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Register a new userShadow and weak deflection angle of extended uncertainty principle black hole surrounded with dark matter
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We discussed the possible effects of dark matter on a Schwarzschild black hole with extended uncertainty principle (EUP) correction such as the parameter $alpha$ and the large fundamental length scale $L_*$. We surrounded the EUP black hole of mass $m$ with a static spherical shell of dark matter described by the parameters mass $M$, inner radius $r_s$, and thickness $Delta r_s$. Considering only the case where the EUP event horizon coincides $r_s$, the study finds that there is no deviation in the event horizon, which readily implies that the black hole temperature due to the Hawking radiation is independent of any dark matter concentration. In addition, we explored the deviations in the innermost stable circular orbit (ISCO) radius of time-like particles, photonsphere, shadow radius, and weak deflection angle. It is found that time-like orbits are sensitive to deviation even for low values of mass M. A greater dark matter density is needed to have considerable deviations to null orbits. Using the analytic expression for the shadow radius and the approximation $Delta r_s>>r_s$ revealed that $L_*$ should not be any lower than $2m$. To broaden the scope of this study, we also calculated the analytic expression for the weak deflection angle using the Ishihara method to improve the dark matter estimate found via shadow radius. As a result, the estimate improved by a factor of $(1+4alpha m^2/L_*^2)$ due to the EUP correction parameters. The estimates for the shadow radius and weak deflection angle are compared using the estimated values of galactic mass, and the mass of the supermassive black hole (SMBH) at the center of Sgr A* and M87 galaxies. We found out that the analytical estimates are not satisfied in these galaxies, which indicates that the notable deviation due to dark matter is only evident and considerable if the dark matter distribution is near the supermassive black hole.
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In this paper, we present the weak deflection angle in a Schwarzschild black hole of mass $m$ surrounded by the dark matter of mass $M$ and thickness $Delta r_{s}$. The Gauss-Bonnet theorem, formulated for asymptotic spacetimes, is found to be ill-behaved in the third-order of $1/Delta r_{s}$ for very large $Delta r_{s}$. Using the finite-distance for the radial locations of the source and the receiver, we derived the expression for the weak deflection angle up to the third-order of $1/Delta r_{s}$ using Ishihara (textit{et al.}) method. The result showed that the required dark matter thickness is $sim2sqrt{3mM}$ for the deviations in the weak deflection angle to occur. Such thickness requirement is better by a factor of 2 as compared to the deviations in the shadow radius ($simsqrt{3mM}$). It implies that the use of the weak deflection angle in detecting dark matter effects in ones galaxy is better than using any deviations in the shadow radius.
A perturbative method to compute the deflection angle of both timelike and null rays in arbitrary static and spherically symmetric spacetimes in the strong field limit is proposed. The result takes a quasi-series form of $(1-b_c/b)$ where $b$ is the impact parameter and $b_c$ is its critical value, with coefficients of the series explicitly given. This result also naturally takes into account the finite distance effect of both the source and detector, and allows to solve the apparent angles of the relativistic images in a more precise way. From this, the BH angular shadow size is expressed as a simple formula containing metric functions and particle/photon sphere radius. The magnification of the relativistic images were shown to diverge at different values of the source-detector angular coordinate difference, depending on the relation between the source and detector distance from the lens. To verify all these results, we then applied them to the Hayward BH spacetime, concentrating on the effects of its charge parameter $l$ and the asymptotic velocity $v$ of the signal. The BH shadow size were found to decrease slightly as $l$ increase to its critical value, and increase as $v$ decreases from light speed. For the deflection angle and the magnification of the images however, both the increase of $l$ and decrease of $v$ will increase their values.
We consider a static, axially symmetric spacetime describing the superposition of a Schwarzschild black hole (BH) with a thin and heavy accretion disk. The BH-disk configuration is a solution of the Einstein field equations within the Weyl class. The disk is sourced by a distributional energy-momentum tensor and it is located at the equatorial plane. It can be interpreted as two streams of counter-rotating particles, yielding a total vanishing angular momentum. The phenomenology of the composed system depends on two parameters: the fraction of the total mass in the disk, $m$, and the location of the inner edge of the disk, $a$. We start by determining the sub-region of the space of parameters wherein the solution is physical, by requiring the velocity of the disk particles to be sub-luminal and real. Then, we study the null geodesic flow by performing backwards ray-tracing under two scenarios. In the first scenario the composed system is illuminated by the disk and in the second scenario the composed system is illuminated by a far-away celestial sphere. Both cases show that, as $m$ grows, the shadow becomes more prolate. Additionally, the first scenario makes clear that as $m$ grows, for fixed $a$, the geometrically thin disk appears optically enlarged, i.e., thicker, when observed from the equatorial plane. This is to due to light rays that are bent towards the disk, when backwards ray traced. In the second scenario, these light rays can cross the disk (which is assumed to be transparent) and may oscillate up to a few times before reaching the far away celestial sphere. Consequently, an almost equatorial observer sees different patches of the sky near the equatorial plane, as a chaotic mirage. As $mrightarrow 0$ one recovers the standard test, i.e., negligible mass, disk appearance.
We have studied the shadows of a Schwarzschild black hole surrounded by a Bach-Weyl ring through the backward ray-tracing method. The presence of Bach-Weyl ring leads to that the photon dynamical system is non-integrable and then chaos would appear in the photon motion, which affects sharply the black hole shadow. The size and shape the black hole shadow depend on the black hole parameter, the Bach-Weyl ring mass and the Weyl radius between black hole and ring. Some self-similar fractal structures also appear in the black hole shadow, which originates from the chaotic lensing. We also study the change of the image of Bach-Weyl ring with the ring mass and the Weyl radius. Finally, we analyze the invariant manifolds of Lyapunov orbits near the fixed points and discuss further the formation of the shadow of a Schwarzschild black hole with Bach-Weyl ring.
We obtain the shadow cast induced by the rotating black hole with an anisotropic matter. A Killing tensor representing the hidden symmetry is derived explicitly. The existence of separability structure implies a complete integrability of the geodesic motion. We analyze an effective potential around the unstable circular photon orbits to show that one side of the black hole is brighter than the other side. Further, it is shown that the inclusion of the anisotropic matter ($Kr^{2(1-w)}$) has an effect on the observables of the shadow cast. The shadow observables include approximate shadow radius $R_s$, distortion parameter $delta_s$, area of the shadow $A_s$, and oblateness $D_{os}$.
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