# Nonsingular black hole

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## Abstract

We consider the Schwarzschild black hole and show how, in a theory with limiting curvature, the physical singularity “inside it” is removed. The resulting spacetime is geodesically complete. The internal structure of this nonsingular black hole is analogous to Russian nesting dolls. Namely, after falling into the black hole of radius \(r_{g}\), an observer, instead of being destroyed at the singularity, gets for a short time into the region with limiting curvature. After that he re-emerges in the near horizon region of a spacetime described by the Schwarzschild metric of a gravitational radius proportional to \(r_{g}^{1/3}\). In the next cycle, after passing the limiting curvature, the observer finds himself within a black hole of even smaller radius proportional to \(r_{g}^{1/9}\), and so on. Finally after a few cycles he will end up in the spacetime where he remains forever at limiting curvature.

## Keywords

Black Hole Small Black Hole Large Black Hole Coordinate Singularity Gravitational Radius## 1 Introduction

The problem of singularity within black holes has remained, since a long time, one of the most intriguing problems in theoretical physics. Although such a singularity is hidden by the event horizon, one can imagine that an observer can decide (at least in a gedanken experiment) to travel inside the black hole and the legitimate physical question which arises is: what will this observer see being inside the black hole and in particular as he approaches the singularity? In the case that the black hole has a huge mass he will have more than enough time to make the needed experiments to measure how the tidal forces are changing. If General Relativity is valid up to arbitrary high curvatures then the theory predicts that, irrespective of what any observer will do, he will finally be destroyed by the infinite curvatures. In fact, assuming universal applicability of Einstein’s theory, and imposing energy dominance conditions on the state of matter, Hawking and Penrose have proved that space-times with black holes cannot be geodesically complete [1]. It is well known that these conditions are not always valid and for instance the condensate of a scalar field or cosmological constant violates some of them. In this case the singularity can, in principle, be avoided and the spacetime can become geodesically complete. For example Ref. [2] considered the possibility of removing the singularity by forcing the contracting space inside the black hole to get to the de Sitter bouncing state. This opens the fascinating possibility of “gedanken travelling” to another universe via a black hole (of course only for those who could survive the extremely high curvatures at which the bounce is supposed to take place). However, although this idea by itself does not contradict any basic physical principles the authors of [2] were not able to provide any concrete example where such an idea could be realized constructively.

Normally the majority of research redirects the question of singularities to the yet unknown nonperturbative quantum gravity (which in turn could well be part of some yet unknown fundamental unified theory). In fact it is clear that quantum corrections to General Relativity become extremely important at Planckian curvatures and could easily modify or resolve the singularities. Therefore, one cannot say that such hopes are completely unjustified. However, until now, the perturbative treatment of these corrections has led to an extremely messy picture and did not give even the slightest constructive hints of how the problem could be treated and solved in a fully nonperturbative quantum gravity. Numerous attempts to address this question did not lead to any reliable progress. Therefore in this paper we will use a completely different approach. Instead of exploiting quantum effects we will try to resolve the problem of singularities fully at the classical level by incorporating the idea of a limiting curvature [3, 4, 5, 6, 7, 8], assuming that Einstein’s equations are modified at curvatures well below the Planckian curvature. There is nothing that forbids this idea because Einstein’s equations have been checked experimentally only for curvatures well below the Planckian ones. If the limiting curvature is below the Planck value the inevitable quantum effects, due to for instance particle production and vacuum polarization, can be ignored and the theory will be under control and would remain completely reliable up to the highest possible curvatures. In particular, in [9] we have suggested a concrete theory with limiting curvature and have shown that cosmological singularities in this theory are fully removed. In this paper we consider how a black hole is modified in our theory and what happens close to the singularity inside a black hole. We would like to point out that removing singularities can have severe consequences for questions broadly discussed in the literature, such as the so-called “information paradox” and the fate of remnants of the minimal mass which can, in principle, survive after the Hawking evaporation is over. We will discuss these questions in more detail after obtaining the solution for a nonsingular black hole.

## 2 Theory with limiting curvature

*A*reflects the time shift symmetry. In this coordinate system

*f*removes singularities in the Friedmann and Kasner universes. In this paper we will consider what happens with singularities for black holes.

*C*is a constant of integration corresponding to mimetic cold matter. Because this matter behaves exactly like dust we can neglect it for the reasons explained above and set \(C=0.\) By subtracting from Eq. (16) one third of its trace we find

*P*. Substituting the function

*f*from (8) into this equation leads to the particularly simple equation

## 3 Schwarzschild solution in general relativity and the boundary conditions for \(\phi \)

*r*and

*t*exchange their roles and

*r*becomes a time-like coordinate while \(t_{S}\) becomes a space-like one. Inside the black hole the decrease of the “radial coordinate” from \(r=r_{g}\) to \(r=0\) corresponds to time increase. Inversely, if we assume that time grows with

*r*then the same Schwarzschild solution describes the white hole, which is just a time reversed black hole. Let us rename the coordinate in (26) as \(r\rightarrow r_{g}\tau ^{2}\) and \(t_{S}\rightarrow R.\) Then inside the Schwarzschild black hole the metric (26) becomes

*space-like singularity*is reached

*at the moment of time*\(\tau =0.\) In fact the spacetime described by metric (27) is obviously non-static and the Riemann squared tensor equals

*but only*if the synchronous coordinate system has

*no coordinate singularities.*Obviously the coordinate system (31) does not satisfy this requirement because \(\gamma =(1-\tau ^{2}) \tau ^{6}r_{g}^{4}\) vanishes as \(\tau ^{2}\rightarrow 1.\)

*t*and

*R*, the new coordinates

*T*and \(\bar{R}\) defined by

*synchronous coordinates*the metric (31) becomes

*a*(

*t*) as it should. Calculating \(\chi \) for solution (39) in the coordinate system (31) we find that it is not equal to \(\dot{\gamma }/2\gamma \) anymore and is now given by

Finally, to complete this section we would like to give the approximate explicit leading order expression for the Schwarzschild metric entirely in terms of time *t*, in the near horizon and close to singularity regions. As we will see this metric will be helpful to understand what happens within the black hole after reaching the limiting curvature and the bounce.

*t*runs in the interval \(-\pi /2<t<0.\) Consider the near horizon region corresponding to \(1+\tau \ll 1.\) Then, as follows from (30),

## 4 Black hole with limiting curvature

*F*(

*t*) consider the times

*t*satisfying

*a*and (30) to express \(\mathrm{d}t/\mathrm{d}\tau ,\) we find

*C*can be found from Eq. (20) for \(\varkappa _{1}^{1},\)

*a*and

*b*from (32) and comparing, we find that

*R*. As a result, after

*the second bounce,*we again re-emerge inside the near horizon region described by the metric

## 5 Summary and speculations

We have shown that in the theory with limiting curvature the internal structure of a black hole is significantly modified compared to a singular Schwarzschild black hole. Namely, the curious observer who decides to travel inside the Schwarzschild eternal black hole after first crossing the horizon will find himself in a non-static space of infinite volume (for eternal black hole), but exists for finite time \(t\sim r_{g}.\) At the beginning the curvature of large black holes is very low but grows and finally, after time \(t\sim r_{g}\), becomes infinite and one ends up in a singularity, which happens not “at the point in the center of black hole” but at the moment of time \(t=0.\) In this sense the evolution and singularity within a black hole is similar to a Kasner universe. The spacetime in this case is not geodesically complete. In our theory with limiting curvature, the Einstein equations are only significantly modified when the curvature starts to approach its limiting value. The singularity is removed and the curvature does not grow indefinitely. In fact, the singularity is replaced by a “time layer” of duration \(\Delta t\sim \varepsilon _{m}^{-1/2},\) which would be of the order of Planck time if the limiting curvature would be the Planckian one. After that the curvature drops down to the value which an observer would find immediately after crossing the horizon of the smaller black hole of radius \(r_{g}^{1/3}\). The subsequent evolution repeats the previous cycle but this time inside a black hole of this smaller radius. Once again, instead of ending at the singularity we pass through a layer of limiting curvature and find ourselves inside a black hole of even smaller radius \(\sim r_{g}^{1/9}\) and so on. Finally when the size of the black hole becomes of the order of the width of a time layer \(\sim \varepsilon _{m}^{-1/2},\) we end inside the black hole of minimal possible mass and stay there forever at limiting curvature. Notice that the number of the “layers” which we have to pass to reach inside this minimal black hole is not big even for large black holes. For instance, for a galactic mass black hole of radius \(r_{g} \sim 10^{49}\) (in Planck units) after the crossing of limiting curvature we find ourselves in black holes of radii \(r_{g}^{1/3}\sim 10^{16},\) \(r_{g} ^{1/9}\sim 10^{5},\) \(r_{g}^{1/27}\sim 10^{2}\) correspondingly. Finally at the fourth layer \(r_{g}^{1/81}\sim O(1)\), and we cannot trust anymore the approximations used to arrive at the above picture and we end up within a minimal black hole at limiting curvature, which after that never drops significantly. The spacetime of a nonsingular black hole is geodesically complete and the singularity problem is resolved.

For an evaporating black hole the derivation of Hawking radiation remains unchanged for a large black hole [13]. However, when it reaches the minimal size of order \(\varepsilon _{m}^{-1/2}\) the near horizon geometry changes and we expect that the minimal remnants of it must be stable. This question obviously requires further investigation [14]. If we take the limiting curvature, which is a free parameter in our theory, to be at least a few orders of magnitude below the Planck scale, the answer to it can be obtained using standard methods of quantum field theory in external gravitational field. In fact, in this case the unknown nonperturbative quantum gravity does not play an essential role and its need in such a case becomes unclear because the uncontrollable Planckian curvatures are never reached. This opens up the possibility of resolving the information paradox without involving the “mysteriously imprinted” correlations in Hawking radiation which is supposed to take care of returning all information back to the Minkowski space after the disappearance of the black hole. In our case the smallest black hole remnant has enough space “inside it” to hide all the information as regards to the original matter from which the black hole was formed together with the information as regards to the negative energy quanta (with respect to an outside observer) which never escapes from the black hole and reduce its mass in the process of Hawking evaporation. The evolution in this case remains unitary on complete Cauchy hypersurfaces which inevitably goes inside the black hole remnant. The picture here is very similar to the one described as a possible option in [2]. The content of the minimal mass black hole can be significantly different depending on the way how the remnant was formed. However, an infinite degeneracy of the black hole remnants is completely irrelevant for an outside observer who calculates, for instance, the scattering processes with participation of these minimal black holes, because this degeneracy is entirely related to events which happen in the absolute future of this observer.

## 6 Appendix

## Notes

### Acknowledgements

The work of A. H. C is supported in part by the National Science Foundation Grant No. Phys-1518371. The work of V.M. is supported in part by Simons Foundation Grant 403033TRR 33 and “The Dark Universe” and the Cluster of Excellence EXC 153 “Origin and Structure of the Universe”.

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