**and peace on Earth to men of good will / Ehre sei Allgemeinen Relativitätstheorie in der Höhe, ина земле мир, в человеках благоволение**

One of the long standing unsolved problem in General Relativity is the problem of spacelike singularities. In fact, assuming that General Relativity is universally valid and imposing rather general conditions on the state of matter, Hawking and Penrose have proven that space-times describing such as, for instance, Friedmann and Kasner universes and black holes, are geodesically incomplete [1]. Gravitational collapse leads to singularities where the curvature invariants grow and become infinite.

...we shall consider a minimal simple modification of Einstein’s theory, where the singularities are avoided at the classical level in Friedmann and Kasner [8] universes irrespective of the matter which fills the universe. In an accompanying paper we will show that black holes in this theory are also nonsingular...This opens the possibility to have a theory of gravity where singularities are absent in general.To construct such a theory we will introduce the constrained scalar field φ, which satisfies

g^{µν}∂_{µ}φ∂_{ν}φ = 1 (1)

As it was shown in [6], this constrained scalar field is not dynamical by itself, but induces mimetic dark matter in Einstein theory making the longitudinal degree of freedom of the gravitational field dynamical. Notice that its emergence can, for example, be justified in noncommutative geometries as a consequence of quantization of thee-dimensional volume[9]...we will assume here the absence of a potential term by requiring invariance with respect to the shift symmetry φ → φ+constant, and instead add to the Einstein action a function f(φ), which is invariant. The appearance of such function f can be easily justified in the spectral action approach of noncommutative geometry[10] [11]. Clearly we cannot derive it in nonperturbative way, but we can try just to find a theory where this function allows to resolve singularities in General Relativity. In particular, we will show that there exist Born-Infeld type of actions with

f (φ) = 1 − √ (1 − (◻φ)^{2}/ ε_{m}) + ....,

and for which singularities are resolved, that is, the contracting universes bounce at the limiting curvature and all curvature invariants always remains regular and bounded by the values, characterized byε_{m}. Once again we would like to stress that introducing φ does not add to the system any new dynamical scalar fields and new degrees of freedom. The “field φ” always remains constrained by (1) and in the synchronous coordinate system it just “serves” as time, making the longitudinal degree of freedom of the gravitational field to be dynamical. Thus,this theory must be viewed as a modification of General Relativity in the longitudinal sector. Because the longitudinal gravitational field induced by matter via the constraint has a “negative energy” by itself, it is not surprising that the proposed modification of Einstein’s theory violates the conditions needed in the proof of singularity theorems and hence these singularities can be avoided. Although noncommutative geometry offers a very strong support for the model considered here [10], we will not require, or use any information or methods from that framework. Instead our purpose is to propose nonsingular classical modification of General Relativity irrespective of its justification from the point of view of so called “fundamental theory”, which is not known at present...

The presence of mimetic field needed to generate Born-Infeld corrections to the Einstein theory adds extra “dust like” degree of freedom to gravity, which can serve as mimetic dark matter. It is rather curious that the non-singular modification of General Relativity delivers for free the realistic candidate for Dark Matter. Besides of the appearance of mimetic matter, Einstein’s equations are significantly modified only at very high curvatures which are close to the limiting one.

The non-singular models we have constructed can be rather important for bouncing cosmologies. In fact, in these models a bounce happens within very short time interval of order t∼ε_{m}^{2}and if the limiting curvature would be of the order of Planck value, this time would be the Planckian time. Outside this time interval Einstein’s theory is well applicable. Using causality one can argue that in this case perturbations generated in a contracting universe on the supercurvature scale can be re-translated to an expanding universe practically without any change. However, this question requires further quantitative investigation.

The idea of limiting curvature can also have rather severe consequences for inflationary universe. First of all, notice that if we want to use the stage of accelerated expansion for amplifying quantum fluctuations, observed in numerous CMB experiments, the limiting curvature cannot be below the inflationary scale, that is, it cannot be less than the Planckian scale more than just by few orders of magnitude. When the limiting curvature is below the self-reproduction scale the multiverse is avoided. The flat inflationary potentials, favoured by observations, also look more natural from the point of view of the idea of limiting curvature.

The theory we have considered here is a classical theory and if the limiting curvature is well below the Planck scale we can safely ignore quantum corrections. However, any field gravitates, and therefore the highly energetic quanta which in Einstein’s theory would produce the curvature exceeding the limiting one, must be either prohibited or modified. This is whywe expect that the natural cut-off in quantum field theories, which is normally taken to be of the order of Planck value, can be well below the Planck scale in theories with limiting curvature. The considerations above are restricted to highly symmetric space-times. One could naturally address the question whether the curvature will remain bounded generically for arbitrary inhomogeneous spaces. In an accompanying paper [12] we show that the singularity is also avoided in the case of a black hole and give the explicit solution for it. Moreover analyzing the expressions for the curvature invariants one could argue that the limiting curvature is the generic property of arbitrary space-time irrespective of their spatial curvature. However these questions require a more serious investigation.

(Submitted on 18 Dec 2016)

**Gedanken trip inside the matrioshka like spacetime structure of a black hole**

..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∼ r_{g}{gravitational radius of the black hole}. At the beginning the curvature of large black holes is very low but grows and finally, after time t∼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, 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. [we find that the spacetime structure inside the nonsingular black hole is similar to “a Russian nesting doll”. Namely, its geometry is a time sequence of the internal Schwarzschild geometries separated by “layers with limiting curvature” of width∆t∼ε_{m}^{-1/2}. The Scharzschild radii characterizing these subsequent geometries decrease and are proportional tor_{g },r_{g}^{1/3}∼10^{16},r_{g}^{1/9}∼10^{5},r_{g}^{1/27},....etc]...Finally when the size of the black hole becomes of the order of the width of a time layer∼ ε_{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 radiusr_{g}∼10^{49}(in Planck units) after the crossing of limiting curvature we find ourselves in black holes of radiir_{g}^{1/3}∼10^{16},r_{g}^{1/9}∼10^{5},r_{g}^{1/27}∼10^{2 }correspondingly. Finally at the fourth layerr_{g}^{1/81}∼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.

(Submitted on 18 Dec 2016)

**Farewell, my lovely information paradox?**

For an evaporating black hole the derivation of Hawking radiation remains unchanged for a large black hole [10]. However, when it reaches the minimal size of orderε_{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[11].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 disappearance of the black hole. In our case the smallest black hole remnant has enough space “inside it” to hide all the information about the original matter from which the black hole was formed together with the information about 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.

Id.

Black Hole Locations in the Milky Way (Milky Way illustration courtesy NASA/JPL-Caltech/R. Hurt (SSC/Caltech)) |