The presence of dark matter around a black hole remarkably affects its spacetime. We consider the effects of dark matter on the shadow of a new solution to the Einstein equations that describes a rotating black hole in the background of perfect dark matter fluid, along with its extension to nonzero cosmological constant $\mathrm{\Lambda }$. Working in Boyer-Lindquist coordinates, we consider the effects of the perfect dark matter fluid parameter $\alpha$ on the shadow cast by a black hole with respect to an observer at position $(r_o,{\theta }_o)$. By applying the Gauss-Bonnet theorem to the optical geometry, we find that notable distortions from a Kerr black hole can occur. We describe their dependence on $\alpha$ and $\mathrm{\Lambda }$.

In this paper we present a new black hole solution surrounded by dark matter (DM) halo in the galactic center using the mass model of M87 and that coming from the universal rotation curve (URC) dark matter profile representing family of spiral galaxies. In both cases the DM halo density is cored with a size $r_0$ and a central density ${\rho }_0$: $\rho (r)={\rho }_0/(1+r/r_0)(1+(r/r_0{)}^2)$. Since $r_0{\rho }_0=120{\mathrm{M}}_{\odot }/{\mathrm{pc}}^2$ [ Mon. Not. R. Astron. Soc. 397, 1169 (2009)], then by varying the central density one can reproduce the DM profile in any spiral. Using the Newman-Jains method we extend our solution to obtain a rotating black hole surrounded by dark matter halo. We find that the apparent shape of the shadow beside the black hole spin $a$, it also depends on the central density of the surrounded dark matter ${\rho }_0$. As a specific example we consider the galaxy M87, with a central density ${\rho }_0=6.9\times {10}^6{\mathrm{M}}_{\odot }/{\mathrm{kpc}}^3$ and a core radius $r_0=91.2\mathrm{kpc}$. In the case of M87, our analyses show that the effect of dark matter on the size of the black hole shadow is almost negligible compared to the shadow size of the Kerr vacuum solution hence the angular diameter $42\mu \mathrm{as}$ remains almost unaltered when the dark matter is considered. For a small totally dark matter dominated spiral such as UGC 7232, we find similar effect of dark matter on the shadow images compared to the M87. However, in specific conditions having a core radius comparable to the black hole mass and dark matter with very high density, we show that the shadow images decreases compared to the Kerr vacuum black hole. The effect of dark matter on the apparent shadow shape can shed some light in future observations as an indirect way to detect dark matter using the shadow images.

In this paper, we construct an effective rotating loop quantum black hole (LQBH) solution, starting from the spherical symmetric LQBH by applying the Newman-Janis algorithm modified by Azreg-Aïnou’s noncomplexification procedure, and study the effects of loop quantum gravity (LQG) on its shadow. Given the rotating LQBH, we discuss its horizon, ergosurface, and regularity as $r\to 0$. Depending on the values of the specific angular momentum $a$ and the polymeric function $P$ arising from LQG, we find that the rotating solution we obtained can represent a regular black hole, a regular extreme black hole, or a regular spacetime without horizon (a non-black-hole solution). We also study the effects of LQG and rotation, and show that, in addition to the specific angular momentum, the polymeric function also causes deformations in the size and shape of the black hole shadow. Interestingly, for a given value of $a$ and inclination angle ${\theta }_0$, the apparent size of the shadow monotonically decreases, and the shadow gets more distorted with increasing $P$. We also consider the effects of $P$ on the deviations from the circularity of the shadow, and find that the deviation from circularity increases with increasing $P$ for fixed values of $a$ and ${\theta }_0$. Additionally, we explore the observational implications of $P$ in comparing with the latest Event Horizon Telescope observation of the supermassive black hole, M87*. The connection between the shadow radius and quasinormal modes in the eikonal limit as well as the deflection of massive particles are also considered.

In this paper, we study the shadow cast by two types of charged and slowly rotating black holes in the Einstein-Æther theory of gravity. These two types of black holes correspond to two specific combinations of the coupling constants of the æther field, i.e., $c_{14}=0$ but $c_{123}\ne 0$ for the first type and $c_{123}=0$ for the second type. For both types of black holes, in addition to the mass and charge of the black holes, we show that the presence of the æther field can also affect the size of the shadow. For the first type of black hole, it is shown that the shadow size increases with the parameter $c_{13}$, while for the second type of black hole, the shadow size still increases with $c_{13}$ but decreases with the parameter $c_{14}$. With these properties of the æther parameters, we also discuss the observational constraints on these parameters by using the data of the first black hole image by the Event Horizon Telescope. In addition, we also explore the effect of the æther field on the deflection angle of light and the time delay by using the Gauss-Bonnet theorem. It is shown that, for a specific combination $c_{123}=0$, the deflection angle/time delay is slightly affected by the æther parameter $c_{13}$ at the leading order.

We investigate the validity of the generalized second law of thermodynamics, in the cosmological scenario where dark energy interacts with both dark matter and radiation. Calculating separately the entropy variation for each fluid component and for the apparent horizon itself, we show that the generalized second law is always and generally valid, independently of the specific interaction form, of the fluids equation-of-state parameters and of the background geometry.