The current manuscript deals with the tidal force effects, geodesic deviation, and shadow constraints of the Schwarzschild-like black hole theorised in Starobinsky–Bel-Robinson gravity exhibiting M-theory compactification. In the current analysis, we explore the radial and angular tidal force effects on a radially in-falling particle by the central black hole, which is located in this spacetime. We also numerically solve the geodesic deviation equation and study the variation of the geodesic separation vector with the radial coordinate for two nearby geodesics using suitable initial conditions. All the obtained results are tested for Sag A* and M87* by constraining the value of the stringy gravity parameter $$\beta $$β using the shadow data from the event horizon telescope observations. All the results are compared with Schwarzschild black hole spacetime. In our study, we found that both the radial and angular tidal forces experienced by a particle switch their initial behaviour and turn compressive and stretching, respectively, before reaching the event horizon. The geodesic deviation shows an oscillating trend as well for the chosen initial condition. For the constrained value of $$\beta $$β , we see that the spacetime geometry generated by Sag A* and M87* is effectively same for both Schwarzschild and Starobinsky–Bel-Robinson black hole. Furthermore, we also calculated the angular diameter of the shadow in Starobinsky–Bel-Robinson black hole and compared with the Schwarzschild black hole. It is observed that the angular diameter of shadow for M87* and Sgr A* in Starobinsky–Bel-Robinson black hole is smaller than the Schwarzschild black hole. The calculated results satisfy the event horizon telescope observational constraints.

We investigate the strong gravitational lensing phenomena caused by a black hole with a dark matter halo in the presence of cloud string and quintessence. This study examines strong gravitational lensing with two significant dark matter models: the universal rotation curve model and the cold dark matter model. To do this, we first numerically estimate the strong lensing coefficients and strong deflection angles for both the universal rotation curve and cold dark matter models. It is observed that the deflection angle, denoted as $$\alpha _D$$α D , increases with the parameter $$\alpha $$α while holding the value of $$2M \gamma $$2 M γ constant. Additionally, it increases with the parameter $$2M \gamma $$2 M γ while keeping the value of $$\alpha $$α constant. The strong deflection angle $$\alpha _D$$α D , for the black hole with a dark matter halo, with parameters $$\alpha $$α and $$2\,M \gamma $$2 M γ , greatly enhances the gravitational bending effect and surpasses the corresponding case of the standard Schwarzschild black hole ( $$A=B=0=\alpha =2M\gamma $$A = B = 0 = α = 2 M γ ). Furthermore, we investigate the astrophysical consequences through strong gravitational lensing observations, using examples of two supermassive black holes, namely $$M87^{*}$$M 87 ∗ and $$Sgr A^{*}$$S g r A ∗ , located at the center of nearby galaxies. It is observed that black holes with dark matter halos in the presence of cloud string and quintessence can be quantitatively distinguished and characterized from the standard Schwarzschild black hole ( $$A=B=0=\alpha =2M\gamma $$A = B = 0 = α = 2 M γ ). The findings in our analysis suggest that observational tests for black holes influenced by dark matter halos, cloud string and quintessence are indeed feasible and viable.

We investigate strong gravitational lensing by quantum-improved charged black holes characterized by an additional parameter denoted as $$\omega $$ω , in addition to the mass M and charge Q. Our findings reveal that when both the parameters Q/2M and $$\omega /4M^2$$ω / 4 M 2 increase simultaneously, various astrophysical consequences, such as the deflection angle $$\alpha _D(u)$$α D ( u ) and angular image separation increase. Concurrently, the angular position $$\theta _{\infty }$$θ ∞ , relative magnification $$r_{mag}$$r mag , and the time delay $$\Delta T_{2,1}$$Δ T 2 , 1 between the first and second relativistic images also decrease with the growing values of the parameters Q/2M and $$\omega /4M^2$$ω / 4 M 2 . It is also observed that the Einstein ring $$\theta _1^E$$θ 1 E for the quantum-improved charged black hole is more significant than those for Schwarzschild, quantum-improved Schwarzschild, and Reisner–Nordström black holes. As with supermassive black holes such as $$M87^*$$M 87 ∗ and $$SgrA^*$$S g r A ∗ , it is observed that to be a viable astrophysical black hole, the EHT results constrain the parameter space ( $$\omega /4M^2$$ω / 4 M 2 , Q/2M). Remarkably, the EHT results for $$SgrA^*$$S g r A ∗ impose more stringent limits on the parameter space of quantum-improved charged black holes compared to those established by the EHT results for $$M87^*$$M 87 ∗ . We investigate the radial profiles of orbital and radial harmonic oscillation frequencies as a function of the dimensionless coupling constants and black hole mass. The main characteristics of test particle quasi-periodic oscillations close to stable circular orbits in the black hole equatorial plane are also examined. We study the positioning of resonant radii in the background of quantum-improved charged black holes for high-frequency quasi-periodic oscillations models: warped disc (WD) models, relativistic precession (RP) and its types, and epicyclic resonance (ER) and its variants.

We study the gravitational lensing of acoustic charged black holes in strong and weak field limit approximations. For this purpose, we first numerically obtain the deflection limit coefficients and deflection angle in the strong field limit. We observe that the strong deflection angle α _D increases with increasing magnitude of the charged parameter Q and that the strong deflection angle α _D of an acoustic charged black hole with tuning parameter ξ = 4 is greater than that of a standard Reissner–Nordström black hole (ξ = 0). We also study the astrophysical consequences via strong gravitational lensing by taking the example of various supermassive black holes in the center of several galaxies and observe that the acoustic charged black hole could be quantitatively distinguished from standard Reissner–Nordström (ξ = 0) and standard Schwarzschild (ξ = 0, Q = 0) black holes. Furthermore, by using the Gauss–Bonnet theorem, we derive the weak deflection angle in the background of an acoustic charged black hole in the curved spacetime. We find that, for fixed values of the charged parameter Q and the tuning parameter (ξ = 0 or 4), the weak deflection angle σ _D decreases with the impact parameter b. We also observe that the weak deflection angle σ _D decreases with increasing magnitude of the charged parameter Q for a fixed value of the tuning parameter (ξ = 0 or 4). Our results suggest that the observational test for an acoustic charged black hole is indeed feasible, and it is generalized to the cases of acoustic Schwarzschild (Q = 0), standard Reissner–Nordström (ξ = 0), and standard Schwarzschild (ξ = 0, Q = 0) black holes.