# Vacuum Semiclassical Gravity Does Not Leave Space for Safe Singularities

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## Abstract

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## 1. Introduction

## 2. Vacuum Energy and the Semiclassical Consistency Test

## 3. Schwarzschild Counterpart in Semiclassical Gravity

## 4. Semiclassical Collapse and Subsequent Evolution

#### 4.1. Evaporation à la Hawking

- Regular black-hole configurations [35]. These configurations contain Cauchy horizons which would be strongly unstable at the semiclassical level [36]. These geometries appear to be semiclassically inconsistent, unless they are just a good approximation to the evolving geometry for a brief transient period. Any semiclassically self-consistent approach should avoid their strict formation.
- Wormhole geometries. We are considering only scenarios of deterministic evolution that start from initial data associated with a regular collapsing star. Thus, we will exclude geometries which involve a topology change (the above case of strict formation of regular black holes would also involve topology change of the putative Cauchy surfaces).
- Geometries with trapped regions touching infinity. We do not have any strong argument against these configurations. However, they can be considered (infinitely) stretched versions of those with trapping regions of finite size. We will not directly contemplate this possibility in the following.

#### 4.2. Regular Black Hole Evaporating Inwards from the Outside

#### 4.3. Regular Black Hole Evaporating Outwards from the Inside

#### 4.4. Time-Symmetric Bounce

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Pictorial representation of the semiclassical Schwarzschild geometry. The horizon is replaced by the neck of an asymmetric wormhole, which is a minimal surface for the areal radius $r\left(l\right)$, l being the proper radial coordinate. Space to the right of this surface connects with the asymptotically flat region, while to the left, it ends in a null singularity at finite proper distance. The red curve represents the redshift function of the geometry, which is positive everywhere but at the singularity. The blue dashed line is the quotient between the Misner–Sharp mass and r, which shows how mass is distributed throughout the spacetime.

**Figure 2.**(

**Left figure**): Penrose diagram of the singular wormhole solution in the Polyakov approximation. The vertical dashed line ${l}_{\mathrm{B}}$ represents the location of the wormhole neck. To its right, the asymptotically flat portion of spacetime is depicted. The left side of the diagram shows the internal past and future null singularities, which are located at finite proper distance from the neck ${l}_{\mathrm{B}}-{l}_{\mathrm{S}}$. The point ${i}_{\mathrm{L}}^{0}$ is singular as well and is reached in finite proper time by spacelike geodesics. (

**Right figure**): Penrose diagram of the singular wormhole solution in the s-wave approximation. The dashed curve represents the position of the wormhole neck. The singularity is timelike and located at finite radial distance from the neck, constituting a naked singularity.

**Figure 3.**Causal diagram of a Hawking-like evaporating black hole. The shaded portion corresponds to the interior of the event horizon, which exists up to the end point of the evaporation process. The blue curves joining ${i}^{-}$ and ${i}^{+}$ qualitatively represent $r=$ const. surfaces. The green wave fronts represent the causal effect of the thunderbolt on future events.

**Figure 4.**Causal diagram of an evaporating regular black hole. The shaded region represents the black hole trapped region. The blue curves joining ${i}^{-}$ and ${i}^{+}$ qualitatively represent $r=$ const. surfaces.

**Figure 5.**Causal diagram of a bouncing geometry. The lower shaded region is a black-hole-like trapped region, while the upper one is a white-hole-like trapped region. The blue curves joining ${i}^{-}$ and ${i}^{+}$ qualitatively represent $r=$ const. surfaces.

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**MDPI and ACS Style**

Arrechea, J.; Barceló, C.; Boyanov, V.; Garay, L.J.
Vacuum Semiclassical Gravity Does Not Leave Space for Safe Singularities. *Universe* **2021**, *7*, 281.
https://doi.org/10.3390/universe7080281

**AMA Style**

Arrechea J, Barceló C, Boyanov V, Garay LJ.
Vacuum Semiclassical Gravity Does Not Leave Space for Safe Singularities. *Universe*. 2021; 7(8):281.
https://doi.org/10.3390/universe7080281

**Chicago/Turabian Style**

Arrechea, Julio, Carlos Barceló, Valentin Boyanov, and Luis J. Garay.
2021. "Vacuum Semiclassical Gravity Does Not Leave Space for Safe Singularities" *Universe* 7, no. 8: 281.
https://doi.org/10.3390/universe7080281