}=15^{\circ} $ ,$ 45^{\circ} $ , and$ 75^{\circ} $ ). The direct images correspond to photons emitted in directions above the equatorial plane, and the secondary images correspond to photons emitted in directions below the equatorial plane. It is evident that as the string cloud parameter (a) increases, the direct image expands outward, increasing the shadow radius. Conversely, the shape and size of the secondary images decrease. Additionally, an increased inclination angle results in a clearer separation between the direct and secondary images.Figure 3. (color online) Direct and secondary images of the thin accretion disk around the Schwarzschild string cloud BH with the following inclination angles of the observer:
$\theta_{0}=15^{\circ},45^{\circ},75^{\circ}$ . Red curves represent the direct image, and blue curves represent the secondary image of the disk. Top Panel – string cloud parameter$a=0.2$ and Bottom Panel – string cloud parameter$a=0.5$ . We set$M=1$ . -
According to Ref. [85], the radial dependence of energy flux radiated by a thin accretion disk around a BH is derived. The flux of the radiant energy is given by [86]
$ F= - \frac{\dot{M}}{4\pi \sqrt{\rm -g}} \frac{\Omega_{,r}}{(E-\Omega L)^{2}} \int_{r_{\rm in}}^{r} (E- \Omega L)L_{,r} {\rm d} r, $
(28) where
$ \dot{M} $ represents the mass accretion rate,$ \rm g $ represents the determinant of the metric, and$ r_{\rm in} $ represents the inner edge of the accretion disk. E, Ω, and L represent the energy, angular momentum, and angular velocity, respectively.For a static spherically symmetric metric
${\rm d}s^{2}=g_{tt} {\rm d}t^{2} + g_{\rm \phi\phi} {\rm d}\phi^{2} + g_{rr}{\rm d}r^{2} + g_{\rm \theta\theta}{\rm d}\theta^{2}$ , E, L, and Ω are expressed as follows:$ E=-\frac{g_{tt}}{\sqrt{-g_{tt}-g_{\rm \phi\phi}\Omega^{2}}},\; \; L=\frac{g_{\rm \phi\phi}\Omega}{\sqrt{-g_{ tt}-g_{\rm \phi\phi}\Omega^{2}}},$
(29) $ \Omega=\frac{{\rm d}\phi}{{\rm d}t}=\sqrt{-\frac{g_{tt,r}}{g_{\rm \phi\phi,r}}}. $
(30) In our situation, the flux of the radiant energy over the disk is given as
$ F = \frac{3 \dot{M} \sqrt{\dfrac{M}{r^{3}}}}{r (24 M \pi - 8 \pi r + 8 a \pi r)} \int_{r_{\rm in}}^{r} \frac{\sqrt{\dfrac{M}{r^{3}}} r \Big(6M + a r - r \Big)}{6 M + 2 r \Big(a - 1 \Big)}{\rm d}r. $
(31) The observed flux
$ F_{\rm obs} $ , differs from the source flux F owing to the effect of redshift. As mentioned in Ref. [20], the observed flux$ F_{\rm obs} $ can be expressed as follows:$ F_{\rm obs} = \frac{F}{(1+z)^{4}}. $
(32) When a photon is emitted by a particle orbiting around a BH, the energy of the photon can be obtained by projecting its 4-momentum, which is denoted as p, onto the 4-velocity, which is denoted as μ, of the emitting particle [25]. Therefore, we have
$ E_{\rm em} = p_{t} \mu^{t} + p_{\rm \phi} \mu^{\rm \phi} = p_{t} \mu^{ t} \Bigg(1 + \Omega \frac{p_{\rm \phi}}{p_{t}}\Bigg), $
(33) where
$p_{t}$ and$ p_{\rm \phi} $ represent the energy ($p_{t}=-E$ ) and angular momentum ($ p_{\rm \phi}=L $ ) of the photon, respectively, which are conserved along the geodesic owing to the Killing symmetries of the spacetime. When the observer is located at infinity, the ratio of$p_{t}$ to$ p_{\rm \phi} $ gives the impact parameter of the photon relative to the z-axis. Using Eq. (24), we have$ \begin{array}{l} \sin \theta_{0} \cos \alpha = \cos \gamma \sin \beta, \end{array} $
(34) and one can obtain
$ \frac{p_{t}}{p_{\rm \phi}} = b \sin \theta_{0} \sin \alpha. $
(35) Hence, the redshift factor is
$ 1 + z = \frac{E_{\rm em}}{E_{\rm obs}} = \frac{1 + b \Omega \cos \beta}{\sqrt{-g_{tt} - 2 g_{t \phi} - g_{\rm \phi\phi}}}. $
(36) For the Schwarzschild BH within the string cloud context, the above equation can be written as
$ 1+z = \frac{\sqrt{1 - a - \dfrac{3M}{r}} r \big(1 + b \sqrt{\dfrac{M}{r^{3}}} \sin \alpha \sin \theta_{0}\big)}{3M + (a-1) r } + 1. $
(37) Figure 4 shows the redshift distribution of the direct images. Owing to the presence of the BH, blueshift and redshift exist simultaneously in the direct image. As illustrated, an increase in the inclination angle
$ \theta_{0} $ results in the blueshift exceeding the gravitational redshift component. Additionally, a decrease in the string cloud parameter a leads to an expansion of the region of the redshift distribution.Figure 4. (color online) Redshift distribution (curves of constant redshift z) of the thin accretion disk around the Schwarzschild string cloud BH with the following inclination angles of the observer:
$\theta_{0}=15^{\circ},45^{\circ},75^{\circ}$ . The inner edge of the disk is at$r_{\rm in}=16M$ , and the outer edge of the disk is at$r=22M$ . Left Panel – string cloud parameter$a=0.2$ and Right Panel – string cloud parameter$a=0.5$ . The BH mass is taken as$M=1$ .Figures 5 and 6 depict the flux distributions of the direct and secondary images, respectively. It can be observed that when the inclination angle is small, the flux distribution is symmetric. As the inclination angle increases, the asymmetry of the flux distribution increases. Furthermore, the decrease in the string cloud parameter a causes an expansion of the region of the flux distribution for the direct and secondary images.
Figure 5. (color online) Total observed flux of the direct image for the thin accretion disk around the Schwarzschild string cloud BH with the following inclination angles of the observer:
$\theta_{0}=15^{\circ},45^{\circ},75^{\circ}$ . The inner edge of the disk is at$r_{\rm in}=16M$ , and the outer edge of the disk is at$r=22M$ . Left Panel – string cloud parameter$a=0.2$ and Right Panel – string cloud parameter$a=0.5$ . The BH mass is taken as$M=1$ .Figure 6. (color online) Total observed flux of the secondary image for the thin accretion disk around the Schwarzschild string cloud BH with the following inclination angles of the observer:
$\theta_{0}=15^{\circ},45^{\circ},75^{\circ}$ . The inner edge of the disk is at$r_{\rm in}=16M$ , and the outer edge of the disk is at$r=22M$ . Left Panel – string cloud parameter$a=0.2$ and Right Panel – string cloud parameter$a=0.5$ . The BH mass is taken as$M=1$ .We generate a Schwarzschild string cloud BH image through a numerical simulation, as shown in Fig. 7, and compare the results with those obtained by the EHT. It is noteworthy that the image of the target BH bears a resemblance to the one portrayed in the Hollywood movie "Interstellar." While the observation of the BH shadow is dependent on the EHT resolution, this basic numerical simulation provides a rough illustration of the EHT's capabilities.
Figure 7. (color online) Numerical simulation image of the Schwarzschild string cloud BH. The inner edge of the disk is at
$r_{\rm in}=16M$ , and the outer edge of the disk is at$r=22M$ . Top Panel – string cloud parameter$a=0.2$ and Bottom Panel – string cloud parameter$a=0.5$ . The BH mass is taken as$M=1$ .
Optical appearance of the Schwarzschild black hole in the string cloud context
- Received Date: 2022-12-05
- Available Online: 2023-06-15
Abstract: The image of a black hole (BH) consists of direct and secondary images that depend on the observer position. We investigate the optical appearance of a Schwarzschild BH in the context of a string cloud to reveal how the BH's observable characteristics are influenced by the inclination angle, string cloud parameter, and impact parameter. Following Luminet's work [Astron. Astrophys. 75, 228 (1979)], we adopt a semi-analytic method to calculate the total bending angle of the light ray and derive the direct and secondary images of the Schwarzschild string cloud BH. Our results show that an increase in the inclination angle leads to a more pronounced separation of the images. We consider the gravitational redshift and present the redshift distribution of the direct image while illustrating the flux distribution. We observe that the direct image exhibits blueshift and redshift simultaneously, and the asymmetry of the flux distribution increases with the inclination angle. Finally, we obtain the Schwarzschild string cloud BH image via a numerical simulation, which provides an approximate illustration of the EHT resolution.