Black hole in cored plummer dark matter environment: Novel solution, light ring, shadow, lensing, lyapunov exponent, eikonal quasinormal modes and thermodynamics-phase transition

Overview

An exact static spherically symmetric black hole solution within a cored Plummer dark matter halo is constructed and analyzed. The investigation encompasses photon dynamics, orbital stability, shadow formation, quasinormal mode behavior, and thermodynamic properties including phase transitions, with systematic comparison to Schwarzschild solutions.

Methods and approach

The black hole geometry is derived from the principle of least action via null geodesics formulation. Orbital stability is characterized through Lyapunov exponent analysis, connecting to quasinormal mode frequency damping in the eikonal limit. Thermodynamic properties including mass function, enthalpy, entropy, temperature, heat capacity, and Gibbs free energy are examined analytically and numerically. Shadow and lensing patterns are computed from null geodesic trajectories in the modified spacetime background.

Key Findings

The cored Plummer dark matter environment modifies photon orbital structure and introduces observable changes in light ring formation and gravitational lensing patterns compared to pure Schwarzschild cases. Lyapunov exponent values are shown to govern the imaginary components of massless quasinormal mode frequencies, establishing the connection between orbital stability and quasi-normal oscillations. Thermodynamic analysis reveals enhanced stability in null geodesics and phase transition behavior absent in standard Schwarzschild solutions, with systematic dependence on dark matter halo parameters affecting the system's Gibbs free energy landscape.

Implications

The analysis demonstrates that dark matter halo profiles substantially modify black hole observables beyond gravitational wave absorption, affecting optical signatures detectable through shadow imaging and lensing observations. The emergence of phase transitions in the thermodynamic structure of black hole-dark matter systems indicates richer thermodynamic behavior than isolated black holes, with potential implications for understanding equilibrium conditions in environments with significant dark matter. The quantitative relationship between Lyapunov exponents and quasinormal mode frequencies provides a dynamical systems perspective on black hole stability metrics within modified gravitational environments.