Can gravitational collapse sustain singularity-free trapped surfaces?

Abstract

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 $r=0$. By following both the expansion parameters $\theta ^{+}$ and $\theta ^{-}$ across the horizon and into the black hole we find that both $\theta ^{+}$ and $\theta ^{+}\theta ^{-}$ have turning points inside the trapped region. Further, we find that deep inside the black hole there is a region $0\leq r<r_{0}$ (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=-\rho $ fluid formation, the resulting black hole may be singularity-free.