A Smoothly Truncated Dark Matter Spike around a Black Hole in an Einstein-Cluster Model

Abstract

We introduce a finite dark matter spike around a static, spherically symmetric black hole, with a general power-law index and an Einstein-cluster stress tensor. Smooth inner depletion and outer truncation ensure that the density and its first radial derivative vanish at both boundaries, while a single normalization preserves the prescribed exterior mass. The resulting spacetime contains an inner vacuum gap, a matter region with vanishing radial pressure, and an outer Schwarzschild vacuum. We derive the enclosed mass, density maximum, local logarithmic slope, and matched metric without surface layers, including exact polynomial-root representations of the matter-region metric for spike indices $7/3$ and $3/2$. Energy conditions and radial stability of the constituent circular orbits constrain the admissible configurations. A sufficient bound guarantees these properties for an inner edge at or beyond the Schwarzschild marginal-stability radius and also ensures positivity of the massless-scalar potential. Comparisons at a common mass budget examine how boundary regularity and radial concentration jointly affect the gravitational response. Shadows and light deflection also depend on the global redshift and mass calibration. Scalar waves matched across the three regions with one infinity-normalized time coordinate distinguish local gap-redshift effects from propagation through the complete geometry. The smoothly truncated profile provides a controlled equilibrium model for investigating how finite support and mass redistribution affect black hole observables; its taper scales, formation, and collective stability require additional physical input.