Can Gravitational Collapse Sustain Singularity-Free Trapped Surfaces?

In singularity generating spacetimes both the out-going and ingoing expansions of null geodesic congruences + and − 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 = − equation of state near r = 0. By following both the expansion parameters + and − across the horizon and into the black hole we find that both + and + − have turning points inside the trapped region. Further, we find that deep inside the black hole there is a region 0 ≤ r \u3c r0 (that includes the black hole center) which is not trapped. Thus the trapped region is bounded both from outside and inside. The spacetime is geodetically complete, a result which violates a condition for singularity formation. It is inferred that in general if gravitational collapse were to proceed with a p = − fluid formation, the resulting black hole may be singularity-free