The Exact Solutions to the Gravitational Contraction in Comoving Coordinate System

The gravitational collapse of a star is a warmly discussed but still puzzling problem, which not only involves the dynamics of the gases, but also the subtle coordinate transformation. In this letter, we give some more detailed investigation on this problem, and reach the results: (I). The comoving coordinate system for the stellar system is only compatible with the zero-pressure free falling particles. (II). For the free falling dust, there are three kind of solutions respectively corresponding to the oscillating, the critical and the open trajectories. The solution of Oppenheimer and Snyder is the critical case. (III). All solutions are exactly derived. There is a new kind singularity in the solution, but its origin is unclear.

💡 Research Summary

The paper revisits the classic problem of stellar gravitational collapse, focusing on the use of comoving (synchronous) coordinates to describe the dynamics of a collapsing star. It begins by clarifying the mathematical structure of a comoving coordinate system, where each fluid element shares a proper time τ and the four‑velocity reduces to uμ=(1,0,0,0). By inserting the corresponding stress‑energy tensor Tμν=ρuμuν into Einstein’s field equations, the author demonstrates that the system remains self‑consistent only when the pressure vanishes. In other words, the comoving frame is compatible exclusively with pressure‑free, freely falling particles (dust).

Assuming spherical symmetry and a dust equation of state (p=0), the metric is written in the familiar form
 ds² = dτ² – a(τ)²