# The Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity

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## Abstract

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## 1. Introduction

- The regularization should be carried out by means of suitable quantum-based modifications of EFE capable of smoothing out all classical BH singularities.
- Such a regularization should have a universal character, i.e., it should hold for arbitrary singular BH solutions.
- The said regularization should not require the introduction of “ad hoc” extra classical or quantum fields.

- discontinuities and singularities associated with discrete quantum theories, which possibly violate, besides continuity, the principle of general covariance and the differential manifold structure of space-time;
- the occurrence of absolute minimum lengths, a feature that by itself implies breaking the principle of general covariance;
- intrinsically frame-dependent theories, such as ADM quantum theory, violating some of the fundamental symmetries characteristic of EFE, i.e., the properties of manifest covariance and gauge invariance.

- Preliminary issue #1: Whether and eventually how quantum gravity models, and specifically CQG-theory, can cure all BH singularities, giving rise to a suitable quantum-modified background metric field tensor (MFT).
- Preliminary issue #2: What is the possible role of the cosmological constant and how its quantum and therefore ubiquitous character could actually be significant for the regularization of singular space-time solutions.
- Preliminary issue #3: What are (if any) the possible large-scale effects produced by the local quantum modifications of MFT.
- Preliminary issue #4: Whether there is a possible connection between the occurrence/prediction of asymptotic/local inflationary regimes, i.e., which are characterized by high values of the cosmological constant, and the expected phenomenon of BH-singularities-quenching.

## 2. CQG-Quantum Hamilton Equations

#### Background Equilibrium Solution of EFE

## 3. Search of Non-Stationary Scale-Transformed Solutions

## 4. Proper-Time Evolution Equation of the Scale-Form Factor $\mathit{N}\left(\mathit{s}\right)$

- First, we notice that $\frac{{d}^{2}N\left(s\right)}{d{s}^{2}}\to 0$ either if ${N}^{2}\left(s\right)\to 1$ or $N\left(s\right)\to 0$. The case $N\left(s\right)=1$ corresponds the standard background solution $\widehat{g}\left(r\right)$ of EFE (see above Equation (29)). Notice that although by assumption $N\left(s\right)\ne 0$, nevertheless it can become infinitesimal (so that $N\left(s\right)\to 0$). This property, as shall be clarified below, will become crucial for the regularization of singular BH solutions.
- Second, the same Equation (46) is conservative. As a consequence it can therefore be reduced by a quadrature to an equivalent first order ODE. In fact it delivers:$$\frac{dN\left(s\right)}{ds}\frac{{d}^{2}N\left(s\right)}{d{s}^{2}}=\frac{d}{ds}\frac{1}{2}{\left[\frac{dN\left(s\right)}{ds}\right]}^{2}=3\sigma \frac{dN\left(s\right)}{ds}\left[{N}^{2}\left(s\right)-1\right]N\left(s\right){\Lambda}_{QM}=\frac{3\sigma}{4}\frac{d}{ds}\left[{N}^{2}\left(s\right)\left({N}^{2}\left(s\right)-2\right){\Lambda}_{QM}\right],$$$$\frac{1}{2}{\left[\frac{dN\left(s\right)}{ds}\right]}^{2}-\frac{1}{2}{\left[\frac{dN\left(s\right)}{ds}\right]}_{s={s}_{o}}^{2}=\frac{3\sigma}{4}\left[{N}^{2}\left(s\right)\left({N}^{2}\left(s\right)-2\right){\Lambda}_{QM}\right]-\frac{3\sigma}{4}\left[{N}^{2}\left({s}_{o}\right)\left({N}^{2}\left({s}_{o}\right)-2\right){\Lambda}_{QM}\right].$$As a consequence, setting the initial constant$$E={\left[\frac{dN\left(s\right)}{ds}\right]}_{s={s}_{o}}^{2}-\frac{3\sigma}{2}\left[{N}^{2}\left({s}_{o}\right)\left({N}^{2}\left({s}_{o}\right)-2\right){\Lambda}_{QM}\right],$$Equation (46) yields the two possible ODE solutions$$\frac{dN\left(s\right)}{ds}=\pm \sqrt{E+\frac{3\sigma}{2}\left[{N}^{2}\left(s\right)\left({N}^{2}\left(s\right)-2\right){\Lambda}_{QM}\right]},$$$${\left[\frac{dN\left(s\right)}{ds}\right]}_{s={s}_{o}}=\pm \sqrt{-\frac{3\sigma}{2}{\Lambda}_{QM}},$$

## 5. Qualitative Properties of the Solutions

- Monotonically decaying solution in the inner BH domain.
- Monotonically growing/decaying solutions in the intermediate domain between two EH’s (${N}^{2}\left({s}_{o}\right)>1$).
- Monotonically decaying solution in the exterior BH domain.

#### 5.1. Monotonically Decaying Solution in the Inner BH Domain

#### 5.2. Monotonically Growing/Decaying Solutions in the Intermediate Domain Between Two EH’s (${N}^{2}\left({s}_{o}\right)>1$)

#### 5.3. Monotonically Decaying Solution in the Exterior BH Domain

## 6. Construction of Background Conformal MFT Solutions

#### 6.1. Conformal Riemann Tensor, Ricci Tensor and 4−Scalar

- The effective quantum CC ${\Lambda}_{QM}({\widehat{g}}^{\left(C\right)}\left(s\right))$ is actually scaled-down by the factor $\frac{1}{N\left(s\right)}$. This means that the effective cosmological constant that characterizes the quantum-modified equilibrium ${\widehat{g}}_{\mu \nu}^{\left(C\right)}\left(s\right)$ actually diverges when $N\left(s\right)\to 0$.
- Conversely, instead, the effective CC ${\Lambda}_{QM}({\widehat{g}}^{\left({C}_{1}\right)}\left(s\right))$ is actually increased by the factor $N\left(s\right)$. This means that the effective cosmological constant that characterizes the alternate (regular) quantum-modified equilibrium ${\widehat{g}}^{\left({C}_{1}\right)\mu \nu}\left(s\right)=N\left(s\right){\widehat{g}}^{\mu \nu}\left(r\left(s\right)\right)$ and its conformally conjugate metric tensor ${\widehat{g}}_{\mu \nu}^{\left({C}_{1}\right)}\left(s\right)=\frac{1}{N\left(s\right)}{\widehat{g}}^{\mu \nu}\left(r\left(s\right)\right)$ actually tends to zero when $N\left(s\right)\to 0$.

#### 6.2. “Conformal” Einstein Field Equations

#### 6.3. “Conformal” Gaussian Quantum PDF and Quantum Continuity Equations

#### 6.4. Conformal Fields as Possible New MFT

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- Both ${\widehat{g}}^{\left(C\right)}\left(s\right)$ and ${\widehat{g}}^{\left({C}_{1}\right)}\left(s\right)$ define conformal fields, which by construction satisfy the required orthogonality conditions.
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- Both fields fulfill suitable Einstein field equations, with suitably (scaled-down or increased) values of the Riemann $4-$scalar and cosmological constant.
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- The prescription of the Gaussian quantum PDF remains unchanged.
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- The quantum continuity equation is fulfilled also in the case of conformal field.

## 7. Physical Interpretation

#### 7.1. Inner BH Domain Behavior

#### 7.2. Intermediate Domain Behavior

- The first one is provided by the conformal solution ${\widehat{g}}^{\left(C\right)}\left(s\right)$. In such a case the corresponding effective cosmological constant is provided by Equation (99). In this case $N\left(s\right)$ is a monotonically increasing function of s but is also bounded from above. This implies that the effective cosmological constant should decrease toward the outer regions of the universe (included in the domain inside the deSitter space-time) but remain bounded from below.
- The second possible realization is provided instead by the conformal solution ${\widehat{g}}^{\left({C}_{1}\right)}\left(s\right)$. In this case the corresponding effective cosmological constant is provided by Equation (100). Again for a monotonically increasing scale form factor $N\left(s\right)$ this means that effective cosmological constant must increase toward the outer regions of the universe (which are inside the deSitter space-time) but remain similarly bounded from above. If $N\left(s\right)\gg 1$ then such a case corresponds to an inflationary solution, i.e., characterized by a strong enhancement of the effective cosmological constant for which ${\Lambda}_{QM}\left({\widehat{g}}^{\left({C}_{1}\right)}\left({s}_{1}\right)\right)$ is larger (or even much larger) than ${\Lambda}_{QM}({\widehat{g}}^{\left({C}_{1}\right)}(r\left({s}_{o}\right))$.

#### 7.3. Exterior BH Domain

- The first one is provided by the conformal solution ${\widehat{g}}^{\left(C\right)}\left(s\right)$. In such a case the corresponding effective cosmological constant is provided by Equation (99). The scale form factor $N\left(s\right)$ is a monotonically decreasing function of s, which in the limit $s\to \infty $ satisfies Equation (111). Such a solution should be viewed as the continuation of the corresponding conformal solution that holds in the intermediate domain. This implies that the corresponding effective cosmological constant should grow monotonically, reaching at infinity a stationary finite value.
- The second possible realization is provided, instead, by the conformal solution ${\widehat{g}}^{\left({C}_{1}\right)}\left(s\right)$. Again this can be regarded as the continuation of the analogous solution holding in the intermediate domain. In this case the initial value of the corresponding effective cosmological constant (provided by Equation (100)) can be expected larger (or even much larger) than ${\Lambda}_{QM}$ (inflationary initial state), while for $s\to \infty $ it decays monotonically reaching again at infinity a stationary value not necessarily identical with the other one indicated above.

#### 7.4. The Initial Conformal Deformation of Space-Time

## 8. Conclusions

- The discovery of a regular conformal representation of the background MFT that holds inside the BH domain in principle for arbitrary singular BH solutions. The regularization effects is purely quantum and arises due to the combined effect of the quantum-produced cosmological constant (${\Lambda}_{QM}$) together with the Hamiltonian character of the underlying quantum hydrodynamic equations.
- The prediction of the large-scale behavior of the corresponding external conformal background MFTs, i.e., occurring in the external domains of the BH. Such predictions are obtained based on the assumption of continuity of the scale form factor $N\left(s\right)$ across event horizons.

- How CQG-theory actually can cure BH singularities, giving rise to a suitable quantum-modified background metric field tensor (MFT).
- The role of the cosmological constant and how its quantum character actually affects the regularization of singular space-time solutions.
- The identification of the possible large-scale effects produced by the local quantum modifications of MFT.
- The possible connection between the occurrence/prediction of asymptotic/local inflationary regimes, characterized by high values of the cosmological constant and the expected phenomenon of BH-singularities-quenching.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Covariant s−Derivative

## Appendix B. Quantum Continuity Equation in the Generalized Lagrangian Path (GLP) Representation

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Tessarotto, M.; Cremaschini, C.
The Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity. *Entropy* **2021**, *23*, 370.
https://doi.org/10.3390/e23030370

**AMA Style**

Tessarotto M, Cremaschini C.
The Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity. *Entropy*. 2021; 23(3):370.
https://doi.org/10.3390/e23030370

**Chicago/Turabian Style**

Tessarotto, Massimo, and Claudio Cremaschini.
2021. "The Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity" *Entropy* 23, no. 3: 370.
https://doi.org/10.3390/e23030370