Based on AdS/CFT correspondence, our study delineates a crucial correlation between the dynamics of Einstein ring and LQG parameter $\textit{a}$, horizon $$\rho _{h},$$ observational positions and the wave source $$\omega ,$$ which aptly reflects the influence of space-time geometry on the Einstein ring. The augmentation of $\textit{a}$ and $$\rho _{h}$$ leads to the growth of the peak of absolute amplitude of the response function, which is generated due to diffraction of the scalar wave on the AdS boundary. The special optical system helps us to construct holographic images on the screen. The results indicate that the smaller $$\rho _{h}$$ shows a series of concentric striped patterns for an observer, and the brightest ring is close to the boundary and gradually shrinks with the aid of $$\rho _{h}.$$ Our analysis reveals that the Einstein ring, which corresponds to the photon ring in geometric optics, constantly exhibits high levels of observational intensity at the north pole. With changes in observation positions, the ring gradually transforms into a bright arc-like shape and ultimately changes into a light spot on the screen’s left. The impact of variations of $\textit{a}$ on the Einstein ring is also presented in the bright curves, where the peak of the curves slightly moves towards the boundary with the increase of $\textit{a}$. The influence of $$\omega $$ reflects on the ring; the width of the ring sharply reduces as $$\omega $$ grows, which can also be confirmed from the corresponding brightness profiles. Finally, the comparison between wave optics and geometric optics outcomes is examined in depth.

China

Xiao-Xiong Zeng

Pakistan

M. Aslam

Rabia Saleem

Xin-Yun Hu

Schwarzschild-like AdS black holes: holographic imaging and LQG influences on Einstein rings

Using the AdS/CFT correspondence, we investigate the holographic image of an AdS black hole in Einstein-power-Yang-Mills gravity. The AdS boundary hosts a Gaussian oscillation source, which induces a lensed response on the opposite side of the boundary during propagation through bulk spacetime. The optical system enables observers at the north pole to continuously capture holographic images that exhibit an axisymmetric bright ring known as the Einstein ring. As the observation position shifts, the bright ring gradually transforms into a luminous arc and eventually transitions into a light point. Additionally, we examine the impact of variations in relevant physical quantities on the ring and present the corresponding brightness curve. The results indicate that, as the temperature $\textit{T}$ and nonlinear Yang Mills charge parameter $\textit{q}$ increase, the ring radius increases, whereas an increase in chemical potential $\textit{u}$ leads to a decrease in ring radius. However, the peak brightness curve of the ring invariably decreases as the values of $\textit{T}$, $\textit{u}$, and $\textit{q}$ increase, albeit to varying degrees. Comparing the outcomes of geometric optics, we observe that the position of the ring in holography images is consistent with that of the photon ring.

Ke-Jian He

Holographic image features of an AdS black hole in Einstein-power-Yang-Mills gravity * Supported by the National Natural Science Foundation (11875095), the Innovation and Development Joint Foundation of Chongqing Natural Science Foundation (CSTB2022NSCQ-LZX0021), the Basic Research Project of Science and Technology Committee of Chongqing, China (CSTB2023NSCQ-MSX0324), and the Natural Science Foundation of Chongqing, China (CSTB2023NSCQ-MSX0594)

In this work, gravitational lensing in the weak and strong field limits is investigated for black hole spacetime within the framework of Hu–Sawicki ( ) gravity. We employ the Ishihara et al. approach for weak lensing and adopt Bozza’s method for strong lensing to explore the impact of Hu–Sawicki model parameters on lensing phenomenon. The deflection angles are computed and analyzed in both the field limits. Our investigation in the weak as well as the strong lensing reveals that in the case of Hu–Sawicki black holes, photons exhibit divergence at smaller impact parameters for different values of the model parameters compared to the Schwarzschild scenario and the photon experiences negative deflection angle when impact parameter moves towards the larger impact parameter values. Additionally, by calculating strong lensing coefficients we study their behavior with model parameters. The strong lensing key observables associated with the lensing effect viz. the angular position $$\vartheta _{\infty }$$ , angular separation and relative magnification $$r_\text {mag}$$ are estimated numerically by extending the analysis to supermassive black holes $$\text {SgrA}^*$$ and $$\text {M87}^*$$ and analyzed their behavior concerning the parameters for each black hole. The analysis shows that $$\text {SgrA}^*$$ demonstrates larger values of $$\vartheta _{\infty }$$ and relative to $$\text {M87}^*$$ .

India

Gayatri Mohan

Nashiba Parbin

Umananda Goswami

Investigating the effects of gravitational lensing by Hu-Sawicki $$\varvec{f(R)}$$ gravity black holes

We study the propagation of massless fermionic fields, implementing a family of special functions: Heun functions, in solving the wave equation in three three-dimensional backgrounds, including the BTZ black hole in string theory and Lifshitz black hole solutions in conformal gravity and Hu–Sawicki ( ) theory. The main properties of the selected black hole solutions is that their line elements are Weyl related to that of a homogeneous spacetime, whose spatial part possesses Lie symmetry, described by Lobachevsky-type geometry with arbitrary negative Gaussian curvature. Using the Weyl symmetry of massless Dirac action, we consider the perturbation equations of fermionic fields in relation to those of the homogeneous background, which having definite singularities, are transformed into Heun’s equation. We point out the existence of quasinormal modes labeled by the accessory parameter of the Heun function. The distribution of the quasinormal modes has been clarified to satisfy the boundary conditions that require ingoing and decaying waves at the event horizon and conformal infinity, respectively. It turned out that the procedure based on the Heun function, beside reproducing the previously known results obtained via hypergeometric function for the BTZ and Lifshitz black hole solution in conformal gravity, brings up new families of quasinormal frequencies, which can also contain purely imaginary modes. Also, the analysis of the quasinormal modes shows that with the negative imaginary part of complex frequencies $$\omega =\omega _{Re}+i\omega _{Im}$$ , the fermionic perturbations are stable in this background.

Iran

F. Naderi

A. Rezaei-Aghdam

Quasinormal modes of three $$(2+1)$$ -dimensional black holes in string theory, conformal gravity, and Hu–Sawicki

We derive novel black hole solutions in a modified gravity theory, namely the Hu–Sawicki model of f(R) gravity. After obtaining the black hole solution, we study the horizon radius of the black hole from the metric and then analyse the dependence of the model parameters on the horizon. We then use the 6th order WKB method to study the quasinormal modes of oscillations (QNMs) of the black hole perturbed by a scalar field. The dependence of the amplitude and damping part of the QNMs are analysed with respect to variations in model parameters and the error associated with the QNMs are also computed. After that we study some thermodynamic properties associated with the black hole such as its thermodynamic temperature as well as greybody factors. It is found that the black hole has the possibility of showcasing negative temperatures and is thermodynamically unstable for feasible values of model parameters. Then we analyse the geodesics and derive the photon sphere radius as well as the shadow radius of the black hole. The photon radius is independent of the model parameters while shadow radius showed fair amount of dependence on the model parameters. We tried to constrain the parameters with the help of Keck and VLTI observational data and obtained some bounds on m and  $$c_{2}$$  <math> <msub> <mi>c</mi> <mn>2</mn> </msub> </math>   parameters.

Ronit Karmakar

Quasinormal modes, thermodynamics and shadow of black holes in Hu–Sawicki  $$\varvec{f(R)}$$  <math> <mrow> <mi>f</mi> <mo>(</mo> <mi>R</mi> <mo>)</mo> </mrow> </math>   gravity theory

With the help of AdS/CFT correspondence, we derive the desired response function of QFT on the boundary of the non-commutative black hole. Using the virtual optical system with a convex lens, we obtain the Einstein rings of the black hole from the response function. All the results show that the holographic ring always appears with the concentric stripe when the observer located at the north pole. With the change of the observation position, the ring changes into a luminosity-deformed ring, or bright spot. We also investigate the effect of the non-commutative parameter on the ring and find the ring radius becomes larger as the parameter increases. The effect of the temperature on the ring radius is also investigated, it is found that the higher the temperature, the smaller the ring radius. In addition, we also obtain the ingoing angle of the photon via geometric optics, as expected, this angle is consistent well with the angle of the Einstein ring obtained via holography.

Li-Fang Li

Peng Xu

Holographic Einstein rings of non-commutative black holes

We adopt the energy momentum relation of charged particles to study the thermodynamics laws and weak cosmic censorship conjecture of D-dimensional f(R) AdS black holes in different phase spaces by considering charged particle absorption. In the normal phase space, it turns out that the laws of thermodynamic and the weak cosmic censorship conjecture are valid. In the extended phase space, though the first law of thermodynamics is valid, the second law of thermodynamics is invalid. More interestingly, the weak cosmic censorship conjecture is shown to be violated only in higher-dimensional near-extremal f(R) AdS black holes. In addition, the magnitudes of the violations for both the second law and weak cosmic censorship conjecture are dependent on the charge Q, constant scalar curvature $$f'(R_0)$$ <math><mrow><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><msub><mi>R</mi><mn>0</mn></msub><mo>)</mo></mrow></mrow></math> , AdS radius l, dimension parameters p, and their variations.

Guo-Ping Li

Violations of the weak cosmic censorship conjecture in the higher dimensional f(R) black holes with pressure

With the Hamilton-Jacobi equation, we obtain the energy-momentum relation of a charged particle as it is absorbed by the Gauss-Bonnet AdS black hole. On the basis of the energy-momentum relation at the event horizon, we investigate the first law, second law, and weak cosmic censorship conjecture in both the normal phase space and extended phase space. Our results show that the first law, second law as well as the weak cosmic censorship conjecture are valid in the normal phase space for all the initial states are black holes. However, in the extended phase space, the second law is violated for the extremal and near-extremal black holes, and the weak cosmic censorship conjecture is violable for the near-extremal black hole, though the first law is still valid. In addition, in both the normal and extended phase spaces, we find the absorbed particle changes the configuration of the near-extremal black hole, while don't change that of the extremal black hole.