Masking Black Hole Spin with a Modified Chaplygin Gas Envelope: Radiative Degeneracies from a Phenomenological Three-Region Spacetime

Abstract

Theoretical interpretations of horizon-scale observations often rely on the idealized assumption of an isolated vacuum Kerr geometry. However, astrophysical black holes are expected to be embedded within dense dark matter distributions that can modify the local spacetime geometry. In this work, we propose a theoretical framework to model a rotating compact object surrounded by a bounded dark matter envelope governed by a Modified Chaplygin Gas (MCG) equation of state. To ensure strict adherence to the Einstein Field Equations, we construct a piece-wise, three-region spacetime using a fully coupled Tolman-Oppenheimer-Volkoff (TOV) integration, allowing the fluid's pressure to taper naturally to zero and dynamically define the outer boundary. Rotation is introduced via a pressure-corrected Kerr-form ansatz where the temporal component is obtained directly from the integrated hydrostatic potential. Using this geometrically rigorous configuration, which explicitly evaluates the exact 4D equatorial metric determinant rather than relying on vacuum approximations, we solve the circular equatorial geodesics and determine the innermost stable circular orbit (ISCO). Evaluating the thin accretion disk thermodynamics via the Novikov-Thorne formalism reveals that the deep gravitational potential well of the MCG envelope acts as a strong driver for viscous dissipation, systematically shifting the peak thermal flux, effective temperature, and multi-colour blackbody spectral luminosity to higher energy bands. Furthermore, we identify a clear structural degeneracy: a static or slowly rotating black hole embedded in a dense MCG structure can elevate radiative efficiencies up to $η\approx 6.5\%$. This framework is presented as a structured proposal to quantify environmental systematic uncertainties in standard black hole spin-estimation techniques.