Abstract
In a classical description the spacetime curvature inside a black hole infinitely grows. In the domain where it reaches the Planckian value and exceeds it the Einstein equations should be modified. In the absence of reliable theory of quantum gravity it is instructive to consider simplified models. We assume that a spacetime curvature is limited by some value (of the order of the Planckian one). We use modified Vaidya metric, proposed by Hayward, to describe the black hole evaporation process. In such a spacetime the curvature near $r=0$ remains finite, it does not have an event horizon and its apparent horizon is closed. If the initial mass of such a `black hole' is much larger than the Planckian one its properties (as seen by an external observer) are practically the same as properties of the `standard' black hole with the event horizon. We study outgoing null rays in the vicinity of the outer apparent and introduce a notion of quasi-horizon. We demonstrate that particles, trapped inside a `black hole' during the evaporation process, finally may return to external space after the evaporation is completed. We also demonstrate that such quanta would have very large blue-shift. The absence of the event horizon makes it possible restoration of the unitarity in evaporating black holes.
Highlights
It was recently proposed a ‘firewall’ model [11]
For slow decrease of the ‘black hole’ mass it is ‘almost null’. It serves as a separatrix between outgoing null rays that reach infinity and those that propagate to the center of the ‘black hole’
In the appendix we demonstrate how a simple massive thin shell model can be used for an effective description of the quantum particle creation by a black hole
Summary
We consider a formation of a spherical black hole in a gravitational collapse of a massive object and its subsequent quantum evaporation. At the initial stage of the evaporation one can use quasi-classical description, so that the Hawking process results in the positive energy flux of created particles to infinity and (in accordance with the conservation law) by the negative energy flux through the horizon. The latter decreases the mass of the black hole. Both incoming and outgoing fluxes are the result of the quantum process of particle creation in the vicinity of the horizon To match these two fluxes and make the model consistent one should introduce a transition region between them. Even consideration of such simple models is quite instructive and allows one to study their robust predictions
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