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Register a new userWeak deflection angle of a dirty 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.
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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.
In this paper, we examine the effect of dark matter to a Kerr black hole of mass $m$. The metric is derived using the Newman-Janis algorithm, where the seed metric originates from the Schwarzschild black hole surrounded by a spherical shell of dark matter with mass $M$ and thickness $Delta r_{s}$. The seed metric is also described in terms of a piecewise mass function with three different conditions. Specializing in the non-trivial case where the observer resides inside the dark matter shell, we analyzed how the effective mass of the black hole environment affects the basic black hole properties. A high concentration of dark matter near the rotating black hole is needed to have considerable deviations on the horizons, ergosphere, and photonsphere radius. The time-like geodesic, however, shows more sensitivity to deviation even at very low dark matter density. Further, the location of energy extraction via the Penrose process is also shown to remain unchanged. With how the dark matter distribution is described in the mass function, and the complexity of how the shadow radius is defined for a Kerr black hole, deriving an analytic expression for $Delta r_{s}$ as a condition for notable dark matter effects to occur remains inconvenient.
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.
A modified Hayward black hole is a nonsingular black hole. It is proposed to form when the pressure generated by quantum gravity can stop matters collapse as the matter reaches Planck density. Strong deflection gravitational lensing happening nearby its event horizon might provide some clues of these quantum effects in its central core. We investigate observables of the strong deflection lensing, including angular separations, brightness differences and time delays between its relativistic images, and estimate their values for the supermassive black hole in the Galactic center. We find that it is possible to distinguish the modified Hayward black hole from a Schwarzschild one, but it demands very high resolution beyond current stage.
The gravitational field of supermassive black holes is able to strongly bend light rays emitted by nearby sources. When the deflection angle exceeds $pi$, gravitational lensing can be analytically approximated by the so-called strong deflection limit. In this paper we remove the conventional assumption of sources very far from the black hole, considering the distance of the source as an additional parameter in the lensing problem to be treated exactly. We find expressions for critical curves, caustics and all lensing observables valid for any position of the source up to the horizon. After analyzing the spherically symmetric case we focus on the Kerr black hole, for which we present an analytical 3-dimensional description of the higher order caustic tubes.
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