We obtain exact solutions to the field equations for five-dimensional locally rotationally symmetric (LRS) Bianchi type-I spacetime in the $f(R,T)$ theory of gravity, where specifically, the following three cases are considered: (i) $f(R,T)=\mu (R+T)$, (ii) $f(R,T)=R\mu +RT{\mu }^2$, and (iii) $f(R,T)=R+\mu R^2+\mu T$, where R and T, respectively, are the Ricci scalar and trace of the energy–momentum tensor. It is found that the equation of state (EOS) parameter w is governed by the parameter $\mu$ involved in the $f(R,T)$ expressions. We fine-tune the parameter $\mu$ to obtain the effect of phantom energy in the model. However, we also restrict this parameter to obtain a stable model of the universe.

This paper investigates the geodesics of the Schwarzschild black hole modified with the quintessence field and modeled as a barotropic fluid with a constant equation of state derived in Boyer Lindquist coordinates with natural units G = c = 1. Hamilton–Jacobi formalization is applied to investigate the geodesics in both static and nonstatic black hole spacetime. It specifically examines the formation of bound orbits of test particles around the black hole, highlighting that these orbits result from the interplay between the gravitational pull of the black hole and a repulsive force associated with quintessence. The thermodynamic properties of a black hole are explored. Angular momentum and energy of the particle in black hole gravitation are also discussed. The turnaround radius is found to be influenced by the value of the quintessential normalization constant (γ), which affects gravitational interactions in various scenarios. Finally, the stability of the orbit is analyzed using the Lyapunov component.

In this work, we introduce an electric charge in anti-de Sitter (AdS) space with a Reissner–Nordstrom–AdS black hole. The primary purpose of the present work is to review and describe the time-like and null geodesic behavior of the black hole. Here, we analyze the lapse function in order to gain some insights about the event horizon of the black hole and the geometrization of the space–time. We also present black hole horizons in 3D form. To have a deeper understanding of the motions of photon-like and massive particles in black hole gravity, we have derived some types of effective potentials. We have also examined the effect of variation of Newtonian constant G in almost every case, which allows us to identify gravity’s nature and its role in the structure of the universe. Finally, we give the Lyapunov exponent for orbital balance for familiarization with it and found that the stability of the circular orbit of the particle moving in the field of gravitation of black hole is conditional and the condition of stability of the orbit is 4 Q² − 9 GM² ⩾ 0 and G > 0.