Abstract
We study analytically the spacetime geometry of the black-hole formation and evaporation. As a simplest model of the collapse, we consider a spherical thin shell, and take the backreaction from the negative energy of the quantum vacuum state. For definiteness, we will focus on quantum effects of s-waves. We obtain an analytic solution of the semiclassical Einstein equation for this model, that provides an overall description of the black hole geometry form the formation to evaporation. As an application of this result, we find its interesting implication that, after the collapsing shell enters the apparent horizon, the proper distance between the shell and the horizon remains as small as the Planck length even when the difference in their areal radii is of the same order as the Schwarzschild radius. The position of the shell would be regarded as the same place to the apparent horizon in the semiclassical regime of gravity.
Highlights
Quantum effects around the black holes have been well studied in the literature
As an application of this result, we find its interesting implication that, after the collapsing shell enters the apparent horizon, the proper distance between the shell and the horizon remains as small as the Planck length even when the difference in their areal radii is of the same order as the Schwarzschild radius
This section is a review of previous results included in Refs. [8,9,12], where the s-wave approximation is used for the matter fields and the vacuum energy-momentum tensor is assumed to be given by the toy model proposed in Refs. [1,13] based on 2-dimensional massless scalar fields.1 (The solution was presented in different coordinate systems in Refs. [8,9], and here we use the same coordinate system as Ref. [12].) As a preliminary of the review, we introduce the vacuum energy-momentum tensor in Refs. [1,13] first
Summary
Quantum effects around the black holes have been well studied in the literature. A common feature of the quantum effects in many models is that there is an incoming negative vacuum energy flux around the apparent horizon [1,2], and the outgoing positive vacuum energy flux (i.e., Hawking radiation) appears well outside the apparent horizon. With the backreaction of quantum effects taken into consideration, the black-hole geometry outside the collapsing matter has a wormholelike structure near the horizon due to the negative vacuum energy [6,7,8,9,10,11]. In the dynamical process of the evaporation, the neck is shrinking with time, and plays the role of the apparent horizon in the geometry This picture has been clearly demonstrated in numerical simulation [5] as well as analytic calculation [12]. Collapsing shell and that at the apparent horizon becomes large as the shell moves to the deeper region, but their proper distance remains as small as the Planck length
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