Binetruy, Pierre; Helou, Alexis; Lamy, Frederic
(2018):
Closed trapping horizons without singularity.
In: Physical Review D, Vol. 98, No. 6, 64058

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Abstract
In gravitational collapse leading to black hole formation, trapping horizons typically develop inside the contracting matter. Classically, an ingoing trapping horizon moves toward the center where it reaches a curvature singularity, while an outgoing horizon moves toward the surface of the star where it becomes an isolated, null horizon. However, strong quantum effects at high curvature close to the center could modify the classical picture substantially, e.g., by deflecting the ingoing horizon to larger radii, until it eventually reunites with the outgoing horizon. We here analyze some existing models of regular "black holes" of finite lifespan formed out of ingoing null shells collapsing from I, after giving general conditions for the existence of (singularityfree) closed trapping horizons. We study the energymomentum tensor of such models by analyzing Einstein's tensor, which describes the geometry and give an explicit form of the metric to model a Hawking radiation reaching I+. A major flaw of the models that aim to describe the formation of black holes (with a Vaidya limit on I) as well as their evaporation is finally exhibited: they necessarily violate the null energy condition up to I i.e., in a noncompact region of spacetime.