Can gravitational collapse sustain singularity-free trapped surfaces?

Understand

In singularity generating spacetimes both the out-going and in-going expansions of null geodesic congruences $\theta ^{+}$ and $\theta ^{-}$ should become increasingly negative without bound, inside the horizon.

• This behavior leads to geodetic incompleteness which in turn predicts the existence of a singularity.
• In this work we inquire on whether, in gravitational collapse, spacetime can sustain singularity-free trapped surfaces, in the sense that such a spacetime remains geodetically complete.
• As a test case, we consider a well known solution of the Einstien Field Equations which is Schwarzschild-like at large distances and consists of a fluid with a $p=-\rho $ equation of state near $r=0$.