Astrophysical reality of black hole thermodynamics and dynamics: Transformative influence of Hernquist dark matter distributions

Abstract

We investigate the thermodynamic properties and orbital dynamics of a Schwarzschild black hole (BH) embedded within a Hernquist dark matter (DM) halo, characterized by a central cusp (rho proportional to r(-1)) and rapid radial decay (rho proportional to r(-4)). By deriving an exact solution for this composite system, we demonstrate that the DM environment fundamentally alters the BH's geometric, thermodynamic, and dynamic characteristics. Our analysis reveals that the Hernquist DM halo modifies the event horizon radius, breaks the Ricci-flat character of spacetime, and reduces the Hawking temperature-findings consistently verified through surface gravity, quantum tunneling, and thermodynamic approaches. The effective potential and force calculations show the DM distribution significantly enhances gravitational attraction in the central region, with critical implications for orbital stability. We further extend our investigation through the Generalized Uncertainty Principle (GUP), deriving quantum-corrected thermodynamic quantities that exhibit logarithmic entropy modifications and enhanced temperature suppression. Additionally, we explore the weak deflection of light in this composite spacetime and examine how plasma presence further modifies gravitational lensing, providing distinctive chromatic signatures for observational tests. Our comprehensive analysis of extended thermodynamic parameters - including internal energy, Helmholtz and Gibbs free energies, pressure, enthalpy, and heat capacity - reveals complex dependencies on both horizon radius and DM density, though the heat capacity remains universally negative, indicating fundamental thermodynamic instability despite environmental modifications. These results demonstrate that realistic astrophysical environments significantly transform BH physics beyond idealized vacuum solutions, with potential observational signatures in gravitational lensing, accretion processes, and BH shadow characteristics.