The exact dynamical solution for two dust shells collapsing towards a black hole

TL;DR

This paper provides an exact solution for two dust shells collapsing into a black hole, challenging the frozen star paradox and suggesting matter can cross the horizon without accumulating, with implications for gravitational wave production.

Contribution

It presents the first exact solution for two dust shells collapsing onto a black hole, revealing new insights into horizon crossing and internal shell dynamics.

Findings

Inner shell dynamics are influenced by the shell's properties, contrary to Newtonian expectations.

The clock inside the shell slows down as it collapses towards the black hole.

Matter can cross the event horizon without accumulating, contradicting the frozen star concept.

Abstract

The gravitational collapse of a star is an important issue both for general relativity and astrophysics, which is related to the well known "frozen star" paradox. Following the seminal work of Oppenheimer and Schneider (1939), we present the exact solution for two dust shells collapsing towards a pre-existing black hole. We find that the inner region of the shell is influenced by the property of the shell, which is contrary to the result in Newtonian theory and and the clock inside the shell becomes slower as the shell collapses towards the pre-existing black hole. This result in principle may be tested experimentally if a beam of light travels across the shell. We conclude that the concept of the "frozen star" should be abandoned, since matter can indeed cross a black hole's horizon according to the clock of an external observer. Since matter will not accumulate around the event horizon of a black hole, we predict that only gravitational wave radiation can be produced in the final stage of the merging process of two coalescing black holes. Our results also indicate that for the clock of an external observer, matter, after crossing the event horizon, will never arrive at the "singularity" (i.e. the exact center of the black hole.