1. 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.
The existing numerous attempts to avoid singularities can be basically classified as follows (see also references therein for each item):
- (a)
In GR, “exotic” sources of gravity are invoked, violating the standard energy conditions, for example, phantom scalar fields; in classical extensions of GR, using quantities of geometric origin (torsion, nonmetricity, extra dimensions) whose effective stress-energy tensors (SETs) can have similar “exotic” properties [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11]; it has also been argued that the effects of rotation in GR can also play the role of exotic matter (see [
12,
13,
14]).
- (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].
- (c)
Diverse models of quantum gravity are also often translated into the language of classical geometry and lead to nonsingular spacetimes describing both regular black hole interiors and early stages of the cosmological evolution [
20,
21,
22,
23].
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 the considerable diversity of their predictions, depending on various leading ideas employed in such models.
Thus, several 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 [
21,
22,
24,
25]. In particular, in [
24,
25], the authors considered quantum corrections to the Oppenheimer-Snyder collapse scenario. Much earlier, a similar scenario with singularity avoidance was considered by [
26] on the basis of a quasiclassical approximation of the Wheeler-DeWitt equation.
Unlike that, application of the so-called polymerization concept to the interior of a Schwarzschild black hole [
27,
28], 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 [
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 leads in [
30] to the 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 [
31] using as guidelines: (i) LQG; (ii) simplified models with concrete results; and (iii) semiclassical effects. The author discussed 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 was provided by Malafarina [
20] in 2017, and we here only mention a few results of interest and some papers that appeared later than this review. However, even this short list shows how diverse the results and conclusions can be 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 ([
17,
18,
19,
32], 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 [
32], 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 [
19], 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
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 , where M is the initial black hole mass; it then follows that the Page time is , and if M is the solar mass, we have years. In other words, any astrophysical black hole (except for very light primordial ones) is at this initial stage of its evaporation. Moreover, 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 or a spacecraft) falls into such a black hole, what is the geometry it meets there?
More specifically, we are considering the neighborhood of a would-be Schwarzschild singularity (
) in the framework of semiclassical gravity and explore a possible emergence of a bounce instead of the singularity. We can recall that in any spacetime 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 showed [
33] 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
and
obtained by variation of the curvature-quadratic invariants
and
in the effective action in agreement with the renormalization methodology of quantum field theory in curved spacetimes [
34,
35]. 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 [
36], and it was shown that its contribution in the vicinity of a bounce is many orders of magnitude smaller than that of
and
.
In the present paper, after a brief representation of the results of [
33,
36], 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.
Section 6 is a brief discussion.
3. 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 is 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 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 express
and
in terms of the Taylor series coefficients in (
5) and equate the coefficients at equal powers of
on different sides of the resulting equations. Let us introduce, for convenience, the following dimensionless parameters:
Since
(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
, reached at bounce should be much larger than the Planck length. Other parameters that should be small are values of the derivatives
, etc. close to the bounce.
An inspection shows that, in the approximation used, it is sufficient to consider the order
in the
component of Equation (
7) (or explicitly (
9)), from which we find
The role of all other equations reduces to expressing the constants , etc. in terms of . Thus, we have a single equation for the three parameters of the bouncing geometry, along with the coefficients . Therefore, we have a broad space of possible solutions.
As stated above, we must assume that
is much larger than the Planck length
, from which it follows that
, or
,
being a small parameter. We can also make the natural assumptions
and
, which means that
and
are of the same order of magnitude as
. Then, since the r.h.s. of Equation (
17) is
while the left-hand side (l.h.s.) is
, to provide the equality, we must require that
and/or
should be large, of the order
.
The remaining Einstein equations
and
at
then show that
and
are of the order
(see (
11)), therefore, the fourth-order derivatives of
and
are of a correct order of smallness with respect to the Planck scale (see (
16)). Similar estimates are obtained for
, etc. if we analyze equations in the order
, and so on. It can also be verified that the curvature invariants
R,
and
are small at bounce (
) as compared to the Planck scale:
Consider a numerical example for illustration. Assuming
,
, and
, a minimum radius
is of
Planck lengths. Since, by construction (see (
6) and (
16)),
and
, we can assume for convenience
. As a result, from Equation (
17) we find
If we substituting this into the
and
components of the Einstein equations at
, with the expressions (
8) and (
12), we can obtain the values of
and
:
From the equations of order one can then determine , and so on.
One can recall that in spherically symmetric spacetimes, if the spherical radius
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
which means violation of the Null Energy Condition (see [
2,
44]). In our model, supposing a bounce at
, we automatically obtain the inequality
.
4. Nonlocal Contribution to the Vacuum SET
To estimate the contribution of the nonlocal term
in the SET (
10), we rewrite the general metric (
1) of a Kantowski–Sachs cosmology as
where the time coordinate
is called “conformal time” and is defined by the condition
, it is convenient when dealing with quantum fields. The black hole under consideration is assumed to have a stellar or larger mass
, and
cm = 1 km is the corresponding gravitational radius. Meanwhile, at bounce (say, at the time
), in agreement with the previous section, we assume that the minimum radius
is
cm. We accordingly introduce the small parameter
(quite different from the parameter
from the previous section). Then, at times not too far from the bounce time we can write
where
according to the definition of
; we have
since
has a minimum and
since
is at maximum at
. The powers of
in
and
characterize the magnitudes of these metric coefficients when approaching to a would-be Schwarzschild singularity, or, which is the same, to the powers of
in (
4), while the factor in
then follows from the definition of conformal time. In other words, the metrics (
4) and (
1) with (
20) coincide at
, while the coefficients
simply correspond to a maximum of
and a minimum of
.
Consider a quantum scalar field satisfying the equation
and its standard Fourier expansion:
where
is a normalization factor,
is a coupling constant,
is a creation operator,
are spherical functions, and each time-dependent mode function
obeys the equation obtained from the original Klein–Gordon-type scalar field equation by separation of variables:
where the dots denote
and
is the effective frequency:
At the bounce time
, we have, due to standard normalization,
and
Further on, to make our estimates, we adhere to a natural assumption, justified by much experience [
34] (Section 3.5), that the most intensive particle production takes place at energies not too far from the curvature scale
. This energy is roughly of the same order of magnitude as the frequency
calculated in terms of proper cosmic time
, which is related to our conformal time by
. Therefore, our assumption means
. Since
, one has
, and, from the relation
, we immediately obtain
, so that
It is now of interest, at which values do the parameters of the model appreciably contribute to
having the order
. These are:
Apparently, momenta k strongly exceeding the Planckian value look quite meaningless, and we can conclude that, at reasonable (that is, sub-Planckian) values of k, their contributions to are negligibly small.
Note that the result
can be obtained in another way using the relations
A similar analysis leads to
. Furthermore, at small values of
, we can assume
The energy density of created particles may be estimated using the standard technique of Bogoliubov coefficients. For of bounce-type metrics similar to ours, the most important Bogoliubov coefficient
can be accurately enough computed by using the formulas [
45]
where
is the initial time instant at which, by assumption,
(that is, assuming that the field is in a vacuum state, without particles). Using Equation (
27) and making the assumption
(which means that
is not very far both from zero value and from
, we obtain
Now, we are ready to estimate the integrals
defined in (
28) at times close to the bounce time
:
Then, assuming
, we arrive at
Thus, the energy density of produced particles is
where, for each factor in (
33), we have taken the following approximate orders of magnitude, in accord with (
26): (i)
since we integrate from
to
; (ii)
, involving a few low multipolarities according to (
25), (
26) (since large multipolarities would mean too large mode energies contrary to our assumption that particle creation occurs most intensively at energies close to the curvature scale); (iii)
; (iv)
; and (v)
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 [
33],
, leads to
, and this value is still smaller if we consider black holes heavier than the Sun. Even if one relaxes some of the requirements in (
33) within a few orders of magnitude (for instance, including larger multipolarities), the smallness of the factor
would safely preserve our qualitative estimate. 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 other physical fields of different spins can hardly change this estimate too strongly.