Motion of a thin spherically symmetric Shell of Dust in the Schwarzschild field

TL;DR

This paper derives a general equation of motion for a thin spherically symmetric dust shell in a Schwarzschild field, providing solutions for parabolic trajectories and analyzing behavior near the horizon and singularity.

Contribution

It presents a new, general derivation of the shell's equation of motion and offers closed-form solutions for specific cases, extending previous special-case analyses.

Findings

Closed-form solutions for parabolic motion

Analysis of shell behavior near horizon and singularity

General equation of motion derived for arbitrary inner metrics

Abstract

The equation of motion announced in the title was already deduced for the cases the inner metric being flat and the shell being negligibly small (test matter), using surface layers and geodesic trajectories resp. Here we derive the general equation of motion and solve it in closed form for the case of parabolic motion. Especially the motion near the horizon and near the singularity are examined.