# Generic features of a polymer quantum black hole

Johannes<EMAIL_ADDRESS>, Alejandro
<EMAIL_ADDRESS>, Simone
<EMAIL_ADDRESS>, Sami
<EMAIL_ADDRESS>(corresponding author)
1Aix-Marseille Université, Université de Toulon, CNRS, CPT,
13288 Marseille, France

###### Abstract

Non-singular black holes models can be described by modified classical
equations motivated by loop quantum gravity. We investigate what happens when
the sine function typically used in the modification is replaced by an
arbitrary bounded function, a generalization meant to study the effect of
ambiguities such as the choice of representation of the holonomy. A number of
features can be determined without committing to a specific choice of
functions. We find generic singularity resolution. The presence and number of
horizons is determined by global features of the function regularizing the
angular components of the connection, and the presence and number of bounces
by global features of the function regularizing the time component. The
trapping or anti-trapping nature of regions inside horizons depends on the
relative location with respect to eventual bounces. We use these results to
comment on some of the ambiguities of polymer black hole models.

###### Contents

  1. 1 Introduction
  2. 2 From the classical black hole to a polymerized black hole
    1. 2.1 Minisuperspace black hole model
    2. 2.2 Ashtekar-Barbero variables
    3. 2.3 BMM polymerisation
  3. 3 Generalised polymerisation
    1. 3.1 Presentation of the setup
    2. 3.2 Dirac observables
    3. 3.3 General expression of the metric
  4. 4 Generic features of the dynamics
    1. 4.1 Evolution and fixed points
    2. 4.2 Bounces and singularity resolution
    3. 4.3 Horizons
    4. 4.4 Trapped and anti-trapped regions
  5. 5 Large distance behaviour
    1. 5.1 Asymptotic behaviour
    2. 5.2 Taming quantum gravity effects outside the horizon
  6. 6 Conclusions
  7. A Boundary term
  8. B Asymptotic expansion of the metric components

## 1 Introduction

An emblematic problem in quantum gravity is to understand the fate of the
black hole singularity predicted by general relativity. Aside from extremely
simplified 2d models, see e.g. [1, 2], explicit calculations are not within
current reach, and this motivates the investigation of minisuperspace models
in order to shed useful light on qualitative aspects of the process. In loop
quantum gravity (LQG), quantum minisuperspace models are constructed using the
key input from the full theory that the fundamental quantum operators are
holonomies of the Ashtekar-Barbero connection. These are defined in a
representation which lacks weak-continuity, thus making the models unitary
inequivalent to those based on metric variables. A compelling result of this
approach is the generic singularity-avoidance [3], with the minisuperspace
dynamics predicting a bounce occurring at a critical energy or curvature
density. This is valid in both cosmological [4, 3] and black hole [5, 6, 7, 8,
9] models. In the latter case, this type of investigations first started in
[10, 11].

One key aspect of the construction is that the resulting quantum corrections
can be equally predicted using an effective classical dynamics, where the
original Hamiltonian is modified in a precise way [12]. This step is referred
to as polymerization, and allows one a simple description of the system, and
an independent exploration of this type of models. In spite of the compelling
singularity resolutions, these models are only a first step towards a clear
understanding of the non-perturbative quantum gravitational effects. To move
forward, it is important to address their limitations and shortcomings. For
instance, the models rely on a specific foliation, and restoring foliation-
independence is far from accomplished [13, 14, 15, 16]. The choice of
polymerization scheme is not identified a priori from fundamental principles
or deductions from the full theory; but rather reconstructed a posteriori
requiring a good semiclassical limit at large scales, and there is no
uniqueness about the procedure used [17]. We will not have much to say about
the first issue, but we would like to focus on the following aspect about the
second. The function used in the scheme is typically picked to be a sine. This
is supposed to represent working with the holonomy in the fundamental
representation in full loop quantum gravity. However it has been known for a
while that there may be quantization ambiguities associated with this choice
already in the full theory [18].555It has been recently pointed out in [20]
that some of the ambiguities can be reduced by a new quantization of the
Hamiltonian constraint of the full theory; nonetheless, ambiguities of the
type analysed here remain in the part of the Hamiltonian constraint
responsible for non-trivial propagation [21]. These ambiguities affect models
like [19] in which the polymerization scheme is derived from the full theory
taking expectation values with respect to symmetry-reduced coherent states.
The effect of such ambiguities where recently analysed in the context of
quantum cosmology in [17], showing that they strongly affect the physics;
here, we investigate how changes in the choice of polymer function affect
static and spherically symmetric black hole models.

To that end, we focus on a specific polymer black hole model, the one proposed
in [8] that we refer to as the BMM (Bodendorfer, Mele, Münch) model from here
on. The model is a spherically symmetric black hole, with a four-dimensional
phase space. The two configuration variables correspond to a time and an
angular component of the metric. The model has the limitations mentioned
above: It is based on a fixed foliation, and uses a specific choice of
variables to be parametrized, identified because they realise a so-called
$\bar{\mu}$-scheme on the Ashtekar-Barbero variables. We will not touch these
choices. We will instead investigate what happens if the sine functions used
in the polymerization [8] are replaced by _arbitrary_ functions. We expect
that our techniques and the implications of our results can be applied to
models of polymer quantum black holes such as [22, 23, 24, 25, 26, 27, 7, 8,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] other than BMM.

At first sight, it may seem better to just consider explicit alternatives,
e.g. changing and/or superimposing frequencies and phase shifts, in order to
mimic the use of different irreducible representations of the holonomies and
of different regularizing paths. But it turns out that many properties of the
polymer black holes are accessible without making an explicit choice for these
functions. This remarkable fact is due to the simplicity of the model, in
particular the fact that one of the two Dirac observables remains simple even
after polymerization. The second does not, and it makes some of ours formulas
implicit, without however hindering our considerations.

The two polymer functions that we keep arbitrary correspond to respectively
the time and angular component of the connection. We show first of all that
the configuration variables of [8] produce a $\bar{\mu}$-scheme for any choice
of polymer functions. Requiring the correct semiclassical limit at spatial
infinity imposes a condition on the first derivatives of both polymer
functions, but also a condition on the second derivative of the angular
polymer function. Regularity of both functions avoids singularities, and
replaces them with bounces. The number of bounces turns out to be determined
by the angular function alone; whereas the number of horizons is determined by
the time function alone. Their relative location depends on both the chosen
functions and the solution considered. We provide a general graphical analysis
to deduce these properties without committing to specific choices, but our
analysis and formulas can be of help also in studying a specific model that
one may be interested in.

Our results shows that there is a valuable richness in the class of polymer
black holes, and that considerable mathematical control can be kept also
relaxing the standard choice of sine polymer functions. We hope that some of
this control can be used to address some of the limitations and help
constructing more robust models.

We use units $G=c=1$.

## 2 From the classical black hole to a polymerized black hole

We will review the classical Schwarzschild solution, based on the Hamiltonian
formulation, in order to introduce the notations commonly used in the
literature. More precisely, after giving the general expression of the line
element compatible with the symmetry considered (i.e. spherical symmetry and
staticity) in term of the geometrodynamic variables. Then we will recall how
the problem can be reformulated in terms of Ashtekar-Barbero variables and how
this reformulation can lead to a new convenient set of variables, the $(v,P)$
variables. From there, we will quickly recall the general ideas of the BMM
polymerisation model which will be useful in the next part. We will highlight
the main result of this model like the resolution of the singularity but we
will also insist on the limitation of this model caused by the particular
choices on which it is built.

### 2.1 Minisuperspace black hole model

We consider the following spherically symmetric and static ansatz

$ds^{2}=-\bar{a}(r)dt^{2}+\frac{\bar{n}(r)}{\bar{a}(r)}dr^{2}+\bar{b}^{2}(r)d\Omega^{2}.$
(2.1)

The spatial diffeo constraint vanishes identically, and we eliminated the
shift vector via our choice for the coordinates $t,\theta$ and $\phi$. On the
other hand, we have not fixed the $r$ coordinate to be the area radius, and
there is still a non-trivial Hamiltonian constraint to be solved associated
with this reparametrization. Plugging this ansatz into the Einstein’s
equations, one finds

$\bar{n}(r)=c_{1}^{2}\dot{\bar{b}}(r)^{2},\qquad\bar{a}(r)=c_{1}^{2}\left(1-\frac{c_{2}}{b(r)}\right).$
(2.2)

The solution space is thus parametrized by two constants of integration, and
$\bar{n}(r)$ is arbitrary. However only one of the constants has a geometrical
interpretation, since $c_{1}$ can always be reabsorbed by rescaling the
coordinate $t$. The metric in this form can be recognized to be the
Schwarzschild solution, with

$M=\frac{c_{2}}{2},$ (2.3)

asymptotic unit proper time $\tau:=c_{1}t$ and area radius $b$. We can freely
choose a constant $\bar{n}(r)\equiv c_{1}^{2}$, and this will make $r$ the
area radius. In other words, one constant of integration is the black hole’s
mass, another is a rescaling of the asymptotic time, and $\bar{n}$ is the
freedom of $r$-reparametrizations.

The symmetry reduction makes the Einstein-Hilbert action, as well as the
ensuing Poisson structure, divergent on a open topology. In the following, we
will study the canonical structure of the foliation by hypersurfaces of
constant $r$. This is because $r$ is a time-like coordinate inside the
horizon, and we are ultimately interested in the dynamics near the
singularity. To regularize the action, one introduces a fiducial cell for the
coordinates $x:=(t,\theta,\phi)$ given by $C:=[0,\ell_{0}]\times S^{2}$. The
physical size $L_{0}$ of the fiducial cell is $r$-dependent. Taking for
instance spatial infinity as reference, we have

$L_{0}:=\int_{0}^{\ell_{0}}\sqrt{|g_{tt}(r=\infty)|}dt=\ell_{0}|c_{1}|$ (2.4)

on-shell. Then,

$S[C]=\frac{1}{16\pi}\int dr\int_{C}d^{3}x\sqrt{-g}R+{\rm boundary\ term}=\int
dr\,L,$ (2.5)

where

$L:=\frac{\ell_{0}\sqrt{\bar{n}}}{2}\left(\frac{\dot{\bar{a}}\dot{\bar{b}}\bar{b}}{\bar{n}}+\frac{\bar{a}\dot{\bar{b}}^{2}}{\bar{n}}+1\right).$
(2.6)

The boundary term to be added to the action is the Gibbons-Hawking-York term,
which takes care of eliminating derivatives on lapse and shift. For the
interested reader, details are reported in Appendix A. It is convenient to
reabsorb the coordinate length $\ell_{0}$ changing variables to:

$\displaystyle\sqrt{a}:=\ell_{0}\sqrt{\bar{a}},\qquad
b:=\bar{b},\qquad\sqrt{n}:=\ell_{0}\sqrt{\bar{n}},$ (2.7)

Notice that $\sqrt{a}$ is the physical length of the fiducial cell in the $t$
direction as a function of $r$. We end up with the following Lagrangian,

$L(a,b,n)=\frac{\sqrt{n}}{2}\left(\frac{\dot{a}\dot{b}b}{n}+\frac{a\dot{b}^{2}}{n}+1\right)$
(2.8)

which will give a well-defined phase space.

At this point the spherical black hole minisuperspace model is effectively
being described by a point-particle mechanical system.666And it is part of a
class of minisuperspace models described by the motion of point particles in a
given supermetric, which in this case reads
$G=\frac{2\sqrt{n}}{b}\left(\begin{array}[]{cc}{-a/b}&{1}\\\
{1}&{0}\end{array}\right).$ See e.g. [42] for a description in these terms of
generic spherical black holes and Bianchi models.

The solution space is the same as before, in particular

$a=\ell_{0}^{2}c_{1}^{2}\left(1-\frac{c_{2}}{b}\right).$ (2.9)

However, there is an important difference about the physical interpretation of
the solutions: in the point-particle description we have no longer access to
the coordinate $t$, hence it is not possible to reabsorb $c_{1}$ with a time
diffeomorphism. In other words, fiducial cells of different physical size
(2.4) correspond to physically distinct points in the solution space.

Turning $c_{1}$ from an irrelevant constant to a physical one is clearly an
artefact of the regularization, but it is also natural in the following sense.
In the full space space, the Schwarzschild solution is a one-parameter
trajectory. The smallest phase space in which this family can be embedded is
2-dimensional, hence we expect a second variable to be present and to be
conjugated to the mass, and $c_{1}$ plays this role. The role of the
regularization in defining this two-dimensional phase space can also be nicely
understood looking at the symplectic potential obtained from the on-shell
variation of (2.5), which in the area-radius gauge gives

$\theta=-\frac{1}{2}\ell_{0}\left(c_{1}{dc_{2}}+2{c_{2}}{dc_{1}}-2r{dc_{1}}\right),$
(2.10)

with symplectic structure

$\omega=\frac{\ell_{0}}{2}dc_{1}\wedge{dc_{2}}.$ (2.11)

Removing the regulator taking $\ell_{0}\to\infty$ gives a divergent symplectic
structure. One can also see that the commutation relation
$\\{c_{1},c_{2}\\}=2/\ell_{0}$ has a vanishing limit, and interpret this as a
sign that one of the coordinates loses its dynamical role, $c_{1}$ in this
case, and can be reabsorbed into the redefinition of asymptotic time .777This
is the same observation already made at the quantum level in [43].

We now review the Hamiltonian of the system associated with the foliation by
$r=$constant hypersurfaces. The signature of this foliation is not fixed,
being time-like outside the horizon, null at the horizon, and space-like
inside. No issues arise from this fact, thanks to the symmetry reduction the
dynamics is well-defined for all values of $r$. The lapse function in the
interior is

$N(r)=\frac{\sqrt{n(r)}}{\sqrt{-a(r)}}.$ (2.12)

Inspection of the Lagrangian shows that one can freely trade between $N$ and
$n$ as Lagrange multipliers. With this choice of time, the conjugate momenta
with respect to this global time parameter are

$\displaystyle p_{a}=\frac{\partial
L}{\partial\dot{a}}=\frac{b\dot{b}}{2\sqrt{n}},\qquad p_{b}=\frac{\partial
L}{\partial\dot{b}}=\frac{2a\dot{b}+\dot{a}b}{2\sqrt{n}},\qquad
p_{n}=\frac{\partial L}{\partial\dot{n}}=0.$ (2.13)

The last equation gives a primary constraint, that should be added to the
Legendre transform of the Lagrangian. Upon doing so, one finds the primary
Hamiltonian

$H(n)=\sqrt{n}\left(\frac{2p_{a}p_{b}}{b}-\frac{2ap_{a}^{2}}{b^{2}}-\frac{1}{2}\right)+\lambda
p_{n},$ (2.14)

where $\lambda$ is a Lagrange multiplier. Stabilizing the primary constraint,
i.e. imposing $\dot{p}_{n}\approx 0$, leads to the secondary constraint

$\psi:=\frac{2p_{a}p_{b}}{b}-\frac{2ap_{a}^{2}}{b^{2}}-\frac{1}{2}\approx 0.$
(2.15)

This is the Hamiltonian constraint generating $r$-diffeomorphisms. It is
automatically stable, and no further constraints arise in the analysis. We
thus obtain a 2-dimensional physical phase space, in agreement with the two-
parameter family of solutions (2.2). Moreover $\dot{n}=\lambda$, hence we can
treat $\sqrt{n}$ as a Lagrange multiplier, remove the pair $(n,p_{n})$ from
the phase space and set $\lambda=0$ without any loss of generality. Then we
have a 4d kinematical phase space with Hamiltonian constraint
$H(n)=\sqrt{n}\psi$. We will use this formulation from now on.

Consider now the following canonical transformation [8],

$\displaystyle v_{1}:=\frac{2}{3}b^{3},\qquad\
P_{1}:=\frac{p_{b}}{2b^{2}}-\frac{ap_{a}}{b^{3}}=\frac{\dot{a}}{4\sqrt{n}b},$
(2.16a) $\displaystyle v_{2}:=2ab^{2},\qquad
P_{2}:=\frac{p_{a}}{2b^{2}}=\frac{\dot{b}}{4\sqrt{n}b},$ (2.16b)
$\displaystyle\\{v_{i},P_{j}\\}=\delta_{ij}.$ (2.16c)

It is the generalization to the black hole model of a similar transformation
used in loop quantum cosmology, and it will be relevant to construct the
effective quantum model below. In terms of the new variables the Hamiltonian
constraints becomes

$H(n)=\sqrt{n}\left(12v_{1}P_{1}P_{2}+4v_{2}P_{2}^{2}-\frac{1}{2}\right),$
(2.17)

and generates the dynamical equations

$\displaystyle\dot{v}_{1}$ $\displaystyle=12\sqrt{n}v_{1}P_{2},$ (2.18a)
$\displaystyle\dot{v}_{2}$
$\displaystyle=12\sqrt{n}v_{1}P_{1}+8\sqrt{n}v_{2}P_{2},$ (2.18b)
$\displaystyle\dot{P}_{1}$ $\displaystyle=-12\sqrt{n}P_{1}P_{2},$ (2.18c)
$\displaystyle\dot{P}_{2}$ $\displaystyle=-4\sqrt{n}P_{2}^{2}.$ (2.18d)

Notice that the equations for $P_{i}$ are decoupled. It is also possible to
easily identify two constants of motion. The first can be deduced by
inspection of the Hamiltonian, and it is given by

$K_{1}=v_{1}P_{1}.$ (2.19)

The second can be found taking the ratio of (2.18c) and (2.18d) and
integrating, giving

$K_{2}=\ln{|P_{1}|}-3\ln{|P_{2}|}.$ (2.20)

In performing the integration we fix the integration constants so that the two
contributions to $K_{2}$ vanish when $P_{1}=1$ and $P_{2}=1$. The two
constants of motion turn out to be canonically conjugated functions,
$\\{K_{1},K_{2}\\}=1$, and commute with the Hamiltonian constraint by
construction. Therefore they provide two Dirac observables for the system.

It is instructive to derive the Schwarzschild solution from this formulation
of the dynamics, because it will help comparison with the polymerized model
below. To solve the equations, let us fix its gauge freedom imposing $n=n_{o}$
constant, Then, (2.18d) is solved by

$P_{2}(r)=\frac{1}{4\sqrt{n_{o}}r},$ (2.21)

up to a constant of integration that can always be absorbed by a shift of the
$r$ coordinate. This solution is valid provided $r\neq 0$. We restrict the
domain to $r>0$, hence $b>0$. Then, using (2.20) directly leads to

$P_{1}(r)=\textrm{sign}(P_{1})\frac{e^{K_{2}}}{(4\sqrt{n_{0}}r)^{3}}.$ (2.22)

From this, (2.19), and the fact that $v_{1}=\frac{2}{3}b^{3}>0$, we derive
that

$P_{1}(r)=\frac{\textrm{sign}(K_{1})e^{K_{2}}}{(4\sqrt{n_{0}}r)^{3}}\quad\textrm{and}\quad
v_{1}(r)=(4\sqrt{n_{o}})^{3}|K_{1}|e^{-K_{2}}r^{3},$ (2.23)

Finally, from the vanishing of the Hamiltonian constraint (2.17) we derive
that

$v_{2}(r)=4\sqrt{n_{o}}\left(\frac{1}{2}\sqrt{n_{o}}-\frac{3K_{1}}{r}\right)r^{2}.$
(2.24)

This general solution is of course equivalent to (2.2) found earlier. To see
this explicitly, we can reconstruct the metric. Let us do so in terms of the
unbarred quantities (2.7), in terms of which it reads

$\mathop{}\mathopen{}\mathrm{d}s^{2}=-\frac{a(r)}{\ell_{0}^{2}}\mathop{}\mathopen{}\mathrm{d}t^{2}+\frac{n(r)}{a(r)}\mathop{}\mathopen{}\mathrm{d}r^{2}+b(r)^{2}\mathop{}\mathopen{}\mathrm{d}\Omega^{2}.$
(2.25)

Inverting (2.16), we obtain

$\displaystyle a$
$\displaystyle=\frac{e^{2K_{2}/3}}{16}\left(\frac{2}{3|K_{1}|}\right)^{2/3}\left(1-\frac{24\left(\frac{3}{2}\right)^{1/3}K_{1}|K_{1}|^{1/3}e^{-K_{2}/3}}{b}\right),$
(2.26) $\displaystyle b$
$\displaystyle=4\sqrt{n_{o}}\left(\frac{3|K_{1}|}{2}\right)^{1/3}e^{-K_{2}/3}r.$
(2.27)

where

$M=2^{5/3}3^{4/3}K_{1}|K_{1}|^{1/3}e^{-K_{2}/3}.$ (2.28)

We see that the metric describes the Schwarzschild solution with mass $M$,
area radius $b$, and asymptotic time

$t_{S}:=\frac{e^{K_{2}/3}}{4\ell_{0}}\left(\frac{2}{3|K_{1}|}\right)^{1/3}t.$
(2.29)

The solution space is spanned by all values $K_{i}\in\mathbb{R}^{2}$. Each
point in this space describes a Schwarzschild black hole (with given
mass—positive or negative—asymptotic time and area radius), except the line
$K_{1}=0$ which describes a degenerate spacetime. The zero-mass limit
corresponds to the limit $K_{i}\rightarrow(0^{\pm},-\infty)$.

The last equation (2.27) shows that if one wants to fix the gauge $b=r$, then
the constant $n_{0}$ has to be chosen in a phase-space dependent manner (note
that we had already stumbled upon this fact below equation (2.3))888There is a
subtlety about $n_{0}$, not relevant for the following, but useful to make
contact with a discussion made in [8]. Under a change of fiducial cell
$\ell_{0}\mapsto\alpha\ell_{0}$, the function $n$ rescales as
$n\mapsto\alpha^{2}n$. Under a time diffeomorphism $t\mapsto t^{\prime}(t)$,
it is invariant—this follows from its definition
$\sqrt{n}:=\ell_{0}\sqrt{\bar{n}}=\int_{0}^{\ell_{0}}\sqrt{-g_{tt}g_{rr}}dt$
which makes it akin to a volume quantity. Therefore, also the constant $n_{0}$
has to be chosen in an $\ell_{0}$-dependent way in order to preserve this
properties. . For convenience, we also report the relation between the
$K_{i}$’s and the $c_{i}$’s,

$K_{1}=\frac{\ell_{0}c_{1}c_{2}}{6}\ \ \ \
K_{2}=\log\left|16\ell_{0}^{4}c_{1}^{4}c_{2}\right|=\log\left|16\ell_{0}^{4}c_{1}c_{2}\right|+3\log{|c_{1}|}.$
(2.30)

From these relations we recover the results of the previous section, in
particular the expression (2.3) for the mass, and $c_{1}$ being the only
parameter that enters the rescaling of the asymptotic time.

Let us add a few remarks useful in the following.

  * •

On the interpretation of the $P_{i}$ variables. Using the on-shell values
(2.21) and (2.23) and the relation between $b$ and $r$ (2.27), one finds [8]

$\displaystyle P_{1}(b)$
$\displaystyle=\frac{M}{b^{3}}\left(\frac{2}{3|K_{1}|}\right)^{1/3}\frac{e^{K_{2}/3}}{8},\qquad
P_{2}(b)=\frac{1}{b}\left(\frac{3|K_{1}|}{2}\right)^{1/3}e^{-K_{2}/3}.$ (2.31)

Thus $P_{1}$ is proportional to the (square root of the) Kretschmann scalar,
and $P_{2}$ to the inverse area radius of the Schwarzschild black hole.

  * •

Any phase space function that is monotonic in $r$ will be a good clock for our
Hamiltonian evolution. From the general solution, we see that all variables
except $v_{2}$ are monotonic functions of $r$. Therefore, all of them provide
good internal clocks. If we choose $P_{2}$ in particular, we can rewrite the
metric in a lapse-independent way,

$\displaystyle ds^{2}=$
$\displaystyle-\frac{1}{16}\left(\frac{2e^{K_{2}}}{3K_{1}}\right)^{2/3}\left(1-{24K_{1}P_{2}}\right)\mathop{}\mathopen{}\mathrm{d}\tau^{2}$
$\displaystyle+\left(\frac{3K_{1}\lambda_{1}}{2\lambda_{1}P_{1}}\right)^{2/3}\left(1-{24K_{1}P_{2}}\right)^{-1}\lambda_{2}{}^{2}\mathop{}\mathopen{}\mathrm{d}P_{2}{}^{2}+\left(\frac{3K_{1}\lambda_{1}}{2\lambda_{1}P_{1}}\right)^{2/3}\mathop{}\mathopen{}\mathrm{d}\Omega^{2},$
(2.32) $\displaystyle\tau:=$ $\displaystyle\frac{t}{\ell_{0}}$ (2.33)

This is the form of the metric that we will write below for the polymerized
model.

### 2.2 Ashtekar-Barbero variables

The first step towards polymerisation is a canonical transformation to
Ashtekar-Barbero variables,

$A_{a}^{i}:=\Gamma^{i}_{a}+\gamma K^{i}_{a},\qquad
E^{a}_{i}=\frac{1}{2}\epsilon_{ijk}\epsilon^{abc}e_{b}^{j}e_{c}^{k},$ (2.34)

where $\gamma$ is the Immirzi parameter. For the interior of a spherically
symmetric and static spacetime we can take them to be[27, 7, 32]999With the
renaming $b(r)\mapsto d(r)$ to avoid confusion with our $b(r)$ defined in
(2.1) and (2.7).

$\displaystyle A_{a}^{i}\tau_{i}dx^{a}$
$\displaystyle=\frac{c(r)}{\ell_{0}}\tau_{3}dt+d(r)\tau_{2}d\theta-d(r)\tau_{1}\sin\theta
d\phi+\tau_{3}\cos\theta d\phi,$ (2.35) $\displaystyle
E^{a}_{i}\tau^{i}\partial_{a}$
$\displaystyle=p_{c}(r)\tau_{3}\sin\theta\partial_{t}+\frac{p_{d}(r)}{\ell_{0}}\tau_{2}\sin\theta\partial_{\theta}-\frac{p_{d}(r)}{\ell_{0}}\tau_{1}\partial_{\phi},$
(2.36)

where $\tau_{i}=(-i/2)\sigma_{i}$ and $\sigma_{i}$ are the Pauli matrices. The
map to the previous variables is given by

$\displaystyle\textrm{sign}(p_{c})\,c$
$\displaystyle=-2\gamma\left(\frac{p_{b}}{2b}-\frac{ap_{a}}{b^{2}}\right)=-2\gamma
P_{1}\left(\frac{3v_{1}}{2}\right)^{1/3},$ (2.37a)
$\displaystyle\textrm{sign}(p_{d})\,d$
$\displaystyle=2\gamma\frac{\sqrt{-a}p_{a}}{b}=4\gamma
P_{2}\sqrt{-\frac{v_{2}}{2}},$ (2.37b)
$\displaystyle\textrm{sign}(p_{c})\,p_{c}$
$\displaystyle=b^{2}=\left(\frac{3v_{1}}{2}\right)^{2/3},$ (2.37c)
$\displaystyle\textrm{sign}(p_{d})\,p_{d}$
$\displaystyle=\sqrt{-a}b=\sqrt{-\frac{v_{2}}{2}}.$ (2.37d)

The overall sign ambiguity is the usual global parity freedom in the triad,
whereas the relative sign ambiguity is a consequence of the symmetry
reduction, and cannot be eliminated [Ashtekar2006]. This map is a canonical
transformation, with non vanishing Poisson brackets given by

$\displaystyle\\{c,p_{c}\\}=2\gamma,\qquad\\{d,p_{d}\\}$
$\displaystyle=\gamma,$ (2.38a)

and Hamiltonian constraint

$H(N)=-\frac{Nd}{2\gamma^{2}\,\textrm{sign}(p_{c})\sqrt{|p_{c}|}}\left(2cp_{c}+\left(d+\frac{\gamma^{2}}{d}\right)p_{d}\right).$
(2.39)

As before lapse is related to our choice of Lagrange multiplier $n$ by (2.12),
a redefinition which does not affect the equations of motion.101010Hamilton’s
equations are modified by the addition of a term proportional to the
constraint and thus zero on-shell.

### 2.3 BMM polymerisation

The BMM model [8] is based on the $(v_{i},P_{i})$ phase space variables we
reviewed above. There are two key observations made in [8] that motivate the
use of these variables. The first is that as we observed above, $P_{1}$ is
related to the Kretschmann scalar and $P_{2}$ to the inverse area radius.
Hence we could intuitively expect that replacing these variables by bounded
functions should remove the classical divergences of the dynamics. The second
is that polymerizing these variables with a non-dynamical regulator induces a
$\bar{\mu}$-scheme on the Ashtekar-Barbero variables, compatible with the
Hamiltonian. Hence these variables are the analogue of the $(v,b)$ variables
in loop quantum cosmology. To see this, consider the replacement

$\displaystyle P_{i}$
$\displaystyle\rightarrow\frac{\sin(\lambda_{i}P_{i})}{\lambda_{i}},\qquad
i=1,2,$ (2.40)

where $\lambda_{1}$ and $\lambda_{2}$ are a priori independent polymerisation
scales (i.e. the scale at which the modification of the dynamic will be
important). We take them to be constants, and of Planckian order. From (2.37)
we have that

$\displaystyle\lambda_{1}P_{1}=\delta_{c}c,\qquad\delta_{c}:=-\textrm{sign}(p_{c})\frac{\lambda_{1}}{2\gamma\sqrt{|p_{c}|}},$
(2.41)
$\displaystyle\lambda_{2}P_{2}=\delta_{d}d,\qquad\delta_{d}:=\textrm{sign}(p_{d})\frac{\lambda_{2}}{4\gamma|p_{d}|}.$
(2.42)

Hence the replacement (2.40) is equivalent to

$\displaystyle c\rightarrow\frac{\sin(\delta_{c}c)}{\delta_{c}},\qquad
d\rightarrow\frac{\sin(\delta_{d}d)}{\delta_{d}d},$ (2.43)

where the quantum parameters $\delta_{c}$ and $\delta_{d}$ have a phase-space
dependance given by (2.41). This phase-space dependence can be used to
introduce the area-gap of the full theory, thus defining a $\bar{\mu}$-scheme
for the Ashtekar-Barbero variables. The replacement (2.40) leads to the
modified Hamiltonian constraint

$H_{\text{eff}}(n)=\sqrt{n}\mathcal{H}_{\text{eff}}\quad,\quad\mathcal{H}_{\text{eff}}=12v_{1}\frac{\sin(\lambda_{1}P_{1})}{\lambda_{1}}\frac{\sin(\lambda_{2}P_{2})}{\lambda_{2}}+4v_{2}\frac{\sin(\lambda_{2}P_{2})^{2}}{\lambda_{2}^{2}}-\frac{1}{2}\approx
0.$ (2.44)

On the other hand, the polymerization of the Ashtekar-Barbero Hamiltonian
(2.39) under (2.43) gives

$H_{\text{eff}}(N)=-\frac{N\sin\left(\delta_{d}d\right)}{\delta_{d}2\gamma^{2}\,\textrm{sign}(p_{c})\sqrt{|p_{c}|}}\left(2\frac{\sin\left(\delta_{c}c\right)}{\delta_{c}}p_{c}+\left(\frac{\sin\left(\delta_{d}d\right)}{\delta_{d}}+\frac{\gamma^{2}\delta_{d}}{\sin\left(\delta_{d}d\right)}\right)p_{d}\right).$

Inserting (2.41), (2.37) and (2.12) in the latter recovers the former,
therefore the polymerization (2.40) is consistent with the $\bar{\mu}$-scheme
also at the dynamical level.

The dynamics generated by the polymerized Hamiltonian is expected to capture
the mean field quantum corrections [12] and it is thus referred to as
effective dynamics. This effective dynamics is obtained from (2.44) and the
original Poisson brackets (2.16c), giving

$\displaystyle\dot{v}_{1}$
$\displaystyle=12\sqrt{n}v_{1}\cos(\lambda_{1}P_{1})\frac{\sin(\lambda_{2}P_{2})}{\lambda_{2}}\
,$ (2.45a) $\displaystyle\dot{v}_{2}$
$\displaystyle=12\sqrt{n}v_{1}\frac{\sin(\lambda_{1}P_{1})}{\lambda_{1}}\cos(\lambda_{2}P_{2})+8\sqrt{n}v_{2}\frac{\sin(\lambda_{2}P_{2})}{\lambda_{2}}\cos(\lambda_{2}P_{2})\
,$ (2.45b) $\displaystyle\dot{P}_{1}$
$\displaystyle=-12\sqrt{n}\frac{\sin(\lambda_{1}P_{1})}{\lambda_{1}}\frac{\sin(\lambda_{2}P_{2})}{\lambda_{2}}\
,$ (2.45c) $\displaystyle\dot{P}_{2}$
$\displaystyle=-4\sqrt{n}\frac{\sin(\lambda_{2}P_{2})^{2}}{\lambda_{2}^{2}}\
,$ (2.45d) $\displaystyle\mathcal{H}_{\text{eff}}$
$\displaystyle=3v_{1}\frac{\sin(\lambda_{1}P_{1})}{\lambda_{1}}\frac{\sin(\lambda_{2}P_{2})}{\lambda_{2}}+v_{2}\frac{\sin(\lambda_{2}P_{2})^{2}}{\lambda_{2}^{2}}-\frac{1}{8}=0\;.$
(2.45e)

It was shown in [8] that the equations can be solved analytically, and the
following properties ensue:

  1. 1.

The singularity at $b=0$ is removed.

  2. 2.

There are two asymptotic spacetime regions, both approaching a Schwarzschild
geometry.

  3. 3.

There is a space-like surface at some $b_{min}\neq 0$ inside the horizon,
which is the a global minimum of the area radius $b$ and corresponds to a
black-to-white hole transition.

  4. 4.

There are two independent Dirac observables, which can be physically
interpreted as the masses of the separate asymptotic Schwarzschild regions,
namely of the black and of the white hole masses respectively.

  5. 5.

The horizon is slightly smaller than the classical result (for black hole
masses large w.r.t. the polymerisation scales).

  6. 6.

For black hole masses comparable or smaller of the polymerisation scales,
there are large quantum effects at the horizon scale.

  7. 7.

There is an upper bound to the curvature, as measured for instance by the
Kretschmann scalar, and this value is obtained at the transition surface and
depends on the black holes mass (as well as the white hole mass), unlike in
the prior polymerization scheme of [7] or in other models of non-singular
black hole [44]

This model has the nice feature that shows how an LQG–inspired quantization
can lead to a resolution of the singularity and its explicit replacement by a
(non-necessarily symmetric) black-hole-to-white-hole transition, a scenario
suggested in [45]. A possible shortcoming of the model is that the scale at
which quantum effects become large depends on the parameters, and a certain
restriction on the masses is necessary if one wants to confine this scale to
inside the horizon.111111This situation arises in other proposals as well
[29]. An attempt to improve this situation has been made in [29, 30]. On the
other hand, the model is based on several arbitrary choices:

  * $(i)$

A specific foliation of the spacetime has been fixed, given by the level sets
of the $r$ coordinate in Schwarzschild coordinates, and it has been shown that
different foliations can give different physical results [46].

  * $(ii)$

A specific $\bar{\mu}$-scheme, based on the polymerisation of the two $P_{i}$
variables, neither more nor less.

  * $(iii)$

The polymerisation function for both variables is a sine function, in analogy
with LQC models.

These choices are common, and indeed motivated by, the literature (see e.g.
[24, 23, 27, 7, 25]). However, more general recent proposals exist, see e.g.
[32]. Testing the robustness of the results with different choices, and
conversely identifying choices that reduce the shortcomings, is crucial to
move forward in the study of such models. In the rest of this paper we focus
on $(iii)$, and study the effect of changing the polymerisation function.

## 3 Generalised polymerisation

### 3.1 Presentation of the setup

To preserve the LQG-inspired idea of a polymer quantisation, it is sufficient
to restrict the polymerisation function to a generic function (for a general
discussion of this ambiguity see [18], for its study in quantum cosmology see
[17]). Consequently, we consider a generic replacement

$\displaystyle P_{i}$
$\displaystyle\rightarrow\frac{f_{i}(\lambda_{i}P_{i})}{\lambda_{i}},$ (3.1)

where $f_{1}$ and $f_{2}$ are real, bounded, $C^{1}$ and periodic functions of
periodicity $2\pi$, and such that $f_{i}(x)=x+o(x)$, that is

$\displaystyle f_{i}(0)=0,$ (3.2a) $\displaystyle f^{\prime}_{i}(0)=1,$ (3.2b)

with the notation
$f^{\prime}_{i}=\frac{\mathrm{d}f_{i}(\lambda_{i}P_{i})}{\mathrm{d}\left(\lambda_{i}P_{i}\right)}$.
Below in Section 5.1 we will prove that these conditions are necessary in
order to recover the Schwarzschild solution in the large distance limit, but
not sufficient: one further needs $f_{2}^{\prime\prime}(0)=0$. This additional
condition however does not affect qualitatively the short distance structure,
hence we will only introduce it when needed explicitly. Following this
replacement, the new polymerized Hamiltonian is given by

$H_{\text{eff}}=\sqrt{n}\mathcal{H}_{eff},\qquad\mathcal{H}_{\text{eff}}:=12v_{1}\frac{f_{1}(\lambda_{1}P_{1})}{\lambda_{1}}\frac{f_{2}(\lambda_{2}P_{2})}{\lambda_{2}}+4v_{2}\frac{f_{2}^{2}(\lambda_{2}P_{2})}{\lambda_{2}^{2}}-\frac{1}{2}\approx
0,$ (3.3)

In the limit $\lambda_{i}P_{i}\rightarrow 0$ we recover the classical
Hamiltonian, thanks to the required linear behaviour around the origin.

The polymerized dynamical equations are

$\displaystyle\dot{v}_{1}$
$\displaystyle=12\sqrt{n}v_{1}f_{1}^{\prime}\frac{f_{2}}{\lambda_{2}},$ (3.4a)
$\displaystyle\dot{v}_{2}$
$\displaystyle=12\sqrt{n}v_{1}\frac{f_{1}}{\lambda_{1}}f_{2}^{\prime}+8\sqrt{n}v_{2}\frac{f_{2}}{\lambda_{2}}f_{2}^{\prime},$
(3.4b) $\displaystyle\dot{P}_{1}$
$\displaystyle=-12\sqrt{n}\frac{f_{1}}{\lambda_{1}}\frac{f_{2}}{\lambda_{2}},$
(3.4c) $\displaystyle\dot{P}_{2}$
$\displaystyle=-4\sqrt{n}\frac{f_{2}^{2}}{\lambda_{2}^{2}},$ (3.4d)
$\displaystyle\mathcal{H}_{\text{eff}}$
$\displaystyle=3v_{1}\frac{f_{1}}{\lambda_{1}}\frac{f_{2}}{\lambda_{2}}+v_{2}\frac{f_{2}^{2}}{\lambda_{2}^{2}}-\frac{1}{8}\approx
0.$ (3.4e)

They generalize (2.45) in a straightforward way. Remarkably, it is possible to
solve these equations in full generality without specifying the functions
$f_{i}$, even though in a partially implicit way. The knowledge of the
implicit general solution will be sufficient to describe a large number of
features of the modified spacetime generated by $f_{1}$ and $f_{2}$. The first
step to construct this solution is to identify the Dirac observables.

### 3.2 Dirac observables

Dirac observables are gauge independent quantities, namely they Poisson-
commute with all the constraints. The only constraint of our model is
$\mathcal{H}_{\text{eff}}$ (cfr. Eq. (3.3)), which also generates the
dynamics. Dirac observables will coincide with constants of motion, thus their
identification leads immediately to the general solution of the equations of
motions. More precisely, the expressions for the Dirac observables plus the
constraints provide sufficient implicit relations to determine all phase space
variables in terms of a chosen internal clock.

As the kinematical phase space is four-dimensional and we have one first class
constraint, there are at most two independent Dirac observables. These were
referred to as $K_{i}$ in the non-polymerized case, and we keep the same
notation. They can be identified proceeding as in the non-polymerized case.
The first can be found by inspection of the Hamiltonian constraint (3.3) to be

$K_{1}:=v_{1}\frac{f_{1}}{\lambda_{1}}.$ (3.5)

Recall further, that $v_{1}$ is closely related to the area radius $b$ (cfr.
Eq. (2.16)). Therefore, this Dirac observable will be useful to analyse the
area radius as a function of $P_{1}$ and to make statements about possible
bounces and black-to-white hole transitions (see Sec. 4.2 below). For the
second Dirac observable again we divide (3.4c) by (3.4d) and integrate,
obtaining as before

$K_{2}\left(P_{1},P_{2}\right)=\lambda_{1}\int_{1}^{P_{1}}\frac{\mathrm{d}P_{1}}{f_{1}(\lambda_{1}P_{1})}-3\lambda_{2}\int_{1}^{P_{2}}\frac{\mathrm{d}P_{2}}{f_{2}(\lambda_{2}P_{2})}.$
(3.6)

This constant of motion is well-defined for any choice of polymer function.
However its explicit form can only be accessed once the functions are
specified. The choice of lower bound in the integrals is made once and for all
and it is needed to fix the freedom of constant shifts of Dirac observables.
Here we picked 1 in agreement with the choice made in (2.20).

### 3.3 General expression of the metric

To solve the dynamics in terms of the Dirac observables, we will start by
noticing that the Hamiltonian constraint (3.3) imposes that $f_{2}$ can never
vanish on solutions. Consequently, the equation (3.4d) implies that $P_{2}$ is
a strictly decreasing function of $r$. It is thus a good internal clock, and
we use it to deparametrise the equations of motion,121212Note that this is
possible at the effective level as here we have access to $P_{2}$ directly. At
the polymer quantum level $P_{2}$ is not available and this de-parametrisation
would have to be reconsidered. and rewrite the metric coefficients as
$a(P_{2})$ and $b(P_{2})$. Notice that with this choice the metric will be
independent of the lapse, see (2.32). The coordinate $r$ and $P_{2}$ are
related via (3.4d)

$dr=-\frac{\lambda_{2}^{2}}{4\sqrt{n}f_{2}^{2}}dP_{2}.$ (3.7)

Next, we want to determine the metric coefficients in (2.25) as functions of
$P_{2}$. To that end, $b$ can be expressed in terms of $P_{1}$ using the first
Dirac observable (3.5) and (2.16a), leading to

$b(P_{1})=\left(\frac{3\lambda_{1}K_{1}}{2f_{1}}\right)^{1/3}.$ (3.8)

This can be turned into a function of $P_{2}$ using the implicit relation
$P_{1}=P_{1}(\lambda_{2}P_{2},K_{2})$ determined by picking a specific value
$K_{2}$ for the second Dirac observable (3.6). To make this step clear, we
will use the following notation,

$f_{1}\big{(}\lambda_{1}P_{1}(\lambda_{2}P_{2},K_{2})\big{)}=g_{1}(\lambda_{2}P_{2},K_{2})\
.$ (3.9)

To determine $a$, we use the Hamiltonian constraint (3.3) to write

$v_{2}=\frac{\lambda_{2}^{2}}{8f_{2}^{2}}-\frac{3K_{1}\lambda_{2}}{f_{2}}.$
(3.10)

Plugging this expression in (2.16) we find

$a=\left(\frac{2f_{1}}{3\lambda_{1}K_{1}}\right)^{2/3}\left(\frac{\lambda_{2}^{2}}{16f_{2}^{2}}-\frac{3K_{1}\lambda_{2}}{2f_{2}}\right).$
(3.11)

This way all metric components are expressed in terms of the Dirac observables
$K_{i}$ and the internal clock $P_{2}$, and the metric (2.25) reads131313Note
that all factors of $n$ dropped out, which was expected as this is a pure
gauge freedom specifying the coordinate $r$, which was replaced by the
intrinsic clock $P_{2}$.

$\displaystyle ds^{2}=$
$\displaystyle-\left(\frac{2g_{1}}{3K_{1}\lambda_{1}}\right)^{2/3}\left(\frac{\lambda_{2}}{4f_{2}}\right)^{2}\left(1-\frac{24K_{1}f_{2}}{\lambda_{2}}\right)\mathop{}\mathopen{}\mathrm{d}\tau^{2}$
$\displaystyle+\left(\frac{3K_{1}\lambda_{1}}{2g_{1}}\right)^{2/3}\left(1-\frac{24K_{1}f_{2}}{\lambda_{2}}\right)^{-1}\lambda_{2}{}^{2}\mathop{}\mathopen{}\mathrm{d}P_{2}{}^{2}+\left(\frac{3K_{1}\lambda_{1}}{2g_{1}}\right)^{2/3}\mathop{}\mathopen{}\mathrm{d}\Omega^{2}\
,$ (3.12)

with $\tau={t}/{\ell_{0}}$ as before.

Note that the Dirac observable $K_{1}$ appears explicitly in this line
element, while $K_{2}$ enters only implicitly via the definition of $g_{1}$,
see Eq. (3.9). Once the polymer functions $f_{i}$ have been specified, one can
compute the integrals (3.6) and obtain $P_{1}(P_{2})$. This will make $g_{1}$
explicit and thus the metric. It is quite remarkable that one can write the
metric line element for completely arbitrary polymer functions, albeit with
the limitation of an implicit function explained above. It means that one can
deduce a great deal of the properties of the models without committing to
specific choices too soon, and this is what we investigate next. As a sanity
check, replacing the polymer functions by their arguments recovers the metric
(2.32), namely the Schwarzschild solution. The polymerized metrics on the
other hand are obviously not solutions of the Einstein’s equations, and as we
show in the next Section, describe spacetimes with a varying number of
horizons and bounces, depending on the polymer functions as well as the Dirac
observables. A feature that remains from the unpolymerized model is that
$K_{1}=0$ describes metrics everywhere degenerate. In the following Section
5.1, we will show that even if the metrics do not describe a classical black
hole, the Schwarzschild metric is recovered in the large radius limit,
provided the condition $f_{2}^{\prime\prime}(0)=0$ is satisfied.

In order to do so, it is convenient to express the metric using the area
radius as coordinate, as opposed to $P_{2}$. However, the polymerization makes
$b(P_{2})$ a non-monotonic function. This is known already for the sine
functions [8], and will become clear during the analysis of (3.4) in the next
Section. The area radius coordinate will have to be independently defined in
different branches. On each of these branches, we have

$\mathop{}\mathopen{}\mathrm{d}s^{2}=\,-\frac{1}{b^{2}}\left(\frac{\lambda_{2}}{4f_{2}}\right)^{2}\left(1-\frac{24K_{1}f_{2}}{\lambda_{2}}\right)\mathop{}\mathopen{}\mathrm{d}\tau^{2}+\frac{1}{f_{1}^{\prime
2}}\left(1-\frac{24K_{1}f_{2}}{\lambda_{2}}\right)^{-1}\mathop{}\mathopen{}\mathrm{d}b^{2}+b^{2}\mathop{}\mathopen{}\mathrm{d}\Omega^{2}$
(3.13)

In other words, (3.12) is the maximal extension, and (3.13) the expression
valid in each area radius patch.

## 4 Generic features of the dynamics

In this Section we study the geometry of the polymer spacetime (3.12). We show
that it is possible to work out several features such horizons, bounces, and
singularity resolution without specifying the polymerisation functions. We
start by analysing the equations of motion as a dynamical system, in order to
identify the asymptotic regions and relative locations of the spacetime
features.

### 4.1 Evolution and fixed points

We proceed as follows. First, we eliminate the $v_{i}$ variables using
respectively $K_{1}$ and the Hamiltonian constraint. We then focus on the
restricted space spanned by $P_{i}$, with equations of motion (3.4c) and
(3.4d). Notice that $P_{2}\neq 0$, otherwise the Hamiltonian constraint is
violated as a consequence of (3.2). We also exclude $P_{1}=0$, because again
from (3.2) it implies $K_{1}=0$ and as shown earlier these points describe
degenerate metrics.

To study the restricted space of the dynamics of the $P_{i}$’s, it is
convenient to fix the lapse to

$\sqrt{n}=\frac{\lambda_{2}}{4f_{2}},$ (4.1)

because it decouples the equations of motion. This gauge choice is always
accessible, as the Hamiltonian constraint (3.3) implies
$f_{2}(\lambda_{2}P_{2})\neq 0$ throughout the evolution. The dynamical flow
is then described by the simple vector field

$V(P_{1},P_{2})=\begin{pmatrix}{\dot{P}_{1}}\\\
{\dot{P}_{2}}\end{pmatrix}=\begin{pmatrix}-3{f_{1}}/{\lambda_{1}}\\\
-{f_{2}}/{\lambda_{2}}\end{pmatrix}.$ (4.2)

We have already argued that $f_{2}$ can never vanish on solutions, because
otherwise the Hamiltonian constraint is violated. On the other hand $f_{1}$
can vanish, but if it does, $K_{1}\equiv 0$, and the motion is then confined
on points which describe only degenerate metrics. Excluding degenerate
metrics, the polymer functions can never vanish along the solutions. This key
property implies that there are no fixed points to the dynamical system, and
that the $P_{i}$’s plane is partitioned into a check-board given by the zeros
of the $f_{i}$’s, with the dynamics being trapped inside each square. See Fig.
1, left panel, for an illustration. Notice that what matters is the zeros, not
the period, of the $f_{i}$’s. If there are zeros before the period, one
obtains an irregular check-board with some squares smaller than others, see
the right panel of the figure. Furthermore, the evolution of the $P_{i}$’s is
_monotonic_ between the extrema allowed. This allows us to get a pretty clear
qualitative picture of the dynamics.

Figure 1: (a) Qualitative picture of generic dynamics. The horizontal and
vertical straight lines are the zeros of $f_{1}$ and $f_{2}$ respectively. The
trajectories are monotonic in both variables within each square. The vertices
of the check-board are alternatively repulsive and attractive asymptotic fixed
points, like the origin and the point $A$, or unstable points like $B$. For
reference, the blue dashed line is the relation in the case of the classical
Schwarzschild solution. (b) Flow generated by the vector field $V$ with
$f_{1}(x)=f_{2}(x)=\left(\sin(x)+\sin(2x)\right)/3$. In this example, an
additional zero is present inside one period of each polymer function, hence
the check-board of confined evolution has squares smaller than the period, and
of different sizes. The (asymptotic) fixed points are marked with red dots.

We focus on the four squares of the check-board connected to the origin, since
they are the only ones that contain the classical regime $\lambda_{i}P_{i}\ll
1$. Even though the origin in itself is not an allowed configuration, it plays
the role of an asymptotic fixed point, as we now show.

Near the origin the $f_{i}$’s have a linear behaviour, hence (4.2) gives
$\mathrm{sgn}(P_{i})=\mathrm{sgn}(\dot{P}_{i})$. Not only the evolution is
monotonic, but it is also tied to the sign of $P_{i}$. Therefore if we trace
it backwards it will tend towards the origin, whichever of the 4 squares we
are on. The origin must thus be a repulsive fixed point. It is actually an
asymptotic fixed point, since the point itself is excluded from the phase
space. The monotonic evolution without fixed points then forces the system to
reach the farthest corner (across the diagonal of the square) for each square,
see again Fig. 1. These corners are thus attractive asymptotic fixed points.

The above description can be completed with an explicit perturbative expansion
around the (asymptotic) fixed points. To that end, we posit

$P_{i}(r)=\overset{\circ}{P}_{i}+\delta P_{i}(r),$ (4.3)

where $\overset{\circ}{P}_{i}$ are zeros of $f_{i}$, and we Taylor-expand the
polymer functions for $\lambda_{i}\delta P_{i}\ll 1$ around
$\overset{\circ}{P}_{i}$. At leading order, (4.2) simplifies to

$\dot{\delta}P_{1}=-3\overset{\circ}{f}{}_{1}^{\prime}\delta
P_{1},\qquad\dot{\delta}P_{2}=-\overset{\circ}{f}{}_{2}^{\prime}\delta P_{2},$
(4.4)

whose solutions are

$\delta P_{1}=\overset{\circ}{\delta
P}_{1}e^{-3\overset{\circ}{f}{}_{1}^{\prime}\,r},\qquad\delta
P_{2}=\overset{\circ}{\delta P}_{2}e^{-\overset{\circ}{f}{}_{2}^{\prime}\,r}.$
(4.5)

From conditions (3.2b) we conclude that the exponents are negative, meaning
that the classical regime is reached for $r\rightarrow\infty$. Conversely, the
farthest corner must have negative derivatives (because it is the next zero of
the function), hence it is reached for $r\rightarrow-\infty.$ The remaining
two corners of each square are unstable fixed points with one positive and one
negative derivative.

This analysis can also be used to show that the area radius is not a monotonic
function of $P_{2}$, and therefore not of $r$. In fact, we see from (3.8), and
the periodicity of $f_{1}$, that $b$ is not a monotonic function of $P_{1}$,
and the monotonicity of the $P_{i}$ as a function of $r$ implies that also
$P_{1}(P_{2})$ is monotonic. Therefore, the inversion $P_{2}(b)$ can only be
done for each of the branches, as illustrated in Fig. 2.

Figure 2: Plot of the area radius as a function of $\lambda_{2}P_{2}$.
Different colours are used to separate the branches where the function
$b(\lambda_{2}P_{2})$ is invertible.

### 4.2 Bounces and singularity resolution

The notion of bounce can be visualized if we keep in mind the parallel between
a black hole and a Kantowski-Sachs cosmological spacetime: a bounce is
associated to a local minimum of the area radius $b(r)$. We will see in this
Section that the location and number of bounces depends only on $f_{1}$. In
the next Section we will see that the location and number of Killing horizon
is determined by $f_{2}$ instead.

The evolution equation for the area radius given by (3.4a) is monotonic until
we hit an extremum of $f_{1}$, namely a point such that

$f_{1}^{\prime}=0,\qquad f_{1}^{\prime\prime}\neq 0.$ (4.6)

At this point the evolution turns around, and we have a bounce if it was
initially decreasing, and a anti-bounce or turning point if it was increasing.
Existence of at least one such point is guaranteed by the requirement of
periodicity of $f_{1}$. So the bounce is guaranteed, but we can have as many
bounces and counter-bounces (or turning points) as we want, choosing the
appropriate $f_{1}$. Furthermore, boundness of $f_{1}$ implies that that $b=0$
is only accessible for the solutions with vanishing $K_{1}$ and those are
everywhere degenerate. Therefore the bounce must occur before reaching zero
area radius. This suggests that the Schwarzschild singularity is resolved. To
look into the question of singularities more precisely, one can evaluate the
Kretschmann scalar associated with (3.12). This can be done explicitly with
the aid of an algebraic manipulator like Mathematica or Maple. The result has
the following form,

$R_{\mu\nu\rho\sigma}^{2}=\frac{F(g_{1},f_{2})}{f_{2}^{4}g_{1}^{8/3}},$ (4.7)

where $F$ is a six-order polynomial in $g_{1}$, $f_{2}$ and their first and
second derivatives. If we take polymerisation functions that are periodic and
$C^{2}$, the numerator in the previous expression is finite. The only
divergences can arise from zeros of $g_{1}$ or $f_{2}$. These do occur, but at
points that we have identified as the asymptotes for $r\rightarrow\pm\infty$.
Let us examine them separately. The first of these is the origin in $P_{i}$
space (see Fig. 1) where we know that the solution describes the Schwarzschild
metric at $r\rightarrow\infty$ with finite (indeed vanishing) Kretschmann
invariant. For the opposite corners, we don’t have a general argument for
finiteness. However these points correspond to $r\rightarrow-\infty$,
therefore in so far as this limit describes the post-bounce behaviour of
another asymptotically flat spacetime, it will be non-singular. We conclude
that the only singularities allowed by these polymer models with smooth
$f_{i}$’s are big-rip-type singularities, and require polymer functions such
that the curvature diverges at $r\mapsto-\infty$.

Of course in the case where the spacetime is also a Schwarzschild spacetime
around this fixed point, the Kretschmann scalar will be finite also near this
fixed point. This case is particularly interesting since it corresponds to a
black to white hole transition. We will see in the following how to choose the
polymerization functions $f_{1}$ and $f_{2}$ to obtain this situation. This
type of argument can in principle be extended to other curvature scalars and
it is an indication of the absence of singularities away from asymptotic
points in these models.

### 4.3 Horizons

The number of horizons depends on $f_{1}$ alone, and their location on both
$f_{i}$. From the general form (2.25) of the metric, we know that
$\partial_{t}$ is a Killing vector, therefore there is a Killing horizon
whenever $a=0$. From (2.16b) and the fact that $b\neq 0$ everywhere, this
condition is equivalent to $v_{2}=0$. Using then the Hamiltonian constraint
(3.3), this translates to

$f_{2}=\frac{\lambda_{1}\lambda_{2}}{24v_{1}f_{1}}=\frac{\lambda_{2}}{24K_{1}}.$
(4.8)

It is clear that it is possible to get any number of horizons, depending on
the choice of $f_{2}$ and the value of $K_{1}$. Fig. 3 shows an example of
polymer function admitting four horizons.

Figure 3: Sketch of an arbitrary example polymerisation function for $f_{2}$.
Only the non-shaded region between $\lambda_{2}P_{2}=0$ and
$\lambda_{2}P_{2}^{o}$ is relevant for the dynamics. The horizons correspond
to the intersection of the function $f_{2}$ and the horizontal line
$\frac{\lambda_{2}}{24K_{1}}$, explicitly depends on the chosen solution and
the value $K_{1}$.

Since $f_{2}$ is bounded, there always exist values of $K_{1}$ for which there
are no horizons. The bound in order to have at least one horizon is

$K_{1}\geq\frac{\lambda_{2}}{24\,{\rm max}f_{2}}.$ (4.9)

It follows that while the unpolymerized solution space describes always
Schwarzschild black holes, of different mass and asymptotic time, the
polymerized solution space always contains (non-singular) black holes as well
as spacetimes without horizons (or exotic quantum stars).

A non-singular black hole scenario well considered in the literature is the
black-to-white transition [47, 45, 48, 8, 27], in which two event horizons sit
on either side of a bounce. Sine polymer functions realize this scenario
already [8]. One can then ask the question if the realization of this scenario
imposes interesting restrictions on the polymer functions, and how much the
scenario can be generalized allowing arbitrary polymer functions. From the
analysis just done, we see that the black-to-white transition corresponds to
having a maximum of $f_{1}(P_{1}(P_{2}))$ between two intersections of
$f_{2}(P_{2})$ with $K_{2}$. No further requirement is needed. Therefore the
freedom/ambiguity in the polymer functions cannot significantly be constrained
by requiring that such scenario be realized. Conversely, all sorts of
modifications of the scenario are possible by playing with global features of
the regularization functions. The simplest qualitative novelty that can be
introduced changing $f_{i}$ from the sine function is an asymmetry between the
contracting phases. More in general, it is possible to combine multiple
bounces and multiple horizons.

In all cases, the relative location of bounces and horizons is not fixed, and
will depend on both the choice of polymer functions and initial conditions. If
the system is explicitly solved, the relative location can straightforwardly
be determined from the functions $P_{i}(r)$. Interestingly, it is possible to
find the relative location of bounces and horizons even without knowing the
explicit solutions $P_{i}(r)$. This follows from the fact that the
trajectories must satisfy the identity

$3\int_{\lambda_{2}P_{2}(r_{0})}^{\lambda_{2}P_{2}(r)}\frac{dx}{f_{2}(x)}=\int_{\lambda_{1}P_{1}(r_{0})}^{\lambda_{1}P_{1}(r)}\frac{dx}{f_{1}(x)},$
(4.10)

as it follows from the constancy of the Dirac observable $K_{2}$ defined in
(3.6). Given two reference values $P_{i}(r_{0})$, this equation can be used to
find the value of say $P_{2}(r)$ as a function of $P_{1}(r)$. To show how this
is done in practise, consider the example of one bounce and two horizons, with
polymer functions depicted in Fig. 4. We want to establish if the bounce
occurs in between the horizons, or outside. To bring out the dependence on the
initial conditions, we can choose a value for $r_{0}$ in the classical domain,
so that an approximate explicit solution is known, given by (2.21) and (2.23).
Equation (4.10) fixes in this way $P_{2}(r_{b})$ as a function of
$P_{1}(r_{b})$ and classical inputs and can be used to determine the position
of the bounce with respect to the horizons present in the effective geometry,
as illustrated on Fig. 5. All this depends explicitly on the choice of the
polymer functions and Dirac observables $K_{1}$ and $K_{2}$. The analysis
holds for multiple bounces and multiple horizons.

Figure 4: (a) Plot of a specific polymerisation function for $f_{1}$ with only
one local maximum between its two first zeros, i.e. only one bounce can appear
(b) Plot of a specific polymerisation function for $f_{2}$ with only one local
maximum between its two first zeros. Depending on the choice made for the
Dirac observable $K_{1}$, at most two horizons will appear.

Figure 5: Plot of the inverse of the polymerisation functions where the dashed
area correspond to the integrals in (4.10).

### 4.4 Trapped and anti-trapped regions

In a black-to-white hole transition, the bounce separates a trapped and anti-
trapped region between the two corresponding horizons. We now show that the
nature of the trapping or anti-trapping region turns out to be very simply
determined by the sign of $f_{1}^{\prime}$. More general cases, for instance
if both horizons are on the same side of the bounce, would require a
systematic analysis.

To see this, we recall that the outgoing and incoming expansions of null
geodesic congruences for the spherical metric (2.1) are given by (see e.g.
[27, 8])

$\displaystyle\theta_{+}$
$\displaystyle=\sqrt{-\frac{2}{N}}\frac{\dot{b}}{b}\text{sign}(a),$ (4.11)
$\displaystyle\theta_{-}$
$\displaystyle=-\sqrt{-\frac{2}{N}}\frac{\dot{b}}{b}.$ (4.12)

The sign of $a$ and $\dot{b}$ determine if the region is trapped, anti-trapped
or free. We can summarize the possibilities in Table 1.

| $\text{sign}(\dot{b})=-$ | $\text{sign}(\dot{b})=+$
---|---|---
$\text{sign}(a)=-$ | Anti-trapped | Trapped
$\text{sign}(a)=+$ | Free | Free
Table 1: Summary of the causal structure of a region in term of the sign of
$a$ and $b$. These results come from (4.11) and (4.12)

At this point, we now that regions outside the horizon ($a>0$) are generically
free regions. Similarly, regions inside the horizon ($a<0$) are trapped or
anti-trapped, as illustrated in Fig. 6. The sign of $a$ is determined uniquely
by the sign of $v_{2}$, see (2.16b), and the latter is positive for all
$P_{2}$ such that

$f_{2}<\frac{\lambda_{2}}{24K_{1}}.$ (4.13)

However, which of both is the case, i.e. if the interior region is trapped or
anti-trapped or if a transition happens as this depends on the position of the
bounce relative to the horizons and thus the polymer functions and the Dirac
observables $K_{1}$ and $K_{2}$. To determine if the region between the
horizon is trapped or anti-trapped, we have to look at $f_{1}$. We recall that
since
$\dot{v}_{1}=2\dot{b}b^{2}=12\sqrt{n}v_{1}f_{1}^{\prime}\frac{f_{2}}{\lambda_{2}}$,
and since $f_{2}>0$ we have
$\text{sign}(\dot{b})=\text{sign}(f_{1}^{\prime})$.

Figure 6: Plot of the polymerisation function $f_{2}$ chosen in Fig. 4. The
horizontal line marks the threshold below which the region is free, and above
which it is trapped/anti-trapped ($T/A$) depending on the sign of $\dot{b}$ as
shown in Table 1. The intersections correspond to two horizons. If $f_{1}$ is
chosen so to have a single bounce, three situations can occur. Starting from
the asymptotic region at $r\rightarrow+\infty$, namely, $\lambda_{2}P_{2}=0$,
the region between the horizon will be trapped (respectively anti-trapped) if
the bounce occurs after the horizon (respectively before the horizon). If it
occurs between the horizons, this region will be trapped and then anti-
trapped.

So in this specific case, since $f_{1}^{\prime}<0$ between the horizons (cf
Fig. 7), which means that $\dot{b}<0$. So referring to Table 1, we deduce that
the region between the horizons is anti-trapped. On the other hand, if we had
chosen $C$ such that the bounce lies within the horizons, the bounce surface
would have been a transition surface between a trapped and an anti-trapped
region. In this case, a proper black-to-white hole transition appears.

Figure 7: Plot of the polymerisation function $f_{1}$ chosen at the beginning
of this section.The property free ($F$), trapped ($T$) or anti-trapped ($A$)
of the spacetime is summarized on this plot.

## 5 Large distance behaviour

We have seen that the polymerized black holes present many non-classical
features, such as multiple horizons and bounces. However, we would still like
these solutions to reproduce the standard Schwarzschild black hole at large
distances. This was the motivation to restrict the polymer functions to
satisfy (3.2). These restriction were based on experience with loop quantum
cosmology and with the standard polymer black hole. In this section we derive
these conditions showing that they are indeed necessary in order to recover
the Schwarzschild solutions at large scales. However, it turns out that they
are not sufficient. One needs a further condition, given by
$f_{2}^{\prime\prime}=0$ which is satisfied by the sine polymerization, for
instance.

Even thought this condition was not explicitly required in the analysis of the
previous Section, a quick inspection shows that it was not used anywhere,
hence all the results there presented apply to the class of polymer functions
correctly reproducing Schwarzschild metric’s at large distances. The analysis
presented in this Section is also somewhat heavier and requires additional
notation, which is our reason to leave it for the end. It will also allows us
to identify the mass of the black hole described by the generic solution
(3.13).

An independent reason to study the large scale behaviour of the solutions
concerns the interpretation of the spacetime _after_ the bounce. If asymptotic
flatness is recovered in that limit as well, it then becomes meaningful to
talk about the mass of the white hole, and discuss the physics of the bouncing
process in terms of asymptotic charges.

Let us now come to the technical aspects. The area radius is given by the
variable $b$, hence the large scale limit is $b\rightarrow\infty$. However,
while we can freely switch between $r$ and $P_{2}$ as time variables,
$b(\lambda_{2}P_{2})$ is not a monotonic function, recall analysis of Section
3.3 and Fig. 2.141414The origin of this is that $b(r)$ is no longer monotonic.
This is why one can have $b$ inverting behaviour going through a bounce while
$r$ runs smoothly over the whole real axis. Notice also that the value
$r_{\scriptscriptstyle\rm bounce}$ can occur for either positive or negative
values, depending on the polymer function and the parameters of the solution.
Therefore, we have to restrict our analysis to each branch of invertibility.
Since we are interested in the asymptotic behaviour, we can focus on the two
branches connected to the $r\rightarrow\pm\infty$ limits which corresponds to
$b\to\pm\infty$. To study this, we will proceed with a perturbative expansion
of the polymer functions with respect to $1/b$. Since the polymer functions
depend implicitly on $b$ via the solution $P_{i}(b)$, this will be a nested
perturbative expansion that requires the use of the equations of motion.

### 5.1 Asymptotic behaviour

Let us consider the regions in $P_{i}$-space connected to the origin $P_{i}=0$
(see Fig. 1). We denote $P_{i}^{\pm}$ the two asymptotic fixed points reached
in the limit $r\to\pm\infty$ respectively (we assume that in the $r\to\infty$
limit we flow to the semiclassical regime, hence $P^{+}_{i}=0$). We introduce
the following notation,

$\displaystyle x:=\frac{1}{b}\ ,$ (5.1a) $\displaystyle
z^{\pm}(x):=\lambda_{1}P_{1}-\lambda_{1}P_{1}^{\pm},$ (5.1b) $\displaystyle
y^{\pm}(x):=\lambda_{2}P_{2}-\lambda_{2}P_{2}^{\pm},$ (5.1c)

so to have $y^{\pm}$ and $z^{\pm}$ vanish at the two fixed points of each
trajectory. We denote the fixed points $x=0^{\pm}$. To treat them at once with
a single perturbative expansion we redefine the functions
$f_{i}(\lambda_{i}P_{i})\mapsto
f_{i}(\lambda_{i}P_{i}-\lambda_{i}P_{i}^{\pm})$, such that $f_{i}(0^{\pm})=0$.
In the following, we will drop the superscript $\pm$ from $y$ and $z$ to
further lighten the notation.

In terms of these variables, the relevant components of (3.13) read

$\displaystyle g_{\tau\tau}(x)=$
$\displaystyle-x^{2}\left(\frac{\lambda_{2}^{2}}{16f_{2}\left(y(x)\right)^{2}}-\frac{3K_{1}\lambda_{2}}{2f_{2}\left(y(x)\right)}\right)\;,$
(5.2a) $\displaystyle g_{xx}(x)=$ $\displaystyle\,g_{bb}(x)/x^{4}\;.$ (5.2b)
$\displaystyle g_{bb}(x)=$
$\displaystyle-\frac{\lambda_{2}^{2}x^{2}}{16f_{1}^{\prime}(z(x))^{2}f_{2}\left(y(x)\right)^{2}g_{\tau\tau}(x)}\;,$
(5.2c)

After some computations that can be found in Appendix B, one can show that
these metric components at first order in $x$ are

$\displaystyle g_{\tau\tau}$
$\displaystyle=-\frac{\lambda_{2}^{2}}{16f_{2}^{\prime}(0^{\pm})^{2}y^{\prime}(0^{\pm})^{2}}\left(1-\frac{2y^{\prime}(0^{\pm})\left(12K_{1}f_{2}^{\prime}(0^{\pm})^{2}+\lambda_{2}f_{2}^{\prime\prime}(0^{\pm})\right)}{\lambda_{2}f_{2}^{\prime}(0^{\pm})}x\right)+O(x^{2})\
,$ (5.3a) $\displaystyle 1/g_{bb}$
$\displaystyle=f_{1}^{\prime}(0^{\pm})^{2}\left(1-\frac{24K_{1}f_{2}^{\prime}(0^{\pm})y^{\prime}(0^{\pm})}{\lambda_{2}}x\right)+O(x^{2})\,.$
(5.3b)

Since we have already fixed the area radius as coordinate, asymptotic flatness
imposes

$f_{1}^{\prime}(0^{\pm})=\pm 1.$ (5.4)

Next, we still have the freedom fo rescaling $\tau$, which we use to reabsorb
the prefactor in $g_{\tau\tau}$, so to have asymptotic flatness in the new
time variable without restrictions on the polymer functions. On the other
hand, area radius also requires that the time and radial components are the
inverse of one another. Inspection of the above equations shows that this
occurs if and only if

$\frac{y^{\prime}(0^{\pm})f^{\prime\prime}_{2}(0^{\pm})}{f^{\prime}_{2}(0^{\pm})}=0\qquad\Rightarrow\qquad
f^{\prime\prime}_{2}(0^{\pm})=0$ (5.5)

by the previous conditions. In that case, the mass of the black hole is given
by

$M=\frac{12K_{1}f_{2}^{\prime}(0^{\pm})y^{\prime}(0^{\pm})}{\lambda_{2}}\,.$
(5.6)

Notice that it depends on $K_{2}$ via $y^{\prime}(0^{\pm})$, since $K_{2}$
appears in the relation between $P_{2}$ and $b$.151515One may ask how this
mass enters the Kretschmann scalar, given schematically by (4.7). The
complexity of that expression makes it however hard to answer this question,
even for the simplest choices of polymer functions.

Consider now separately the two solutions for $r\to\pm\infty$. The
corresponding values of $M$ will generically be different, because
$f_{2}^{\prime}(0^{\pm})$ and $y^{\prime}(0^{\pm})$ can take different values.
Since $f_{2}^{\prime}(0^{\pm})$ and $y^{\prime}(0^{\pm})$ are both positive
(respectively negative) in the black hole side (respectively in the white hole
side), then the sign of the mass is only determined by the sign of $K_{1}$, as
in the unpolymerized case.

In summary, in order to obtain the Schwarzschild solution asymptotically at
the fixed point, we need to impose the conditions (B.5), (5.4) and (5.5), i.e.

$f_{1}^{\prime}(0^{\pm})=f_{2}^{\prime}(0^{\pm})\quad\text{and}\quad
f_{1}^{\prime}(0^{\pm})^{2}=1\quad\text{and}\quad
f_{2}^{\prime\prime}(0^{\pm})=0\;.$ (5.7)

This fixes of the three of the four derivatives of $f_{1}$ and $f_{2}$ up to
the second order expansion. Note that in the case of the sine polymerisation,
these conditions are automatically satisfied.

Figure 8: Sketch of the parametric solution $y(x)$. $y_{-}^{\prime}(0^{\pm})$
uniquely fixes the trajectory $y(x)$ and thus the final derivative
$y_{+}^{\prime}(0^{\pm})$.

If we impose (5.7) at both fixed points, we obtain an evolution from an
asymptotic Schwarzschild region to another asymptotic Schwarzschild region.
Let us call the masses in the two regions $M_{BH}$ and $M_{WH}$.161616These
names have absolutely no meaning in terms of a black or white hole. Both
asymptotic regions are Schwarzschild so both of these regions contain both, a
black and a white hole. However, for the initial region, the black hole lies
in the future and for the final region the white hole lies in the past.
Depending on where the bounce is located w.r.t. the horizons (see the
discussion in Sec. 4.4), an observer falling radially and freely starting in
the initial region, will experience a black hole first, then see a transition
from a trapped to an anti-trapped region, which the observer would call a
white hole region until, she finds herself in an asymptotic exterior region of
a Schwarzschild spacetime. This motivates the given notation, but there is no
deeper physical meaning. They are given by

$M_{BH}=\frac{12K_{1}f_{2}^{\prime}(\lambda_{2}P_{2}^{-})y^{\prime}(0^{\pm})}{\lambda_{2}}\quad,\quad
M_{WH}=\frac{12K_{1}f_{2}^{\prime}(\lambda_{2}P_{2}^{+})y^{\prime}(0^{\pm})}{\lambda_{2}}\,.$
(5.8)

From (5.7) we see that
$\left|f_{2}^{\prime}(\lambda_{2}P_{2}^{-})\right|=\left|f_{2}^{\prime}(\lambda_{2}P_{2}^{+})\right|=1$.
The evolution studied in Sec. 4.1 is such that no zeros of $f_{2}$ are
crossed. Therefore,
$f_{2}^{\prime}(\lambda_{2}P_{2}^{-})=-f_{2}^{\prime}(\lambda_{2}P_{2}^{+})$
and

$\frac{M_{WH}}{M_{BH}}=-\frac{y^{\prime}(0^{\pm})}{y^{\prime}(0^{\pm})}.$
(5.9)

Moreover, as it is illustrated in Figure 8, $y^{\prime}(0^{\pm})$ is positive
in the black hole side and negative in the white hole side. This implies that
the black hole mass and the white hole mass have the same sign. It is also
possible to locally171717The change of coordinates is given by (5.8) (recall
that $y^{\prime}(0^{\pm})$ depends on $K_{2}$), and this may not be globally
bijective. Consequently, we are limited to saying that this change of
coordinates exists locally. change coordinates in phase space from $K_{i}$ to
$(M_{BH},M_{WH})$.

Because the two asymptotic regions are disconnected, different asymptotic
masses does not mean a violation of energy conservation: it only means that
the time-like Killing vector cannot be normalized to one at both asymptotes.
If one were interested in solutions with matching masses, one would be looking
at a 1-dimensional subspace of the 2-dimensional phase space, determined by
(5.9) once the polymer functions are chosen.

### 5.2 Taming quantum gravity effects outside the horizon

An obvious issue with non-singular black holes is to contain the deviations
from general relativity not to spoil the macroscopic behaviour. To that end,
we look at the next order term in the expansion (B.8a), and impose bounds on
it. This requires $y^{(3)}(0^{\pm})$, which can be computed as before from the
equation of motion (B.3b) and taking the limit $x\rightarrow 0$. This leads to
an recursive relation with $y^{(3)}(0^{\pm})$ on both sides, which can be
solved to obtain

$y^{(3)}(0^{\pm})=\frac{y^{\prime}(0^{\pm})^{3}f_{2}^{(3)}(0^{\pm})}{2f_{2}^{\prime}(0^{\pm})},$
(5.10)

where we use explicitly the condition $f_{2}^{\prime\prime}(0^{\pm})=0$. We
then find

$\displaystyle
g_{\tau\tau}=-\frac{\lambda_{2}^{2}}{16f_{2}^{\prime}(0^{\pm})^{2}y^{\prime}(0^{\pm})^{2}}\left(1-2Mx-\text{sign}\left(f_{2}^{\prime}(0^{\pm})\right)\frac{y^{\prime}(0^{\pm})^{2}f_{2}^{(3)}(0^{\pm})}{2\lambda_{2}}x^{2}\right)+O\left(x^{3}\right).$
(5.11a)

The $O(x^{2})$ term of $1/g_{bb}$ vanishes, as a consequence of
$z^{\prime\prime}(0^{\pm})=0$ (cfr. (B.1)) and using the condition
$f_{2}^{\prime\prime}(0^{\pm})=0$.

To keep the quantum corrections small outside of the horizon, we need the
third term to be much smaller than the second for $b=x^{-1}\geq 2M$, namely

$\frac{y^{\prime}(0^{\pm})^{2}f_{2}^{(3)}(0^{\pm})}{8\lambda_{2}M^{2}}\ll 1.$
(5.12)

This means that for given polymer functions, the condition is always satisfied
for large enough black holes. Moreover, the expansion (5.11) allows us to
obtain a criterion on the value of the area radius for which the deviations
from the Schwarzschild metric become order one, given by

$b\leq\sqrt{\left|\frac{y^{\prime}(0)^{2}f_{2}^{(3)}(0)}{2\lambda_{2}}\right|}.$
(5.13)

A more precise characterization of the quantum corrections can be obtained
studying the violations of the energy conditions, like for example in [49].

## 6 Conclusions

We have shown that it is possible to investigate generic features of effective
models describing spherically symmetric black hole models inspired by loop
quantum gravity taking into account the large freedom in choosing the
polymerization functions. Most of the relevant global features of the
solutions, as the location of horizons and the existence of bounces, are
directly related to global features of the polymerization functions and the
values of some Dirac observables or constants of motion. More precisely, the
features of the polymerization function $f_{1}$ encode the physics of bounces,
while $f_{2}$ and its relation to certain constants of motion determines the
location of the Killing horizons. Consistency with the classical regime does
not strongly restrict the infinite dimensional ambiguity in the polymerization
procedure. However, it is worth pointing out that in addition to the usual
requirements on $f_{i}(0^{\pm})$ and $f_{i}^{\prime}(0^{\pm})$ that are
derived from the naive continuum limit, our asymptotic analysis shows that in
addition one needs to have $f_{2}^{\prime\prime}(0^{\pm})=0$. The number of
bounces and the number of horizons can be freely specified by tuning the
polymer functions and suitably choosing constants of motion. With respect to
the list of properties of the BMM model (listed in Section 2.3) we conclude
that all of them hold for an arbitrary polymerization except the shrinking of
horizon. The deformation of the horizon can be freely controlled playing with
the choice of polymer function. We notice in passing that the model also
allows for solutions with no horizons at all, where quantum effects produce
violations of the classical Einstein’s equations that can be seen as quantum
stars made of ‘matter’ of an entirely quantum nature. The existence of such a
large ambiguity in the construction of these models should not be surprising
on general grounds [17, 18]. On the upper side this freedom could prove useful
in the investigation of off-shell hypersurface deformation algebra [50, 51,
52]. Our analysis can also be applied to specific choices of non-sine polymer
functions. It would be interesting to see whether it could be extended to
different $\bar{\mu}$-schemes like the one proposed in [19], and then be able
to study analytically the behaviour of the geometry post-bounce [9].

### Acknowledgements

The work of JM was made possible through the support of the ID# 61466 grant
from the John Templeton Foundation, as part of the The Quantum Information
Structure of Spacetime (QISS) Project (qiss.fr). The opinions expressed in
this publication are those of the author(s) and do not necessarily reflect the
views of the John Templeton Foundation.

##  Appendix A Boundary term

In this Appendix we recall the derivation of the minisuperspace Lagrangian
(2.6). We consider a single boundary given by a $r$=constant hypersurfaces
outside the horizon. The total action including the Gibbons-Hawking-York
boundary term is

$S[C]=\frac{1}{16\pi}\int
dr\int_{C}d^{3}x\sqrt{-g}R-\frac{1}{8\pi}\int_{C}d^{3}x\sqrt{|h|}K.$ (A.1)

With the ansatz (2.1), we have

$\sqrt{-g}R=\sin(\theta)\left(2\sqrt{\bar{n}}-\frac{4\bar{b}\dot{\bar{a}}\dot{\bar{b}}}{\sqrt{\bar{n}}}-\frac{2\bar{a}\dot{\bar{b}}^{2}}{\sqrt{\bar{n}}}+\frac{\bar{b}^{2}\dot{\bar{a}}\dot{\bar{n}}}{2\bar{n}^{3/2}}+\frac{2\bar{a}\bar{b}\dot{\bar{b}}\dot{\bar{n}}}{\bar{n}^{3/2}}-\frac{\bar{b}^{2}\ddot{\bar{a}}}{\sqrt{\bar{n}}}-\frac{4\bar{a}\bar{b}\ddot{\bar{b}}}{\sqrt{\bar{n}}}\right),$
(A.2)

hence the Einstein-Hilbert term gives

$\frac{1}{16\pi}\int_{C}d^{3}x\sqrt{-g}R=\frac{\ell_{0}\sqrt{\bar{n}}}{2}\left(\frac{\dot{\bar{a}}\dot{\bar{b}}\bar{b}}{\bar{n}}+\frac{\bar{a}\dot{\bar{b}}^{2}}{\bar{n}}+1\right)-\ell_{0}\frac{\mathop{}\mathopen{}\mathrm{d}}{\mathop{}\mathopen{}\mathrm{d}r}\left(\frac{\dot{\bar{a}}\bar{b}^{2}}{4\sqrt{\bar{n}}}+\frac{\bar{a}\bar{b}\dot{\bar{b}}}{\sqrt{\bar{n}}}\right).$
(A.3)

Coming to the boundary term, the unit-normal to the $r$ foliation is, outside
the horizon,

$n=\sqrt{\frac{\bar{n}(r)}{\bar{a}(r)}}dr,\qquad n^{2}=1.$ (A.4)

To this we associate the induced metric and extrinsic curvature

$q_{\mu\nu}=g_{\mu\nu}-n_{a}n_{b},\qquad
K_{\mu\nu}=q_{\mu}^{\rho}\nabla_{\rho}n_{\nu}=\frac{1}{2}\pounds_{n}q_{\mu\nu}.$
(A.5)

With the ansatz (2.1), we find

$q=-\bar{a}\bar{b}^{4}\sin(\theta)^{2},\qquad
K=\frac{\dot{\bar{a}}}{2\sqrt{\bar{n}\bar{a}}}+\frac{2\sqrt{\bar{a}}\dot{\bar{b}}}{\sqrt{\bar{n}}\bar{b}}.$
(A.6)

Hence the boundary term gives

$\frac{1}{8\pi}\int
d^{3}x\sqrt{h}K=\ell_{0}\left(\frac{\dot{\bar{a}}\bar{b}^{2}}{4\sqrt{\bar{n}}}+\frac{\bar{a}\bar{b}\dot{\bar{b}}}{\sqrt{\bar{n}}}\right).$
(A.7)

Adding up, we recover (2.6).

##  Appendix B Asymptotic expansion of the metric components

In this Appendix, we provide details on the asymptotic expansion of the metric
components (5.2) leading to (5.3). First, one can note that the only functions
whose expansions are needed are $f_{2}(y(x))$ and
$f_{1}^{\prime}(z(x))=\frac{\mathop{}\mathopen{}\mathrm{d}f_{1}}{\mathop{}\mathopen{}\mathrm{d}z}(z(x))$,
while all dependencies of $f_{1}(z(x))$ have been replaced by the use of the
Dirac observable $K_{1}$ and Eq. (3.8). This also fixed the function

$z(x)=f_{1}^{-1}\left(3\lambda_{1}K_{1}x^{3}/2\right)\,.$ (B.1)

With these notations, the formal Taylor expansions are given by

$\displaystyle f_{2}(y(x))$
$\displaystyle=f_{2}^{\prime}(0^{\pm})y^{\prime}(0^{\pm})\,x+\frac{1}{2}\left(f_{2}^{\prime\prime}(0^{\pm})y^{\prime}(0^{\pm})^{2}+f_{2}^{\prime}(0^{\pm})y^{\prime\prime}(0^{\pm})\right)\,x^{2}+O\left(x^{3}\right),$
(B.2a) $\displaystyle f_{1}^{\prime}(z(x))$
$\displaystyle=f_{1}^{\prime}(0^{\pm})+f_{1}^{\prime\prime}(0^{\pm})z^{\prime}(0^{\pm})\,x+\frac{1}{2}\left(f_{1}^{(3)}(0^{\pm})z^{\prime}(0^{\pm})^{2}+f_{1}^{\prime\prime}(0^{\pm})z^{\prime\prime}(0^{\pm})\right)\,x^{2}+O\left(x^{3}\right)$
(B.2b)
$\displaystyle\stackrel{{\scriptstyle\eqref{zofx}}}{{=}}f_{1}^{\prime}(0^{\pm})+O\left(x^{3}\right),$
(B.2c)

with $f_{i}^{(n)}$ denoting the $n$-th derivative w.r.t. $y$ or $z$,
respectively, while
$y^{\prime}=\mathop{}\mathopen{}\mathrm{d}y/\mathop{}\mathopen{}\mathrm{d}x$
and respectively, $y^{\prime\prime}$, $z^{\prime}$ and $z^{\prime\prime}$. It
was assumed and will be in the following that $f_{i}^{\prime}(0^{\pm})\neq 0$,
which would also violate the classical limit. These expressions are formal
however, as the derivatives $y^{\prime}(0^{\pm})$, $y^{\prime\prime}(0^{\pm})$
are not specified. The derivatives $z^{\prime}$ and $z^{\prime\prime}$ follow
from (B.1) and we see that linear and quadratic order vanish exactly. From the
equations of motion (3.4) follow the expressions

$\displaystyle y^{\prime}(x)=\frac{f_{2}(y(x))}{xf_{1}^{\prime}(z(x))}\ ,$
(B.3a) $\displaystyle
y^{\prime\prime}(x)=\frac{f_{2}^{\prime}(y(x))f_{2}(y(x))}{x^{2}f_{1}^{\prime}(z(x))^{2}}-\frac{f_{2}(y(x))}{x^{2}f_{1}^{\prime}(z(x))}-\frac{9K_{1}\lambda_{1}f_{1}^{\prime\prime}(z(x))f_{2}(y(x))x}{2f_{1}^{\prime}(z(x))^{3}}\
,$ (B.3b)

Extracting $y^{\prime}(0^{\pm})$ requires then

$y^{\prime}(0^{\pm})=\lim_{x\rightarrow
0}y^{\prime}(x)=\frac{f_{2}^{\prime}(0^{\pm})}{f_{1}^{\prime}(0^{\pm})}y^{\prime}(0^{\pm}).$
(B.4)

This implies

$f_{1}^{\prime}(0^{\pm})=f_{2}^{\prime}(0^{\pm}),$ (B.5)

with $y^{\prime}(0^{\pm})$ unconstrained.181818Recall that the fixed points
are not part of the phase space but only reached asymptotically. This
eliminates the possibility of solving (B.4) taking $y^{\prime}(0^{\pm})\equiv
0$. Similarly, for the second derivative, we get

$\displaystyle y^{\prime\prime}(0^{\pm})=\lim_{x\rightarrow
0}y^{\prime\prime}(x)=\frac{y^{\prime}(0^{\pm})^{2}f_{2}^{\prime\prime}(0^{\pm})+f_{2}^{\prime}(0^{\pm})y^{\prime\prime}(0^{\pm})}{2f_{1}^{\prime}(0^{\pm})}\,,$
(B.6)

which can be solved to obtain

$y^{\prime\prime}(0^{\pm})=\frac{y^{\prime}(0^{\pm})^{2}f_{2}^{\prime\prime}(0^{\pm})}{f_{2}^{\prime}(0^{\pm})}\;.$
(B.7)

Here we again used the constraint
$f_{1}^{\prime}(0^{\pm})=f_{2}^{\prime}(0^{\pm})$. This makes the expansions
(B.2) explicit and allows to compute the metric functions, leading to

$\displaystyle g_{\tau\tau}$
$\displaystyle=-\frac{\lambda_{2}^{2}}{16f_{2}^{\prime}(0^{\pm})^{2}y^{\prime}(0^{\pm})^{2}}\left(1-\frac{2y^{\prime}(0^{\pm})\left(12K_{1}f_{2}^{\prime}(0^{\pm})^{2}+\lambda_{2}f_{2}^{\prime\prime}(0^{\pm})\right)}{\lambda_{2}f_{2}^{\prime}(0^{\pm})}x\right)+O(x^{2})\
,$ (B.8a) $\displaystyle 1/g_{bb}$
$\displaystyle=f_{1}^{\prime}(0^{\pm})^{2}\left(1-\frac{24K_{1}f_{2}^{\prime}(0^{\pm})y^{\prime}(0^{\pm})}{\lambda_{2}}x\right)+O(x^{2})\,.$
(B.8b)

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