# Can static regular black holes form from gravitational collapse?

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## Abstract

Starting from the Oppenheimer–Snyder model, we know how in classical general relativity the gravitational collapse of matter forms a black hole with a central spacetime singularity. It is widely believed that the singularity must be removed by quantum-gravity effects. Some static quantum-inspired singularity-free black hole solutions have been proposed in the literature, but when one considers simple examples of gravitational collapse the classical singularity is replaced by a bounce, after which the collapsing matter expands for ever. We may expect three possible explanations: (i) the static regular black hole solutions are not physical, in the sense that they cannot be realized in Nature, (ii) the final product of the collapse is not unique, but it depends on the initial conditions, or (iii) boundary effects play an important role and our simple models miss important physics. In the latter case, after proper adjustment, the bouncing solution would approach the static one. We argue that the “correct answer” may be related to the appearance of a ghost state in de Sitter spacetimes with super Planckian mass. Our black holes have indeed a de Sitter core and the ghost would make these configurations unstable. Therefore we believe that these black hole static solutions represent the transient phase of a gravitational collapse but never survive as asymptotic states.

## Keywords

Black Hole Form Factor Black Hole Solution Gravitational Collapse Spacetime Singularity## 1 Introduction

In classical general relativity, under the main assumptions of the validity of the strong energy condition and of the existence of global hyperbolicity, the collapse of matter inevitable produces a singularity of the spacetime [1, 2]. At a singularity, predictability is lost and standard physics breaks down. According to the weak cosmic censorship conjecture, spacetime singularities formed from collapse must be hidden behind an event horizon and the final product of the collapse must be a black hole [3]. In 4-dimensional general relativity, the only uncharged black hole solution is the Kerr metric [4, 5], which reduces to the Schwarzschild solution in the spherically symmetric case. The Oppenheimer–Snyder model is the simplest fully analytic example of gravitational collapse, describing the contraction of a homogeneous spherically symmetric cloud of dust [6]. It clearly shows how the collapse produces a spacetime singularity and the final product is a Schwarzschild black hole.

In analogy with the appearance of divergent quantities in other classical theories, it is widely believed that spacetime singularities are a symptom of the limitations of classical general relativity, to be removed by quantum-gravity effects. While we do not have yet any robust and reliable theory of quantum gravity, the resolution of spacetime singularities has been investigated in many quantum-gravity inspired models. Very different approaches have studied corrections to the Schwarzschild/Kerr solution, finding black hole metrics in which the curvature invariants are always finite [7, 8, 9, 10, 11, 12, 13].^{1} In the same spirit, one can study the modifications to the Oppenheimer–Snyder solution and to other models of collapse. In this case, the singularity is replaced by a bounce, after which the cloud starts expanding [14, 15, 16, 17, 18]. It is therefore disappointing that the quantum-gravity corrected model of collapse does not reproduce the quantum-gravity corrected black hole solution.

In this paper, we want to investigate this apparent contradictory result. First, we determine both the quantum-gravity corrected static black hole metric and the quantum-gravity corrected homogeneous collapse solution within the same theoretical framework, since the ones reported in the literature come from different models. We find that the problem indeed exists. Second, we try to figure out the possible reason. One possibility is that the static regular black hole spacetimes are ad hoc solutions, but they cannot be created in a collapse and therefore they are physically irrelevant. The collapse always produces an object that bounces. Another possible explanation is that the final product of the collapse depends on the initial conditions. The collapse of a homogeneous cloud creates an object that bounces, while with other initial conditions (not known at present) the final product is a static regular black hole. Lastly, it is possible that the simple homogeneous collapse oversimplifies the model, ingoing and outgoing energy fluxes between the interior and the exterior solutions are important, and, after proper readjustment that seems to be difficult to have under control within an analytic approach, the collapsing model approaches the static regular black hole solution. Our quantum-gravity inspired theories are unitary, super-renormalizable or finite at quantum level, and there are no extra degrees of freedom at perturbative level around flat spacetime. This should rule out the possibility that the explanation of our puzzle is due to the fact that these models may not be consistent descriptions of quantum gravity. However, these theories display a ghost state in de Sitter spacetime when the cosmological constant exceeds the square of the Planck mass. This fact may be responsible for our finding and answers the question in the title of this paper. Our black holes have indeed a de Sitter core with an effective cosmological constant larger than the square of the Planck mass when the black hole mass exceeds the Planck mass. The presence of a ghost makes the solutions unstable and therefore they cannot be the final product of the gravitational collapse.

The content of the paper is as follows. In the next section, we briefly review the classical homogeneous and spherically symmetric collapse model. In Sect. 3, we derive the spherically symmetric black hole solutions in a super-renormalizable and asymptotically free theory of gravity with the family of form factors proposed by Krasnikov [19] and Tomboulis [20]. In Sect. 4, we study the spherically symmetric homogeneous collapse in the same models. Summary and conclusions are in Sect. 5.

## 2 Black holes and gravitational collapse in classical general relativity

For a homogeneous perfect fluid, \(p_r = p_\theta = p(t)\). The simplest case is the gravitational collapse of a cloud of dust, \(p=0\), which is the well-known Oppenheimer–Snyder model [6]. From Eq. (3), we see that \(F\) is proportional to the amount of matter enclosed within the shell labeled by \(r\) at the time \(t\). For dust, from Eq. (4) it follows that \(F\) is independent of \(t\), so there are no inflows and outflows through any spherically symmetric shell of radial coordinate \(r\). If \(r_\mathrm{b}\) denotes the comoving radial coordinate of the boundary of the cloud, \(F(r_\mathrm{b})=2M_\mathrm{Sch}\), where \(M_\mathrm{Sch}\) is the Schwarzschild mass of the vacuum exterior. Let us note that in the general case of a perfect fluid this is not true, and for a non-vanishing pressure the homogeneous spherically symmetric interior must be matched with a non-vacuum Vaidya exterior spacetime. Equation (5) reduces to \(\nu ' = 0\) and one can always choose the time gauge in such a way that \(\nu =0\). From Eq. (6), we find that \(G\) is independent of \(t\) and we can write \(G=1+f(r)\). In the homogeneous marginally bound case (representing particles that fall from infinity with zero initial velocity), \(f=0\) and therefore \(G=1\).

## 3 Quantum-gravity inspired black holes

## 4 Nonlocal gravity inspired collapse

## 5 Summary and conclusions

In the present paper, we have studied both the static black hole solution and the homogeneous spherically symmetric collapse of a cloud of matter in a super-renormalizable and asymptotically free theory of gravity. The spacetime singularity predicted in classical general relativity is removed in both cases. In the literature there were so far some scattered results in different theoretical frameworks. Here we have studied this issue in more detail within the Krasnikov and Tomboulis models.

Static and spherically symmetric singularity-free black hole solutions have been obtained. At the origin, the effective energy density is always finite and positive, independently of the exact expression of the form factor \(V(z)\). In other words, these black holes have a de Sitter core in their interior, where the effective cosmological constant is of order \(\kappa ^2 M \Lambda \), \(\kappa ^2 = 32\pi G_\mathrm{N}\), \(M\) is the black hole mass, and \(\Lambda \) is the energy scale of the theory which is naturally to expect to be close to the Planck mass. The singularity of the spacetime is therefore avoided due to the repulsive behavior of the gravitational force. For a large family of form factors, the effective energy density can be negative in some regions, which eventually provides the possibility of having *multi-horizon black holes*. In the homogeneous and spherically symmetric collapse of a cloud of matter, the formation of the singularity is always replaced by a bounce. Far from the bounce, the collapse follows the classical solution, while it departs from it at high densities. Strictly speaking, asymptotic freedom is sufficient to remove the singularity, but the presence of a bounce requires also a repulsive character for gravitational field in the high energy regime.

- 1.
Static regular black holes cannot be created in any physical process. In this case, even if they are solution of a theory, they are much less interesting than their classical counterparts that can be created in a collapse.

- 2.
The final product of the gravitational collapse is not unique. The collapse of a homogeneous and spherically symmetric cloud of matter does not produce a static regular black hole, but the collapsing matter bounces and then expands. With different initial conditions, not known at the moment, static regular black holes may form.

- 3.
The simple example of a homogeneous cloud of matter oversimplifies the picture and misses important physics. As discussed in Sect. 2, in the classical dust case we have a homogeneous interior and a Schwarzschild exterior without ingoing or outgoing flux through any spherical shell of comoving radial coordinate \(r\). However, that is not true in general, and the exterior spacetime is a generalized Vaidya solutions with ingoing or outgoing flux of energy. This means that the homogeneous solution is not stable and must evolve to an inhomogeneous model. While the bounce can still occur, after it the collapsing matter may not expand forever. The boundary effects are important and, after proper readjustment that can unlikely be described without a numerical strategy, the collapse approaches the static black hole solution.

## Footnotes

- 1.
We note that it may also be possible that the quantum corrections that smooth out the singularity may be intrinsically quantum and not reducible to the metric form. In such a case, the metric description would simply break down.

## Notes

### Acknowledgments

This work was supported by the NSFC grant No. 11305038, the Shanghai Municipal Education Commission grant for Innovative Programs No. 14ZZ001, the Thousand Young Talents Program, and Fudan University.

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