Matter accretion versus semiclassical bounce in Schwarzschild interior

We discuss the properties of the previously constructed model of a Schwarzschild black hole interior where the singularity is replaced by a regular bounce, ultimately leading to a white hole. The model is semiclassical in nature and uses as a source of gravity the effective stress-energy tensor (SET) corresponding to vacuum polarization of quantum fields, and a minimum spherical radius is a few orders of magnitude larger than the Planck length, so that the effects of quantum gravity should be still negligible. We estimate the other quantum contributions to the effective SET, caused by a nontrivial topology of spatial sections and particle production from vacuum due to a nonstationary gravitational field and show that these contributions are negligibly small as compared to the SET due to vacuum polarization. The same is shown for such classical phenomena as accretion of different kinds of matter to the black hole and its further motion to the would-be singularity. Thus, in a clear sense, our model of a semiclassical bounce instead of a Schwarzschild singularity is stable under both quantum and classical perturbations.


Introduction
The existence of singularities in various solutions of general relativity (GR) as well as many alternative classical theories of gravity, describing black holes or the early Universe, is an undesirable but apparently inevitable feature. On the other hand, one can hardly believe that the curvature invariants or the densities and temperatures of matter that appear in such singularities can really reach infinite values. There is therefore a more or less common hope that a future theory of gravity valid at very large curvatures, high energies, small length and time scales will be free of singularities, and that such a theory should take into account quantum phenomena.
(b) In semiclassical gravity, where the geometry is treated classically and obeys the equations of GR or an alternative classical theory, using averages of quantum fields of matter as sources of gravity with possible "exotic" properties [15][16][17][18][19][20].
(c) Diverse models of quantum gravity are also often translated into the language of classical geometry and lead to nonsingular space-times describing both regular black hole interiors and early stages of the cosmological evolution [21][22][23][24][25][26][27][28][29][30][31] (see also references therein for each item). One can notice that the singularity problems in black hole physics and Big Bang cosmology are quite similar. For example, the Schwarzschild singularity is located in a nonstationary "T-region", where the metric can be written as that of a homogeneous anisotropic cosmology, a special case of Kantowski-Sachs models. It is therefore natural that the same tools are used in attempts to attack these problems.
Classical nonsingular models in cosmology, black hole and wormhole physics are quite popular, but the "exotic" components that are necessarily present in those models require certain conjectures so far not confirmed by observations or experiments, and their consideration is often justified as a kind of phenomenological description of underlying quantum effects.
Many models of quantum gravity, in their representations in the language of classical geometry, lead to nonsingular cosmologies and black hole models, but most frequently such models reach the values of curvatures and densities close to the Planck scale. However, more surprising is a considerable diversity of their predictions, depending on various leading ideas employed in such models.
Thus, a number of scenarios in the framework of Loop Quantum Gravity (LQG) predict a bounce close to the Planck scale and a transition from a black hole to a white hole [22][23][24][25]. In particular, in [24,25] the authors consider quantum corrections to the Oppenheimer-Snyder collapse scenario.
Unlike that, application of the so-called polymerization concept to the interior of a Schwarzschild black hole [26,27], also removing the singularity, leads to a model with a single horizon and a Kantowski-Sachs cosmology with an asymptotically constant spherical radius at late times. (This geometry is partly similar to the classical black universes with a phantom scalar [8][9][10], but in the latter the late-time Kantowski-Sachs cosmology tends to de Sitter isotropic expansion.) Some of the scenarios (see, e.g., [29]) even lead to a quantum-corrected effective metric with an unconventional asymptotic behavior, although it is claimed that the quantum correction to the black hole temperature is quite negligible for sufficiently large black holes, and that the metric is asymptotically flat in a precise sense.
A consideration of homogeneous gravitational collapse of dust and radiation with LQG effects has led in [31] to avoidance of both a final singularity and an event horizon, so that the outcome is a dense compact object instead of a black hole.
Let us also mention a study of black hole evaporation process by Ashtekar [30] using as guidelines (i) LQG, (ii) simplified models with concrete results, and (iii) semiclassical effects. The author discusses various issues concerning the information loss problem and the final fate of evaporating black holes; one of his conclusions is that LQG effects do not appreciably change the semiclassical picture outside macroscopic black holes.
A comprehensive review of quantum gravity effects in gravitational collapse and black holes has been provided by Malafarina [21] in 2017, and we here only mention a few results of interest and some papers that appeared later than this review. But even this short list shows how diverse can be the results and conclusions depending on the particular approach. All that may be a manifestation of a so far uncertain status of quantum gravity.
Since matter can manifest its quantum properties at the atomic or macroscopic scales (as exemplified by lasers or the Casimir effect), one may hope that singularities in cosmology or black holes may be prevented at length scales much larger than the Planck one. This would look more attractive both from the observational viewpoint and also theoretically since the corresponding results, at least today, look more confident than those obtained with quantum gravity.
The black hole studies in the framework of semiclassical gravity, such as [17][18][19][20] and many others, mostly focus on the consequences of the Hawking black hole evaporation and the related information paradox. Their conclusions seem promising from the viewpoint of singularity avoidance. Thus, in [17] it is concluded that the black hole evaporation ultimately leads to emergence of an inner macroscopic region that hides the lost information and is separated from the external world. According to [20], the evaporation process even prevents the emergence of an event horizon. Thus, after formation of a large spherically symmetric black hole by gravitational collapse, the classical r = 0 singularity is replaced by an initially small regular core, whose radius grows with time due to increasing entanglement between Hawking radiation quanta outside and inside the black hole, and by the Page time (when half the black hole mass has evaporated), all quantum information stored in the interior is free to escape to the outer space.
However, there remains a question of what is happening inside a large black hole when it has just formed, and the evaporation process is too slow to immediately launch the above processes. Indeed, an approximate expression for the full evaporation time is t evap ∝ M 3 , where M is the initial black hole mass; it then follows that the Page time is 7 8 t evap , and if M is the solar mass, we have t evap ≈ 2.1 × 10 67 years. In other words, any astrophysical black hole (except for very light primordial ones) is at this initial stage of its evaporation. Even more than that: under realistic conditions, its mass much faster grows due to accretion than decreases by evaporation.
In our study we try to answer the following question: what is the internal geometry of such a large and "young" black hole if its Hawking evaporation can be neglected, but the impact of quantum fields that are present in a vacuum form is taken into account? In other words: if a body (a particle, a planet, a spacecraft) falls into such a black hole, what is the geometry it will meet there?
More specifically, we are considering the neighborhood of a would-be Schwarzschild singularity (r = 0) in the framework of semiclassical gravity and explore a possible emergence of a bounce instead of the singularity. We can recall that in any space-time region there always exist quantum oscillations of all physical fields. We do not assume any particular composition of these fields, considering only their vacuum polarization effects. In such a simplified statement of the problem, we have shown [32] that there is a wide choice of the free parameters of the model that provide a possible implementation of such a scenario. The SET used to describe the vacuum polarization of quantum fields is taken in the form of of a linear combination of the tensors (1) H ν µ and (2) H ν µ obtained by variation of the curvature-quadratic invariants R 2 and R µν R µν in the effective action in agreement with the renormalization methodology of quantum field theory in curved space-times [33,34]. In this scenario, in the internal Kantowski-Sachs metric, the spherical radius r evolves to a regular minimum instead of zero, while its longitudinal scale has a regular maximum instead of infinity. The free parameters of the model can be chosen so that the curvature scale does not reach the Planck scale but remains a few orders smaller (for example, on the GUT scale), sufficiently far from the necessity to include quantum gravity effects. The whole scenario is assumed to be time-symmetric with respect to the bouncing instant, therefore, as in many other papers, we are describing a smooth transition from black to white hole.
The nonlocal part of the effective SET of quantum fields in the Schwarzschild interior, depending on the whole history and mainly represented by particle production from vacuum, was estimated in [35], and it was shown that its contribution in the vicinity of a bounce is many orders of magnitude smaller than that of (1) H ν µ and (2) H ν µ . In the present paper, after a brief representation of the results of [32,35], we try to find out whether or not there are classical phenomena that could potentially destroy the bounce, namely, accretion of different kinds of matter which is always present near astrophysical black holes and whose density increases as it further moves inside the horizon towards the would-be singularity. It turns out that this accretion is also unable to affect the bounce due to its negligibly small contribution to the total SET. The paper is structured as follows. Section 2 summarizes the problem statement and the assumptions made. In Section 3 we describe the bouncing solution to the field equations, in Section 4 we estimate the nonlocal contribution to the effective SET, Section 5 is devoted to calculations of the spherically symmetric accretion of the CMB radiation and massive matter to a Schwarzschild black hole, and Section 6 is a brief discussion.

Field equations and assumptions 2.1 Near-bounce geometry
Considering a generic static, spherically symmetric black hole in its interior region (beyond the horizon), also called a T-region, we can write its metric in the general Kantowski-Sachs form where τ is the natural time coordinate in the corresponding reference frame, and x is a spatial coordinate that "inherits" the time coordinate of the static region after crossing the horizon; dΩ 2 is, as usual, the metric on a unit sphere S 2 . It is a homogeneous anisotropic cosmological model with the topology R × S 2 of its spatial sections.
Assuming that quantum effects can appreciably change the space-time geometry only if the latter is very strongly curved, while at smaller curvatures, even in a T-region (r < 2m), we can use with sufficient accuracy the Schwarzschild solution, which then takes the form where m = GM , G being Newton's constant of gravity, M the black hole mass, and we use the units = c = 1). Compared to the conventional expression, we have changed the notation, r → T , to emphasize that in the T-region the coordinate r is temporal. Furthermore, at T ≪ 2m, passing on to the Kantowski-Sachs cosmological time by putting T /(2m)dT = dτ , we obtain an asymptotic form of the metric in the notations of (1): which is valid at τ /m ≪ 1. It is the Schwarzschild metric at approach to the singularity τ → 0, at which the scale along the x axis is infinitely stretched while the spheres x = const are shrinking to zero. In this study, our basic assumption will be that quantum field effects do not allow the space time to approach too close to the singularity r ≡ e β = 0 (or τ = 0 in (3)) but, instead, stop the contraction of r at τ = 0 at some regular minimum value r = r 0 > 0, while the scale factor e γ along the x axis simultaneously turns to a regular maximum. Then, at small τ , in agreement with (2) and (3), the metric takes the form where r 0 ,b,c are positive constants with appropriate dimensions. In addition to these assumptions, let us also suppose that the time evolution of the metric is symmetric with respect to the bouncing instant τ = 0. Then, in the notations of (1), we can present the functions β(τ ) and γ(τ ) as Taylor expansions with only even powers of τ , where β i , γ i (i = 0, 2, 4, 6, ...) are constants. Then, according to (4), To explain the behavior (4) of the metric, we invoke the semiclassical approach, writing the Einstein equations as where the r.h.s. represents a renormalized stress-energy tensor (SET) T ν µ of quantum fields, containing, in general, both local and nonlocal contributions.
In the general metric (1), the Einstein tensor G ν µ has the following nonzero components: Substituting the Taylor expansions (5), we can explicitly present these components up to O(τ 2 ) as follows:

The stress-energy tensor
In agreement with the vast literature on quantum field theory in curved space-times, including the books [33,34], the renormalized vacuum SET T µ ν of quantum fields may be presented as a linear combination of two tensors of geometric origin (i) H ν µ (i = 1, 2) (which can be obtained by variation of actions containing R 2 and R µν R µν , i.e., the Ricci scalar and tensor squared), with some phenomenological constants N 1 , N 2 , and two other contributions, (c) H µ ν and P µ ν : where (1) and = g µν ∇ µ ∇ ν . The tensor (c) H µ ν is of local nature and depends on the space-time topology and/or on boundary conditions (e.g., the Casimir effect [36,37]), while P µ ν is nonlocal, it depends on the particular quantum states of the constituent fields and, in particular, describes particle production in a nonstationary metric. Its nonlocal nature means that it is not a function of a space-time point but depends, in general, on the whole history. Its calculation is rather a complex task and requires additional assumptions on quantum states of different fields. We will temporarily assume that the contribution of P µ ν is small as compared to the other terms in (10) (at least under a suitable choice of quantum state) and try to justify this assumption in Section 4.
The components of the tensors (i) H µ ν (which turn out to be diagonal) can be easily calculated from the ansatz (1) with the Taylor expansions (5). At the very instant τ = 0 (at bounce) they are (1) As will be clear further on, their higher orders in τ will be unnecessary in our calculations.
What is known about the numerical coefficients N 1 and N 2 in (10)? According to [33,34], their values should be found from experiments or observations. The orders of magnitude of these coefficients may be roughly estimated by recalling that they appear in higher-derivative theories of gravity where the action has the form the tensors (1,2) H µν resulting from variation of the corresponding terms. The upper bounds on these parameters are N 1,2 10 60 (see, e.g., [42]), as follows from observations performed at very small curvatures, at which any possible effects of terms quadratic in the curvature are extremely weak. However, the factors N 1,2 may be estimated in another way if such theories of gravity are used to describe the early (inflationary) Universe with much larger curvatures, for example, N 1 ∼ 10 10 [38 -40]. For our purposes, we will keep in mind this order of magnitude.
Concerning the Casimir contribution, there are arguments indicating that it must be much smaller than the contribution of (i) H ν µ . If we consider, for instance, the static counterpart of the metric (1) with e β = r = r 0 , something treatable as a description of an infinitely long wormhole throat, we can use the result obtained in [41] for a conformally coupled massless scalar field, which reads for this geometry where a 0 is some fixed length to be determined by experiment. Note that the quantity (14) is obtained for a single massless scalar, and the total Casimir contribution must take into account all existing fields with different spins and masses, hence this contribution may be two or three orders of magnitude larger than (14). On the other hand, for the same space-time geometry, Therefore, if N 1 and/or N 2 are at least of the order of unity (as we will consider in what follows), the tensors (i) H µν contribute much stronger to T ν µ in the Einstein equations (7) than (c) H µν , unless the uncertain length a 0 in (14) is unreasonably high, or the total number of fields is so large as to overcome the denominator which is ∼ 10 4 .
In our further consideration we will assume that (c) H µ ν can also be neglected in our geometry (4) and take into account only the contributions (i) H ν µ .

The semiclassical bounce
In this section we consider the Einstein equations (7) with the SET (10), taking into account only the first two terms. Our task will be to find out whether or not there are solutions consistent with the bouncing metric (4), and if it is the case, what are the requirements to the free parameters of the model that would justify the semiclassical nature of the equations. In the subsequent sections we will analyze the influence of other effects that could in principle destroy the model thus constructed: the nonlocal contribution to the SET (10) and the possible influence of matter surrounding the black hole and falling to its interior region. For our purpose, we will express G µ ν and (i) H µ ν in terms of the Taylor series coefficients in (5) and equate the coefficients at equal powers of τ on different sides of the resulting equations. Let us introduce, for convenience, the following dimensionless parameters Since κ ≈ l 2 Pl (the Planck length squared), it is evident that our system remains on the semiclassical scale only if all parameters (16) are much smaller than unity. Hence, in particular, the minimum spherical radius r = r 0 , reached at bounce, should be much larger than the Planck length. Other parameters that should be small are values of the derivativesβ,γ , etc. close to the bounce.
An inspection shows that, in the approximation used, it is sufficient to consider the order O(1) in the 0 0 component of Eqs. (7), from which we find The role of all other equations reduces to expressing the constants B 4 , C 4 , etc. in terms of A, B 2 , C 2 . Thus we have a single equation for the three parameters A, B 2 , C 2 of the bouncing geometry, along with the coefficients N 1 , N 2 . Therefore, we have a broad space of possible solutions.
As stated above, we must assume that r ) is much larger than the Planck length l Pl ∼ √ κ , from which it follows that A ≪ 1, or A = O(ε), ε being a small parameter. We can also make the natural assumptions B 2 = O(ε) and C 2 = O(ε), which means thatβ andγ are of the same order of magnitude as 1/r 2 0 . Then, since the r.h.s. of Eq. (17) is O(ε 2 ) while the l.h.s. is O(ε), to provide the equality, we must require that N 1 and/or N 2 should be large, of the order O(1/ε).
The remaining Einstein equations 1 1 and 2 2 at τ = 0 then show that B 4 and C 4 are of the order O(ε 2 ) (see (11)), therefore, the 4th order derivatives of β and γ are of a correct order of smallness with respect to the Planck scale (see (16)). Similar estimates are obtained for B 6 , C 6 , etc. if we analyze equations in the order O(τ 2 ), and so on. It can also be verified that the curvature invariants R, R µν R µν and K ≡ R µνρσ R µνρσ are small at bounce (τ = 0) as compared to the Planck scale: Consider a numerical example for illustration. Assuming N 1 = 0, N 2 = 10 10 , and A = 10 −10 , a minimum radius a is of 10 5 Planck lengths. Since, by construction (see (6) and (16)), B 2 > 0 and C 2 < 0, we can assume for convenience B 2 + C 2 = 0. As a result, from Eq. (17) we find From the equations of order O(τ 2 ) one can then determine B 6 , C 6 , and so on. One can recall that in spherically symmetric space-times, if the spherical radius e β = r has a regular minimum (it is a wormhole throat if the minimum is in an R-region and a bounce if it is in a T-region), then the SET must satisfy the condition T 0 0 − T 1 1 < 0 which means violation of the Null Energy Condition, see, e.g., [2,43]. In our model, supposing a bounce at τ = 0, we automatically obtain the inequality T 0 0 − T 1 1 < 0.

Nonlocal contribution to the vacuum SET
To estimate the contribution of the nonlocal term P ν µ in the SET (10), we rewrite the general metric (1) of a Kantowski-Sachs cosmology as where the time coordinate η is the so-called "conformal time," defined by the condition 3α(η) = 2β(η) + γ(η), being convenient for considering quantum fields. We assume that the black hole has a stellar (or larger) mass m Sch , and µ = 2Gm Sch 10 5 cm = 1 km is the corresponding gravitational radius. Meanwhile, at bounce (say, at the time η = 0), in agreement with the previous section, we assume that the minimum radius is r 0 = µ e β(0) is ∼ 10 5 l Pl ∼ 10 −28 cm. Introducing the small parameter ǫ = r 0 /µ 10 −33 (not to be confused with ε from the previous section), for times close to the bounce we can write with 3a = 2b + c according to the definition of η , b > 0 since e β has a minimum, and c < 0 since e γ has a maximum at η = 0. The powers of ǫ correspond to magnitudes of the metric coefficients at approach to a would-be Schwarzschild singularity. Consider a quantum scalar field satisfying the equation ( + M 2 + ξR)Φ = 0 and its standard Fourier expansion: where N is a normalization factor, ξ is a coupling constant, c + klm is a creation operator, Y lm are spherical functions, and each mode function g klm (η) ≡ g obeys the equation obtained by separation of variables in the original Klein-Gordon-type equation: where the dot stands for d/dη , and Ω is the effective frequency: At bounce time η = 0 we have, due to standard normalization, |g| ∼ Ω −1/2 and Now, for estimation purposes, we will make a natural assumption, justified by experience [33,34], that particle production takes place most intensively at energies close to the curvature scale ∼ r −1 0 . This energy is of the order of the frequencyΩ(τ ) in terms of the proper cosmic time τ related to our conformal time by dτ = e α dη . Therefore our assumption meansΩ ∼ 1/r 0 . Since e α ∼ √ ǫ , one has τ ∼ √ ǫη , and from the relation Ωη =Ωτ we obtainΩ = Ω/ √ ǫ, so that Ω 2 (0) = k 2 ǫ + l(l + 1) + 2ξ It is of interest, at which values do the parameters of the model appreciably contribute toΩ 2 having the order ∼ r −2 0 = (µǫ) −2 . They are: Apparently, momenta k strongly exceeding the Planckian value look meaningless, and we can conclude that at reasonable (sub-Planckian) values of k , their contributions toΩ are negligibly small. Note that the result a ∼ ǫ/r 2 0 can be obtained in another way using the relations A similar analysis leads to b, c ∼ ǫ/r 2 0 . Furthermore, at small η we can assume The energy density of created particles may be estimated using the standard technique of Bogoliubov coefficients. For the case of bounce-type metrics, the crucial Bogoliubov coefficient β kl can be computed with necessary accuracy by using the formulas [44] where η 1 is the initial time at which, by assumption, β kl = 0 (that is, assuming a vacuum state of the field, with no particles). Using Eq. (27) and making the assumption Bη O(1) (which means that η is not very far both from zero and from η 1 ), we obtain Now we can calculate the integrals I ± involved in (28) at times close to bounce (η = 0): Then, assuming Bη 1 O(1), we arrive at Thus the energy density of created particles is where we have employed the following approximate orders of magnitude for each factor in (33), in agreement with (26): (i) dk ∼ 2m Pl = 10 5 /r 0 since we integrate from −m Pl to +m Pl ; (ii) l (2l + 1) ∼ 10 2 , involving a few low multipolarities (since large multipolarities would mean too large mode energies); (iii) e −4α /µ 2 ∼ 1/r 2 0 ; (iv) Ω ∼ √ ǫ/r 0 ; (v) |β kl | 2 ∼ 1 as a very rough upper bound.
A comparison of the estimate (33) with that of the local energy density contribution from vacuum polarization obtained in the previous section and [32], ρ loc ∼ 10 10 r −4 0 , leads to ρ nonloc /ρ loc ∼ 10 −21 , and this value is still smaller if we consider black hole masses larger than that of the Sun. We conclude that the nonlocal contribution to the vacuum energy density due to particle production is negligibly small in the regime of semiclassical bounce, and a more accurate calculation including more physical fields of different spins can hardly change this estimate too strongly.

CMB accretion
Black holes in the real Universe are surrounded by various kinds of matter: interstellar or intergalactic gas, dust and stellar matter if the black hole gravity destroys approaching stars. Depending on specific astrophysical circumstances, the ambient matter may form an accretion disk or experience spherical or close to spherical accretion. The falling matter crosses the horizon and should ultimately approach the black hole singularity, if the latter really exists. Or, if the theory predicts a bouncing region instead of a singularity, it is natural to ask: will the gravity of the accreted matter strongly change the geometry of the bouncing region? Can it happen that this falling matter will destroy the bounce (whatever be its origin) and restore the singularity?
We will try to answer this question for a Schwarzschild black hole with a semiclassical bounce described in [32] and in the previous sections. Thus we assume that the space-time metric is approximately Schwarzschild, everywhere except for a region close to bounce, that is, r nr 0 , where, say, n 10, and r 0 is the minimum radius at bounce. In this subsection, we consider spherical accretion of the kind of matter that exists anywhere in the Universe, the Cosmic Microwave Background (CMB). Thus our calculation can correspond to an isolated Schwarzschild black hole in intergalactic space, surrounded by the CMB only, and the accretion consists in capture of CMB photons. It is thus a minimum possible environment of any black hole. At each point of the black hole's ambient space, there is a flow of photons to be captured: these are photons whose path gets into the so-called photon sphere with the radius r ph = 3m. Such photons may be considered as those forming a radiation flow with the SET where the null vector k µ is, in a reasonable approximation, radially directed, so that where e γ = 1 − 2m/r . Then the conservation law ∇ ν T ν µ = 0 in the metric (34) gives for Φ = ρ flow (the flow energy density) The constant Φ 0 should be determined by the CMB energy density and the black hole mass, taking into account bending of photon paths in the black hole's gravitational field. Fortunately, there is no necessity to carry out such a computation anew: we can use, for example, the result obtained by Bisnovatyi-Kogan and Tsupko [45]. They showed that if a source of radiation is located at r = 10 4 m in Schwarzschild space-time, then the black hole will capture radiation emitted inside a cone with an angular radius α ≈ 0.0298 • ≈ 5.203×10 −4 . If the source radiates isotropically, then the fraction ∆(r) of the emitted radiation energy captured by the black hole will be equal to the part of the complete solid angle of 4π contained in the spot of πα 2 , that is, ∆(r) = πα 2 /(4π) = α 2 /4 ≈ 6.768×10 −8 for r = 10 4 m.
At r = 10 4 m or larger, the space-time may be regarded approximately flat, therefore, due to flux conservation, the fraction ∆ should be proportional to r −2 ; on the other hand, since the area of a sphere from which the flux is collected, is ∝ m 2 , it should be also ∆ ∝ m 2 . As a result, we can write, using (38), On the other hand, at such distances from the black hole, the CMB can be safely regarded homogeneous and isotropic, and we can conclude that the accretion flow will have the energy density where the CMB density ρ CMB is nowadays where ρ Pl = l −4 Pl is the Planck density. Thus we know the SET (35) with (37) and (40) in the external region of the black hole, but the quantity (37) diverges at the horizon r = 2m. This looks natural since in our static reference frame the radiation is infinitely blueshifted at the horizon, where this reference frame in no more valid. However, our purpose is to find out how this radiation behaves deeply beyond the horizon. To extend the expression (35) to r < 2m, let us transform it to the Kruskal coordinates valid at all r . To do that, it is convenient to use at r > 2m the so-called tortoise radial coordinate (note that r * → −∞ as r → 2m). This coordinate belongs to the same static reference frame, hence the flow energy density is T 0 0 = Φ. However, the null vector k µ is now, instead of (36), where, as before, e γ = 1 − 2m/r , and the nonzero covariant SET components have the form convenient for the transformation. The Kruskal coordinates R, T , in which the metric has the form are related to r * , t by Using this, we transform T µν to the Kruskal coordinates and find the nonzero components In (45) and (47) the horizon r * = −∞ → R 2 = T 2 is a regular surface, the static region r > 2m corresponds to R 2 > T 2 , while at r < 2m instead of the coordinates r * , t or r, t we can introduce their counterparts x (analog of t) and τ (analog of r * ) by putting, for T > R > 0 (the upper quadrant in Kruskal's diagram), so that the metric acquires the Kantowski-Sachs form the two timelike coordinates r and τ being related by The horizon corresponds to r = 2m or τ → −∞, while the singularity r = 0 occurs at τ = 0. Using (48), we transform the tensor (47) to the Kantowski-Sachs coordinates τ, x, obtaining from which it follows that the energy density is .
We see that in the Kantowski-Sachs reference frame, in which the Schwarzschild metric looks very similar to its usual appearance in the static region, the expression for ρ flow also looks very similar. It is the density in the same reference frame that was used for describing the bounce and can thus be compared with the vacuum polarization density ρ vac ∼ 10 10 r −4 0 ∼ 10 −10 ρ Pl at bounce. Assuming that the internal Schwarzschild metric (49) is the true metric up to r r 0 ≪ 2m, using (40) and (41), we obtain for such small radii and, since ∆ 0 is of the order of unity, we conclude that the flow density at small radii is larger than ρ CMB approximately by a factor of m/r . For a black hole of stellar mass, m ∼ 10 5 cm and r ∼ r 0 ∼ 10 5 l Pl this factor is ∼ 10 33 , so that, with ρ vac ∼ 10 −10 ρ Pl and recalling (41), we obtain ρ flow /ρ vac ∼ 10 −85 . This ratio will certainly be larger for heavier black holes and for earlier epochs when ρ CMB was larger by a factor of (a 0 /a) 4 , where a is the cosmological scale factor and a 0 its present value. Assuming the existence of supermassive black holes with m ∼ 10 9 solar masses at scale factors a ∼ 10 −3 a 0 (that is, at z ∼ 1000, close to the recombination time), the above ratio gains 21 orders of magnitude, resulting in ρ flow /ρ vac ∼ 10 −64 .
We conclude that CMB accretion cannot exert any influence on the model dynamics at small radii close to bounce or a would-be singularity inside a Schwarzschild black hole. Very probably, accretion of ambient matter can be much more important, and our next task is to estimate its impact.

Dust accretion
Matter falling onto a black hole has in general the form of a hot gas, but close to the horizon this gas is nearly in a state of free fall [46], therefore the approximation of dust freely radially moving to the horizon looks quite adequate, and it is reasonable to assume that the same regime well describes its further motion in the T-region.
Thus we consider the Schwarzschild space-time with the metric (34) or, in terms of the tortoise coordinate r * , (42). In this metric, we consider matter with the SET where the components of the 4-velocity vector u µ for radial motion may be written, in terms of the radial coordinate r * , in the form where v = e −γ dr * /ds (s is proper time along the world line), so that u µ u ν = 1.
The constant K can be found if we know the dust density at some r in the R-region. To this end, we can recall that, according to [46] (page 324), under typical conditions the falling matter density is ρ ≃ (6×10 −12 g/cm 3 )(2m/r) 3/2 . Thus, say, at r = 10m we obtain ρ ∼ 10 −12 g/cm 3 which approximately equals 2×10 −106 ρ Pl . 5 We thus have With this value of K , let us estimate the dust energy density ρ E at the radius r = r 0 = 10 5 l Pl , the supposed bounce radius. According to (62), For the black hole mass m ≈ m ⊙ , we have (m/r) 3/2 ≈ 10 50 , so that if we assume ρ vac ≈ 10 −10 ρ Pl . We conclude that the influence of the accretion flow on the hypothetic semiclassical bounce is quite negligible. The situation does not change if we assume, say, the initial dust density 5 orders of magnitude larger and a supermassive black hole of 10 9 m ⊙ : we thus gain about 18 orders of magnitude in (65), and there still remains a difference of 24 orders.

Conclusion
We have constructed a simple model [32] describing a possible geometry that can exist deeply inside a sufficiently large black hole at its sufficiently early stage of evolution, when the Hawking radiation is negligible due to its extremely low temperature, and one could not yet feel the influence of quantum entanglement phenomena. The model is semiclassical in nature and is governed by vacuum polarization leading to the emergence of quadratic curvature invariants in the effective action. We have assumed that the free constants appearing at these invariants have values of the same order as in some well-known models of the inflationary universe, and showed that the corresponding terms in the effective Einstein equations lead to solutions in which the Schwarzschild singularity is replaced by a regular bounce, ultimately leading to a white hole. Furthermore, we have argued that other quantum effects such as the Casimir effect, caused by the spherical topology of a subspace in the Kantowski-Sachs cosmology inside the black hole, and particle production from vacuum caused by a nonstationary nature of the metric, make only negligible contributions to the total effective SET and therefore cannot destroy the bouncing geometry. The same has been shown for possible classical phenomena that could interfere, namely, accretion of different kinds of matter and its further motion to the black hole interior. It can be said that, in a sense, our simple bouncing model is stable under both quantum and classical perturbations.
It would be of substantial interest to study how this model will be modified if Hawking radiation at its early stages is included into consideration. Another subject of future studies can be concerned with using similar assumptions for black holes with charge and spin, where the nature of singularities is quite different and where Cauchy horizons take place. As is mentioned in [30], according to the stability analysis of Kerr and Reissner-Nordström space-times, their Cauchy horizons are unstable under small perturbations, from which it follows that a generic black hole singularity must be null rather than spacelike as in the Schwarzschild metric, and the analysis of such singularities and their possible avoidance should be a promising field of research.