In this paper, we investigate the optical behaviors of a quantum Schwarzschild black hole with a spacetime solution including a parameter $\lambda$ that encodes its discretization. Concretly, we derive the effective potential of such solution. In particular, we study the circular orbits around the quantum black hole. Indeed, we find that the effective potential is characterized by a minimum and a maximum yielding a double photon spheres denoted by $r_{p_1}, r_{p_2}$ respectively. Then, we analyse the double shadow behaviors as a function of the parameter $\lambda$ where we show that it controles the shadow circular size. An inspection of the Innermost Stable Circular Orbits (ISCO) shows that the radius $r_{ISCO}$ increases as a function of $\lambda$. Besides, we find that such radius is equal to $6M$ for an angular momentum $L=2\sqrt{3}$ independently of $\lambda$. A numerical analysis shows that the photon sphere of radius $r_{p_1}$ generates a shadow with a radius larger than $r_{ISCO}$. Thus, a truncation of the effective potential is imposed to exclude such behavior. Finally, the $\lambda$-effect is inspect on the deflection angle of such a black hole showing that it increases when higher values of the parameter $\lambda$ are considered. However, such an increase is limited by an upper bound given by $\frac{6 M}{b}$.

In this paper, we generate a rotating solution of the reduced Kiselev black hole through the Newman-Janis formalism. Based on such solution, we remark different shadow behaviors by varying the involved parameters r_k, a, α. Concretely, we observe that the allowed values of the spin parameter a are much less than the usual rotating black holes. By deeply analysing the shadow shapes, we show that comparable shadow shapes emerge for the same ratio a/r_k. On the other hand, we recognize that the parameters a and α governs the shadow geometry while the parameter r_k rules the size of such a quantity. Besides, we notice that an elliptic shadow geometry appears for certain range of relevant parameters. By making contact with the observational side, we provide a constraint on the rotating reduced Kiselev (RRK) black hole parameters. In particular, we find a good compatibility between the theoretical and experimental results. Regarding Hawking radiation, we note that the Kiselev radius r_ k shows a similar behavior to the quintessence filed intensity 𝐜. Concerning the light motion in the vicinity of a RRK black hole, we investigate deeply the deflection by varying the relevant parameters. In particular, we remark that such a quantity decreases by increasing the parameters a and α while the opposite effect is observed when increasing r_k.

In this paper, we construct the slowly rotating case of an asymptotically flat supermassive black hole embedded in dark matter using Newman-Janis procedure. Our analysis is carried with respect to the involved parameters including the halo total mass $M$ and the galaxy's lengthscale $a_0$. Concretly, we investigate the dark matter impact on the effective potential and the photon sphere. In particular, we find that the lengthscale $a_0$ controles such potential values. Indeed, for low $a_0$ values, we find that the halo total mass $M$ decreases the potential values significantly while for high $a_0$ values such impact is diluted. Regarding the shadow aspects, we show that the shadow size is much smaller for high values of $a_0$ while the opposite effect is observed when the halo total mass $M$ is increased. By comparing our case to the slowly rotating case, we notice that the former exhibits a shadow shifted from its center to the left side. Finally, we compute the deflection angle in the weak-limit approximation and inspect the dark matter parameters influence. By ploting such quantity, we observe that one should expect lower bending angle values for black holes in galactic nuclei.

Motivated by recent astrophysical observations, we investigate the shadow behaviors of four-dimensional charged rotating black holes with a cosmological constant. This study is made in terms of a reduced moduli space parameterized by the charge and the rotation parameters. For fixed observers, we analyse in some details the shadow behaviors and the corresponding naked singularities of Kerr–Newman and Kerr–Sen four-dimensional black holes in Anti-de Sitter backgrounds. Then, a comparative discussion is provided by computing the geometrical observables and the energy emission rate.

We study [Formula: see text]-dimensional black holes surrounded by dark energy (DE), embedded in [Formula: see text]-dimensional M-theory/superstring inspired models having [Formula: see text] space–time where [Formula: see text]. We focus first on the thermodynamical Hawking–Page phase transitions, whose microscopical origin is linked to [Formula: see text] coincident [Formula: see text]-branes supposed to live in such inspired models. Interpreting the cosmological constant as the number of colors, we compute various thermodynamical quantities in terms of the brane number, the entropy and the DE contribution. Calculating the ordinary chemical potential conjugated to the number of colors, we show that a generic black hole is more stable for a larger number of branes in lower dimensions. In the presence of DE, however, we find that the DE state parameter [Formula: see text] takes particular values, for [Formula: see text] models, providing nontrivial phase transitions. Then, we examine some optical properties. Concretely, we investigate shadow behaviors of quintessential black holes in terms of [Formula: see text]-brane physics. In terms of certain ratios, we find similar behaviors for critical quantities and shadow radius.

We study the shadows of four-dimensional black holes in M-theory inspired models. We first inspect the influence of M2-branes on such optical aspects for nonrotating solutions. In particular, we show that the M2-brane number can control the circular shadow size. This geometrical behavior is distorted for rotating solutions exhibiting cardioid shapes in certain moduli space regions. Implementing a rotation parameter, we analyze the geometrical shadow deformations. Among others, we recover the circular behaviors for a large M2-brane number. Investigating the energy emission rate at high energies, we find, in a well-defined approximation, that the associated peak decreases with the M2-brane number. Moreover, we investigate a possible connection with observations (from Event Horizon Telescope or future devices) from a particular M-theory compactification by deriving certain constraints on the M2-brane number in the light of the M87* observational parameters.

Motivated by M-theory/superstring inspired models, we investigate certain behaviors of the deflection angle and the shadow geometrical shapes of higher dimensional quintessential black holes associated with two values of the dark energy (DE) state parameter, being omega=-13 omega=-23<i. Concretely, we derive the geodesic equation of photons on such backgrounds. Thanks to the Gauss-Bonnet theorem corresponding to the optical metric, we compute the leading terms of the deflection angle in the so-called weak-limit approximation. After that, we inspect the effect of DE and the space-time dimensiondon the calculated optical quantities. Introducing DE via the field intensitycand the state parameter omega, we find that the shadow size and the deflection angle increase by increasing values of the field intensityc. However, we observe that the high dimensions decrease such quantities for omega-models exhibiting similar behaviors. Then, we consider the effect of the black hole charge, on these optical quantities, by discussing the associated behaviors. The present investigation recovers certain known results appearing in ordinary four dimensional models.

We study $d$-dimensional $AdS$ black holes surrounded by Dark Energy (DE), embedded in $D$-dimensional M-theory/superstring inspired models having $AdS_d \times \mathbb{S}^{d+k}$ space-time with $D=2d+k$. We focus on the thermodynamic Hawking-Page phase transitions of quintessential DE black hole solutions, whose microscopical origin is linked to $N$ coincident $(d-2)$-branes supposed to live in such $(2d+k)$-dimensional models. Interpreting the cosmological constant as the number of colors $\propto N^{\frac{d-1}{2}}$, we calculate various thermodynamical quantities in terms of brane number, entropy and DE contributions. Computing the chemical potential conjugated to the number of colors in the absence of DE, we show that a generic black hole is more stable for a larger number of branes for lower dimensions $d$.In the presence of DE, we find that the DE state parameter $\omega_q$ should take particular values, for $(D,d,k)$ models, providing a non trivial phase transition structure.

We investigate the critical behaviors of four-dimensional Kerr-AdS black holes from quintessential Dark Energy (DE) contributions. Using a moduli space, coordinated by the DE state parameter [Formula: see text] and the quintessence field intensity [Formula: see text], we deal with three different [Formula: see text]-models. By elaborating analytical formulas of relevant thermodynamical quantities denoted by [Formula: see text], we find significant similarities and distinctions. Precisely, for the [Formula: see text]-model, we show that DE contributions stabilize such black holes. For the [Formula: see text]-model, however, we get a reversed DE effect. In the [Formula: see text]-model, Kerr-AdS black holes reveal a resistance regarding the usual DE effects. Exploiting the explicit formulas of such thermodynamical quantities, we give certain physical interpretations for thermal behaviors. Although such relevant distinctions, we show that the [Formula: see text]-models involve similar universal ratios associated with certain critical thermodynamical quantities. Then, we analyze the photon orbits in the presence of DE.

We investigate the thermodynamic properties of the Hawking-Page phase transition of AdS black holes. We present evidence for the existence of two universal critical constants associated with the Hawking-Page (HP) and minimum black hole thermodynamical transition points. These constants are defined by CS=SHP−SminSmin and CT=THP−TminTmin where Smin(SHP) and Tmin(THP) are the minimal (HP phase transition) entropy and temperature, respectively, below which no black hole can exist. For a large class of four dimensional non-rotating black holes, we find CS=2 and CT=2−33. For the rotating case, however, such universal ratios are slightly affected without losing the expected values. Taking small values of the involved rotating parameter, we recover the same constants. Higher dimensional models, with other universal constants, are also discussed in some details.

We investigate phase transitions and critical behaviors of the Kerr-Sen black hole in four dimensions. Computing the involved thermodynamical quantities including the specific heat and using the Ehrenfest scheme, we show that such a black hole undergoes a second-order phase transition. Adopting a new metric form derived from the Gibss free energy scaled by a conformal factor associated with extremal solutions, we calculate the geothermodynamical scalar curvature recovering similar phase transitions. Then, we obtain the scaling laws and the critical exponents, matching perfectly with mean field theory.

Using the canonical typicality method, we reconsider the study of dark energy effects on four-dimensional black holes. Concretely, we investigate the associated influences on the spectrum of various black hole backgrounds including the charged and the rotating ones. For such black hole solutions, we first elaborate analytically the corresponding radiation spectrum, the Hawking temperature and the dark information. Then, we discuss and analyze the corresponding findings. This work, recovering the results of the Schwarzschild black hole, confirms that dark energy can be considered as a cooling system surrounding the black holes providing a colder radiation and a slower Hawking radiation process.