# Effective Quantum Gravitational Collapse in a Polymer Framework

Lorenzo Boldorini<EMAIL_ADDRESS>Department of
Physics, “La Sapienza” University of Rome, P.le Aldo Moro 5, 00185 Rome, Italy
Giovanni Montani<EMAIL_ADDRESS>ENEA, Fusion and Nuclear Safety
Department, C. R. Frascati,
Via E. Fermi 45, 00044 Frascati (RM), Italy  Department of Physics, “La
Sapienza” University of Rome, P.le Aldo Moro 5, 00185 Rome, Italy

###### Abstract

We study how the presence of an area gap, different than zero, affects the
gravitational collapse of a dust ball. The implementation of such discreteness
is achieved through the framework of polymer quantization, a scheme inspired
by loop quantum gravity (LQG). We study the collapse using variables which
represent the area, in order to impose the non-zero area gap condition. The
collapse is analyzed for both the flat and spherical Oppenheimer-Snyder
models. In both scenarios the formation of the singularity is avoided, due to
the inversion of the velocity at finite values of the sphere surface. This
happens due to the presence of a negative pressure, with origins at a quantum
level. When the inversion happens inside the black hole event horizon, we
achieve a geometry transition to a white hole. When the inversion happens
outside the event horizon, we find a new possible astrophysical object. A
characterization of such hypothetical object is done. Some constraints on the
value for the area gap are also imposed in order to maintain the link with our
already established physical theories.

††preprint: APS/123-QED

##  I INTRODUCTION

One of the most notorious features of the theory of General Relativity (GR) is
the fact that it predicts the formation of black holes during gravitational
collapse of stars[1][2]. The theory predicts that when the internal pressure
of a star is not strong enough to counterbalance the gravitational pull of its
mass, the star collapses to become a black hole. Sometimes this collapse could
be halted by an internal pressure with quantum origins: electron degeneracy
pressure is responsible for the formation of white dwarfs, while neutron
degeneracy pressure, with the aid of the strong force, is responsible for the
formation of a neutron star. More details can be found in [3].
This is a remarkable result to keep in mind for the scope of this paper: a
quantum effect induces a pressure which has macroscopic consequences.
When the mass is too high the internal quantum pressure is not enough to
prevent the formation of a black hole. The mass range after which collapse is
inevitable is called Tolman-Oppenheimer-Volkoff Limit, the most recent
theoretical esteem being $2.2-2.9M_{\odot}$[4], while constraints from
gravitational wave signals put the limit at $2.01-2.16M_{\odot}$[5].
When such limits are passed, the collapsing solution present also one of the
most critically discussed points of the theory: the singularity.
Singularities are mostly though to be a sign of incompleteness for the theory
of GR, invoking for the search of a more complete theory which encompass
quantum mechanics. The first approach to achieve the quantization of gravity
was the Wheeler-DeWitt formulation [6]. This approach was rather conservative,
since it was based off canonical quantization of a constrained system:
gravity.
The evolution of this canonical quantization procedure is today called Loop
Quantum Gravity (LQG), a canonical quantization scheme based on the Ashtekar-
Barbero-Immirzi’s connection variables. This approach more closely resembles
the usual gauge theories encountered in theoretical physics, but possessess
also very peculiar features, such as a discrete spectrum for the operator
which represents the area, or like a diffeomorphisms invariant Hilbert space.
A very detailed account for such a quantization procedure for gravity is given
in [7].
Our analysis in this work is based on such a canonical approach. We will adopt
the Oppenheimer-Snyder[8] model (OS) for gravitational collapse, which is the
first model developed to describe the gravitational collapse of a sphere
without internal pressure.
This model is described internally by an evolving FLRW metric, while outside
the geometry is fixed to be the Schwarzschild spacetime. Our quantization
procedure is applied then to the internal dynamical FLRW metric.
We make use of the so called polymer quantization[9], a canonical quantization
scheme which implements the non-vanishing area gap and the diffeomorphisms
invariance of the Hilbert space, like in the LQG framework.
The main difference is that the scale factor $a(t)$ of the FLRW solution does
not act as a field but rather as a single degree of freedom, so the polymer
technique is to be understood as a LQG-like quantization for a single degree
of freedom.
We will build an effective Hamiltonian for the OS model where it gets
corrections from the polymer quantization procedure. The meaning of such a
method of analysis is that we consider the classical trajectories to be the
trajectories for the expectation values of the quantum operators, which we
assume to be peaked around the classical trajectory.
Our aim in this paper is to understand whether or not the implementation of
the quantum nature of gravity could prevent the formation of the above
mentioned singularity. This phenomenon is already known in the canonical
quantization approach for the collapse of shells [10] and for the Lemaitre-
Tolman-Bondi (LTB) model [11][12].
As main issue of our study we demonstrate that the collapse singularity is
removed in favor of a possible guess on the existence of a quantum black-to-
white hole transition.
Furthermore, as most interesting phenomenological outcome, we see that under
suitable conditions, but independent of the object mass, the collapse is
unable to cross the horizon. As a result, we deal with a new hypothetical
astrophysical object, which could also be over-critic but radially oscillating
between two super-horizon configurations.
Finally we show how the emergence of the polymer phenomenology can be
interpreted as a negative quantum pressure, which form is explicitly
determined.
The paper is organized as follows. In Section II we first give a review of the
Hamiltonian formulation of the Oppenheimer-Snyder model, then we introduce a
new pair of canonical variables and we use them to study the classical model.
In Section III we introduce the effective model, built using the polymer
corrections, and solve the equations of motion for both the area and momentum.
Some relevant aspects are discussed after presenting each solution. In Section
IV the phenomenology of the effective model is discussed: we will characterize
a kind of quantum gravitational pressure generated by polymer quantization,
and macroscopic and quantum effects will be analyzed.
Section V closes the paper with a summary, some remarks and future
perspectives for this line of research.
Throughout all this paper we will use geometrized $(G=c=1)$ units.

##  II HAMILTONIAN OPPENHEIMER-SNYDER MODEL

In this section we are going to analyze the geometry behind the gravitational
collapse model.
First we will introduce the solution found by Oppenheimer and Snyder[8], then
we will see how an Hamiltonian description of the very same model can be
formulated [13] [14] [15], giving rise to a constrained model.
We will later introduce a new set of canonical variables, well suited for the
effective analysis to be made in the next section. This new variables will
then be used to solve the classical model, in order to compare it to effective
one developed in section III.

###  II.1 The Oppenheimer-Snyder Geometry

Let’s start our analysis with the description of the Oppenheimer-Snyder model
geometry.
The spacetime manifold is composed of mainly three pieces[16]: an interior
region, where a dust is present, an exterior vacuum region and at last a
boundary, which will be set to infinity.
The manifold, denoted by $\Omega$, is then split into two portions: the matter
region $\Omega_{-}$ and the vacuum region
$\Omega_{+}=\Omega\;\backslash\;\Omega_{-}$. The timelike boundary between the
two portions of the manifold is denoted by $\Gamma$ while the boundary of the
manifold is denoted by $\Gamma_{+}$ .
We are willing to give an Hamiltonian description of the model, so we need to
apply the same subdivision also to the spacelike hypersurfaces.
Denoted the spacelike hypersurfaces by $\Sigma_{t}$, the spacelike interior
matter region is given by $S_{-}=\Omega_{-}\cap\Sigma_{t}$ while the exterior
vacuum is given by $S_{+}=\Omega_{+}\cap\Sigma_{t}$. The boundary between the
two regions is where the spatial surface of the dust sphere lies, given by
$\Sigma_{S}=\Gamma\cap\Sigma_{t}$.
The boundary of the manifold is given by
$\Sigma_{+}=\Gamma_{+}\cap\Sigma_{t}$.
Since we are dealing with a spherical symmetric distribution of matter, the
geometry of the spacelike hypersurface is well described by the set of
spherical coordinates $x^{i}=(r,\theta,\phi)$ where $r\geq 0$ has the meaning
of a radial coordinate, with $r=0$ in the center of the dust ball. The
boundary surface $\Sigma_{S}$ is held fixed at a finite constant coordinate
radius $r=r_{S}$ while the boundary of the manifold $\Sigma_{+}$ is placed at
a coordinate radius of $r=r_{+}$.
It is very important to note that the coordinates are not observables here,
since observables are given by functions on the phase space. The coordinates
thus are just a parametrization of the manifold, useful to relate canonical
variables.
Let’s see now how the geometry in each manifold portion is described.
The geometry inside the region $\Omega_{-}$ is determined by the Friedmann-
Lemaître-Robertson-Walker (FLRW) metric[1]:

$ds^{2}_{(-)}=-d\tau^{2}+a^{2}(\tau)\left(\frac{dr^{2}}{1-\epsilon
r^{2}}+r^{2}d\Omega^{2}\right)$ (1)

Where $\epsilon\in\\{0,+1,-1\\}$ respectively for flat, spherical and
hyperbolic $\Omega_{-}$. The dimensions are such that
$\big{[}\epsilon\big{]}=L^{-2}$ and $a(\tau)$ is dimensionless.
$a(\tau)$ is the scale factor, while the coordinate $\tau$ is the proper time
measured by an observer comoving with the dust in $\Omega_{-}$.
The flat geometry describes collapse starting from infinity with zero
velocity, the hyperbolic geometry describes collapse starting from infinity
with a given initial velocity and, at last, the spherical geometry describes
collapse starting from an initial finite radius with zero initial
velocity[17][18].
The geometry inside the region $\Omega_{+}$ is determined by the Birkhoff
theorem and is the Schwarzschild spacetime geometry[1] given by:

$ds^{2}_{(+)}=-F(R)dT^{2}+F^{-1}(R)dR^{2}+R^{2}d\Omega^{2}$ (2)

The function $F(R)$ is given by the usual:

$F(R)=1-\frac{2M_{S}}{R}$ (3)

Where $M_{S}$ is the dust sphere mass, namely the ADM mass of the system.
The vector $\partial/\partial T$ is a Killing vector of the line element (2)
and it’s orthogonal to the constant $T$ hypersurfaces, while $R$ is the radial
Schwarzschild coordinate.
From now on will only cover the flat and spherical cases, since they appear to
be the most interesting from an astrophysical point of view.

###  II.2 Hamiltonian Model

Let’s turn our attention now to the Hamiltonian description of said model,
which is portrayed in [19][20].
The dust model adopted is the one provided by Brown and Kuchař[21] of an
isotropic, homogeneous pressureless dust. The canonical variable $\tau$
describes the dust proper time, while its conjugate momentum $P_{\tau}$ is the
energy inside the dust sphere.
Before the derivation of the other canonical variables we define the useful
quantity:

$\begin{split}V_{S}&=\int_{0}^{r_{S}}dr\frac{r^{2}}{\sqrt{1-\epsilon
r^{2}}}=\\\ &=\begin{cases}r_{S}^{3}/3&\epsilon=0\\\ \frac{-\sqrt{\varepsilon
r_{S}^{2}-r_{S}^{4}\varepsilon^{2}}+\text{arctan}\left(\frac{\sqrt{\varepsilon
r_{S}^{2}}}{\sqrt{1-\varepsilon
r_{S}^{2}}-1}\right)+\frac{\pi}{2}}{\varepsilon^{3/2}}&\epsilon=+1\varepsilon\end{cases}\end{split}$
(4)

Where $\varepsilon\equiv 1[L]^{-2}$ for dimensional reasons. This quantity is
such that $4\pi V_{S}$ is the parametric volume of a ball of coordinate radius
$r=r_{S}$ in the two geometries.
We also identify the lapse function to be
$N_{\tau}^{2}(t)=\left(\frac{d\tau}{dt}\right)^{2}$.
Defining $\rho$ as the dust energy density, the momentum $P_{\tau}$ is given
by:

$P_{\tau}=4\pi V_{S}a^{3}(t)\rho$ (5)

The canonical variables which describe the interior geometry are the scale
factor $a$ and its conjugate momentum $P_{a}$, while the exterior geometry is
described by the canonical variables $(R,T)$, the Schwarzschild coordinates,
and their conjugated momenta $(\tilde{P}_{R},P_{T})$.
To obtain the Hamiltonian description of the model the following canonical
transformation is made:

$R_{s}=ar_{S}\qquad P_{s}=P_{a}/r_{S}$ (6)

Those are the proper radius and momentum of the surface of the dust cloud.
Having also defined:

$\Xi=\frac{r_{S}^{3}}{3V_{S}}$ (7)

Which is the ratio between the volume of a flat sphere and the proper volume
of a sphere of equal radius, we see that the relation between the sphere mass
and the dust energy is given by:

$M_{S}=\Xi P_{\tau}$ (8)

The last ingredient needed to obtain an Hamiltonian formulation of the theory
is a parametrization of the Schwarzschild time on the dust surface, labeled
$T_{S}(t)$, which is required to make the action canonical.
Following [19] we see that we have two choices.
For the flat case, i.e. $\epsilon=0$, the function is given by:

$T_{S}^{PG}=t+\int dr\frac{\sqrt{2M_{S}/R}}{1-\frac{2M_{S}}{R}}$ (9)

Those are called Painlevé-Gullstrand[22][23] coordinates. For the spherical
case, i.e. $\epsilon=+1$, the function is given by:

$T_{S}^{GH}=\frac{1}{1-\frac{2M_{S}}{R}}\left(t+\int_{R_{i}}^{R}dy\frac{\sqrt{\frac{2M_{S}}{y}-\frac{2M_{S}}{R_{i}}}}{1-\frac{2M_{S}}{y}}\right)$
(10)

Where:

$R_{i}=\frac{2M_{S}}{R^{\phi}_{\theta\phi\theta}}$ (11)

Those are called Gautreau-Hoffman[24] coordinates.
For increased readability we define:

$\kappa_{S}=\epsilon r_{S}^{2}$ (12)

Which is a dimensionless parameter.
We are now ready to write down the action for the Oppenheimer-Snyder model:

$\begin{split}&S=\int
dt\left\\{\left(P_{\tau}\dot{\tau}+P_{s}\dot{R}_{s}-H_{-}\right)\right.\\\
&\left.+\int_{r_{S}}^{\infty}dr\left[\left(\tilde{P}_{R}\dot{R}+P_{T}\dot{T}\right)-\left(\beta^{T}P_{T}+\beta^{R}\tilde{P}_{R}\right)\right]\right\\}\end{split}$
(13)

With the Hamiltonian:

$\begin{split}&H_{-}=N_{\tau}\mathcal{H}_{-}\\\
&\mathcal{H}_{-}=-\frac{\Xi}{2}\left(\frac{P_{s}^{2}}{R_{s}}+\frac{\kappa_{S}}{\Xi^{2}}R_{s}-2\frac{P_{\tau}}{\Xi}\right)\end{split}$
(14)

The constraints are:

$\mathcal{H}_{-}\approx 0\qquad P_{T}\approx 0\qquad\tilde{P}_{R}\approx 0$
(15)

This means that the only dynamical degrees of freedom are $\tau$ and $R_{s}$,
or more properly $a$ since $R_{s}$ is just one of the possible rescaled
versions. The variables $(R,T)$ are then fully constrained, with no dynamics:
the external geometry really is fixed to be Schwarzschild up to the
diffeomorphisms generetad by the last two constraints in (15).

###  II.3 Area Variables

We now aim to develop a new formulation in order to solve both the classical
and the effective dynamics. We will build a new set of variables and use them
to recast the Hamiltonian OS model from a different point of view.
We start by making a canonical transformation, adopting the new set of
variables more suited for the effective model.
This new change of variables is made through the following canonical
transformation and its inverse:

$\displaystyle\begin{cases}\mathcal{A}(R_{s},P_{s})=4\pi R_{s}^{2}\\\
P_{\mathcal{A}}(R_{s},P_{s})=\frac{P_{s}}{8\pi R_{s}}\end{cases}$ (16)
$\displaystyle\begin{cases}R_{s}(\mathcal{A},P_{\mathcal{A}})=\sqrt{\frac{\mathcal{A}}{4\pi}}\\\
P_{s}(\mathcal{A},P_{\mathcal{A}})=P_{\mathcal{A}}\sqrt{16\pi\mathcal{A}}\end{cases}$
(17)

Where $\mathcal{A}$ is the proper surface area of a sphere of proper radius
$R_{s}$ and $P_{\mathcal{A}}$ is its conjugate momentum.
It is clear that $P_{s}\dot{R}_{s}=P_{\mathcal{A}}\dot{\mathcal{A}}$ so that
the action in the new variables reads:

$\begin{split}&S=\int
dt\left\\{\left(P_{\tau}\dot{\tau}+P_{\mathcal{A}}\dot{\mathcal{A}}-N_{\tau}\tilde{\mathcal{H}}\right)\right.\\\
&\left.+\int_{r_{S}}^{\infty}dr\left[\left(\tilde{P}_{R}\dot{R}+P_{T}\dot{T}\right)-\left(\beta^{T}P_{T}+\beta^{R}\tilde{P}_{R}\right)\right]\right\\}\end{split}$
(18)

With the new Hamiltonian given by:

$\begin{split}&\tilde{H}=N_{\tau}\tilde{\mathcal{H}}\\\
&\tilde{\mathcal{H}}=-\frac{\Xi}{2}\left(32\pi^{\frac{3}{2}}P_{\mathcal{A}}^{2}\sqrt{\mathcal{A}}+\frac{\kappa_{S}}{\Xi^{2}}\frac{\sqrt{\mathcal{A}}}{\sqrt{4\pi}}-2\frac{P_{\tau}}{\Xi}\right)\end{split}$
(19)

The constraints become:

$\tilde{\mathcal{H}}\approx 0\qquad P_{T}\approx 0\qquad\tilde{P}_{R}\approx
0$ (20)

We will choose now the comoving gauge, which corresponds to the choice of
$N_{\tau}=1$, from which the following equations of motion are deduced:

$\displaystyle\dot{\mathcal{A}}=\frac{\delta\tilde{H}}{\delta
P_{\mathcal{A}}}=-32\pi^{3/2}\Xi P_{\mathcal{A}}\sqrt{\mathcal{A}}$ (21)
$\displaystyle\dot{P}_{\mathcal{A}}=-\frac{\delta\tilde{H}}{\delta\mathcal{A}}=\frac{\kappa_{S}}{8\Xi\sqrt{\pi\mathcal{A}}}+\frac{8\Xi\pi^{3/2}P_{\mathcal{A}}^{2}}{\sqrt{\mathcal{A}}}$
(22) $\displaystyle\dot{\tau}=\frac{\delta\tilde{H}}{\delta P_{\tau}}=1$ (23)
$\displaystyle\dot{P}_{\tau}=-\frac{\delta\tilde{H}}{\delta\tau}=0$ (24)

We now start to develop the classical dynamics.

###  II.4 Classical Dynamics

We now aim to solve the classical equations of motion in this new variables.
Equations (23) and (24) are easily solved, yielding:

$\displaystyle\tau(t)$ $\displaystyle=t$ (25) $\displaystyle P_{\tau}(t)$
$\displaystyle=const=\frac{M_{S}}{\Xi}$ (26)

Now we get onto the equations of motion for the area and its conjugate
momentum.

####  II.4.1 Classical Surface Equation

Starting from the one relative to the area, we first try to express the
equation for the dust surface as only a function of its area. In order to do
so we first evaluate $P_{\mathcal{A}}^{2}(\mathcal{A})$ solving the constraint
$\tilde{\mathcal{H}}\approx 0$, which gives:

$P_{\mathcal{A}}^{2}=\frac{4M_{S}\sqrt{\pi\mathcal{A}}-\kappa_{S}\mathcal{A}}{64\pi^{2}\Xi^{2}\mathcal{A}}$
(27)

Extrapolating $P_{\mathcal{A}}$ from this and substituting it into
$\dot{\mathcal{A}}$ we obtain the equations of motion:

$\dot{\mathcal{A}}^{\pm}=\pm
4\sqrt{\pi}\sqrt{4M_{S}\sqrt{\pi\mathcal{A}}-\kappa_{S}\mathcal{A}}$ (28)

Here $\dot{\mathcal{A}}^{-}$ is the equation for the in-falling dust ball
while $\dot{\mathcal{A}}^{+}$ is the one for the out-rising dust sphere.
Before solving this we seek for the maxima and minima of the canonical
variable.
In order to achieve this, we impose the condition $\dot{\mathcal{A}}^{2}=0$,
which we can solve easily for $\mathcal{A}$ after the substitution of
$P_{\mathcal{A}}^{2}$ into $\dot{\mathcal{A}}^{2}$.
The solutions of the equation are:

$\mathcal{A}_{max}=\frac{16\pi
M_{S}^{2}}{\kappa_{S}^{2}}\qquad\mathcal{A}_{min}=0$ (29)

Here $\mathcal{A}_{max}$ is a maximum while $\mathcal{A}_{min}$ is a minimum,
as shown by the domain restrictions on the equations of motion.
Analyzing the maximum will shed some light on the parameter $\kappa_{S}$: if
we identify the maximum with the initial surface of the sphere, namely
$\mathcal{A}_{max}=\mathcal{A}_{0}$, we see that the $\kappa_{S}=0$ case is
equivalent to saying that the maximum for the flat case is at
$\mathcal{A}_{max}=+\infty$: this is consistent with the previous statement
that $\epsilon=0$ is associated with the collapse from infinity, so the
surface has no maximum value.
To study the spherical case we start by defining the Schwarzschild-surface as
the surface of the event horizon:

$\mathcal{A}_{SC}=4\pi\left(2M_{S}\right)^{2}=16\pi M_{S}^{2}$ (30)

Defining also $\alpha$ as the ratio between the initial surface and the
Schwarzschild-surface of the dust ball:

$\alpha=\frac{\mathcal{A}_{0}}{\mathcal{A}_{SC}}$ (31)

We immediately see that the parameter $\kappa_{S}$ is related to the initial
value of the surface via:

$\kappa_{S}=\sqrt{\frac{1}{\alpha}}=\frac{2M_{S}}{R_{max}}$ (32)

Since for any physical situation $\alpha>1$, it means that $\kappa_{S}<1$
always.
The $\kappa_{S}=0$ case then corresponds to the choice of $\alpha=\infty$, so
the collapse starts at an initial radius infinitely far from the event
horizon.
We now turn back to solve our equations of motion (28).
The equations are separable, so that they become:

$\frac{d\mathcal{A}^{\pm}}{\sqrt{4M_{S}\sqrt{\pi\mathcal{A}}-\kappa_{S}\mathcal{A}}}=\pm
4\sqrt{\pi}dt$ (33)

Those are two different sets of equations, two for the flat case and two for
the spherical case. With the help of the integrals defined in Appendix A, and
remembering that $t=\tau$, the solutions read:

$\displaystyle\mathcal{A}^{-}:-4\sqrt{\pi}\;\tau=C^{-}+\mathcal{I}_{\epsilon}^{CA}(\mathcal{A})$
(34)
$\displaystyle\mathcal{A}^{+}:4\sqrt{\pi}\;\tau=C^{+}+\mathcal{I}_{\epsilon}^{CA}(\mathcal{A})$
(35)

We now need to obtain the constants of integration: for the in-falling branch
we want that, at $\tau=0$, the surface has value
$\mathcal{A}=\mathcal{A}_{0}$. This is also valid for the collapses coming
from infinity: since they need to start at $\tau=-\infty$ we can choose a
finite value for $\mathcal{A}$ when the comoving time is zero.
This means that:

$C^{-}=-\mathcal{I}_{\epsilon}^{CA}(\mathcal{A}_{0})$ (36)

For the out-rising solution, in order to have a continuous behaviour, we would
like to join the two branches at the minimum $\mathcal{A}_{min}=0$. This is
impossible because both branches are singular at $\mathcal{A}_{min}=0$.
If instead we take $C^{+}=\mathcal{I}_{\epsilon}^{CA}(\mathcal{A}=0^{+})$ we
obtain the time reversal of the collapse.
The solution then reads:

$\displaystyle\mathcal{A}^{-}:-4\sqrt{\pi}\;\tau=-\mathcal{I}_{\epsilon}^{CA}(\mathcal{A}_{0})+\mathcal{I}_{\epsilon}^{CA}(\mathcal{A})$
(37)
$\displaystyle\mathcal{A}^{+}:4\sqrt{\pi}\;\tau=\mathcal{I}_{\epsilon}^{CA}(\mathcal{A}=0^{+})+\mathcal{I}_{\epsilon}^{CA}(\mathcal{A})$
(38)

Clearly the time reversal is not a physical solution due to the presence of
the singularity in the conjunction point.
We show in Fig. 1 a plot of the classical in-falling trajectory in this new
variables.

Figure 1: In-falling trajectories against comoving time $\tau$ for the two
scenarios. The parameters chosen are $\alpha=2$ and $M_{S}=M_{\odot}$.

Now its the time to solve the momentum equations.

####  II.4.2 Classical Momentum Equation

We now solve the equation for the dust sphere momentum.
We will not solve the equation in function of the comoving time $\tau$ but
solve for $P_{\mathcal{A}}$ as a function of $\mathcal{A}$. The reason why
this appears more useful is because it gives the momentum when the sphere has
surface $\mathcal{A}$, and since we have $\tau(\mathcal{A})$ we can evaluate
$P_{\mathcal{A}}(\tau)$ from such relation. We have that:

$P_{\mathcal{A}}^{\prime}:=\frac{dP_{\mathcal{A}}}{d\mathcal{A}}=\frac{\dot{P}_{\mathcal{A}}}{\dot{\mathcal{A}}}=-\frac{\frac{\kappa_{S}}{\pi^{2}\Xi^{2}}+64P_{\mathcal{A}}^{2}}{256P_{\mathcal{A}}\mathcal{A}}$
(39)

Which yields the separable equation:

$\frac{256P_{\mathcal{A}}dP_{\mathcal{A}}}{\frac{\kappa_{S}}{\pi^{2}\Xi^{2}}+64P_{\mathcal{A}}^{2}}=-\frac{d\mathcal{A}}{\mathcal{A}}$
(40)

This is solved exactly with the aid of the integrals in Appendix A:

$\displaystyle\epsilon=0\;:\qquad
P_{\mathcal{A}}(\mathcal{A})=\frac{e^{-\frac{C_{0}}{4}}}{\sqrt[4]{\mathcal{A}}}$
(41) $\displaystyle\epsilon=+1\;:\qquad
P_{\mathcal{A}}(\mathcal{A})=\pm\sqrt{\frac{\frac{e^{-\frac{C_{1}}{2}}}{\sqrt{\mathcal{A}}}-\kappa_{S}}{64\pi^{2}\Xi^{2}}}$
(42)

The constants of integration are obtained by setting the momentum to be zero
at the initial surface for the spherical case, while for the flat case,
according to equations (21) and (28), is recovered by setting at some given
surface $\mathcal{A}_{0}$:

$P_{\mathcal{A}}^{\pm}(\mathcal{A}_{0})=\frac{-\dot{\mathcal{A}}^{\pm}(\mathcal{A}_{0})}{32\pi^{3/2}\sqrt{\mathcal{A}_{0}}}=\frac{\mp\sqrt{M_{S}}}{4\pi^{3/4}\sqrt[4]{\mathcal{A}_{0}}}$
(43)

This is because the momentum is zero only at infinity in this scenario. Note
that the momentum switches sign when the branch changes, since (28) has
opposite signs in the two branches.
The final solutions, after simplifying, read:

$\displaystyle\epsilon=0\;:\qquad
P_{\mathcal{A}}^{\pm}(\mathcal{A})=\mp\sqrt[4]{\frac{M_{S}^{2}}{256\pi^{3}\mathcal{A}}}$
(44) $\displaystyle\epsilon=+1\;:\qquad
P_{\mathcal{A}}^{\pm}(\mathcal{A})=\mp\sqrt{\frac{\kappa_{S}\left(-1+\sqrt{\frac{\mathcal{A}_{0}}{\mathcal{A}}}\right)}{64\pi^{2}\Xi^{2}}}$
(45)

The $P_{\mathcal{A}}^{+}$ is the equation for the out-rising solution and
$P_{\mathcal{A}}^{-}$ for the in-falling surfaces, and we will only keep the
in-falling solution as before, since here the out-rising is unphysical for the
same motivations. The signs are inverted since, from equation (21), the
momentum has always the opposite sign of the velocity.
Fig. 2 shows the behaviour of the momentum against the surface. As it
approaches the singularity it clearly diverges.

Figure 2: In-falling momentum trajectory for both scenarios. The parameters
chosen are $\alpha=2$ and $M_{S}=M_{\odot}$.

Now we tackle the effective model.

##  III EFFECTIVE OS MODEL

For this section our work will be centered on the construction of an effective
dynamical model, where the effective dynamics is built to take into accounts
the quantum effects rooted into the implementation of a polymer quantization
of the theory[9].
First we will briefly resume what this quantization procedure implies, and
then we will develop a new formulation for the OS model, where we apply
quantum corrections to the classical dynamics and solve the corresponding
equations for the sphere surface and momentum.
Non trivial effects will emerge in both the solutions, hinting at what a fully
quantization of the model could imply.
Let’s now introduce the polymer quantization scheme.

###  III.1 Polymer Effective Theory

The framework of Polymer Quantization falls naturally into the quantization
procedure referred as ”Weyl Quantization”, a generalization of the usual
Heisenberg-Dirac procedure[25].
In this framework the operators associated to the canonical variables $(q,p)$
are obtained by differentiating the corresponding one-parameter exponential
operators, namely $U(\alpha)=e^{i\alpha q}$ and $V(\beta)=e^{i\beta p}$. When
one of the exponential operators isn’t continuous with respect to the
parameter, the operator corresponding to its canonical variable is not
defined, since the operation of derivation is ill defined. The polymer
quantization is then the Weyl quantization associated to the invariance under
the diffeomorphisms group[26], and contains multiple schemes.
We choose the polymer scheme in which the momentum operator in the momentum
polarization is not defined, since this corresponds to a choice of discrete
areas. This choice is inspired by the literature on loop quantum
gravity[27][28].
Since the Hamiltonian in Eq. (19) contains the squared momentum
$P_{\mathcal{A}}^{2}$, we need to regularise it by introducing a graph
structure on the phase space, regularizing the $\mathcal{A}$ variable with a
lattice of step $\mu_{0}$.
The introduction of this regularization implies that there is a cutoff on the
momentum values, which we will define as $p_{0}=\frac{\hbar}{\mu_{0}}$. To see
how to implement such a structure into our effective model we show how it
affects the quantum dynamics.
The operator action of the squared momentum is defined by:

$\hat{P}_{\mathcal{A}}^{2}\phi(P_{\mathcal{A}})=P_{\mathcal{A}}^{2}\phi(P_{\mathcal{A}})$
(46)

Since we don’t have such an operator, we approximate $P_{\mathcal{A}}$ with a
translation operator $\hat{T}$:

$\hat{P}_{\mathcal{A}}\simeq\frac{\hbar}{\mu_{0}}\frac{1}{2i}\left(\hat{T}(\mu_{0})-\hat{T}(-\mu_{0})\right)$
(47)

When this operator acts on wave functions we get:

$\hat{P}_{\mathcal{A}}\phi(P_{\mathcal{A}})\simeq\frac{\hbar}{\mu_{0}}\frac{1}{2i}\left(e^{i\frac{\mu_{0}}{\hbar}P_{\mathcal{A}}}-e^{-i\frac{\mu_{0}}{\hbar}P_{\mathcal{A}}}\right)\phi(P_{\mathcal{A}})$
(48)

This means that we have the following relation:

$\hat{P}_{\mathcal{A}}^{2}\phi(P_{\mathcal{A}})\simeq
p_{0}^{2}\sin^{2}\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)\phi(P_{\mathcal{A}})$
(49)

So we can obtain an effective model, with the natural inclusion of a momentum
cutoff, by substituting the squared momentum with its corresponding polymer
eigenvalue, namely by allowing $P_{\mathcal{A}}^{2}\rightarrow
p_{0}^{2}\sin^{2}\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)$.
After this substitution into the Hamiltonian, we can derive the effective
equations of motion.
The effective Hamiltonian, in the comoving gauge, is given by:

$H_{pol}=-\frac{\Xi}{2}\left(32\pi^{\frac{3}{2}}p_{0}^{2}\sin^{2}\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)\sqrt{\mathcal{A}}+\frac{\kappa_{S}}{\Xi^{2}}\frac{\sqrt{\mathcal{A}}}{\sqrt{4\pi}}-2\frac{P_{\tau}}{\Xi}\right)$
(50)

The equations for the variables $P_{\tau}$ and $\tau$ are unchanged, so their
solutions remain the same as equations (25) and (26). The equations for the
dust momentum and area are instead modified by our quantum correction, and
those read:

$\displaystyle\dot{\mathcal{A}}=\frac{\delta H_{pol}}{\delta
P_{\mathcal{A}}}=-32\pi^{3/2}\Xi\;p_{0}\sin\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)\cos\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)\sqrt{\mathcal{A}}$
(51) $\displaystyle\dot{P}_{\mathcal{A}}=-\frac{\delta
H_{pol}}{\delta\mathcal{A}}=\frac{\kappa_{S}}{8\Xi\sqrt{\pi\mathcal{A}}}+\frac{8\Xi\pi^{3/2}p_{0}^{2}\sin^{2}\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)}{\sqrt{\mathcal{A}}}$
(52)

It is clear that now the momentum plays the role of an angle variable, with a
period of $p_{0}\pi$.
This is also clear if we look at the phase space portraits.

Figure 3: Phase Portraits in Geometrical Units for both the Flat and Spherical
models. In red, lines of constant $P_{\mathcal{A}}=kp_{0}\pi$,
$k\in\mathbb{Z}$.

The phase space shown in Fig. 3 is clearly periodic, with the period
previously given by $p_{0}\pi$. Being periodic, we can identify in the phase
space $P_{\mathcal{A}}=P_{\mathcal{A}}+p_{0}\pi$, so that the momentum takes
values on a circle.
This reduced phase space is shown in Fig. 4.

Figure 4: Reduced Phase Portraits in Geometrical Units for both the Flat and
Spherical models. In red the line of constant $P_{\mathcal{A}}=p_{0}\pi$.

It is immediately clear that the spherical case oscillates between a finite
maximum and a finite minimum, while the flat case possesses a finite minimum
but has its maximum at infinity. The analysis of those features is done in the
subsections III.2 and III.3.
We know seek the solutions of the effective equations (51) and (52).

###  III.2 Sphere Surface Equation of Motion

We start by studying the equation of motion for the area variable. In order to
do so, we adopt the same technique we used in the previous section to solve
the classical equations of motion.
The first step is again to obtain
$\sin^{2}\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)$ as a function of
$\mathcal{A}$ from the constraint, solving $\mathcal{H}_{poly}\approx 0$.
Defining:

$\kappa_{p}=64\pi^{2}\Xi^{2}p_{0}^{2}$ (53)

Such procedure gives:

$\sin^{2}\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)=\frac{4M_{S}\sqrt{\pi\mathcal{A}}-\kappa_{S}\mathcal{A}}{\kappa_{p}\mathcal{A}}$
(54)

From this it is immediate to obtain
$\cos^{2}\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)$, which is just:

$\cos^{2}\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)=1-\sin^{2}\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)$
(55)

Setting for readability:

$\mathfrak{a}=\frac{4M_{S}\sqrt{\pi\mathcal{A}}-\kappa_{S}\mathcal{A}}{\kappa_{p}\mathcal{A}}$
(56)

We see that:

$\sin\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)\cos\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)=\pm\sqrt{\mathfrak{a}}\sqrt{\mathfrak{1-a}}$
(57)

Substituting this into $\dot{\mathcal{A}}$, the equation of motion simply
becomes:

$\dot{\mathcal{A}}^{\pm}=\pm
32\pi^{3/2}\Xi\;p_{0}\sqrt{\mathcal{A}(\mathfrak{a}-\mathfrak{a}^{2})}$ (58)

The plus and minus signs represent again the in-falling and out-rising
solutions.
Making explicit $\mathfrak{a}$ yields:

$\begin{split}\dot{\mathcal{A}}^{\pm}=\pm
32\pi^{3/2}\Xi\;p_{0}&\left[\frac{4M_{S}\sqrt{\pi\mathcal{A}}-\kappa_{S}\mathcal{A}}{\kappa_{p}}\right.\\\
&\left.-\frac{\left(4M_{S}\sqrt{\pi\mathcal{A}}-\kappa_{S}\mathcal{A}\right)^{2}}{\kappa_{p}^{2}\mathcal{A}}\right]^{\frac{1}{2}}\end{split}$
(59)

As before, instead of solving directly this couple of equations, we search for
the maxima and minima for the dust ball surface.
We do this by setting $\dot{\mathcal{A}}^{2}=0$ and solving for $\mathcal{A}$
after the substitution of the constraint solution (54).
The one for $\epsilon=+1$ the solutions are:

$\mathcal{A}_{max}=\frac{16\pi
M_{S}^{2}}{\kappa_{S}^{2}}\qquad\mathcal{A}_{min}=\frac{16\pi
M_{S}^{2}}{\left(\kappa_{S}+\kappa_{p}\right)^{2}}$ (60)

From the domain restrictions in equation (59) we see that $\mathcal{A}_{max}$
is a maximum, the same as in the classical case, while $\mathcal{A}_{min}$ is
a minimum, now different from zero. The most important aspect is that on the
minimum now our solution will not be singular anymore, allowing us to match
the two branches through such point.
The solution for the $\epsilon=0$ case yields:

$\mathcal{A}_{m}=\frac{16\pi M_{S}^{2}}{\kappa_{p}^{2}}$ (61)

Which again, from the domain restrictions in equation (59), we see that
$\mathcal{A}_{m}$ is a regular non-zero minimum, while the sphere surface has
no maximum, just like in the classical case.
A more precise analysis of the physical meaning of such findings is given
below in the sub-subsection III.2.1.
With the maxima and minima in our hands we now turn to solve the equations of
motion.
The equations (59) are separable, so the differential problem can be cast as:

$\begin{split}&\frac{d\mathcal{A}^{\pm}}{\sqrt{\left(4M_{S}\sqrt{\pi}-\kappa_{S}\sqrt{\mathcal{A}}\right)\left(\sqrt{\mathcal{A}}\left(\kappa_{S}+\kappa_{p}\right)-4M_{S}\sqrt{\pi}\right)}}\\\
&=\pm\frac{1}{2\Xi p_{0}\sqrt{\pi}}dt\end{split}$ (62)

Setting $t=\tau$ from the comoving time solution, and making use of the
integrals defined in Appendix A, we get the solutions:

$\mathcal{A}^{-}:-\frac{1}{2\Xi
p_{0}\sqrt{\pi}}\tau=C^{-}+\mathcal{I}^{EA}_{\epsilon}(\mathcal{A})$ (63)
$\mathcal{A}^{+}:\frac{1}{2\Xi
p_{0}\sqrt{\pi}}\tau=C^{+}+\mathcal{I}_{\epsilon}^{EA}(\mathcal{A})$ (64)

The integration constant for the in-falling surface is found just like before,
by setting $\mathcal{A}(\tau=0)=\mathcal{A}_{0}$, yielding:

$C^{-}=-\mathcal{I}^{EA}_{\epsilon}(\mathcal{A}_{0})$ (65)

An important note: since we can choose for the $\kappa_{S}=0$ case whatever
initial surface we desire, it is important to choose one which is always
bigger than the minimum. For the $\epsilon=+1$ case this is always assured by
the fact that $\mathcal{A}_{max}>\mathcal{A}_{min}$ always.
Since now the minimum is not a singular point, we can ask for a continuity
condition to find the constant $C^{+}$ inside the out-rising solution:

$\mathcal{A}^{-}(\tau=\tau_{min})=\mathcal{A}^{+}(\tau=\tau_{min})$ (66)

Where $\tau_{min}$ is given by:

$\tau_{min}=-2\Xi
p_{0}\sqrt{\pi}\left[-\mathcal{I}^{EA}_{\epsilon}(\mathcal{A}_{0})+\mathcal{I}^{EA}_{\epsilon}(\mathcal{A}_{min})\right]$
(67)

This gives us:

$C^{+}=+\mathcal{I}^{EA}_{\epsilon}(\mathcal{A}_{0})-2\mathcal{I}^{EA}_{\epsilon}(\mathcal{A}_{min})$
(68)

By substitution in the above integrals of the respective minima, it is easy to
see that $\mathcal{I}^{EA}_{\epsilon}(\mathcal{A}_{min})=0$ for both cases.
The constant for the out-rising solution then become:

$C^{+}=+\mathcal{I}^{EA}_{\epsilon}(\mathcal{A}_{0})$ (69)

The meaning of such an integration condition lies in the fact that $\tau$, the
comoving time, is continuous as a function of $\mathcal{A}$. This implies that
there is no reason to restart the clock once the dust ball has reached the
minimum.
Finally the equations of motion read:

$\displaystyle\mathcal{A}^{-}$ $\displaystyle:-\frac{1}{2\Xi
p_{0}\sqrt{\pi}}\tau=-\mathcal{I}^{EA}_{\epsilon}(\mathcal{A}_{0})+\mathcal{I}^{EA}_{\epsilon}(\mathcal{A})$
(70) $\displaystyle\mathcal{A}^{+}$ $\displaystyle:\frac{1}{2\Xi
p_{0}\sqrt{\pi}}\tau=+\mathcal{I}^{EA}_{\epsilon}(\mathcal{A}_{0})+\mathcal{I}^{EA}_{\epsilon}(\mathcal{A})$
(71)

A plot of the trajectory, against comoving time, is shown in Fig. 5 and Fig. 6
for the two cases.

Figure 5: Bouncing object for the $\epsilon=+1$ scenarios. The parameters
chosen are $\alpha=2$ , $M_{S}=M_{\odot}$ and $\mu_{0}=6\pi\;\ell_{p}^{2}$
The minimum is $\mathcal{A}_{min}=40.6\;\text{Km}^{2}$. Figure 6: Bouncing
object for the $\epsilon=0$ scenarios. The parameters chosen are $\alpha=2$ ,
$M_{S}=M_{\odot}$ and $\mu_{0}=6\pi\;\ell_{p}^{2}$
The minimum is $\mathcal{A}_{m}=35.1\;\text{Km}^{2}$.

The trajectories show clearly that the dust ball bounces, so that after the
minimum it starts to expand again, eventually crossing the event horizon.
In the regime where the quantum effects become important, namely the high
momentum regime, a change in the geometry is induced by quantum effects, and a
black hole geometry makes a transition to a white hole geometry. This will be
analyzed clearly in the sub-subsection III.3.1.
Now let’s review the physical meaning of the minima we have found.

####  III.2.1 Minima and Cut-Offs

Now we analyze more critically the significance of the minima we have found.
The graph structure on the phase space that we introduced was meant to
regularize the $P_{\mathcal{A}}^{2}$ operator, since this is not a sensible
operator in the full quantum theory. The quantum theory suggests that only
finite area shift operators exist, and that the infinitesimal version, the
$P_{\mathcal{A}}$ operator, is just a classical approximation which emerges
for low momenta. This means that a full quantum theory could possess some sort
of minimal area gap, which is small but not infinitesimal, which we emulated
in our effective theory with the parameter $\mu_{0}$[29].
The minima that we have found in this work are much bigger than simply
$\mu_{0}$, in fact they also strictly depend on the physical parameters of the
model and on the specific Hamiltonian which describe the collapse. What this
means is that our newly found minima arise dynamically and are not the
fundamental area gap $\mu_{0}$.
The presence of such dynamical minima is nonetheless related to the existence
of a fundamental discreteness of the area, which enables some kind of quantum
gravitational pressure to slow the collapse and halt it at a non zero value of
the surface, ultimately creating a bouncing object. Like in the case of the
neutron stars or white dwarfs, a quantum pressure has macroscopical effects on
the collapse.
The minima clearly go to zero when we take the limit $\mu_{0}\rightarrow 0$,
equivalent to the $p_{0}\rightarrow\infty$ limit, so $\mu_{0}$ is supposed to
be the source of such quantum gravitational pressure. In subsection IV.1 we
will characterize such phenomenon more accurately.
We also want to notice that the points $\mathcal{A}_{min}$ and
$\mathcal{A}_{m}$ could lie outside the Schwarzschild-surface for some values
of the minimum area gap.
Before analyzing such conditions, we recall that $\hbar$ is expressed in
geometrized units and it corresponds to $\hbar=\ell_{p}^{2}$, so
$p_{0}=\ell_{p}^{2}/\mu_{0}$ is the rateo between the Planck area and the
minimum area gap.
We also define $\mathcal{A}_{p}=4\pi\ell_{p}^{2}=4\pi\hbar$ to be the Planck
area of a sphere of Planck radius.
We now see for what values of the parameter $\mu_{0}$ we have a dynamical
minimum bigger than the Schwarzschild-surface.
In the $\epsilon=0$ case the condition $\mathcal{A}_{m}>\mathcal{A}_{SC}$ for
a minimum outside the event horizon holds only if:

$\frac{16\pi M_{S}^{2}}{\kappa_{p}^{2}}>16\pi
M_{S}^{2}\Longrightarrow\mu_{0}>8\pi\ell_{p}^{2}$ (72)

For the $\epsilon=+1$ case the condition $\mathcal{A}_{min}>\mathcal{A}_{SC}$
holds only if:

$\frac{16\pi M_{S}^{2}}{\left(\kappa_{S}+\kappa_{p}\right)^{2}}>16\pi
M_{S}^{2}\Longrightarrow\frac{\ell_{p}^{2}}{\mu_{0}}<\frac{\sqrt{\frac{1-\kappa_{S}}{\Xi^{2}}}}{8\pi}$
(73)

So, after a quick substitution, the conditions to have a dynamical minimum
outside the event horizon, in both cases, are given by:

$\begin{split}\epsilon=0:&\qquad\mu_{0}>2\mathcal{A}_{p}\\\
\epsilon=+1:&\qquad\mu_{0}>2\mathcal{A}_{p}\frac{\Xi}{\sqrt{1-\kappa_{S}}}\end{split}$
(74)

Those are reasonable parameters for a quantum cut-off, being of the same order
of the Planck-sphere area, and those may have a macroscopic effect on the dust
ball collapse, since for such values the formation of an event horizon is
avoided.
Such conditions could then pose a constraint on the minimum area gap.
We now solve the equation of motion for the momentum.

###  III.3 Sphere Momentum Equation of Motion

As in the classical section, we will not solve the equation in function of the
comoving time $\tau$ but again we aim to solve for $P_{\mathcal{A}}$ as a
function of $\mathcal{A}$.
We recall that
$P_{\mathcal{A}}^{\prime}=\frac{dP_{\mathcal{A}}}{d\mathcal{A}}=\frac{\dot{P}_{\mathcal{A}}}{\dot{\mathcal{A}}}$,
so we substitute the constraint solution (54) inside $\dot{\mathcal{A}}$ and
$\dot{P}_{\mathcal{A}}$, thus obtaining the equation:

$P_{\mathcal{A}}^{\prime}=\pm\frac{P_{\tau}}{64\pi^{3/2}\Xi\;p_{0}\mathcal{A}\sqrt{\frac{4M_{S}\sqrt{\pi\mathcal{A}}-\kappa_{S}\mathcal{A}}{\kappa_{p}}-\frac{\left(4M_{S}\sqrt{\pi\mathcal{A}}-\kappa_{S}\mathcal{A}\right)^{2}}{\kappa_{p}^{2}\mathcal{A}}}}$
(75)

The plus and minus equations are associated to the in-falling and out-rising
velocity equations.
We can directly integrate those two equations without the need of defining
some external integrals and without treating separately the flat and spherical
case, since if we substitute $\kappa_{S}\rightarrow 0$ and $\Xi\rightarrow 1$
in the solutions we obtain the same expression as if we evaluate the two
integrals separately.
The solution then is:

$\begin{split}P_{\mathcal{A}}^{\pm}(\mathcal{A})=&\pm
p_{0}\text{arctan}\left(\sqrt{\frac{-4M_{S}\sqrt{\pi}+\sqrt{\mathcal{A}}\left(\kappa_{S}+\kappa_{p}\right)}{4M_{S}\sqrt{\pi}-\kappa_{S}\sqrt{\mathcal{A}}}}\right)\\\
&+C_{\pm}\end{split}$ (76)

To find the integration constant $C_{-}$ we set the momentum to be zero at the
initial surface $\mathcal{A}_{0}$, which is $\mathcal{A}_{max}$ for the
spherical case and $\mathcal{A}=+\infty$ for the flat case. In both the cases
the argument of the arctangent goes to infinity.
This yields:

$0=-p_{0}\frac{\pi}{2}+C_{-}\Longrightarrow C_{-}=p_{0}\frac{\pi}{2}$ (77)

We now need to impose the continuity on the minimum for the out-rising branch:

$P_{\mathcal{A}}^{-}(\mathcal{A}_{min})=P_{\mathcal{A}}^{+}(\mathcal{A}_{min})$
(78)

The arctangent is zero on the minimum, so we obtain:

$p_{0}\frac{\pi}{2}=C_{+}$ (79)

So the two constants are $C_{-}=p_{0}\frac{\pi}{2}=C_{+}$ for both flat and
spherical cases, and the solution reads:

$\displaystyle P_{\mathcal{A}}^{-}$
$\displaystyle=-p_{0}\left[\text{arctan}\left(\sqrt{\frac{-4M_{S}\sqrt{\pi}+\sqrt{\mathcal{A}}\left(\kappa_{S}+\kappa_{p}\right)}{4M_{S}\sqrt{\pi}-\kappa_{S}\sqrt{\mathcal{A}}}}\right)-\frac{\pi}{2}\right]$
(80) $\displaystyle P_{\mathcal{A}}^{+}$
$\displaystyle=+p_{0}\left[\text{arctan}\left(\sqrt{\frac{-4M_{S}\sqrt{\pi}+\sqrt{\mathcal{A}}\left(\kappa_{S}+\kappa_{p}\right)}{4M_{S}\sqrt{\pi}-\kappa_{S}\sqrt{\mathcal{A}}}}\right)+\frac{\pi}{2}\right]$
(81)

Where $P_{\mathcal{A}}^{-}$ is the in-falling momentum and
$P_{\mathcal{A}}^{+}$ is the out-rising one.
We show in Fig. 7 and Fig. 8 the trajectories for the momentum in both the
spherical and flat case.
As can be seen, in the spherical case the sphere has zero momentum at surface
$\mathcal{A}_{0}$, and the same is true for the flat case, which has it at
$\mathcal{A}=\infty$, so at finite surface $\mathcal{A}_{0}$ it possesses a
non-zero momentum.
Since the arctangent is zero on the minima, the momentum value
$P_{\mathcal{A}}(\mathcal{A}_{min})=p_{0}\frac{\pi}{2}$ is the one associated
to the velocity inversion.
We also notice that while in equation (21) there is a minus sign, so that the
momentum needs always to have the opposite sign of the velocity, in the
effective case this is not true, due to the velocity inversion after reaching
the minimum.
We will tackle the meaning of such inversion in subsubsection III.3.1.

Figure 7: Momentum as a function of the surface for the $\epsilon=+1$
scenario. The parameters chosen are $\alpha=2$ , $M_{S}=M_{\odot}$ and
$\mu_{0}=6\pi\ell_{p}^{2}$ Figure 8: Momentum as a function of the surface for
the $\epsilon=0$ scenario. The parameters chosen are $\alpha=2$ ,
$M_{S}=M_{\odot}$ and $\mu_{0}=6\pi\ell_{p}^{2}$

We now aim to analyze the inversion behaviour.

####  III.3.1  Velocity Inversion and Transition Amplitudes

We are ready to understand the implications of the inversion of the velocity.
To understand the physical meaning behind this behaviour, we will firstly
analyze the time evolution of the system, to check whether it is continuous or
not.
The first thing we need to do is to evaluate $\ddot{\mathcal{A}}$. Using the
fact that:

$\ddot{\mathcal{A}}=\frac{d\dot{\mathcal{A}}}{dt}=\frac{\partial\dot{\mathcal{A}}}{\partial
t}+\left\\{\dot{\mathcal{A}},H\right\\}=\dot{\mathcal{A}}\frac{\partial\dot{\mathcal{A}}}{\partial\mathcal{A}}+\dot{P}_{\mathcal{A}}\frac{\partial\dot{\mathcal{A}}}{\partial
P_{\mathcal{A}}}$ (82)

We can obtain the surface acceleration:

$\ddot{\mathcal{A}}=-4\pi\kappa_{S}+8\pi\left(\kappa_{S}+\kappa_{p}/2\right)\sin^{2}\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)$
(83)

We now show in Fig. 9 and Fig. 10 the trajectories against time of
$\mathcal{A},\;\dot{\mathcal{A}},\;\ddot{\mathcal{A}}$ and $P_{\mathcal{A}}$.
The acceleration in the spherical case has an inflection point:

$\mathcal{A}_{IN}=\mathcal{A}_{max}\frac{\left(\kappa_{S}+\kappa_{p}/2\right)^{2}}{\left(\kappa_{S}+\kappa_{p}\right)^{2}}$
(84)

In the flat case there are no inflection points. In the flat case is also
clearly visible that both the momentum and the velocity are not zero on some
surface $\mathcal{A}_{0}$ at a finite time, except for the zero velocity on
the minimum: that is compatible with the fact that the zero velocity condition
is imposed at infinity.

Figure 9: Time evolution of the system in the $\epsilon=+1$ scenario. The
parameters chosen are $\alpha=2$ , $M_{S}=M_{\odot}$ and
$\mu_{0}=6\pi\ell_{p}^{2}$.
The vertical lines show the time when the surface reaches the minimum, the
inflection point and the Schwarzschild surface. Figure 10: Time evolution of
the system in the $\epsilon=0$ scenario. The parameters chosen are $\alpha=2$
, $M_{S}=M_{\odot}$ and $\mu_{0}=6\pi\ell_{p}^{2}$.
The vertical lines show the time when the surface reaches the minimum and the
Schwarzschild surface.

The inversion has to be treated differently depending on whether it happens
inside or outside the event horizon. When it happens outside the event
horizon, there are no problems, since it means that the dust particles
’bounce’ at a finite radius, like in the presence of a potential barrier, and
turn away from the event horizon.
The problem appears if such an inversion happens after entering the event
horizon: inside the Schwarzschild surface the velocity can only point in the
decreasing radial direction, so the meaning of the inversion is that of an
inversion of causality, namely a transition to a white hole geometry with an
anti-horizon.
This could happen due to a quantum effect yet to be explored, and that make
possible the transition between the two geometries: that is what in the
literature is known as a black-to-white hole geometry
transition[30][31][32][33][34].
This transition should happen when the corresponding location in the phase
space for the canonical variables is in a neighborhood of the point
$(\mathcal{A}_{min},\;p_{0}\frac{\pi}{2})$, associated to the minimal surface
and inversion value of the momentum, with
$\mathcal{A}_{min}<\mathcal{A}_{SC}$. The inversion value of the momentum is
clearly Planckian, being of the same order of $M_{p}/t_{p}$.
This is where we expect the quantum effects to have a strong influence on the
dynamics of the system.
We also suppose that the bounce could be altered in a more accurate model.
This is because the dust model neglects all the internal thermodynamical
properties, as well as all the internal degrees of deformation and
compressibility. Those could result, in a more realistic model, in a different
transition mechanism between the two regions, where the momentum interacts
with the internal degrees of freedom of the star to over-compress it and
change its thermodynamical properties. This, together with the inclusion of
dissipative effects, could avoid the bounce once inside the event horizon and
stabilize the sphere at some fixed surface less than the Schwarzschild
surface, but still not singular.
Alternative bounce models with and without dissipative effects can be found at
[35][36][37][38][39][40].
Also in string models the singularity is avoided due to T-duality effects
[41].
Inhomogeneous density profiles show a very interesting property, that is, the
formation of shock-waves due to the most inner shells, which bounce first,
colliding with the outer layers of the cloud which are still collapsing. As a
result, the horizon disappear when the shock reaches said surface.
Those effects are not present is the OS model, being homogeneous.
The presence of such phenomena in an LQG setting was studied extensively in
[42] [43] [44].
We expect similar effects to take place also in the polymer model, when
pressure and inhomogeneities are introduced, and those should alter
significantly the bounce dynamics.
We are ready to analyze some phenomenological aspects of this effective model.

##  IV PHENOMENOLOGY

The scope of this section is to analyze the implications of the newly found
model.
First we are gonna characterize how the polymer quantization effects could be
understood as a form of quantum gravitational pressure.
Then we will analyze the implications, divided in two categories: the
phenomena which arise already at a macroscopical scale, before the sphere
reaches the quantum regime, and the effects that take place only after the
system has entered the quantum region. Both regimes could have observational
consequences.

###  IV.1 Polymer Pressure

Before diving into the description of possibly observable effects of this
model, we wish to give a proper description of the polymer quantum pressure
mentioned to be the microscopic cause of the modified collapse.
A similar quantum effective pressure was also found in [45] and [46].
Clearly there is no actual macroscopical pressure inside the dust, but as we
will see, the effective dynamics generated by the polymer implementation of
the area gap can be thought as the introduction of an internal pressure
generated by quantum effects of the gravitational field itself.
To achieve this result, we first recall the Friedmann acceleration equation
for a homogeneous isotropic fluid, which reads:

$\frac{\ddot{a}}{a}=-\frac{4\pi}{3}\left(\rho+3p\right)$ (85)

We can rewrite this equation as:

$\frac{\ddot{R}_{s}}{R_{s}}=-\frac{4\pi}{3}\rho-4\pi p$ (86)

Since the LHS is given by:

$\frac{\ddot{R}_{s}}{R_{s}}=\frac{\ddot{\mathcal{A}}}{2\mathcal{A}}-\frac{\dot{\mathcal{A}}^{2}}{4\mathcal{A}^{2}}$
(87)

This can be evaluated using equation (82) for the classical case, and yields:

$\left.\frac{\ddot{R}_{s}}{R_{s}}\right|_{CL}=-\frac{8\pi^{3/2}M_{S}}{\mathcal{A}^{3/2}}=-\frac{4\pi}{3}\rho$
(88)

So there is no pressure in the classical case, as expected. For the polymer
case the situation is different.
Using equation (83) we see that:

$\left.\frac{\ddot{R}_{s}}{R_{s}}\right|_{POL}=-\frac{8\pi^{3/2}M_{S}}{\mathcal{A}^{3/2}}+\frac{M_{S}^{2}}{\mathcal{A}^{2}\Xi^{2}p_{0}^{2}}-\frac{\left(\frac{\kappa_{S}M_{S}\sqrt{\mathcal{A}}}{\sqrt{\pi}}\right)}{4\mathcal{A}^{2}p_{0}^{2}}$
(89)

So there are extra terms, coming from the introduction of the polymer
corrections, which we will call polymer pressure:

$-4\pi\mathcal{P}_{\mu_{0}}=+\frac{M_{S}^{2}}{\mathcal{A}^{2}\Xi^{2}p_{0}^{2}}-\frac{\left(\frac{\kappa_{S}M_{S}\sqrt{\mathcal{A}}}{\sqrt{\pi}}\right)}{4\mathcal{A}^{2}p_{0}^{2}}$
(90)

Rewriting this in a more clean way gives:

$\mathcal{P}_{\mu_{0}}=\frac{\mathcal{A}_{SC}}{\kappa_{p}}\frac{1}{\mathcal{A}^{2}}\left(\sqrt{\frac{\mathcal{A}}{\mathcal{A}_{0}}}-1\right)$
(91)

Now let’s analyze the pressure $\mathcal{P}_{\mu_{0}}$.
The quantum origin is very clear: if we take $\mu_{0}=0$ then
$\frac{1}{\kappa_{p}}=0$, so the pressure vanishes identically. Then we see
that the pressure vanishes also at the start of the collapse, when
$\mathcal{A}=\mathcal{A}_{0}$. This is good since we have seen that quantum
effects arise with the increasing of the momentum.
The last thing to notice is that since $\mathcal{A}<\mathcal{A}_{0}$ always,
the pressure is always negative: this means that $\mathcal{P}_{\mu_{0}}$ has
to be interpreted as a kind of 3D tension, which tries to restore the surface
to its initial value when it gets compressed.
An heuristic tentative explanation for that phenomenon is that the quanta of
surface make resistance when compressed against each other, due to
gravitational self interactions. In the classical scenario those quanta are
not present and hence this new pressure is absent.
For this reasons from now on we will refer to this quantum gravitational
pressure as ”polymer tension”.
With the nature of this tension made clear, we are now ready to study the
phenomenological aspects of this model.

###  IV.2 Macroscopic Phenomena

The first thing we wish to accomplish is to understand whether we need to
impose some consistencies conditions on the newly introduced parameter
$\mu_{0}$.
To do such a thing we need to check if the physical motion of the sphere
exceeds the light speed limit. Since we are working with a ball of
pressureless homogeneous dust, the velocity of the dust particles corresponds
to the radial velocity of the star.
We then proceed to check if this velocity is consistent with the light speed
limit. The radial velocity must be obtained from the surface velocity, since
after the introduction of the effective correction we have lost the ability to
make canonical transformations from $P_{\mathcal{A}}$ to $P_{s}$, but we can
still obtain $R_{s}$ from $\mathcal{A}$.
We then have:

$\dot{\mathcal{A}}=\frac{d\mathcal{A}}{dR_{s}}\frac{dR_{s}}{dt}=\frac{d\mathcal{A}}{dR_{s}}\dot{R}_{s}$
(92)

Since $\frac{d\mathcal{A}}{dR_{s}}=8\pi R_{s}$, after the substitution of
$\mathcal{A}(R_{s})$, we get:

$\dot{R}_{s}=\frac{\dot{\mathcal{A}}}{8\pi R_{s}}=-8\pi\Xi
p_{0}\sin\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)\cos\left(\frac{P_{\mathcal{A}}}{p_{0}}\right)$
(93)

We then make the trigonometric substitution $2\sin(x)\cos(x)=\sin(2x)$, so we
end up with the equation of motion:

$\dot{R}_{s}=-4\pi\Xi p_{0}\sin\left(\frac{2P_{\mathcal{A}}}{p_{0}}\right)$
(94)

Clearly the velocity is dependent on the value of the parameter $\mu_{0}$, so
we need to impose by hand the subluminal collapse speed condition. It is easy
to see that the velocity has its maximum value when the sine is equal to one,
or minus one equivalently.
This means that the absolute maximum value of the velocity is:

$\dot{R}_{s}^{MAX}=4\pi\Xi p_{0}$ (95)

To impose $\dot{R}_{s}^{MAX}$ to be subluminal, in geometric units, means that
we need $\dot{R}_{s}^{MAX}<1$, yielding the following constraint on the
parameter $\mu_{0}$:

$\frac{4\pi\Xi\ell_{p}^{2}}{\mu_{0}}<1\Longrightarrow\mu_{0}>\mathcal{A}_{p}\Xi$
(96)

This means that we have a lower constraint on the polymer area gap given by
$\mu_{0}>\mathcal{A}_{p}\Xi$. Remember that in the flat case $\Xi=1$.
Equivalently the constraint reads:

$p_{0}<\frac{1}{4\pi\Xi}$ (97)

With those constraints established, we see that the values for which we have a
minimum outside the event horizon in equations (74) are compatible with what
we have found.
Regarding only the spherical case, which is a model more physically relevant
than the flat one, we can investigate deeper into the consequences of such
scenario.

####  IV.2.1 A New Astrophysical Object

Let us fix a value of the parameter $\frac{\mu_{0}}{\mathcal{A}_{p}}=\gamma$.
Then the condition for a minimum outside the event horizon reads:

$\frac{2\Xi}{\sqrt{1-\kappa_{S}}}<\gamma$ (98)

To see when this condition is satisfied we explicit the $r_{S}$ parameter in
the LHS, which yields the function:

$f(\kappa_{S})=\frac{4\kappa_{S}^{3/2}}{3\sqrt{1-\kappa_{S}}\left(2\text{arctan}\left(\frac{-1+\sqrt{\kappa_{S}}}{\sqrt{1-\kappa_{S}}}\right)+\pi-\sqrt{\kappa_{S}-\kappa_{S}^{2}}\right)}$
(99)

Recall that the $r_{S}$ dependence is recovered by $\kappa_{S}$ since
$\sqrt{\kappa_{S}}=r_{S}\sqrt{\epsilon}$, where $\epsilon=1$ is kept for
dimensional reasons.
We show in Fig. 11 the behaviour of such function with $r_{S}\in(0,1)$, as
imposed by the condition (32).

Figure 11: Behaviour of $f(r_{S})$ as a function of $r_{S}$. The function
clearly diverges as $r_{S}$ approaches 1.
$\gamma$ is set to 2 for illustrative reasons.

The plot clearly shows that the closer to the event horizon the collapse
start, i.e. the closer we get to $\kappa_{S}=1$, the greater the value of the
area gap must be in order to avoid the formation of the event horizon.
The plot gives the values of $\kappa_{S}$, far enough from the event horizon
given a value of $\gamma$, for which the sphere will oscillate between the
initial collapsing size $\mathcal{A}_{0}$ and the minimum value
$\mathcal{A}_{min}$, outside its event horizon.
This suggests that some astrophysical objects, massive enough that should form
a black hole instead of a neutron star, could instead become some new radially
pulsating objects, given the right conditions on the size and the polymer
parameter.
The pulsation/oscillation is evident both from the phase space portrait in
Fig. 3 and from the fact that in Fig. 9, after a full collapsing/expanding
cycle, the system comes back at the initial values for all the four variables
$(\mathcal{A},\dot{\mathcal{A}},\ddot{\mathcal{A}},P_{\mathcal{A}})$ and hence
the motion is periodic, with a comoving period of:

$T_{Cyc}=2\tau_{min}=4\sqrt{\pi}\Xi
p_{0}\mathcal{I}^{EA}_{+1}(\mathcal{A}_{0})$ (100)

Substituting the value of $\mathcal{I}^{EA}_{+1}(\mathcal{A}_{0})$ yields:

$T_{Cyc}=\frac{M_{S}}{2\Xi
p_{0}}\;\frac{\kappa_{p}\left(\kappa_{S}+\kappa_{p}/2\right)}{\left[\kappa_{S}\left(\kappa_{S}+\kappa_{p}\right)\right]^{3/2}}$
(101)

For Sun-like parameters the comoving time period is of order $10^{-5}\;s$, to
which gravitational time dilation should be applied in order to evaluate the
cycle period for a distant observer.
We will call this new objects periodically radially pulsating objetcs (PRPOs).

###  IV.3 Quantum Effects

The dominant effect which arise in the quantum regime, as said in the previous
discussion on the inversion behaviour, is that of a quantum transition between
the two geometries of a black and a white hole.
The two geometries under evaluation are depicted by the Penrose diagrams[47]
in Fig. 13 and Fig. 13. We show only the flat model, since the spherical model
cannot be described by a single patch of coordinates at infinity.
In the transition we have a sudden change of geometry, but some of the
spacetime properties are identified between the two geometries. What we need
to do is identify the null-like infinity and the time-like infinity of the two
geometries. Also the space-like infinity and the minimal dynamical radius are
to be identified. Those identifications are such that the event horizon of the
collapsing sphere effectively become an anti-horizon in the out-rising
picture, so the out-rising geometry is recognised to be the one of a white
hole.
This means that, in the quantum regime, the transition actually transforms the
horizon, while the asymptotic geometry remains the same.

Figure 12: Carter-Penrose Diagram of a Collapsing Black Hole Geometry. In grey
the area excluded from the external geometry, in green the sphere surface
trajectory, in red the dynamically removed area with $R_{S}<R_{min}$. The
horizontal purple line is the event horizon.

Figure 13: Carter-Penrose Diagram of a Collapsing White Hole Geometry. In grey
the area excluded from the external geometry, in green the sphere surface
trajectory, in red the dynamically removed area with $R_{S}<R_{min}$. The
horizontal purple line is the event anti-horizon.

We now try to put some bounds on the transition rate $\Gamma_{B\rightarrow W}$
with some simplified arguments.
The probability of seeing a transition from a black to a white hole could be
approximated by:

$P(t)=1-e^{-\Gamma_{B\rightarrow W}t}$ (102)

Since we know that the rate is the inverse of the mean time for the
transition, namely $\Gamma_{B\rightarrow W}=\frac{1}{\langle t\rangle}$, we
can infer the transition rate from the knowledge of the mean time.
Assuming that we have not seen yet an astrophysical event recognizable as a
white hole, we can infer that the mean time for the transition should be
greater than the age of the universe, so we obtain:

$\langle t\rangle>t_{univ}\Longrightarrow\Gamma_{B\rightarrow
W}<\frac{1}{t_{univ}}$ (103)

The Planck 2018 data[48] given an estimate of $t_{univ}=(13.787\pm
0.020)10^{9}$yrs, so an estimate of the transition rate could be given by:

$\Gamma_{B\rightarrow W}\lesssim 7.25\cdot 10^{-11}\;\text{Events}/\text{yrs}$
(104)

This transition rate should be evaluated carefully in a full quantization of
the theory, which is beyond the scope of this work. We can only constrain it
to the experimental facts that we are confronted with, which state that the
probability of such transition needs to be quite low.
As also stated in subsubsection III.3.1, in a more realistic model is possible
that no transition occurs at all if the star is stabilized inside the event
horizon, so in that scenario the transition rate $\Gamma_{B\rightarrow W}$
should be zero.

##  V CONCLUSIONS

In this paper we analyzed the role played by a non-zero area gap in the
crucial setting of gravitational collapse.
In Section III we have found that, thanks to the polymer tension (91) caused
by the discrete nature of the area, the collapse is regularized both in the
flat and in the spherical case, avoiding the formation of a singularity. This
implies the existence of a dynamical minimum surface that the sphere can
reach, given by equations (60 \- 61). In equations (74) we showed that exist
values of the polymer parameter for which said minima lie outside of the event
horizon. Those are compatible with the subluminal collapsing speed constraint
imposed on $\mu_{0}$ in equation (96).
In Section IV we characterized such polymer tension, and we have found that a
pressure of quantum origin has macroscopic consequences for the collapse, so
conceptually it is not very different from the various degeneracy pressures
which appear in the usual treatment of compact objects.
We have also found that on the new regular minima there is a velocity
inversion, which has two important consequences depending on whether or not it
happens inside or outside the event horizon.
We found that when such an inversion happens inside the event horizon, the
associated process has to be understood as a quantum transition from a black
hole geometry to a white hole geometry, in accordance with the literature
cited. We also considered the possibility that such process is avoided
completely in a more realistic star model, and this is more plausible
according to our astrophysical data.
More work is needed in order to implement the polymer/LQG procedure for models
with rotating stars or internal pressure, which could provide a mechanism to
dissipate the quantum gravitational pressure into internal degrees of freedom
of the star and stabilize it without the formation of a white hole.
We then found the conditions for the bounce to happen outside the event
horizon, finding a new class of objects, PRPOs, which exhibit a periodic
radially pulsating behaviour, with the comoving pulsation period given by
equation (101).
We state that those objects, due to the presence of the new polymer tension,
could have a different deformability with respect to neutron stars or black
holes, and hence they could leave a different signature in gravitational waves
emitted by binary mergers.
A careful study of such signatures, in the case of more accurate star models,
is left for future investigations in the hope for new tests of quantum
gravity.

###### Acknowledgements.

We wolud like to thank Paolo Pani for fruitful discussions on this subject.

##  Appendix A Integrals of the Equations of Motion

We make use of this Appendix to define the integrals defined throughout the
sections II and III.

###  A.1 Classical Dynamics Integrals

####  A.1.1 Surface Integrals

We start with the classical integrals for the surface equations of motion.
Let’s define the classical area (CA) integrals as:

$\mathcal{I}_{\epsilon}^{CA}(\mathcal{A})=\int{\frac{d\mathcal{A}}{\sqrt{4M_{S}\sqrt{\pi\mathcal{A}}-\kappa_{S}\mathcal{A}}}}$
(105)

Those are two integrals, one for each value of $\epsilon$, and are equivalent
to:

$\displaystyle\mathcal{I}_{0}^{CA}(\mathcal{A})=$
$\displaystyle\frac{2\mathcal{A}}{3\sqrt{M_{S}\sqrt{\pi\mathcal{A}}}}$ (106)
$\displaystyle\begin{split}\mathcal{I}_{+1}^{CA}(\mathcal{A})=&\frac{8M_{S}\sqrt{\pi}\;\text{atan}\left(\frac{\sqrt{\kappa_{S}\mathcal{A}}}{\sqrt{4M_{S}\sqrt{\pi\mathcal{A}}-\kappa_{S}\mathcal{A}}}\right)}{\left(\kappa_{S}\right)^{3/2}}\\\
&-\frac{2\sqrt{4M_{S}\sqrt{\pi\mathcal{A}}-\kappa_{S}\mathcal{A}}}{\kappa_{S}}\end{split}$
(107)

Those can be directly put into the solutions of the equations of motion.

####  A.1.2 Momentum Integrals

Let’s define the classical momentum (CM) integrals as:

$\mathcal{I}_{\epsilon}^{CM}(P_{\mathcal{A}})=\int\frac{256P_{\mathcal{A}}dP_{\mathcal{A}}}{\frac{\kappa_{S}}{\pi^{2}\Xi^{2}}+64P_{\mathcal{A}}^{2}}$
(108)

Those are two integrals, one for each value of $\epsilon$, evaluated to be:

$\displaystyle\mathcal{I}_{0}^{CM}(P_{\mathcal{A}})$
$\displaystyle=4\ln\left(P_{\mathcal{A}}\right)$ (109)
$\displaystyle\mathcal{I}_{+1}^{CM}(P_{\mathcal{A}})$
$\displaystyle=2\ln\left(\kappa_{S}+64\pi^{2}\Xi^{2}P_{\mathcal{A}}^{2}\right)$
(110)

The integral in the area variable instead yields:

$-\int\frac{d\mathcal{A}}{\mathcal{A}}=-\ln\left(\mathcal{A}\right)$ (111)

Plugging those integrals in the classical momentum equations we can invert the
logarithms to obtain $P_{\mathcal{A}}(\mathcal{A})$.

###  A.2 Effective Dynamics Integrals

####  A.2.1 Surface Integrals

We continue this appendix with the effective integrals for the surface
equations of motion.
Let’s define the effective area (EA) integrals as:

$\begin{split}&\mathcal{I}^{EA}_{\epsilon}(\mathcal{A})=\int{\frac{d\mathcal{A}}{\sqrt{\left(4M_{S}\sqrt{\pi}-\kappa_{S}\sqrt{\mathcal{A}}\right)\left(\sqrt{\mathcal{A}}\left(\kappa_{S}+\kappa_{p}\right)-4M_{S}\sqrt{\pi}\right)}}}\end{split}$
(112)

That evaluates to:

$\mathcal{I}_{0}^{EA}(\mathcal{A})=\frac{\left(M_{S}+8\pi^{3/2}p_{0}^{2}\sqrt{\mathcal{A}}\right)}{384\pi^{7/2}p_{0}^{4}}\sqrt{\frac{\left(16\pi^{3/2}p_{0}^{2}\sqrt{\mathcal{A}}-M_{S}\right)}{M_{S}}}$
(113)
$\begin{split}\mathcal{I}_{+1}^{EA}(\mathcal{A})&=\frac{16M_{S}\sqrt{\pi}\left(\kappa_{S}+\kappa_{p}/2\right)\text{arctan}\left(\frac{\sqrt{\kappa_{S}\left(\sqrt{\mathcal{A}}\left(\kappa_{S}+\kappa_{p}\right)-4M_{S}\sqrt{\pi}\right)}}{\sqrt{-\left(\kappa_{S}+\kappa_{p}\right)\left(\kappa_{S}\sqrt{\mathcal{A}}-4M_{S}\sqrt{\pi}\right)}}\right)}{\left[\kappa_{S}\left(\kappa_{S}+\kappa_{p}\right)\right]^{3/2}}-2\frac{\sqrt{\left(4M_{S}\sqrt{\pi}-\kappa_{S}\sqrt{\mathcal{A}}\right)\left(\sqrt{\mathcal{A}}\left(\kappa_{S}+\kappa_{p}\right)-4M_{S}\sqrt{\pi}\right)}}{\kappa_{S}\left(\kappa_{S}+\kappa_{p}\right)}\end{split}$
(114)

We got two different integrals, just like before, since those evaluate to
different functions for different $\epsilon$ parameters.

## References

  * Misner  _et al._ [1973] C. W. Misner, K. S. Thorne, and J. A. Wheeler,  _Gravitation_ (W. H. Freeman, San Francisco, 1973).
  * Weinberg [2008] S. Weinberg,  _Cosmology_ (Oxford University Press, London, England, 2008).
  * Shapiro and Teukolsky [1983] S. L. Shapiro and S. A. Teukolsky,  _Black holes, white dwarfs, and neutron stars_ (John Wiley & Sons, Nashville, TN, 1983).
  * Kalogera and Baym [1996] V. Kalogera and G. Baym, The maximum mass of a neutron star, The Astrophysical Journal 470, L61 (1996).
  * Rezzolla  _et al._ [2018] L. Rezzolla, E. R. Most, and L. R. Weih, Using gravitational-wave observations and quasi-universal relations to constrain the maximum mass of neutron stars, The Astrophysical Journal Letters 852, L25 (2018).
  * DeWitt [1967] B. S. DeWitt, Quantum theory of gravity. i. the canonical theory, Phys. Rev. 160, 1113 (1967).
  * Thiemann [2007] T. Thiemann,  _Modern Canonical Quantum General Relativity_ (Cambridge University Press, 2007).
  * Oppenheimer and Snyder [1939] J. R. Oppenheimer and H. Snyder, On continued gravitational contraction, Phys. Rev. 56, 455 (1939).
  * Corichi  _et al._ [2007] A. Corichi, T. Vukašinac, and J. A. Zapata, Polymer quantum mechanics and its continuum limit, Phys. Rev. D 76, 044016 (2007).
  * Hájíček and Kiefer [2001] P. Hájíček and C. Kiefer, Singularity avoidance by collapsing shells in quantum gravity, International Journal of Modern Physics D 10, 775 (2001).
  * Kiefer and Schmitz [2019] C. Kiefer and T. Schmitz, Singularity avoidance for collapsing quantum dust in the lemaître-tolman-bondi model, Phys. Rev. D 99, 126010 (2019).
  * Malafarina [2017] D. Malafarina, Classical collapse to black holes and quantum bounces: A review, Universe 3, 10.3390/universe3020048 (2017).
  * Gourgoulhon [2012] E. Gourgoulhon,  _3+1 Formalism in General Relativity: Bases of Numerical Relativity_ (Springer Berlin Heidelberg, 2012).
  * Arnowitt  _et al._ [1959] R. Arnowitt, S. Deser, and C. W. Misner, Dynamical structure and definition of energy in general relativity, Phys. Rev. 116, 1322 (1959).
  * Poisson [2004] E. Poisson,  _A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics_ (Cambridge University Press, 2004).
  * Hájíček and Kijowski [1998] P. Hájíček and J. Kijowski, Lagrangian and hamiltonian formalism for discontinuous fluid and gravitational field, Phys. Rev. D 57, 914 (1998).
  * Kanai  _et al._ [2011] Y. Kanai, M. Siino, and A. Hosoya, Gravitational Collapse in Painlevé-Gullstrand Coordinates, Progress of Theoretical Physics 125, 1053 (2011).
  * Casadio [1998] R. Casadio, Hamiltonian formalism for the oppenheimer-snyder model, Phys. Rev. D 58, 064013 (1998).
  * Kiefer and Mohaddes [2023] C. Kiefer and H. Mohaddes, From classical to quantum oppenheimer-snyder model: Nonmarginal case, Phys. Rev. D 107, 126006 (2023).
  * Schmitz [2020] T. Schmitz, Towards a quantum oppenheimer-snyder model, Phys. Rev. D 101, 026016 (2020).
  * Brown and Kuchař [1995] J. D. Brown and K. V. Kuchař, Dust as a standard of space and time in canonical quantum gravity, Phys. Rev. D 51, 5600 (1995).
  * Painlevé [1921] P. Painlevé, La mécanique classique et la théorie de la relativité, C. R. Acad. Sci. 173, 677 (1921).
  * Gullstrand [1922] A. Gullstrand,  _Allgemeine Lösung des statischen Einkörperproblems in der Einsteinschen Gravitationstheorie_ , Arkiv för matematik, astronomi och fysik, Vol. 16,8 (Almqvist & Wiksell, Stockholm, 1922).
  * Gautreau and Hoffmann [1978] R. Gautreau and B. Hoffmann, The schwarzschild radial coordinate as a measure of proper distance, Phys. Rev. D 17, 2552 (1978).
  * Strocchi [2016] F. Strocchi,  _Gauge Invariance and Weyl-polymer Quantization_ (Springer International Publishing, 2016).
  * Morchio and Strocchi [2007] G. Morchio and F. Strocchi, Quantum mechanics on manifolds and topological effects, Lett. Math. Phys. 82, 219 (2007).
  * Rovelli and Smolin [1995a] C. Rovelli and L. Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys. B 442, 593 (1995a), [Erratum: Nucl.Phys.B 456, 753–754 (1995)].
  * Rovelli and Smolin [1995b] C. Rovelli and L. Smolin, Spin networks and quantum gravity, Phys. Rev. D 52, 5743 (1995b).
  * Ashtekar  _et al._ [2003] A. Ashtekar, S. Fairhurst, and J. L. Willis, Quantum gravity, shadow states, and quantum mechanics, Class. Quant. Grav. 20, 1031 (2003).
  * Haggard and Rovelli [2015] H. M. Haggard and C. Rovelli, Quantum-gravity effects outside the horizon spark black to white hole tunneling, Physical Review D 92, 10.1103/physrevd.92.104020 (2015).
  * Münch [2021] J. Münch, Effective quantum dust collapse via surface matching, Classical and Quantum Gravity 38, 10.1088/1361-6382/ac103e (2021).
  * Kelly  _et al._ [2020] J. G. Kelly, R. Santacruz, and E. Wilson-Ewing, Black hole collapse and bounce in effective loop quantum gravity, Classical and Quantum Gravity 38, 10.1088/1361-6382/abd3e2 (2020).
  * Bianchi  _et al._ [2018] E. Bianchi, M. Christodoulou, F. D’Ambrosio, H. M. Haggard, and C. Rovelli, White holes as remnants: a surprising scenario for the end of a black hole, Classical and Quantum Gravity 35, 10.1088/1361-6382/aae550 (2018).
  * Achour  _et al._ [2020a] J. B. Achour, S. Brahma, S. Mukohyama, and J.-P. Uzan, Towards consistent black-to-white hole bounces from matter collapse, Journal of Cosmology and Astroparticle Physics 2020 (09).
  * Rovelli and Vidotto [2018] C. Rovelli and F. Vidotto, Small black/white hole stability and dark matter, Universe 4, 10.3390/universe4110127 (2018).
  * De Lorenzo and Perez [2016] T. De Lorenzo and A. Perez, Improved black hole fireworks: Asymmetric black-hole-to-white-hole tunneling scenario, Physical Review D 93, 10.1103/physrevd.93.124018 (2016).
  * Husain  _et al._ [2022a] V. Husain, J. G. Kelly, R. Santacruz, and E. Wilson-Ewing, Quantum gravity of dust collapse: Shock waves from black holes, Physical Review Letters 128, 10.1103/physrevlett.128.121301 (2022a).
  * Achour  _et al._ [2020b] J. B. Achour, S. Brahma, and J.-P. Uzan, Bouncing compact objects. part i. quantum extension of the oppenheimer-snyder collapse, Journal of Cosmology and Astroparticle Physics 2020 (03).
  * Ben Achour and Uzan [2020] J. Ben Achour and J.-P. Uzan, Bouncing compact objects. ii. effective theory of a pulsating planck star, Physical Review D 102, 10.1103/physrevd.102.124041 (2020).
  * Barca and Montani [2024] G. Barca and G. Montani, Non-singular gravitational collapse through modified Heisenberg algebra, Eur. Phys. J. C 84, 261 (2024).
  * Jusufi [2023] K. Jusufi, Avoidance of singularity during the gravitational collapse with string t-duality effects, Universe 9, 10.3390/universe9010041 (2023).
  * Fazzini  _et al._ [2023] F. Fazzini, C. Rovelli, and F. Soltani, Painlevé-gullstrand coordinates discontinuity in the quantum oppenheimer-snyder model, Phys. Rev. D 108, 044009 (2023).
  * Fazzini  _et al._ [2024] F. Fazzini, V. Husain, and E. Wilson-Ewing, Shell-crossings and shock formation during gravitational collapse in effective loop quantum gravity, Phys. Rev. D 109, 084052 (2024).
  * Husain  _et al._ [2022b] V. Husain, J. G. Kelly, R. Santacruz, and E. Wilson-Ewing, Fate of quantum black holes, Phys. Rev. D 106, 024014 (2022b).
  * Bambi  _et al._ [2013] C. Bambi, D. Malafarina, and L. Modesto, Non-singular quantum-inspired gravitational collapse, Phys. Rev. D 88, 044009 (2013).
  * Liu  _et al._ [2014] Y. Liu, D. Malafarina, L. Modesto, and C. Bambi, Singularity avoidance in quantum-inspired inhomogeneous dust collapse, Phys. Rev. D 90, 044040 (2014).
  * Penrose [1963] R. Penrose, Asymptotic properties of fields and space-times, Physical Review Letters 10, 10.1103/physrevlett.10.66 (1963).
  * Planck Collaboration  _et al._ [2020] Planck Collaboration, Aghanim, N., Akrami, Y., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A. J., Barreiro, R. B., Bartolo, N., Basak, S., Battye, R., Benabed, K., Bernard, J.-P., Bersanelli, M., Bielewicz, P., Bock, J. J., Bond, J. R., Borrill, J., Bouchet, F. R., Boulanger, F., Bucher, M., Burigana, C., Butler, R. C., Calabrese, E., Cardoso, J.-F., Carron, J., Challinor, A., Chiang, H. C., Chluba, J., Colombo, L. P. L., Combet, C., Contreras, D., Crill, B. P., Cuttaia, F., de Bernardis, P., de Zotti, G., Delabrouille, J., Delouis, J.-M., Di Valentino, E., Diego, J. M., Doré, O., Douspis, M., Ducout, A., Dupac, X., Dusini, S., Efstathiou, G., Elsner, F., Enßlin, T. A., Eriksen, H. K., Fantaye, Y., Farhang, M., Fergusson, J., Fernandez-Cobos, R., Finelli, F., Forastieri, F., Frailis, M., Fraisse, A. A., Franceschi, E., Frolov, A., Galeotta, S., Galli, S., Ganga, K., Génova-Santos, R. T., Gerbino, M., Ghosh, T., González-Nuevo, J., Górski, K. M., Gratton, S., Gruppuso, A., Gudmundsson, J. E., Hamann, J., Handley, W., Hansen, F. K., Herranz, D., Hildebrandt, S. R., Hivon, E., Huang, Z., Jaffe, A. H., Jones, W. C., Karakci, A., Keihänen, E., Keskitalo, R., Kiiveri, K., Kim, J., Kisner, T. S., Knox, L., Krachmalnicoff, N., Kunz, M., Kurki-Suonio, H., Lagache, G., Lamarre, J.-M., Lasenby, A., Lattanzi, M., Lawrence, C. R., Le Jeune, M., Lemos, P., Lesgourgues, J., Levrier, F., Lewis, A., Liguori, M., Lilje, P. B., Lilley, M., Lindholm, V., López-Caniego, M., Lubin, P. M., Ma, Y.-Z., Macías-Pérez, J. F., Maggio, G., Maino, D., Mandolesi, N., Mangilli, A., Marcos-Caballero, A., Maris, M., Martin, P. G., Martinelli, M., Martínez-González, E., Matarrese, S., Mauri, N., McEwen, J. D., Meinhold, P. R., Melchiorri, A., Mennella, A., Migliaccio, M., Millea, M., Mitra, S., Miville-Deschênes, M.-A., Molinari, D., Montier, L., Morgante, G., Moss, A., Natoli, P., Nørgaard-Nielsen, H. U., Pagano, L., Paoletti, D., Partridge, B., Patanchon, G., Peiris, H. V., Perrotta, F., Pettorino, V., Piacentini, F., Polastri, L., Polenta, G., Puget, J.-L., Rachen, J. P., Reinecke, M., Remazeilles, M., Renzi, A., Rocha, G., Rosset, C., Roudier, G., Rubiño-Martín, J. A., Ruiz-Granados, B., Salvati, L., Sandri, M., Savelainen, M., Scott, D., Shellard, E. P. S., Sirignano, C., Sirri, G., Spencer, L. D., Sunyaev, R., Suur-Uski, A.-S., Tauber, J. A., Tavagnacco, D., Tenti, M., Toffolatti, L., Tomasi, M., Trombetti, T., Valenziano, L., Valiviita, J., Van Tent, B., Vibert, L., Vielva, P., Villa, F., Vittorio, N., Wandelt, B. D., Wehus, I. K., White, M., White, S. D. M., Zacchei, A., and Zonca, A., Planck 2018 results - vi. cosmological parameters, A&A 641, A6 (2020).


* Wehrhahn (2021) Wehrhahn, A. 2021, in The 20.5th Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun (CS20.5), Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun, 1, doi: 10.5281/zenodo.4537913
  * Wijbenga & Zwaan (1972) Wijbenga, J. W., & Zwaan, C. 1972, Sol. Phys., 23, 265, doi: 10.1007/BF00148089
  * Worley et al. (2020) Worley, C. C., Jofré, P., Rendle, B., et al. 2020, A&A, 643, A83, doi: 10.1051/0004-6361/201936726
  * Zhao et al. (2012) Zhao, G., Zhao, Y., Chu, Y., Jing, Y., & Deng, L. 2012, arXiv e-prints, arXiv:1206.3569.  https://arxiv.org/abs/1206.3569