Matter density distribution and gravitational collapse around a Schwarzschild black hole

Abstract

Star formation near black holes is governed by multiple physical factors, including the interplay between gravity and matter density. This work employs a toy model of the radial density distribution around a Schwarzschild black hole to identify regions that may be more favorable for gravitational instability. The density profile is parameterized by ζ, which controls the steepness of the central cusp. Results show that lower ζ leads to stronger central concentration, while higher ζ produces more extended distributions. Jeans mass analysis indicates that regions of higher density correspond to lower Jeans mass, suggesting reduced gravitational stability in those zones. A logarithmic scale is used to resolve both the inner cusp and outer diffuse structure. Overall, the results highlight the coupled roles of spacetime curvature and density in regulating gravitational stability near compact objects.