# Black Hole Evaporation: A Perspective from Loop Quantum Gravity

## Abstract

**:**

## 1. Introduction

## 2. Black Hole Evaporation: Inclusion of Back-Reaction

#### 2.1. Dynamical Horizons

#### 2.2. No Violation of Semi-Classical Expectations

#### 2.3. Singularity Resolution and the Quantum Region

#### 2.3.1. Kruskal Space–Time in Lqg

#### 2.3.2. Beyond the Semi-Classical Region

- (i)
- A semi-classical phase which is expected to provide an excellent approximation to full quantum dynamics in the part of the trapped region that lies to past of the surface ${\mathcal{S}}_{\mathrm{sc}}$. This is the part of the trapped region to the past of the shaded (pink) portion in Figure 2b. During this phase dynamics would be well approximated by quantum field theory of the field $\widehat{\Phi}$ on the Vaidya background. During this phase, space–time
- (ii)
- An adiabatic quantum gravity phase in which the space–time curvature is larger and enters the Planck regime but the evaporation process is adiabatic. This phase, depicted by the shaded pink region of Figure 2b, could be well-described by a dressed effective metric ${\tilde{g}}_{ab}$. The task is to calculate this ${\tilde{g}}_{ab}$ and use it to describe the propagation of the field $\widehat{\Phi}$. This region is expected to constitute a neighborhood of the transition surface, $\tau $, to the past of which we have a trapped region and to the future of which, an anti-trapped region, both defined using ${\tilde{g}}_{ab}$. The metric ${\tilde{g}}_{ab}$ will incorporate two distinct sets of quantum effects. The first set is from LQG proper—such as negative effective energy density which is very large in the Planck domain and reduces the mass of the infalling scalar field enormously even at the left end, $v=0$, of the shaded (pink) region, even though the influx of the Hawking partner modes is negligible there. As v increases, we slide to the right in this region, and then the negative energy density due to quantum geometry effects decreases but is compensated by the negative energy carried by more and more partner modes. The second set of quantum effects is precisely that from the dynamics of the scalar field. It includes both, the evolution of the incoming shell and the infalling partner modes. The actual calculation will incorporate both these types of effects in one go. But conceptually it is useful to separate the two contributions, since the first has been absent in most other approaches as they focus just on the partner modes and also ignore the quantum geometry effects. Finally, the Kruskal results depicted in Figure 4a suggest that the first set of effects will decay rapidly as we move away from Planck curvature into the semi-classical region. Therefore in the semi-classical region, ${\tilde{g}}_{ab}$ will be well approximated by the semi-classical Vaidya metric used there.
- (iii)
- A full quantum gravity phase that is localized in the region depicted by a dark (red) blob at the right end of the shaded region in Figure 2b. This is the region in which not only is the curvature of Planck scale, but it is varying rapidly because of the evaporation process; dynamical nature of the back reaction is now significant. Therefore, description in terms of ${\tilde{g}}_{ab}$ would now be inadequate. In this phase, one would have to use the LQG quantum state for the combined system, adapted to the spherical collapse under consideration. In terms of space–time geometry, the time-like piece of TDH continues to shrink in the semi-classical region and terminates in the full quantum region as depicted in Figure 2b.

#### 2.4. Summary: The Global Picture

## 3. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1. | I would like to emphasize that there is neither a black hole nor a white hole, because both of them have singularities, while the quantum space–time has none. Therefore we will refer to these regions simply as trapped and anti-rapped. The past boundary of the trapped region is a dynamical horizon and its future boundary is the transition surface $\tau $. The past boundary of the anti-trapped region is $\tau $ and the future boundary is another dynamical horizon. See Figure 2b. |

2. | |

3. | In fact, in this specific case, space–time metric is completely regular across the last ray; there is no ‘thunderbolt’ suggested by Hawking and Stewart [39]. Therefore, in a more complete, quantum theory, one can expect space–time to extend further. If the singularity were to be resolved in full quantum gravity, then the quantum space–time would have a longer ${\mathcal{I}}^{+}$ as suggested in Figure 3b. General conditions under which unitarity can be restored in this space–time are discussed in [38]. |

4. | Another geometrically natural family of 3-surfaces is defined by constancy of the trace of the extrinsic curvature is constant [43]. But they do not provide a foliation of the entire trapped region because if we start from the right end—the time-like piece of T-DH—the leaves ‘pile up’, all approaching the 2-sphere $r=1.5{M}_{\odot}$ on the left end. But the main phenomenon of developing enormously long necks occurs also on these 3-surfaces. This foliation was motivated by a calculation of the growth of the volume of 3-surfaces inside the event horizon of a collapsing shell in classical GR [44]. |

5. | If the spatial topology is non-compact, the scale factor a does not have an invariant meaning. In this case, the spatial region with a fixed comoving volume has the minimum physical volume at the LC bounce. |

6. | Illustrative numbers: The expectation is that one can use the adiabatic approximation discussed below during the long process in which $m\left(v\right)$ decreases from ${M}_{\odot}$ to, say, $100{m}_{\mathrm{Pl}}$ (when $\dot{m}\left(v\right)\sim {10}^{-4}$), and the full quantum treatment would be necessary only at every end when $m\left(v\right)$ decreases further. |

7. | |

8. | This and other open issues in this program (described below) are being analyzed in collaboration with Tommaso De Lorenzo. Several steps in narrowing down the a priori freedom in the program grew out of our discussions. |

9. | The second of these reasons—space–time geometry considerations—seem to apply also to the left end of $\tau $, suggesting that one may have to use quantum geometry also in a blob near the left end. However, since the first and the third considerations do not apply there, in the main text I focused on the right end. |

**Figure 1.**(

**a**) Gravitational collapse of a star. While the past boundary of space–time consists only of ${\mathcal{I}}^{-}$, the future boundary is the union of ${\mathcal{I}}^{+}$ and the black hole singularity. There is information loss in the future evolution because the singularity can soak up part of the information. (

**b**) Commonly used Penrose diagram to depict black hole evaporation, including back reaction. Modes are created in pairs, one escaping to ${\mathcal{I}}^{+}$ and its partner falling into the black hole. The dashed line is the continuation of the event horizon that meets ${\mathcal{I}}^{+}$ at retarded time ${u}_{EH}$. If this were an accurate depiction, the future singularity would again act a sink of information.

**Figure 2.**(

**a**) Part of space–time in which the semi-classical approximation holds. The shaded region depicts an infalling massless scalar field that would give rise to a singularity in classical GR. But loop quantum gravity (LQG) quantum geometry effects intervene to prevent the formation of this singularity. This quantum geometry region and its future are excised (dashed line with scissors). A trapping dynamical horizon (TDH) develops which is space-like with increasing area (in the outward direction) during the collapse, and time-like with decreasing area (in the future direction) during evaporation. Null rays from the contracting part of the TDH go out to meet ${\mathcal{I}}^{+}$. Hawking radiation starts in earnest at $u={u}_{0}$. The dashed line with scissors that includes the last ray $u={u}_{LR}$ represents the future boundary of the semi-classical region. Evolution from the initial Cauchy surface ${\Sigma}_{i}$ to generic Cauchy surfaces $\Sigma $ that lie entirely in the semi-classical region is unitary. However, evolution from $\Sigma $ to the portion of ${\Sigma}_{f}$ that lies within the semi-classical region is not. (

**b**) Proposed quantum extension of the space–time. The classical singularity is replaced by a transition surface $\tau $, to the past of which we have a trapped region, bounded in the past by a trapping dynamical horizon T-DH, and to the future of which we have an anti-trapped region bounded by an anti-trapping dynamical horizon (AT-DH). Cauchy surfaces $\Sigma $ develop astronomically long necks already in the semi-classical region and the long necks persist also in the quantum region. Partner modes that fall in the trapped region get stretched enormously and are slowly released to the future of the quantum region. Thus, ‘most of the purification’ occurs to the future of the transition surface $\tau $. The dark (red) blob at the right end of $\tau $ is a genuinely quantum region discussed in Section 2.3 and Section 2.4.

**Figure 3.**(

**a**) Classical collapse of a narrow pulse of a massless scalar field, incident from ${\mathcal{I}}^{-}$. Note that during the collapse a dynamical horizon (DH) is formed. It is space-like and grows in area as the collapse continues and smoothly joins on to the (null) event horizon (EH) in the distant future. (

**b**) Artist’s depiction of a semi-classical Callen–Giddings–Harvey–Strominger (CGHS) black hole that was analyzed in detail in [33,34]. A Delta-distribution pulse of a scalar field from ${\mathcal{I}}^{-}$ would lead to a black hole in the classical theory. In the semi-classical theory, the singularity is weakened (the metric is ${C}^{0}$ but not ${C}^{1}$) but persists. The dynamical horizon becomes time-like, shrinks in its ‘area’ due to emission of quantum radiation, terminates and meets the singularity. The dynamical horizon together with the last ray from its endpoint to ${\mathcal{I}}^{+}$ constitutes the future boundary of the semi-classical space–time.

**Figure 4.**(

**a**) LQG extension of the Schwarzschild-interior. In classical GR we only have the lower triangular region that is bounded below by the trapping event horizon T-EH and ends with a future space-like singularity at $r=0$. Quantum geometry effects of LQG resolve this singularity and replace it with a transition surface, labeled $\tau $, to the future of which we have an anti-trapped region with a future boundary that constitutes an anti-trapping event horizon AT-EH. Quantum geometry effects are important only in a neighborhood of $\tau $, shown by a shaded (pink) region. (

**b**) LQG extension of the Kruskal space–time [18,19]. Quantum geometry effects resolve both black hole and white hole singularities and the quantum corrected LQG space–time extends indefinitely. The central diamond corresponds to panel (a).The dashed (blue) arrows depict integral curves of the Killing vector. Just as in only a part of the full Kruskal space–time is physically relevant for gravitational collapse in classical GR, only a portion of this infinite extension is relevant for the black hole evaporation process (see Figure 2b).

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Ashtekar, A.
Black Hole Evaporation: A Perspective from Loop Quantum Gravity. *Universe* **2020**, *6*, 21.
https://doi.org/10.3390/universe6020021

**AMA Style**

Ashtekar A.
Black Hole Evaporation: A Perspective from Loop Quantum Gravity. *Universe*. 2020; 6(2):21.
https://doi.org/10.3390/universe6020021

**Chicago/Turabian Style**

Ashtekar, Abhay.
2020. "Black Hole Evaporation: A Perspective from Loop Quantum Gravity" *Universe* 6, no. 2: 21.
https://doi.org/10.3390/universe6020021