We study some exact solutions in a $D(\ge 4)$-dimensional Einstein-Born-Infeld theory with a cosmological constant. These solutions are asymptotically de Sitter or anti-de Sitter, depending on the sign of the cosmological constant. Black hole horizon and cosmological horizon in these spacetimes can be a positive, zero or negative constant curvature hypersurface. We discuss the thermodynamics associated with black hole horizon and cosmological horizon. In particular we find that for the Born-Infeld black holes with Ricci flat or hyperbolic horizon in AdS space, they are always thermodynamically stable, and that for the case with a positive constant curvature, there is a critical value for the Born-Infeld parameter, above which the black hole is also always thermodynamically stable, and below which a unstable black hole phase appears. In addition, we show that although the Born-Infeld electrodynamics is nonlinear, both black hole horizon entropy and cosmological horizon entropy can be expressed in terms of the Cardy-Verlinde formula. We also find a factorized solution in the Einstein-Born-Infeld theory, which is a direct product of two constant curvature spaces: one is a two-dimensional de Sitter or anti-de Sitter space, the other is a ($D-2$)-dimensional positive, zero or negative constant curvature space.

The nonminimal coupling of the nonzero vacuum expectation value of the self-interacting antisymmetric Kalb-Ramond field with gravity leads to a power-law hairy black hole having a parameter $s$, which encompasses the Reissner-Nordstrom black hole ($s=1$). We obtain the axially symmetric counterpart of this hairy solution, namely, the rotating Kalb-Ramond black hole, which encompasses, as special cases, Kerr ($s=0$) and Kerr-Newman ($s=1$) black holes. Interestingly, for a set of parameters ($M$, $a$, and $\mathrm{\Gamma }$), there exists an extremal value of the Kalb-Ramond parameter ($s=s_e$), which corresponds to an extremal black hole with degenerate horizons, while for $s<s_e$, it describes a nonextremal black hole with Cauchy and event horizons, and no black hole for $s>s_e$. We find that the extremal value $s_e$ is also influenced by these parameters. The black hole shadow size decreases monotonically and the shape gets more distorted with an increasing $s$; in turn, shadows of rotating Kalb-Ramond black holes are smaller and more distorted than the corresponding Kerr black hole shadows. We investigate the effect of the Kalb-Ramond field on the rotating black hole spacetime geometry and analytically deduced corrections to the light deflection angle from the Kerr and Schwarzschild black hole values. The deflection angle for Sgr A* and the shadow caused by the supermassive black hole M87* are included and compared with analogous results of Kerr black holes. The inferred circularity deviation $\mathrm{\Delta }C\le 0.10$ for the M87* black hole merely constrains the Kalb-Ramond field parameter, whereas shadow angular diameter ${\theta }_d=42\pm 3\mu \mathrm{as}$, within the $1\sigma$ region, places bounds $\mathrm{\Gamma }\le 0.09205$ for $s=1$ and $\mathrm{\Gamma }\le 0.02178$ for $s=3$.

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.

We discuss the horizon properties, shadow cast, and the weak gravitational lensing of charged rotating regular black holes, which in addition to mass ($M$) and rotation parameter ($a$) have an electric charge ($Q$) and magnetic charge ($g$). The considered regular black holes are the generalization of the Kerr ($Q=g=0$) and Kerr-Newman ($g=0$) black holes. Interestingly, for a given parameter set, the apparent size of the shadow monotonically decreases and the shadow gets more distorted with increasing charge parameter $Q$. We put constraints on the black hole parameters with the aid of recent M87* shadow observation. The conserved quantities associated with the rotating regular black holes are calculated and also a brief description of the weak gravitational lensing using the Gauss-Bonnet theorem is presented. Interestingly, the deflection angle decreases with the charge of the black hole. Our results vis-à-vis go over to the Kerr and Kerr-Newman black holes in the appropriate limits.