Exploring the impact of dark energy in Finslerian black hole dynamics and observational features

This study explores black hole dynamics within the framework of Finsler geometry, emphasizing the influence of the Finslerian constant $\eta$ and the quintessence dark energy parameter ${\omega }_s$. By extending the conventional black hole metric with Finslerian corrections, we employ the osculating Riemannian approach and Barthel connection to derive a novel black hole metric, termed the Finsler Black Hole metric. The research investigates the effects of these modifications on photon trajectories, gravitational lensing, and shadow formation. The findings indicate that increasing $\eta$ enhances gravitational lensing, reduces the black hole shadow size, and produces distinct photon ring patterns. The analysis also reveals that the effective potential $V\left(r\right)$ significantly influences photon orbits, with higher $\eta$ values pulling the photon sphere closer to the black hole. Furthermore, the study examines static and infalling spherical accretion models, showing that variations in $\eta$ and ${\omega }_s$ substantially impact the intensity distribution and geometry of black hole shadows. The inclusion of the Finslerian vector field $\beta$ adds complexity to the gravitational dynamics. These modifications introduce observable features that distinguish Finslerian black holes from those predicted by General Relativity. Additionally, the incorporation of dark energy through ${\omega }_s$ is shown to affect gravitational behavior and observable phenomena such as bending angles and critical impact parameters. This work provides a geometrical framework for understanding the interplay between Finsler geometry, dark energy, and black hole physics.