Singularities and Collision Criteria in the Gravitational Straight-Segment Problem with Arbitrary Positive Density
We study singularities and collision criteria for a massless particle around a fixed finite straight segment with an arbitrary positive continuous density. We first show that every singularity forces the trajectory to approach the material segment. Using the axial symmetry of the potential, we prove that any trajectory approaching the segment is confined to an invariant plane containing it. Moreover, we show that every singularity is either a collision singularity or an ejection singularity. Finally, in the invariant plane, we introduce an angular-cone condition for positions and momenta directed toward the segment. We prove that if the initial position-momentum pair belongs to the angular cone, then the trajectory collides with the segment in finite time, and we derive an upper bound for the collision time. We further prove that the final arc of any collision satisfies the angular-cone condition.