# Dirac observables in the 4-dimensional phase space of Ashtekar’s variables
and spherically symmetric loop quantum black holes

Geeth Ongole a<EMAIL_ADDRESS>Hongchao Zhang b,c
<EMAIL_ADDRESS>Tao Zhub,c<EMAIL_ADDRESS>Anzhong Wanga
111Corresponding author<EMAIL_ADDRESS>Corresponding author Bin
Wangd,e<EMAIL_ADDRESS>a GCAP-CASPER, Physics Department, Baylor
University, Waco, Texas 76798-7316, USA
bInstitute for Theoretical Physics & Cosmology, Zhejiang University of
Technology, Hangzhou, 310023, China
c United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of
Technology, Hangzhou, 310023, China
d Center for Gravitation and Cosmology, College of Physical Science and
Technology, Yangzhou University, Yangzhou, 225009, China
e Shanghai Frontier Science Center for Gravitational Wave Detection, Shanghai
Jiao Tong University, Shanghai 200240, China

###### Abstract

In this paper, we study a proposal put forward recently by Bodendorfer, Mele
and Münch and García-Quismondo and Marugán, in which the two polymerization
parameters of spherically symmetric black hole spacetimes are the Dirac
observables of the four-dimensional Ashtekar’s variables. In this model, black
and white hole horizons in general exist and naturally divide the spacetime
into the external and internal regions. In the external region, the spacetime
can be made asymptotically flat by properly choosing the dependence of the two
polymerization parameters on the Ashtekar variables. Then, we find that the
asymptotical behavior of the spacetime is universal, and, to the leading
order, the curvature invariants are independent of the mass parameter $m$. For
example, the Kretschmann scalar approaches zero as $K\simeq A_{0}r^{-4}$
asymptotically, where $A_{0}$ is generally a non-zero constant and independent
of $m$, and $r$ the geometric radius of the two-spheres. In the internal
region, all the physical quantities are finite, and the Schwarzschild black
hole singularity is replaced by a transition surface whose radius is always
finite and non-zero. The quantum gravitational effects are negligible near the
black hole horizon for very massive black holes. However, the behavior of the
spacetime across the transition surface is significantly different from all
loop quantum black holes studied so far. In particular, the location of the
maximum amplitude of the curvature scalars is displaced from the transition
surface and depends on $m$, so does the maximum amplitude. In addition, the
radius of the white hole is much smaller than that of the black hole, and its
exact value sensitively depends on $m$, too.

##  I Introduction

Loop quantum gravity (LQG) has burgeoned in an effort to quantize gravity. It
is a non-perturbative and background independent approach to canonically
quantizing Einstein’s general relativity (GR) AALQGreport ; Thiemann:2007zz ;
GP11 ; MB11 ; RV15 . Loop quantum cosmology (LQC) is an application of the LQG
techniques by first performing the symmetry reduction of the homogeneous and
isotropic spacetimes at the classical level, and then quantizing it by using
the canonical Dirac quantization for systems with constraints, the so-called
minisuperspace approach AS11 . Singularities are one of the major predictions
by GR, which appear (classically) in the very early cosmological epoch and the
interior regions of black holes. Classical GR becomes invalid when such
singularities appear. One usually expects that in such high curvature regimes
quantum gravitational effects will take over and become dominant, whereby the
singularities are smoothed out and finally replaced by regions with the Planck
scale curvatures. Because of the quantum nature of geometry in LQG,
cosmological singularities can be naturally resolved in LQC models, without
any additional constraints on matter fields AS11 . Although the full theory is
still under construction, symmetry reduced models constructed from LQG have
received great attention.

Since the Schwarzschild interior is isometric to the homogeneous but
anisotropic (vacuum) Kantowski-Sachs cosmological model, techniques of LQC can
be used to study black hole(BH) singularities in the spherically symmetric
spacetimes. In the treatment of LQC, the full quantum evolution is well
approximated by quantum corrected effective equations. Similar treatment is
applied to the interior of the Schwarzschild spacetime to get the quantum
corrected Schwarzschild spacetime, which cures the black hole singularity.
Recently, such works have received lot of attention Ashtekar:2005qt ;
Modesto:2005zm ; BV07 ; GP08 ; Campiglia:2007pb ; Brannlund:2008iw ;
Modesto:2008im ; Chiou:2008nm ; Chiou:2008eg ; Perez:2012wv ; Gambini:2013exa
; GOP14 ; GP14 ; Haggard:2014rza ; Joe:2014tca ; Corichi:2015xia ;
Dadhich:2015ora ; Olmedo:2017lvt ; Cortez:2017alh ; CR17 ; AP17 ; BMM18 ;
RMD18 ; BCDHR18 ; BMM19a ; MDR19 ; Assanioussi:2019twp ; AAN20 ; Ashtekar20 ;
Zhang:2020qxw ; Gambini:2020nsf ; GSSW2020 ; Kelly:2020uwj ; Liu:2020ola ;
BMM21a ; Giesel:2021dug ; BMM21 ; SG21 ; GOP21 ; Li:2021snn ; LFSW21 ;
Giesel:2021rky ; HL22 ; ZMSZ22 ; RD22 ; Gan22 .

A particular model proposed recently is the Ashtekar-Olmedo-Singh (AOS) loop
quantum black hole (LQBH) AOS18a ; AOS18b ; AO20 , in which AOS constructed
the effective Hamiltonian that governs the dynamics of spherically symmetric
loop quantum black holes in the semi-classical limit. This effective
Hamiltonian contains two polymerization parameters
$\left(\delta_{b},\delta_{c}\right)$, characterizing the quantum gravitational
effects. In some of the previous approaches, they were simply taken as
constantsAshtekar:2005qt ; Assanioussi:2019twp ; Campiglia:2007pb ; BMM19a ,
similar to the $\mu_{0}$ scheme first introduced in LQC AS11 . However, in LQC
it was found APS06  that the $\mu_{0}$ scheme leads to large quantum geometric
effects even in regions much lower than the Planck curvatures. To remand this
problem, Ashtekar, Pawlowski and Singh (APS) APS06  proposed that the
polymerization parameter should depend on phase variables, the so-called
$\bar{\mu}$ scheme222It should be noted that in LQC, there exists only one
parameter $\bar{\mu}$ corresponding to the area operator $p$, and APS set it
to $\bar{\mu}^{2}|p|=\left(4\sqrt{3}\pi\gamma\right)\ell_{p}^{2}\equiv\Delta$
APS06 , where $\ell_{p}$ denotes the Planck scale.. It turns out that so far
this is the only scheme that leads to consistent results in LQC AS11 .

On the other hand, in the AOS model AOS18a ; AOS18b ; AO20 , instead of
treating $\left(\delta_{b},\delta_{c}\right)$ as arbitrary functions of the
phase variables, they consider them as Dirac observables, that is, they are
particular functions of the phase variables, such that along the trajectories
of the effective Hamiltonian equations they become constants. Similar
treatments have also been adopted in BV07 ; Chiou:2008nm ; Chiou:2008eg ;
Corichi:2015xia ; Olmedo:2017lvt . But the AOS approach is different as they
considered $\left(\delta_{b},\delta_{c}\right)$ as Dirac observables in the
8-dimensional extended phase space $\Gamma_{\text{ext}}$ of the variables
$\left(b,c,p_{b},p_{c};\delta_{b},\delta_{c},p_{\delta_{b}},p_{\delta_{c}}\right)$,
instead of the 4-dimensional phase space $\Gamma$ of the variables
$\left(b,c,p_{b},p_{c}\right)$. Another key feature that differentiates the
AOS approach is the imposition of the minimum area condition of LQG on the
plaquettes that tesselate the transition surface. This treatment helped
resolve the long standing problems in LQBH such as the dependence of the
system on the fiducial structure and non-negligible quantum corrections at low
curvatures, to name a few.

Despite the success of the AOS model, some questions have been raised BBCCY20
; MB20 . In particular, Bodendorfer, Mele and Münch (BMM) BMM19  argued that
the polymerization parameters can be treated canonically as Dirac observables
directly in the 4-dimensional phase space $\Gamma$, so that
$\delta_{i}=f_{i}\left(O_{i}\right),\;(i=b,c)$, where $O_{i}$’s are the two
independent Dirac observables that can be constructed in the spherically
symmetric spacetimes, and are given explicitly by Eq.(2.10) below in terms of
the four Ashtekar variables $(b,c,p_{b},p_{c})$. Then, the corresponding
dynamics of the effective Hamiltonian is different from that of AOS. More
recently, García-Quismondo and Marugán (GM) GQ21  argued that in the BMM
approach, the two polymerization parameters in general should depend on both
$O_{b}$ and $O_{c}$, that is, $\delta_{i}=f_{i}\left(O_{b},O_{c}\right)$, and
the BMM choice can be realized as a special case. GM also derived the
corresponding dynamical equations.

In this paper, we shall study the main properties of the LQBH spacetimes
resulting from the BMM/GM proposal. In particular, the paper will be organized
as follows: In Section II we will briefly review the AOS model, so readers can
clearly see the difference between the AOS and BMM/GM approaches. In Section
III, we first introduce the BMM/GM model and then restrict ourselves to the
external region of the BMM/GM LQBH spacetime. By requiring that the spacetime
in this region be asymptotically flat, we find that the parameter
$\Omega_{b}\;[\equiv\omega_{bb}+\omega_{bc}]$ must be non-negative
$\Omega_{b}\geq 0$, where $\omega_{ij}\equiv\partial f_{i}/\partial O_{j}$
[cf. Eq.(3.19)]. This excludes the BMM choice
$\delta_{i}=f_{i}\left(O_{i}\right)$ BMM19 , which is also the choice made by
AOS AOS18a ; AOS18b ; AO20 , but it must be noted that AOS did it in the
extended phase space. With this condition, we find that the asymptotical
behavior of the spacetime is universal and independent of the mass parameter
$m$ for the curvature invariants [cf. Eqs.(III.2) and (III.3)]. In particular,
the Kretschmann scalar behaves as $K\rightarrow A_{0}r^{-4}$ as
$r\rightarrow\infty$, where $A_{0}$ is a constant and independent of $m$, and
$r$ the geometric radius of the two-spheres. Similar behavior is also found in
the AOS model.

In Section IV, we analyze the properties of the BMM/GM model in the internal
region and find that all the physical quantities are finite, and the
Schwarzschild black hole singularity is replaced by a transition surface whose
radius is always finite and non-zero. However, the behavior of the spacetime
across the transition surface is significantly different from all LQBHs
studied previously. In particular, the curvature invariants, such as the
Kretschmann scalar, achieve their maxima not at the transition surface but
right after or before crossing it. Detailed investigations of the metric
components reveal that this is because of the fact that now $\delta_{i}$’s are
the Dirac observables in the 4D phase space, which considerably modify the
structure of the spacetime. Due to such modifications, the location of the
white hole horizon is also very near to the transition surface, and the ratio
of the white and black hole horizon radii is much smaller than one, and
sensitively depends on the mass parameter $m$. Finally, in Section V, we
summarize our main conclusions.

To distinguish the AOS and BMM/GM approaches, in this paper, we shall refer
them as to the extended and canonical phase space approaches, respectively.

Before proceeding further, we would also like to note that parts of the
results presented in this paper had been reported in the APS April meeting,
April 9 - 12, 2022, New York, as well as in the 23rd International Conference
on General Relativity and Gravitation (GR23), Liyang, China, July 3 - 8, 2022.

##  II Extended Phase Space Approach

The starting point of LQG is the introduction of the Ashtekar variables. In
the spherically symmetric spacetimes, they are the metric components $p_{b}$
and $p_{c}$ and their moment conjugates $b$ and $c$ with the canonical
relations

$\\{b,p_{b}\\}=G\gamma,\quad\\{c,p_{c}\\}=2G\gamma,$ (2.1)

where $\gamma$ is the Barbero-Immirzi parameter and $G$ is the Newtonian
gravitational constant.

In terms of $p_{b}$ and $p_{c}$, the four-dimensional spacetime line element
takes the form,

$ds^{2}=-N^{2}dT^{2}+\frac{p_{b}^{2}}{|p_{c}|L_{o}^{2}}dx^{2}+|p_{c}|d\Omega^{2},$
(2.2)

where $N$ is the lapse function, and $L_{o}$ is a constant, denoting the
length of the fiducial cell in the $x$-direction with
$x\in\left(0,L_{o}\right)$, and $d\Omega^{2}\equiv d\theta^{2}+\sin^{2}\theta
d\phi^{2}$ with $\theta$ and $\phi$ being the two angular coordinates defined
on the two spheres $T,x=$ Constant.

In the internal region of a classical black hole, $(N,p_{b},p_{c})$ are all
functions of $T$ only (so are $b$ and $c$), and the corresponding spacetimes
are of the Kantowski-Sachs cosmological model, which allows one to apply LQC
techniques to such homogeneous but anisotropic spacetimes. As a result, the
internal region of the Schwarzschild has been extensively studied in the
framework of LQC.

On the other hand, in the external region, the coordinates $T$ and $x$
exchange their rules, and the spacetime becomes static. However, such changes
can be also carried out by the replacement $N\rightarrow iN$ and
$p_{b}\rightarrow ip_{b}$, as shown explicitly below, while keeping the
dependence of the Ashtekar variables still only on $T$.

With the above in mind, we can see that in general the metric (2.2) has the
gauge freedom,

$T^{\prime}=T^{\prime}(T),\quad x^{\prime}=\alpha x+x_{0},$ (2.3)

in both external and internal regions, where $T^{\prime}(T)$ is an arbitrary
function of $T$ only, and $\alpha$ and $t_{0}$ are real constants. To see the
AOS approach more clearly, let us consider the AOS effective Hamiltonian
inside and outside the LQBH, separately.

###  II.1  AOS Internal Solution

With the gauge freedom of (2.3), AOS chose $T^{\prime}(T)$ so that

$N=\frac{\gamma\delta_{b}\;{\text{sgn}}\left(p_{c}\right)\sqrt{|p_{c}|}}{\sin{\left(\delta_{b}b\right)}}.$
(2.4)

Then, the effective Hamiltonian in the interior of LQBHs reads AOS18a ; AOS18b
; AO20

$\displaystyle\begin{aligned}
H_{\mathrm{eff}}=-\frac{1}{2G\gamma}\Bigg{[}&2\frac{\sin\left(\delta_{c}c\right)}{\delta_{c}}\left|p_{c}\right|\\\
&+\left(\frac{\sin\left(\delta_{b}b\right)}{\delta_{b}}+\frac{\gamma^{2}\delta_{b}}{\sin\left(\delta_{b}b\right)}\right)p_{b}\Bigg{]},\end{aligned}$
(2.5)

where $\delta_{b}$ and $\delta_{c}$ are two Dirac observables, appearing in
the polymerizations

$b\rightarrow\frac{\sin\left(\delta_{b}b\right)}{\delta_{b}},\quad
c\rightarrow\frac{\sin\left(\delta_{c}c\right)}{\delta_{b}}.$ (2.6)

That is, replacing $b$ and $c$ by Eq.(2.6) in the classical Hamiltonian

$H_{\text{cl}}=-\frac{1}{2G\gamma}\Bigg{[}2c\left|p_{c}\right|+\left(b+\frac{\gamma^{2}}{b}\right)p_{b}\Bigg{]},$
(2.7)

whereby the effective Hamiltonian (2.5) is obtained, provided that the
classical lapse function is chosen as

$N_{\text{cl}}=\frac{\gamma\;{\text{sgn}}\left(p_{c}\right)\sqrt{|p_{c}|}}{b}.$
(2.8)

To fix $\delta_{b}$ and $\delta_{c}$, AOS first noticed that the above
effective Hamiltonian can be written as

$\displaystyle H_{\mathrm{eff}}$
$\displaystyle=\frac{L_{o}}{G}\left(O_{b}-O_{c}\right),$ (2.9)

where

$\displaystyle O_{b}$ $\displaystyle\equiv$
$\displaystyle-\frac{p_{b}}{2\gamma
L_{o}}\left(\frac{\sin\left(\delta_{b}b\right)}{\delta_{b}}+\frac{\gamma^{2}\delta_{b}}{\sin\left(\delta_{b}b\right)}\right),$
(2.10) $\displaystyle O_{c}$ $\displaystyle\equiv$
$\displaystyle\frac{\left|p_{c}\right|}{\gamma
L_{o}}\frac{\sin\left(\delta_{c}c\right)}{\delta_{c}},$ (2.11)

are two Dirac observables. Then, AOS proceeded as follows:

  * •

First extend the 4-dimensional (4D) phase space $\Gamma$ spanned by
$\left(b,c;p_{b},p_{c}\right)$ to 8-dimensional (8D) phase space
$\Gamma_{\text{ext}}$ spanned by
$\left(b,c,\delta_{b},\delta_{c};p_{b},p_{c},p_{\delta_{b}},p_{\delta_{c}}\right)$.
In $\Gamma_{\text{ext}}$ the variables $\delta_{b}$ and $\delta_{c}$ are
independent, so they are in particular not functions of
$\left(b,c,p_{b},p_{c}\right)$ and instead Poisson commute with all of them.

  * •

Lift $H_{\mathrm{eff}}$ given by Eq.(2.5) to $\Gamma_{\text{ext}}$, and then
consider its Hamiltonian flow. Since $O_{b}$ and $O_{c}$ are the Dirac
observables of this flow, the following choice can be made

$\delta_{b}=\delta_{b}\left(O_{b}\right),\quad\delta_{c}=\delta_{c}\left(O_{c}\right),$
(2.12)

so that $\left(\delta_{b},\delta_{c}\right)$ are also the Dirac observables.

  * •

Introduce these dependences as two new first-class constraints

$\displaystyle\Phi_{b}\equiv O_{b}-F_{b}\left(\delta_{b}\right)\simeq 0,$
$\displaystyle\Phi_{c}\equiv O_{c}-F_{c}\left(\delta_{c}\right)\simeq 0,$
(2.13)

so that the four-dimensional reduced $\hat{\Gamma}$ corresponding to these
constraints is symplectomorphic to the original phase space $\Gamma$. Since
$O_{b}$ and $O_{c}$ are the Dirac observables, Eq.(• ‣ II.1) implies

$\delta_{b}=F_{b}^{-1}\left(O_{b}\right),\quad\delta_{c}=F_{c}^{-1}\left(O_{c}\right),$
(2.14)

are also constants on the trajectories of the effective Hamiltonian
$H_{\mathrm{eff}}$ given by Eq.(2.5).

  * •

To fix $\delta_{b}$ and $\delta_{c}$, AOS assumed that at the transition
surface, (where $T={\cal{T}}$), the physical areas of the ($x,\theta$)- and
($\theta,\phi$)-planes are respectively equal to the minimal area $\Delta$
AOS18a

$\displaystyle 2\pi\delta_{c}\delta_{b}\left|p_{b}({\cal{T}})\right|=\Delta,$
(2.15) $\displaystyle 4\pi\delta_{b}^{2}p_{c}({\cal{T}})=\Delta.$ (2.16)

With all the above, AOS found that the corresponding Hamilton equations are
given by

$\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\frac{\sin\left(\delta_{b}b\right)}{\delta_{b}}+\frac{\gamma^{2}\delta_{b}}{\sin\left(\delta_{b}b\right)}\right),$
(2.17) $\displaystyle\dot{p}_{b}$ $\displaystyle=$
$\displaystyle\frac{1}{2}p_{b}\cos\left(\delta_{b}b\right)\left(1-\frac{\gamma^{2}\delta_{b}^{2}}{\sin^{2}\left(\delta_{b}b\right)}\right),$
(2.18)

and

$\displaystyle\dot{c}$ $\displaystyle=$
$\displaystyle-2\frac{\sin\left(\delta_{c}c\right)}{\delta_{c}},$ (2.19)
$\displaystyle\dot{p}_{c}$ $\displaystyle=$
$\displaystyle{2}p_{c}\cos\left(\delta_{c}c\right).$ (2.20)

It is remarkable to note that in the above equations, no cross terms exists
between the equations for $(b,p_{b})$ and the ones for $(c,p_{c})$. As a
result, we can solve the two sets of equations independently, and the
corresponding solutions are given by AOS18a ; Gan22

$\displaystyle\cos\left(\delta_{b}b\right)$ $\displaystyle=$ $\displaystyle
b_{o}\frac{1+b_{o}\tanh\left(\frac{b_{o}T}{2}\right)}{b_{o}+\tanh\left(\frac{b_{o}T}{2}\right)},$
$\displaystyle=$ $\displaystyle
b_{o}\frac{b_{+}e^{b_{o}T}-b_{-}}{b_{+}e^{b_{o}T}+b_{-}},$ $\displaystyle
p_{b}$ $\displaystyle=$
$\displaystyle-\frac{mL_{o}}{2b_{o}^{2}}\left(b_{+}+b_{-}e^{-b_{o}T}\right){\cal{A}},$
(2.21) $\displaystyle\sin\left(\delta_{c}c\right)$ $\displaystyle=$
$\displaystyle\frac{2a_{o}e^{2T}}{a^{2}_{o}+e^{4T}},$ $\displaystyle p_{c}$
$\displaystyle=$ $\displaystyle 4m^{2}\left(a_{o}^{2}+e^{4T}\right)e^{-2T},$
(2.22)

where

$\displaystyle{\cal{A}}$ $\displaystyle\equiv$
$\displaystyle\Big{[}2\left(b_{o}^{2}+1\right)e^{b_{o}T}-b_{-}^{2}-b_{+}^{2}e^{2b_{o}T}\Big{]}^{1/2},$
$\displaystyle a_{o}$ $\displaystyle\equiv$
$\displaystyle\frac{\gamma\delta_{c}L_{o}}{8m},\quad
b_{o}\equiv\left(1+\gamma^{2}\delta_{b}^{2}\right)^{1/2},$ $\displaystyle
b_{\pm}$ $\displaystyle\equiv$ $\displaystyle b_{o}\pm 1,$ (2.23)

with

$\displaystyle\delta_{b}b\in\left(0,\pi\right),\quad\delta_{c}c\in\left(0,\pi\right),$
$\displaystyle p_{b}\leq 0,\quad p_{c}\geq 0,\quad-\infty<T<0.$ (2.24)

The parameter $m$ is an integration constant, related to the mass parameter of
the AOS solution. From the above solution, it can be shown that the two Dirac
observables on-shell are given by

$\displaystyle O_{b}=m=O_{c}.$ (2.25)

In the large mass limit, $m\gg m_{p}$, from Eqs.(2.15) and (2.16) AOS found
that

$\displaystyle\delta_{b}=\left(\frac{\sqrt{\Delta}}{\sqrt{2\pi}\gamma^{2}m}\right)^{\frac{1}{3}},\;\;\;L_{0}\delta_{c}=\frac{1}{2}\left(\frac{\gamma\Delta^{2}}{4\pi^{2}m}\right)^{\frac{1}{3}},~{}~{}~{}~{}~{}$
(2.26)

where $m_{p}$ denotes the Planck mass.

It should be noted that in AOS18a  two solutions for $c$ were given, and here
in this paper we only consider the one with “+” sign, as physically they
describe the same spacetime.

Figure 1: The Penrose diagram for the AOS LQBH. The dashed horizontal lines
$ab$ and $cd$ represent the transition surfaces (throats), and the regions
marked with B is the BH interior, and the regions marked with W is the WH
interior, but there are no spacetime singularities, so the extensions are
infinite along the vertical line in both directions. Regions marked with I,
I’, II, II’, III, and III’ are asymptotically flat regions but with a falling
rate slower than that of the Schwarzschild black hole AO20 .

From Eq.(II.1), it can be seen that the transition surface is located at
$\left.\partial p_{c}\left(T\right)/\partial T\right|_{T={\cal{T}}}=0$, which
yields

${\cal{T}}=\frac{1}{2}\ln\left(\frac{\gamma\delta_{c}L_{o}}{8m}\right)<0.$
(2.27)

There also exist two horizons, located respectively at

$T_{\text{BH}}=0,\quad
T_{\text{WH}}=-\frac{2}{b_{o}}\ln\left(\frac{b_{o}+1}{b_{o}-1}\right),$ (2.28)

at which we have ${\cal{A}}(T)=0$, where $T=T_{\text{BH}}$ is the location of
the black hole horizon, while $T=T_{\text{WH}}$ is the location of the white
hole. In the region ${\cal{T}}<T<0$, the 2-spheres are all trapped, while in
the one $T_{\text{WH}}<T<{\cal{T}}$, they are all anti-trapped. Therefore, the
region ${\cal{T}}<T<0$ behaves like the BH interior, while the one
$T_{\text{WH}}<T<{\cal{T}}$ behaves like the WH interior, denoted,
respectively, by Region B and Region W in Fig. 1. This explains the reason why
we call them the black hole and white hole regions, although the geometric
radius $\sqrt{p_{c}}$ of the two-sphere ($T,x$ = Const) is always finite and
non-zero, so spacetime singularities never appear.

Finally, we note that in this region the lapse function reads

$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{\gamma\delta_{b}\;\text{sgn}\left(p_{c}\right)\left|p_{c}\right|^{1/2}}{\sin\left(\delta_{b}b\right)}$
(2.29) $\displaystyle=$
$\displaystyle\frac{2m}{{\cal{A}}}\left(b_{+}e^{b_{o}T}+b_{-}\right)\left(a_{o}^{2}e^{-2T}+e^{2T}\right)^{1/2},~{}~{}~{}~{}~{}~{}$

where ${\cal{A}}$ is given in Eq.(II.1).

###  II.2 AOS External Solution

At the two horizons (2.28), we have ${\cal{A}}(T)=0$, and the metric becomes
singular, so extensions beyond these surfaces are needed in order to obtain a
geodesically complete spacetime. AOS showed that such extensions can be
obtained from (2.5) by the following replacements

$\displaystyle b\rightarrow ib,\quad p_{b}\rightarrow ip_{b},\quad
c\rightarrow c,\quad p_{c}\rightarrow p_{c},$ (2.30)

for which the canonical relations (2.1) now become

$\\{b,p_{b}\\}=-G\gamma,\quad\\{c,p_{c}\\}=2G\gamma,$ (2.31)

while the effective Hamiltonian in the external space of the LQBH is given by

$\displaystyle H_{\mathrm{eff}}^{\mathrm{ext}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2G\gamma}\Bigg{[}2\frac{\sin\left(\delta_{c}c\right)}{\delta_{c}}p_{c}$
(2.32)
$\displaystyle-\left(\frac{\sinh\left(\delta_{b}b\right)}{\delta_{b}}-\frac{\gamma^{2}\delta_{b}}{\sinh\left(\delta_{b}b\right)}\right)p_{b}\Bigg{]}$
$\displaystyle=$ $\displaystyle\frac{L_{o}}{G}\left(O_{b}-O_{c}\right),$

but now with

$\displaystyle O_{b}$ $\displaystyle\equiv$ $\displaystyle\frac{p_{b}}{2\gamma
L_{o}}\left(\frac{\sinh\left(\delta_{b}b\right)}{\delta_{b}}-\frac{\gamma^{2}\delta_{b}}{\sinh\left(\delta_{b}b\right)}\right),$
(2.33) $\displaystyle O_{c}$ $\displaystyle\equiv$
$\displaystyle\frac{p_{c}}{\gamma
L_{o}}\frac{\sin\left(\delta_{c}c\right)}{\delta_{c}},$ (2.34)

which can be obtained directly from Eqs.(2.10) and (2.11) with the replacement
(2.30). Then, the corresponding Hamilton equations for $(c,p_{c})$ are still
given by Eqs.(2.19) and (2.20), while the ones for $(b,p_{b})$ now are
replaced by

$\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\frac{\sinh\left(\delta_{b}b\right)}{\delta_{b}}-\frac{\gamma^{2}\delta_{b}}{\sinh\left(\delta_{b}b\right)}\right),$
(2.35) $\displaystyle\dot{p}_{b}$ $\displaystyle=$
$\displaystyle\frac{1}{2}p_{b}\cosh\left(\delta_{b}b\right)\left(1+\frac{\gamma^{2}\delta_{b}^{2}}{\sinh^{2}\left(\delta_{b}b\right)}\right).$
(2.36)

Then, the corresponding solutions of the Hamilton equations are given by

$\displaystyle\cosh\left(\delta_{b}b\right)$ $\displaystyle=$ $\displaystyle
b_{o}\frac{1+b_{o}\tanh\left(\frac{b_{o}T}{2}\right)}{b_{o}+\tanh\left(\frac{b_{o}T}{2}\right)},$
$\displaystyle p_{b}$ $\displaystyle=$ $\displaystyle-2m\gamma
L_{o}\delta_{b}\frac{\sinh\left(\delta_{b}b\right)}{\gamma^{2}\delta_{b}^{2}-\sinh^{2}\left(\delta_{b}b\right)}$
(2.37) $\displaystyle=$
$\displaystyle-\frac{mL_{o}}{2b_{o}^{2}}\left(b_{+}+b_{-}e^{-b_{o}T}\right){\cal{A}},$
$\displaystyle\sin\left(\delta_{c}c\right)$ $\displaystyle=$
$\displaystyle\frac{2a_{o}e^{2T}}{a^{2}_{o}+e^{4T}},$ $\displaystyle p_{c}$
$\displaystyle=$ $\displaystyle 4m^{2}\left(e^{2T}+a_{o}^{2}e^{-2T}\right),$
(2.38)

but now with

${\cal{A}}\equiv\Big{[}b_{-}^{2}+b_{+}^{2}e^{2b_{o}T}-2\left(b_{o}^{2}+1\right)e^{b_{o}T}\Big{]}^{1/2},$
(2.39)

which can be obtained from Eq.(II.1) by the replacement ${\cal{A}}\rightarrow
i{\cal{A}}$ (or ${\cal{A}}^{2}\rightarrow-{\cal{A}}^{2}$), so that
$g_{xx}\rightarrow-g_{xx}$, and the coordinate $x$ now becomes timelike in the
external region ($T>0$) of the black hole horizon, located at $T=0$. It can be
shown that for the above solution, we have $O_{b}=m=O_{c}$, which shows
clearly that $O_{b}$ and $O_{c}$ defined by Eqs.(2.33) and (2.34) are two
Dirac observables.

We also note that the replacement of Eq.(2.30) leads to

$\displaystyle N^{2}$ $\displaystyle=$
$\displaystyle-\frac{\gamma^{2}\delta_{b}^{2}\left|p_{c}\right|}{\sinh^{2}\left(\delta_{b}b\right)}$
(2.40) $\displaystyle=$
$\displaystyle-\frac{4m^{2}}{{\cal{A}}^{2}}\left(b_{+}e^{b_{o}T}+b_{-}\right)^{2}\left(a_{o}^{2}e^{-2T}+e^{2T}\right),~{}~{}~{}~{}~{}$

so that, in terms of $N$, $p_{b}$ and $p_{c}$, the metric now takes the form
AOS18a

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-N^{2}dT^{2}-\frac{p_{b}^{2}}{|p_{c}|L_{o}^{2}}dx^{2}+|p_{c}|d\Omega^{2}$
(2.41) $\displaystyle=$
$\displaystyle-\frac{p_{b}^{2}}{|p_{c}|L_{o}^{2}}dx^{2}+\frac{\gamma^{2}\delta_{b}^{2}\left|p_{c}\right|}{\sinh^{2}\left(\delta_{b}b\right)}dT^{2}+|p_{c}|d\Omega^{2},~{}~{}~{}~{}~{}$

which shows clearly that now $T$ is spacelike, while $x$ becomes timelike, so
the spacetime outside of the LQBH is static.

In addition, AOS showed that the two metrics (2.2) and (2.41) are analytically
connected to each other across the two horizons, and as a result, the
extensions are unique. The global structure of the spacetime is given by the
Penrose diagram of Fig. 1, from which we can see that the extensions along the
vertical direction are infinite, quite similar to the charged spherically
symmetric Reissner-Nordström solutions HE73 , but without spacetime
singularities, as now the geometric radius $\sqrt{p_{c}}$ never becomes zero.

Before proceeding to the next section, we also note that technically the AOS
extended space approach can be realized directly by taking $\delta_{b}$ and
$\delta_{c}$ to be constants in the phase space of $(b,c,p_{b},p_{c})$, and
then impose the conditions (2.15) and (2.16), as by definition constants over
the whole phase space are also Dirac observables.

##  III Canonical Phase Space Approach

Instead of extending the 4D physical phase space to 8D phase space, and then
considering $\delta_{b}$ and $\delta_{c}$ as the Dirac observables of the
extended phase space, Bodendorfer, Mele, and Münch (BMM) pointed out BMM19
that they can be considered directly as the Dirac observables in the 4D
physical phase space of $(b,c,p_{b},p_{c})$, as those given by Eq.(2.12).
Lately, García-Quismondo and Marugán argued GQ21  that $\delta_{b}$ and
$\delta_{c}$ should in general depend on both of the two Dirac observables
$O_{b}$ and $O_{c}$,

$\delta_{i}=f_{i}\left(O_{b},O_{c}\right),\;(i=b,c),$ (3.1)

while Eq.(2.12) only represents a particular choice of the general case.
Eq.(3.1) shows clearly that now $\delta_{b}$ and $\delta_{c}$ all depend on
the four variables, $(b,c,p_{b},p_{c})$, through Eqs.(2.10) and (2.11) [or
Eqs.(2.33) and (2.34) when outside of the LQBH]. Then, the corresponding
Hamilton equations are given by GQ21

$\displaystyle\partial_{T}i$ $\displaystyle=$ $\displaystyle
C_{ij}\left[s_{i}\frac{L_{o}}{G}\left\\{i,p_{i}\right\\}\frac{\partial
O_{i}}{\partial p_{i}}\right],$ (3.2) $\displaystyle\partial_{T}p_{i}$
$\displaystyle=$ $\displaystyle
C_{ij}\left[-s_{i}\frac{L_{o}}{G}\left\\{i,p_{i}\right\\}\frac{\partial
O_{i}}{\partial i}\right],$ (3.3)

where $i,j=b,c$, $i\neq j$, $s_{b}=1$, $s_{c}=-1$, and

$\displaystyle C_{ij}$ $\displaystyle\equiv$
$\displaystyle\frac{1-\Delta_{jj}-\Delta_{ji}}{\left(1-\Delta_{ii}\right)\left(1-\Delta_{jj}\right)-\Delta_{ij}\Delta_{ji}},$
(3.4) $\displaystyle\Delta_{ij}$ $\displaystyle\equiv$
$\displaystyle\frac{\partial O_{i}}{\partial\delta_{i}}\frac{\partial
f_{i}}{\partial O_{j}}.$ (3.5)

It is interesting to note that, introducing two new variables,
$t_{i},(i=b,c)$, via the relations

$dt_{i}\equiv C_{ij}dT,\;(i\not=j),$ (3.6)

Eqs.(3.2) and (3.3) take the forms,

$\displaystyle\partial_{t_{i}}i$ $\displaystyle=$ $\displaystyle
s_{i}\frac{L_{o}}{G}\left\\{i,p_{i}\right\\}\frac{\partial O_{i}}{\partial
p_{i}},$ (3.7) $\displaystyle\partial_{t_{i}}p_{i}$ $\displaystyle=$
$\displaystyle-s_{i}\frac{L_{o}}{G}\left\\{i,p_{i}\right\\}\frac{\partial
O_{i}}{\partial i},$ (3.8)

which will lead to the same Hamilton equations as those given by AOS, if we
replace $T$ by $t_{b}$ in the equations for $b$ and $p_{b}$, and $T$ by
$t_{c}$ in the equations for $c$ and $p_{c}$, as first noted in GQ21 . This
observation will significantly simplify our following discussions.

To proceed further, in the rest of this section, let us consider the above
equations only in the external region, while the ones in the internal region
will be considered in the next section.

###  III.1 Dynamics of the external LQBH Spacetimes

In the external region, the Hamilton equations take the form

$\displaystyle\frac{db}{dt_{b}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\frac{\sinh\left(\delta_{b}b\right)}{\delta_{b}}-\frac{\gamma^{2}\delta_{b}}{\sinh\left(\delta_{b}b\right)}\right),$
(3.9) $\displaystyle\frac{dp_{b}}{dt_{b}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}p_{b}\cosh\left(\delta_{b}b\right)\left(1+\frac{\gamma^{2}\delta_{b}^{2}}{\sinh^{2}\left(\delta_{b}b\right)}\right),$
(3.10)

for $(b,p_{b})$, and

$\displaystyle\frac{dc}{dt_{c}}$ $\displaystyle=$
$\displaystyle-2\frac{\sin\left(\delta_{c}c\right)}{\delta_{c}},$ (3.11)
$\displaystyle\frac{dp_{c}}{dt_{c}}$ $\displaystyle=$
$\displaystyle{2}p_{c}\cos\left(\delta_{c}c\right),$ (3.12)

for $(c,p_{c})$. Then, the corresponding solutions for $b$ and $p_{b}$ will be
given by Eqs.(II.2) and (2.39) by simply replacing $T$ by $t_{b}$, that is,

$\displaystyle\cosh\left(\delta_{b}b\right)=b_{o}\frac{1+b_{o}\tanh\left(\frac{b_{o}t_{b}}{2}\right)}{b_{o}+\tanh\left(\frac{b_{o}t_{b}}{2}\right)},$
$\displaystyle
p_{b}=-\frac{mL_{o}}{2b_{o}^{2}}\left(b_{+}+b_{-}e^{-b_{o}t_{b}}\right){\cal{A}},$
$\displaystyle{\cal{A}}\equiv\Big{[}b_{-}^{2}+b_{+}^{2}e^{2b_{o}t_{b}}-2\left(b_{o}^{2}+1\right)e^{b_{o}t_{b}}\Big{]}^{1/2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(3.13)

while the solutions for $c$ and $p_{c}$ will be given by Eqs.(II.2) with the
replacement $T$ by $t_{c}$, i.e.

$\displaystyle\sin\left(\delta_{c}c\right)=\frac{2a_{o}e^{2t_{c}}}{a^{2}_{o}+e^{4t_{c}}},$
$\displaystyle p_{c}=4m^{2}\left(e^{2t_{c}}+a_{o}^{2}e^{-2t_{c}}\right).$
(3.14)

The relation between $t_{b}$ and $t_{c}$ is given by Eq.(3.6), from which we
find that

$dt_{c}=\frac{C_{cb}}{C_{bc}}dt_{b}.$ (3.15)

To study the above relation, let us first note that Eqs.(2.33) and (2.34) lead
to

$\displaystyle\frac{\partial O_{b}}{\partial\delta_{b}}$ $\displaystyle=$
$\displaystyle\frac{p_{b}}{2\gamma
L_{o}\delta_{b}^{2}}\left(1+\frac{\gamma^{2}\delta_{b}^{2}}{\sinh^{2}\left(\delta_{b}b\right)}\right)\big{[}\left(\delta_{b}b\right)\cosh\left(\delta_{b}b\right)-\sinh\left(\delta_{b}b\right)\big{]},$
$\displaystyle\frac{\partial O_{c}}{\partial\delta_{c}}$ $\displaystyle=$
$\displaystyle\frac{p_{c}}{\gamma
L_{o}\delta_{c}^{2}}\big{[}\left(\delta_{c}c\right)\cos\left(\delta_{c}c\right)-\sin\left(\delta_{c}c\right)\big{]}.$
(3.16)

Then, we find that

$\displaystyle C_{bc}$ $\displaystyle=$
$\displaystyle\frac{1}{\cal{D}}\left(1-\Omega_{c}\frac{\partial
O_{c}}{\partial\delta_{c}}\right)=\frac{1}{\cal{D}}\left\\{1-\frac{\Omega_{c}p_{c}}{\gamma
L_{o}\delta_{c}^{2}}\Big{[}\left(\delta_{c}c\right)\cos\left(\delta_{c}c\right)-\sin\left(\delta_{c}c\right)\Big{]}\right\\},$
$\displaystyle C_{cb}$ $\displaystyle=$
$\displaystyle\frac{1}{\cal{D}}\left(1-\Omega_{b}\frac{\partial
O_{b}}{\partial\delta_{b}}\right)=\frac{1}{\cal{D}}\Bigg{\\{}1-\frac{\Omega_{b}p_{b}}{2\gamma
L_{0}\delta_{b}^{2}}\left(1+\frac{\gamma^{2}\delta_{b}^{2}}{\sinh^{2}\left(\delta_{b}b\right)}\right)\Big{[}\left(\delta_{b}b\right)\cosh\left(\delta_{b}b\right)-\sinh\left(\delta_{b}b\right)\Big{]}\Bigg{\\}},$
(3.17)

where

$\displaystyle{\cal{D}}$ $\displaystyle\equiv$ $\displaystyle
1-\omega_{cc}\frac{\partial
O_{c}}{\partial\delta_{c}}-\omega_{bb}\frac{\partial
O_{b}}{\partial\delta_{b}}+\left(\omega_{bb}\omega_{cc}-\omega_{bc}\omega_{cb}\right)\frac{\partial
O_{b}}{\partial\delta_{b}}\frac{\partial O_{c}}{\partial\delta_{c}}$ (3.18)
$\displaystyle=$ $\displaystyle 1-\frac{\omega_{cc}p_{c}}{\gamma
L_{o}\delta_{c}^{2}}\big{[}\left(\delta_{c}c\right)\cos\left(\delta_{c}c\right)-\sin\left(\delta_{c}c\right)\big{]}-\frac{\omega_{bb}p_{b}}{2\gamma
L_{0}\delta_{b}^{2}}\left(1+\frac{\gamma^{2}\delta_{b}^{2}}{\sinh^{2}\left(\delta_{b}b\right)}\right)\left[\left(\delta_{b}b\right)\cosh\left(\delta_{b}b\right)-\sinh\left(\delta_{b}b\right)\right]$
$\displaystyle~{}+\frac{\omega_{bb}\omega_{cc}-\omega_{bc}\omega_{cb}}{2\gamma^{2}L_{o}^{2}\delta_{b}^{2}\delta_{c}^{2}}p_{b}p_{c}\left(1+\frac{\gamma^{2}\delta_{b}^{2}}{\sinh^{2}\left(\delta_{b}b\right)}\right)\big{[}\left(\delta_{c}c\right)\cos\left(\delta_{c}c\right)-\sin\left(\delta_{c}c\right)\big{]}\left[\left(\delta_{b}b\right)\cosh\left(\delta_{b}b\right)-\sinh\left(\delta_{b}b\right)\right],~{}~{}~{}$
$\displaystyle\omega_{ij}$ $\displaystyle\equiv$ $\displaystyle\frac{\partial
f_{i}}{\partial
O_{j}},\quad\Omega_{c}\equiv\omega_{cc}+\omega_{cb},\quad\Omega_{b}\equiv\omega_{bb}+\omega_{bc}.$
(3.19)

It should be noted that the numerator of $C_{bc}$ is a function of $t_{c}$ and
the one of $C_{cb}$ is a function of $t_{b}$, where $t_{b}$ and $t_{c}$ are
related one to the other through Eq.(3.15). In particular, for $t_{b},t_{c}\gg
1$, from Eq.(3.15) we find

$\displaystyle\begin{aligned} t_{c}-2\beta
m^{2}a_{o}^{3}+{\cal{O}}\left(e^{-4t_{c}}\right)=\left(1+\alpha_{2}\right)t_{b}+\frac{\alpha_{1}}{b_{o}}e^{b_{o}t_{b}}+\frac{\alpha_{3}}{2b_{o}}+{\cal{O}}\left(e^{-b_{o}t_{b}}\right),\;\left(t_{b},t_{c}\gg
1\right),\end{aligned}$ (3.20)

where

$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\frac{m(b_{o}+1)^{2}}{2\gamma
b_{o}^{2}\delta_{b}^{2}}\left(b_{o}\cosh^{-1}{b_{o}}-\gamma\delta_{b}\right)\Omega_{b},$
$\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle-\frac{m\gamma^{2}\delta_{b}}{\gamma^{2}\delta_{b}^{2}+1}\Omega_{b},$
(3.21)

$\beta$ and $\alpha_{3}$ are other constants, and their explicit expressions
will not be given here, as they will not affect our following discussions.

It is interesting to note that for the BMM choice, $f_{i}=f_{i}(O_{i})$ [cf.
Eq.(• ‣ II.1)], and $\delta_{i}$ given by Eq.(2.26) together with the fact
that on-shell we have $O_{b}=m=O_{c}$, we find that

$\displaystyle\omega_{bb}^{\text{BMM}}=-\frac{\delta_{b}}{3m},\quad\omega_{cc}^{\text{BMM}}=-\frac{\delta_{c}}{3m},\quad\omega_{bc}^{\text{BMM}}=\omega_{cb}^{\text{BMM}}=0,$
$\displaystyle\Omega^{\text{BMM}}_{b}=-\frac{\delta_{b}}{3m},\quad\Omega^{\text{BMM}}_{c}=-\frac{\delta_{c}}{3m}.$
(3.22)

To study the external spacetimes further, in the following let us consider the
two cases, $\alpha_{1}=0$ and $\alpha_{1}\not=0$, separately.

###  III.2 External Spacetimes with $\alpha_{1}\not=0$

If $\alpha_{1}\not=0$, from Eq.(3.20) we find that

$\displaystyle t_{c}\approx\frac{\alpha_{1}}{b_{o}}e^{b_{o}t_{b}}.$ (3.23)

Then, from Eq.(3.6) we find that $dT=dt_{b}/C_{bc}$, and in terms of $t_{b}$
the metric (2.41) becomes

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{{p}_{b}^{2}}{\left|{p}_{c}\right|L_{o}^{2}}dx^{2}+\frac{\gamma^{2}\left|{p}_{c}\right|\delta_{{b}}^{2}}{\sinh^{2}\left(\delta_{{b}}{b}\right)C_{bc}^{2}}dt_{b}^{2}+\left|{p}_{c}\right|\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (3.24)

where $C_{bc}$ is given by Eq.(III.1), and

$\displaystyle g_{xx}$ $\displaystyle\equiv$
$\displaystyle\frac{{p}_{b}^{2}}{\left|{p}_{c}\right|L_{o}^{2}}\simeq\left(c_{1}e^{2b_{o}t_{b}}+c_{2}e^{b_{o}t_{b}}+c_{3}+\cdots\right)\exp\left(-\frac{2\alpha_{1}}{b_{o}}e^{b_{o}t_{b}}\right),$
$\displaystyle g_{t_{b}t_{b}}$ $\displaystyle\equiv$
$\displaystyle\frac{\gamma^{2}\left|{p}_{c}\right|\delta_{{b}}^{2}}{\sinh^{2}\left(\delta_{{b}}{b}\right)C_{bc}^{2}}\simeq\left(d_{1}e^{2b_{o}t_{b}}+d_{2}e^{b_{o}t_{b}}+d_{3}+\cdots\right)\exp\left(\frac{2\alpha_{1}}{b_{o}}e^{b_{o}t_{b}}\right),$
$\displaystyle g_{\theta\theta}$ $\displaystyle\equiv$
$\displaystyle\left|{p}_{c}\right|\simeq
4m^{2}\exp\left(\frac{2\alpha_{1}}{b_{o}}e^{b_{o}t_{b}}\right),$ (3.25)

where ($c_{i}$, $d_{i}$) are constants defined as

$\displaystyle c_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{\left(b_{o}+1\right)^{4}}{16b_{o}^{4}},\quad
c_{2}\equiv-\frac{\left(b_{o}+1\right)^{2}}{4b_{o}^{4}},\quad
c_{3}\equiv-\frac{\gamma^{2}\delta_{b}^{2}\left(\gamma^{2}\delta_{b}^{2}+4\right)}{8b_{o}^{4}},$
$\displaystyle d_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{\omega_{bb}^{2}f^{2}m^{4}}{\gamma^{2}\delta_{b}^{4}b_{o}^{4}}\left(b_{o}+1\right)^{4},\quad
d_{2}\equiv\frac{4\omega_{bb}fm^{3}}{\gamma^{2}b_{o}^{4}\delta_{b}^{4}}\left(b_{o}+1\right)^{2}\left\\{\gamma
b_{o}^{2}\delta_{b}^{2}-m\omega_{bb}\left(\gamma^{3}\delta_{b}^{3}-b_{o}^{2}f\right)\right\\},$
$\displaystyle d_{3}$ $\displaystyle\equiv$ $\displaystyle
2m^{2}\left(\frac{m\omega_{bb}}{\gamma^{2}\delta_{b}^{4}b_{o}^{4}}\left(m\omega_{bb}\left(2\gamma^{6}\delta_{b}^{6}+f^{2}\left(\gamma^{4}\delta_{b}^{4}+4\gamma^{2}\delta_{b}^{2}\left(b_{o}^{2}+1\right)+8b_{o}^{2}+2\right)+2\gamma^{3}f\delta_{b}^{3}\left(1-4b_{o}^{2}\right)\right)\right.\right.$
(3.26) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\left.\left.+8\gamma
f\delta_{b}^{2}b_{o}^{4}-4\gamma^{4}\delta_{b}^{5}b_{o}^{2}\right)+2\right),$

with

$\displaystyle f(\gamma\delta_{b})$ $\displaystyle\equiv$ $\displaystyle
b_{o}\cosh^{-1}b_{o}-\gamma\delta_{b}=b_{o}\ln\left(b_{o}+\gamma\delta_{b}\right)-\gamma\delta_{b}$
(3.27) $\displaystyle=$
$\displaystyle\frac{1}{3}\gamma^{3}\delta_{b}^{3}+{\cal{O}}\left(\gamma^{5}\delta_{b}^{5}\right),$

which is always non-zero for $\gamma\delta_{b}>0$, as shown in Fig. 2. The
function $f(\gamma\delta_{b})$ defined above must not be confused with the
Dirac observables $f_{i}(O_{b},O_{c})\;(i=b,c)$ introduced in Eq.(3.1).

Figure 2: The function $f(x)$ defined by Eq.(3.27) vs
$x\equiv\gamma\delta_{b}$.

From the above expressions, it is clear that $\alpha_{1}$ must be positive, in
order to have the spacetime asymptotically flat as $t_{b}\gg 1$. This is also
consistent with Eq.(3.23), as we assumed that $t_{b},\;t_{c}\gg 1$
asymptotically. Therefore, in the rest of this subsection we assume
$\alpha_{1}>0$, which requires

$\Omega_{b}>0.$ (3.28)

It is interesting to note that, corresponding to the BMM choices of
$f_{i}=f_{i}(O_{i})$ and $\delta_{i}$ given by Eq.(2.26), we have

$\displaystyle\Omega^{\text{BMM}}_{b}=-\frac{\delta_{b}}{3m}<0,$ (3.29)

as given in Eq.(III.1). Therefore, the BMM choices cannot be realized in this
case.

It is also interesting to note that the spacetimes described by
Eqs.(3.24)-(3.27) actually have similar asymptotic behavior as the AOS
solution does, although the two metrics, given respectively by Eqs.(3.24) and
(2.41), look quite different. To show this claim, let us first introduce a new
spacelike coordinate $\xi$ via the relation, $\xi=e^{b_{o}t_{b}}$, and then we
find that the metric (3.24) becomes

$\displaystyle ds^{2}$ $\displaystyle\simeq$
$\displaystyle-\left(c_{1}\xi^{2}+c_{2}\xi+c_{3}+\cdots\right)e^{-\alpha_{0}\xi}dT^{2}+\left(d_{1}+\frac{d_{2}}{\xi}+\frac{d_{3}}{\xi^{2}}+\cdots\right)e^{\alpha_{0}\xi}\frac{d\xi^{2}}{b_{o}^{2}}+4m^{2}e^{\alpha_{0}\xi}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),~{}~{}~{}~{}$ (3.30)

where $T\equiv x$ and $\alpha_{0}\equiv{2\alpha_{1}}/{b_{o}}>0$. Then, the
corresponding curvature invariants of the above metric are given by

$\displaystyle g^{\mu\nu}R_{\mu\nu}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{2m}{r}\right)^{2}\left[\frac{1}{2}\left(\frac{1}{m^{2}}-\frac{b_{o}^{2}\alpha_{0}^{2}}{d_{1}}\right)+\frac{b_{o}^{2}\alpha_{0}(2d_{1}+d_{2}\alpha_{0})}{2d_{1}^{2}\xi}+\mathcal{O}\left(\frac{1}{\xi^{2}}\right)\right],$
$\displaystyle R^{\mu\nu}R_{\mu\nu}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{2m}{r}\right)^{4}\left[\frac{1}{8}\left(\frac{1}{m^{4}}+\frac{2b_{o}^{4}\alpha_{0}^{4}}{d_{1}^{2}}\right)-\frac{b_{o}^{2}\alpha_{0}(d_{1}^{2}+3b_{o}^{2}d_{1}m^{2}\alpha_{0}^{2}+b_{o}^{2}d_{2}m^{2}\alpha_{0}^{3})}{2(d_{1}^{3}m^{2})\xi}+\mathcal{O}\left(\frac{1}{\xi^{2}}\right)\right],$
$\displaystyle R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}$
$\displaystyle\simeq$
$\displaystyle\left(\frac{2m}{r}\right)^{4}\left[\frac{d_{1}^{2}-2b_{o}^{2}d_{1}m^{2}\alpha_{0}^{2}+7b_{o}^{4}m^{4}\alpha_{0}^{4}}{4d_{1}^{2}m^{4}}+\frac{b_{o}^{2}\alpha_{0}^{2}(d_{1}d_{2}-b_{o}^{2}m^{2}\alpha_{0}(16d_{1}+7d_{2}\alpha_{0}))}{2d_{1}^{3}m^{2}\xi}+\mathcal{O}\left(\frac{1}{\xi^{2}}\right)\right],$
$\displaystyle C^{\mu\nu\alpha\beta}C_{\mu\nu\alpha\beta}$
$\displaystyle\simeq$
$\displaystyle\left(\frac{2m}{r}\right)^{4}\left[\frac{(d_{1}-4b_{o}^{2}m^{2}\alpha_{0}^{2})^{2}}{12d_{1}^{2}m^{4}}+\frac{2b_{o}^{2}\alpha_{0}(2d_{1}+d_{2}\alpha_{0})(d_{1}-4b_{o}^{2}m^{2}\alpha_{0}^{2})}{3d_{1}^{3}m^{2}\xi}+\mathcal{O}\left(\frac{1}{\xi^{2}}\right)\right],$
(3.31)

where $r\left(\equiv 2me^{\alpha_{0}\xi/2}\right)$ is the geometric radius of
the two spheres $\xi,\;T=$ constant. Comparing the above with the ones
presented in AO20 , we can see that now the metric approaches asymptotically
to the Minkowski spacetime as $r^{-4}$, which is the same as that of the AOS
solution.

It is also remarkable to note that for the AOS choices of $\delta_{b}$ and
$\delta_{c}$ given by Eq.(2.26), we find that $\alpha_{1}\propto m^{2/3}$ and
$d_{1}\propto m^{10/3}$. Then, the above expressions show that they are all
independent of $m$ asymptotically. In particular, we have

$R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}\simeq\frac{A_{0}}{r^{4}}+{\cal{O}}\left(\frac{1}{r^{4}\xi}\right),$
(3.32)

where $\xi=\frac{2}{\alpha_{0}}\ln\left(\frac{r}{2m}\right)$, and $A_{0}$ is
independent of $m$ given by

$A_{0}\equiv\frac{28\Omega_{b}^{4}}{\omega_{bb}^{4}}-\frac{8\Omega_{b}^{2}}{\omega_{bb}^{2}}+4.$
(3.33)

This is sharply in contrast to the relativistic case, in which the Kretschmann
scalar is given by

$\left.R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}\right|_{\text{GR}}=\frac{48m^{2}}{r^{6}}.$
(3.34)

It is also very interesting to note that the leading order of the Kretschmann
scalar of the AOS solution also behaves like $r^{-4}$ as $r\rightarrow\infty$
BBCCY20 . In the current case, even the Dirac observables $f_{i}$ are chosen
so that $A_{0}$ given by Eq.(3.33) is zero, the next leading order is
${\cal{O}}\left(\frac{1}{r^{4}\xi}\right)$, which approaches zero still not as
fast as $r^{-6}$. In fact, it is even slower than $r^{-5}$.

To understand the solutions further, we first note that to the leading order
the metric takes the form

$ds^{2}\simeq-\frac{c_{1}b_{o}^{2}}{\alpha_{1}^{2}}\left(\frac{\ln\left(\frac{r}{2m}\right)}{\frac{r}{2m}}\right)^{2}dT^{2}+\frac{d_{1}}{4m^{2}\alpha_{1}^{2}}dr^{2}+r^{2}d\Omega^{2}.$
(3.35)

for $r\gg 2m$. On the other hand, the AOS solution takes the asymptotic form
BBCCY20

$ds_{\text{AOS}}^{2}\simeq-r^{2(b_{o}-1)}dT^{2}+dr^{2}+r^{2}d\Omega^{2},$
(3.36)

which is identical to the global monopole solution found in a completely
different content BV89 . Then, the corresponding effective energy-momentum
tensor is given by

$T_{\nu\nu}=u_{\mu}u_{\nu}\rho+p_{r}r_{\mu}r_{\nu}+p_{\bot}\left(\theta_{\mu}\theta_{\nu}+\phi_{\mu}\phi_{\nu}\right),$
(3.37)

where $u_{\mu}$ denotes the unit timelike vector along $T$-direction, and
$r_{\mu}$, $\theta_{\mu}$ and $\phi_{\mu}$ are the spacelike unity vectors
along, respectively, $r-$, $\theta-$, and $\phi-$directions, and $\rho$,
$p_{r}$ and $p_{\bot}$ are the energy density and pressures along the radial
and tangential directions. To the leading order, they are given by

$\displaystyle\rho\simeq\frac{4\alpha_{1}^{2}m^{2}-d_{1}}{d_{1}r^{2}},\;\;p_{r}\simeq-\frac{d_{1}+4\alpha_{1}^{2}m^{2}}{d_{1}r^{2}},\;\;p_{\bot}\simeq\frac{4\alpha_{1}^{2}m^{2}}{d_{1}r^{2}},$

which all approach zero as $r^{-2}$. This is also consistent with the
asymptotical behaviors of the quantities given in Eq.(III.2).

It should be also noted that, despite these differences, the spacetimes of the
current solutions are also asymptotically flat and the corresponding ADM
masses are as well defined as that of the AOS solution AO20 .

###  III.3 External Spacetimes with $\alpha_{1}=0$

When $\alpha_{1}=0$, from Eq.(III.1) and Fig. 2 we find that this can be the
case only when

$\displaystyle\Omega_{b}\equiv\omega_{bb}+\omega_{bc}=0.$ (3.39)

It is clear that the BMM choices of $f_{i}$ and $\delta_{i}$, given by Eqs.(•
‣ II.1) and (2.26), are not compatible with this case, too.

Then, to the leading order, Eq.(3.20) yields

$\displaystyle t_{b}\simeq t_{c}\equiv t,$ (3.40)

as $t_{c}\rightarrow\infty$. With Eqs.(3.39) and (3.40) we find that

$\displaystyle g_{xx}$ $\displaystyle\simeq$ $\displaystyle
e^{-2t}\left(c_{1}e^{2b_{o}t}+c_{2}e^{b_{o}t}+c_{3}+\cdots\right)$
$\displaystyle g_{\theta\theta}$ $\displaystyle\simeq$ $\displaystyle
4m^{2}e^{2t},$ (3.41)

where $c_{n}$’s are still given by Eq.(III.2). Finding the asymptotic limit of
$g_{tt}$ is not so straightforward, and this is mainly because of the term
$C_{bc}$ seen in the expression

$\displaystyle
g_{tt}=\frac{\gamma^{2}\left|{p}_{c}\right|\delta_{{b}}^{2}}{\sinh^{2}\left(\delta_{{b}}{b}\right)C_{bc}^{2}}.$
(3.42)

The numerator of $C_{bc}$ in Eq.(3.18) is equal to $1$ with the choice of
$\omega_{bb}+\omega_{bc}=0$ and the remainder of $g_{tt}$ is evaluated with
the help of Mathematica, and is given by

$\displaystyle\begin{aligned}
g_{tt}=e^{2t}\Big{(}&d_{1}e^{2b_{o}t}+d_{2}e^{b_{o}t}+d_{3}+\cdots\Big{)},\end{aligned}$
(3.43)

where $d_{n}$’s are also given by Eq.(III.2). Introducing the new coordinates,

$r=2me^{t},\quad x=\frac{4b_{o}^{2}}{(b_{o}+1)^{2}}\tau,$ (3.44)

we find

$\displaystyle ds^{2}=-g_{\tau\tau}d\tau^{2}+g_{rr}dr^{2}+r^{2}d\Omega^{2},$
(3.45)

where

$\displaystyle g_{\tau\tau}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{r}{2m}\right)^{2\left(b_{o}-1\right)}\left(1+\frac{c_{2}}{c_{1}}\left(\frac{2m}{r}\right)^{b_{o}}+\frac{c_{3}}{c_{1}}\left(\frac{2m}{r}\right)^{2b_{o}}\right),$
$\displaystyle g_{rr}$ $\displaystyle\simeq$
$\displaystyle\frac{d_{1}}{4m^{2}}\left(\frac{r}{2m}\right)^{2b_{o}}+\frac{d_{2}}{4m^{2}}\left(\frac{r}{2m}\right)^{b_{o}}+\frac{d_{3}}{4m^{2}}.$
(3.46)

Then, we find

$\displaystyle g^{\mu\nu}R_{\mu\nu}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{2m}{r}\right)^{2}\left[\frac{1}{2m^{2}}+\frac{2(2b_{o}-1)}{d_{1}\xi^{2}}+\mathcal{O}\left(\frac{1}{\xi^{3}}\right)\right],$
$\displaystyle R^{\mu\nu}R_{\mu\nu}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{2m}{r}\right)^{4}\left[\frac{1}{8m^{4}}-\frac{b_{o}(c_{1}d_{2}-c_{2}d_{1})}{2c_{1}d_{1}^{2}m^{2}\xi^{3}}+\mathcal{O}\left(\frac{1}{\xi^{4}}\right)\right],$
$\displaystyle R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}$
$\displaystyle\simeq$
$\displaystyle\left(\frac{2m}{r}\right)^{4}\left[\frac{1}{4m^{4}}-\frac{2}{d_{1}m^{2}\xi^{2}}+\mathcal{O}\left(\frac{1}{\xi^{3}}\right)\right],$
$\displaystyle C^{\mu\nu\alpha\beta}C_{\mu\nu\alpha\beta}$
$\displaystyle\simeq$
$\displaystyle\left(\frac{2m}{r}\right)^{4}\left[\frac{1}{12m^{4}}+\frac{4(b_{o}-2)}{3d_{1}m^{2}\xi^{2}}+\mathcal{O}\left(\frac{1}{\xi^{3}}\right)\right],$
(3.47)

here $\xi\equiv(r/2m)^{b_{o}}$. Interestingly the spacetime is again
asymptotically flat, and to the leading order has the same asymptotic behavior
as that in the case $\alpha_{1}\not=0$. In particular, all these scalars are
asymptotically independent of the mass parameter $m$, and approach zero as
$r^{-4}$, sharply in contrast to the relativistic case given by Eq.(3.34).
However, different from the case $\alpha_{1}\not=0$, to the next leader order
the Kretschmann scalar behaves like ${\cal{O}}\left(1/r^{2(2+b_{o})}\right)$.

To study this case in more details, let us first note that to the leading
order the metric takes the form

$\displaystyle ds^{2}$ $\displaystyle\simeq$ $\displaystyle-
c_{1}\left(\frac{r}{2m}\right)^{2(b_{o}-1)}dT^{2}+\frac{d_{1}}{4m^{2}}\left(\frac{r}{2m}\right)^{2b_{o}}dr^{2}$
(3.48) $\displaystyle~{}~{}~{}~{}~{}~{}~{}+r^{2}d\Omega^{2},$

for $r\gg 2m$. Clearly, this is still different from Eq.(3.36) for the AOS
solution, despite the fact that to the leading order, the Kretschmann scalar
approaches zero like $r^{-4}$ in both cases. However, because the
$r-$dependence of the $g_{rr}$ component, to the next leading order, the
Kretschmann scalar approaches zero like
${\cal{O}}\left(1/r^{2(2+b_{o})}\right)$. Recall that
$b_{o}\equiv\sqrt{1+\gamma^{2}\delta_{b}^{2}}\geq 1$. This can be further
understood by the analysis of the corresponding effective energy-momentum
tensor, which can be also cast in the form of Eq.(3.37), but now with

$\displaystyle\rho$
$\displaystyle=-\frac{1}{r^{2}}\left({1+\frac{4m^{2}\left(2b_{o}-1\right)}{d_{1}\left(\frac{r}{2m}\right)^{2b_{o}}}}\right),$
$\displaystyle p_{r}$
$\displaystyle=\frac{1}{r^{2}}\left({-1+\frac{4m^{2}\left(2b_{o}-1\right)}{d_{1}\left(\frac{r}{2m}\right)^{2b_{o}}}}\right),$
$\displaystyle p_{\bot}$
$\displaystyle=-\frac{4m^{2}\left(2b_{o}-1\right)}{d_{1}r^{2}\left(\frac{r}{2m}\right)^{2b_{o}}},$
(3.49)

which are consistent with the behaviors of the quantities given in Eq.(III.3).
Following AO20 , it is not difficult to see that the spacetimes of the current
solutions are also asymptotically flat and the corresponding ADM masses are as
well defined as that of the AOS solution.

##  IV Canonical Phase Space Approach: Internal Spacetimes

In the internal region of the LQBH, the dynamical equations (3.2) and (3.3)
take the form

$\displaystyle\frac{db}{dt_{b}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\frac{\sin\left(\delta_{b}b\right)}{\delta_{b}}+\frac{\gamma^{2}\delta_{b}}{\sin\left(\delta_{b}b\right)}\right),$
(4.1) $\displaystyle\frac{dp_{b}}{dt_{b}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}p_{b}\cos\left(\delta_{b}b\right)\left(1-\frac{\gamma^{2}\delta_{b}^{2}}{\sin^{2}\left(\delta_{b}b\right)}\right),$
(4.2)

for the variables $(b,p_{b})$, and

$\displaystyle\frac{dc}{dt_{c}}$ $\displaystyle=$
$\displaystyle-2\frac{\sin\left(\delta_{c}c\right)}{\delta_{c}},$ (4.3)
$\displaystyle\frac{dp_{c}}{dt_{c}}$ $\displaystyle=$
$\displaystyle{2}p_{c}\cos\left(\delta_{c}c\right),$ (4.4)

for $(c,p_{c})$. Eqs.(4.1) and (4.2) are identical with Eqs.(2.17) and (2.18),
if we replace $T$ by $t_{b}$, while Eqs.(4.3) and (4.4) are identical with
Eqs.(2.19) and (2.20), if we replace $T$ by $t_{c}$. Then, the corresponding
solutions can be obtained directly from Eqs.(II.1) - (II.1) by the above
replacements, which lead to

$\displaystyle\cos\left(\delta_{b}b\right)$ $\displaystyle=$ $\displaystyle
b_{o}\frac{1+b_{o}\tanh\left(\frac{b_{o}t_{b}}{2}\right)}{b_{o}+\tanh\left(\frac{b_{o}t_{b}}{2}\right)}$
$\displaystyle=$ $\displaystyle
b_{o}\frac{b_{+}e^{b_{o}t_{b}}-b_{-}}{b_{+}e^{b_{o}t_{b}}+b_{-}},$
$\displaystyle p_{b}$ $\displaystyle=$
$\displaystyle-\frac{mL_{o}}{2b_{o}^{2}}\left(b_{+}+b_{-}e^{-b_{o}t_{b}}\right){\mathfrak{A}},$
(4.5) $\displaystyle\sin\left(\delta_{c}c\right)$ $\displaystyle=$
$\displaystyle\frac{2a_{o}e^{2t_{c}}}{a^{2}_{o}+e^{4t_{c}}},$ $\displaystyle
p_{c}$ $\displaystyle=$ $\displaystyle
4m^{2}\left(a_{o}^{2}+e^{4t_{c}}\right)e^{-2t_{c}},$ (4.6)

but now with

$\displaystyle{\mathfrak{A}}$ $\displaystyle\equiv$
$\displaystyle\Big{[}2\left(b_{o}^{2}+1\right)e^{b_{o}t_{b}}-b_{-}^{2}-b_{+}^{2}e^{2b_{o}t_{b}}\Big{]}^{1/2},$
(4.7)

where $a_{o}$ and $b_{\pm}$ are still given by Eq.(II.1), and the range of the
variables is given by Eq.(II.1). Then, it can be seen that the two Dirac
observables $O_{b}$ and $O_{c}$ are also given by Eq.(2.25) along the
dynamical trajectories. However, instead of imposing the conditions (2.26),
now we shall leave the choice of $\delta_{b}$ and $\delta_{c}$ open, as we did
in the last section. Thus, the corresponding internal spacetimes are described
by the metric

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-N^{2}dT^{2}+\frac{p_{b}^{2}}{|p_{c}|L_{o}^{2}}dx^{2}+|p_{c}|d\Omega^{2}$
(4.8) $\displaystyle=$
$\displaystyle-\left(\frac{N}{C_{cb}}\right)^{2}dt_{c}^{2}+\frac{p_{b}^{2}}{|p_{c}|L_{o}^{2}}dx^{2}+|p_{c}|d\Omega^{2},~{}~{}~{}$

where

$\displaystyle N$ $\displaystyle\equiv$
$\displaystyle\frac{\gamma\delta_{b}\;\text{sgn}\left(p_{c}\right)\left|p_{c}\right|^{1/2}}{\sin\left(\delta_{b}b\right)}$
(4.9) $\displaystyle=$
$\displaystyle\frac{2m}{{\mathfrak{A}}}\left(b_{+}e^{b_{o}t_{b}}+b_{-}\right)\left(a_{o}^{2}e^{-2t_{c}}+e^{2t_{c}}\right)^{1/2}.~{}~{}~{}~{}~{}~{}$

In the following, let us study the above spacetimes near the horizons
($\mathfrak{A}=0$) and throat ($\partial p_{c}/\partial t_{c}=0$), separately.

###  IV.1 Spacetimes near the Horizons

The horizons now are located at $\mathfrak{A}=0$, which yields two solutions

$t_{b}^{\text{BH}}=0,\quad
t_{b}^{\text{WH}}=-\frac{2}{b_{o}}\ln\left(\frac{b_{o}+1}{b_{0}-1}\right).$
(4.10)

Now to find the relation between $t_{b}$ and $t_{c}$ the following expression
has to be integrated

$dt_{c}=\frac{C_{cb}}{C_{bc}}dt_{b},$ (4.11)

where the expressions of $C_{bc}$ and $C_{cb}$ in the interior are

$\displaystyle C_{bc}$ $\displaystyle=$
$\displaystyle\frac{1}{\cal{D}}\left(1-\Omega_{c}\frac{\partial
O_{c}}{\partial\delta_{c}}\right),$ $\displaystyle C_{cb}$ $\displaystyle=$
$\displaystyle\frac{1}{\cal{D}}\left(1-\Omega_{b}\frac{\partial
O_{b}}{\partial\delta_{b}}\right),$ (4.12)

but now with

$\displaystyle{\cal{D}}$ $\displaystyle\equiv$ $\displaystyle
1-\omega_{cc}\frac{\partial
O_{c}}{\partial\delta_{c}}-\omega_{bb}\frac{\partial
O_{b}}{\partial\delta_{b}}+\left(\omega_{bb}\omega_{cc}-\omega_{bc}\omega_{cb}\right)\frac{\partial
O_{b}}{\partial\delta_{b}}\frac{\partial O_{c}}{\partial\delta_{c}},$
$\displaystyle\frac{\partial O_{b}}{\partial\delta_{b}}$ $\displaystyle=$
$\displaystyle-\frac{p_{b}}{2\gamma
L_{o}\delta_{b}^{2}}\left(1-\frac{\gamma^{2}\delta_{b}^{2}}{\sin^{2}\left(\delta_{b}b\right)}\right)\big{[}\delta_{b}b\cos\left(\delta_{b}b\right)-\sin\left(\delta_{b}b\right)\big{]},$
$\displaystyle\frac{\partial O_{c}}{\partial\delta_{c}}$ $\displaystyle=$
$\displaystyle\frac{p_{c}}{\gamma
L_{o}\delta_{c}^{2}}\big{[}\delta_{c}c\cos\left(\delta_{c}c\right)-\sin\left(\delta_{c}c\right)\big{]}.$
(4.13)

Similar to the previous subsection, in the following section we consider the
cases $\alpha_{1}=0$ and $\alpha_{1}\not=0$, separately.

####  IV.1.1 $\alpha_{1}=0$

In this case, it is remarkable to note that by integrating Eq.(3.15) we find
the following explicit solution,

$\displaystyle t_{b}$ $\displaystyle=$ $\displaystyle
t_{b}^{0}+t_{c}+\frac{m\Omega_{c}}{\delta_{c}}\left\\{\cosh\left(2{\cal{T}}\right)\tan^{-1}\left(e^{2{\cal{T}}}\right)\right.$
(4.14)
$\displaystyle\left.-\cosh\left[2\left(t_{c}-{\cal{T}}\right)\right]\tan^{-1}\left[e^{-2\left(t_{c}-{\cal{T}}\right)}\right]\right\\},~{}~{}~{}$

which holds for any $t_{c}$, including the region $t_{c}\geq 0$, outside the
black hole horizon, where $t_{c}={\cal{T}}$ is the location of the transition
surface, defined by Eq.(2.27). And $t_{b}^{0}$ is an integration constant
which will be set to zero in the following discussions. When $t_{c}=0$ the
second term in the right-hand side of the above expression vanishes
identically, and as $t_{c}\rightarrow\infty$ it goes to zero as
${\cal{O}}\left(e^{-2t_{c}}\right)$. This is consistent with Eq.(3.20).

Figure 3: Plots of $t_{b}$ vs $t_{c}$ for $\alpha_{1}=0$ defined by Eq.(4.14).
Depending on the signs of $\Omega_{c}$, the dependence of $t_{b}$ on $t_{c}$
is different. Curves $b$, $c$ and $d$ are all for $\Omega_{c}=0.5$ but with
different choices of $(m,\delta_{c})$. In particular, they correspond to
$(m,\delta_{c})=\\{(10^{6},10^{-7}),(10^{6},0.1),(1,0.1)\\}$, respectively.
Curves $b^{\prime}$, $c^{\prime}$ and $d^{\prime}$ are all for
$\Omega_{c}=-0.5$ but with the same choice of $(m,\delta_{c})$ as that of the
unprimed curves in the respective order.

In Fig. 3, we plot the curves of $t_{b}$ vs $t_{c}$ of Eq. (4.14) for
different choices of parameters involved. In particular, we find that the
properties of $t_{b}$ across the transition surface sensitively depend on the
signs of $\Omega_{c}$. More specifically, when $\Omega_{c}>0$, $t_{b}$
decreases exponentially right after crossing the transition surface, as
$t_{c}$ becomes more and more negative, as shown by Curves $b$, $c$ and $d$
with the choice $\Omega_{c}=0.5$, where the dots on the curves mark the
locations of the transition surfaces. On the other hand, when $\Omega_{c}<0$,
$t_{b}$ increases exponentially right after crossing the transition surface,
as shown by Curves $b^{\prime}$, $c^{\prime}$ and $d^{\prime}$ with
$\Omega_{c}=-0.5$. However, the locations of the transition surface indeed
depend on the choices of the parameters $(m,\;L_{0}\delta_{c})$, as shown by
Eq.(2.27). In particular, Curves $b$, $c$ and $d$ respectively correspond to

$\left(\frac{m}{m_{p}},\frac{L_{o}\delta_{c}}{\ell_{p}}\right)=\left\\{\left(10^{6},10^{-7}\right),\left(10^{6},0.1\right),\left(1,0.1\right)\right\\},$

while Curves $b^{\prime}$, $c^{\prime}$ and $d^{\prime}$ are all for the same
choices of $(m,\delta_{c})$, as that of the unprimed curves in respective
order. Curves $b$ and $c$ share the same mass, i.e. $m/m_{p}=10^{6}$, but with
different $\delta_{c}$’s. Meanwhile, the locations of the throats (the gray
dots) move from the left-hand side to the right-hand side in the direction
closer to the horizon, which means that the quantum effects increase as
$\delta_{c}$ increases. Curves $c$ and $d$ share the same $\delta_{c}=0.1$,
but different masses. Comparing their throat positions, we find that the
smaller mass also means the more significant quantum effects. On the other
hand, outside the horizon, no matter what the parameters are, $t_{b}\simeq
t_{c}$, which is consistent with our previous conclusion for large $t_{b}$ and
$t_{c}$, as shown by Eq. (3.40). To understand this point further, let us
expand the above expression around the horizon, for which we find

$\displaystyle t_{b}$ $\displaystyle=$
$\displaystyle\beta_{1}t_{c}+\beta_{2}t_{c}^{2}+\beta_{3}t_{c}^{3}+{\cal{O}}\left(t_{c}^{4}\right),$
(4.15)

where

$\displaystyle\beta_{1}$ $\displaystyle\equiv
1+\frac{m\Omega_{c}}{a_{o}\delta_{c}}\left[a_{o}+\left(a_{o}^{2}-1\right)\tan^{-1}(a_{o})\right],$
$\displaystyle\beta_{2}$
$\displaystyle\equiv-\frac{m\Omega_{c}}{a_{o}\left(a_{o}^{2}+1\right)\delta_{c}}\left[a_{o}\left(a_{o}^{2}-1\right)\right.$
$\displaystyle\left.+\left(a_{o}^{2}+1\right)^{2}\tan^{-1}(a_{o})\right],$
$\displaystyle\beta_{3}$
$\displaystyle\equiv\frac{2m\Omega_{c}}{3a_{o}\left(a_{o}^{2}+1\right)^{2}\delta_{c}}\left[a_{o}+6a_{o}^{3}+a_{o}^{5}\right.$
$\displaystyle+\left.\left(a_{o}^{2}-1\right)\left(a_{o}^{2}+1\right)^{2}\tan^{-1}(a_{o})\right].$
(4.16)

For macroscopic black holes, we have $m/m_{p}\gtrsim M_{\odot}\simeq 10^{38}$,
while the semi-classical limit requires $L_{0}\delta_{c}\ll 1$. Then,
expanding $\beta_{n}$ in terms of $a_{o}$, we find that

$\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle
1+\frac{m\Omega_{c}}{a_{o}\delta_{c}}\left[a_{o}+\left(a_{o}^{2}-1\right)\tan^{-1}(a_{o})\right]$
$\displaystyle\simeq$ $\displaystyle
1+\frac{\gamma^{2}L_{o}^{2}\delta_{c}\Omega_{c}}{48m}+{\cal{O}}\left(a_{o}^{4}\right)\simeq
1,$ $\displaystyle\beta_{2}$ $\displaystyle=$
$\displaystyle-\frac{\gamma^{2}L_{o}^{2}\delta_{c}\Omega_{c}}{24m}+{\cal{O}}\left(a_{o}^{4}\right)\simeq
0,$ $\displaystyle\beta_{3}$ $\displaystyle=$
$\displaystyle\frac{\gamma^{2}L_{o}^{2}\delta_{c}\Omega_{c}}{18m}+{\cal{O}}\left(a_{o}^{4}\right)\simeq
0.$ (4.17)

Therefore, for macroscopic black holes, the relation $t_{b}\simeq t_{c}$ near
the horizon is well justified. Then, we find that the metric components take
the form

$\displaystyle g_{xx}$ $\displaystyle=$
$\displaystyle\frac{e^{2t_{c}}\zeta\left(t_{c}\right)^{2}}{16b_{o}^{4}\left(a_{o}^{2}+e^{4t_{c}}\right)}\left(\left(b_{o}-1\right)e^{-b_{o}t_{c}}+b_{o}+1\right)^{2},$
$\displaystyle g_{t_{c}t_{c}}=$
$\displaystyle\\!\\!\\!\\!\\!4m^{2}e^{-2t_{c}}\left(a_{o}^{2}+e^{4t_{c}}\right)\frac{\beta_{-}\left(t_{c}\right)^{2}}{\zeta\left(t_{c}\right)^{2}}\Bigg{\\{}1-\omega_{cc}\frac{\partial
O_{c}}{\partial\delta_{c}}+(\Omega_{c}\frac{\partial
O_{c}}{\partial\delta_{c}}-1)m\omega_{bb}e^{-b_{o}t_{c}}\frac{\left(2b_{o}^{2}e^{b_{o}t_{c}}-\zeta\left(t_{c}\right)^{2}\right)}{2\gamma\delta_{b}^{2}b_{o}^{2}\zeta\left(t_{c}\right)}$
(4.18)
$\displaystyle\times\left[\gamma\delta_{b}\zeta\left(t_{c}\right)-b_{o}\beta_{-}\left(t_{c}\right)\cos^{-1}\left(\frac{b_{o}\beta_{-}\left(t_{c}\right)}{\beta_{+}\left(t_{c}\right)}\right)\right]\Bigg{\\}}^{2},$
$\displaystyle\zeta(t_{c})$ $\displaystyle\equiv$
$\displaystyle\left[2\left(b_{o}^{2}+1\right)e^{b_{o}t_{c}}-\left(b_{o}-1\right)^{2}-\left(b_{o}+1\right)^{2}e^{2b_{o}t_{c}}\right]^{1/2},$
$\displaystyle\beta_{\pm}(t_{c})$ $\displaystyle\equiv$
$\displaystyle\left(b_{o}+1\right)e^{b_{o}t_{c}}\pm\left(b_{o}-1\right).$
(4.19)

(a)

(b)

Figure 4: Plots of the relative difference between the Kretschmann scalars $K$
and $K_{\text{GR}}$ in the $\alpha_{1}=0$ case, for (a) $m=10^{6}$, and (b)
$m=10^{12}$. Here $K_{\text{GR}}(\equiv 48m^{2}/p_{c}^{3})$ is the
corresponding Kretschmann scalar given by GR.

To quantify the quantum effects near the horizon, let us compute the Hawking
temperature at the horizon. Given a metric of the form

$ds^{2}=-g_{tt}dt^{2}+g_{xx}dx^{2}+p_{c}d\Omega^{2},$ (4.20)

the Hawking temperature of the black hole is given by AO20 ,

$\displaystyle
T_{H}=\frac{\hbar}{k_{B}\mathcal{P}},\quad\mathcal{P}=\lim_{t\to
0}\frac{4\pi\left(g_{tt}g_{xx}\right)^{\frac{1}{2}}}{\partial_{t}g_{xx}},$
(4.21)

where $k_{B}$ is the Boltzmann constant. Then, for the metric coefficients
given by Eq.(IV.1.1) we find

$\displaystyle
T_{H}=\frac{T_{H}^{\text{GR}}}{\left(1+a_{o}^{2}\right)\left(1+\varepsilon_{T}\right)},$
(4.22)

where $T_{H}^{\text{GR}}=\hbar/(8k_{B}\pi m)$ denotes the Hawking temperature
of the Schwarzschild black hole calculated in GR, and

$\varepsilon_{T}\equiv\frac{m\omega_{cc}}{a_{o}\delta_{c}}\left[a_{o}+\left(a_{o}^{2}-1\right)\tan^{-1}\left(a_{o}\right)\right].$
(4.23)

For a BH of mass $10^{6}$, we find that

$a_{o}^{2}=\left(\frac{\gamma\delta_{c}L_{o}}{8m}\right)^{2}\simeq 10^{-22},$

and

$\varepsilon_{T}=\left(\frac{4m\omega_{cc}}{3\delta_{c}}\right)\left(1-\frac{2}{5}a_{o}^{2}\right)a_{o}^{2}+{\cal{O}}\left(a_{o}^{6}\right).$

For the AOS choice of Eq.(• ‣ II.1), we find that
$4m\omega_{cc}/(3\delta_{c})\simeq{\cal{O}}(1)$, so that
$\varepsilon_{T}\lesssim 10^{-44}$, that is, for macroscopic black holes, the
quantum effects are negligible. This is consistent with what was concluded by
AOS AOS18b ; AO20 .

The above conclusion can be further verified by comparing the Kretschmann
scalars $K$ with its relativistic counterpart $K_{\text{GR}}\equiv
48m^{2}/p_{c}^{3}$. In particular, in Fig. 4 we plot the relative difference
of $K$ and $K_{\text{GR}}$ for $m=10^{6}m_{pl}$ and $m=10^{12}m_{pl}$, which
indicate negligible quantum corrections near the horizon for massive LQBHs.

####  IV.1.2 $\alpha_{1}\not=0$

When $\alpha_{1}\not=0$, we find that

$\displaystyle
t_{b}\left(1+\frac{m\Omega_{b}}{\delta_{b}b_{o}^{2}}\right)+\frac{2m\Omega_{b}}{\delta_{b}b_{o}^{2}}-\frac{m\Omega_{b}}{2\gamma\delta_{b}^{2}b_{o}^{2}}e^{-b_{o}t_{b}}\beta_{+}\left(t_{b}\right)\zeta\left(t_{b}\right)\cos^{-1}\left(\frac{b_{o}\beta_{-}\left(t_{b}\right)}{\beta_{+}\left(t_{b}\right)}\right)$
(4.24) $\displaystyle+$
$\displaystyle\frac{m\Omega_{b}}{2\delta_{b}b_{o}^{3}}\left(\left(b_{o}-1\right)^{2}e^{-b_{o}t_{b}}-\left(b_{o}+1\right)^{2}e^{b_{o}t_{b}}-2b_{o}^{3}t_{b}\right)+t_{b}^{0}$
$\displaystyle=$ $\displaystyle
t_{c}+\frac{m\Omega_{c}}{\delta_{c}}\left(\cosh\left(2{\cal{T}}\right)\tan^{-1}\left(e^{2{\cal{T}}}\right)-\cosh\left(2\left(t_{c}-{\cal{T}}\right)\right)\tan^{-1}\left(e^{-2\left(t_{c}-{\cal{T}}\right)}\right)\right),$

where $t_{b}^{0}$ is an integration constant and will be set to zero as done
previously in the $\alpha_{1}=0$ case.

Figure 5: Plots of Eq.(4.24) for various choices as given by Eq.(IV.1.2).

(a)

(b)

Figure 6: Plots of the relative difference between the Kretschmann scalars $K$
and $K_{\text{GR}}$ in the $\alpha_{1}\neq 0$ case, for (a) $m=10^{6}$, and
(b) $m=10^{12}$.

Notice that the $t_{c}$ part of the above expression is precisely the right-
hand side of Eq.(4.14), and we showed explicitly in the last subsection that
near the horizon $t_{c}=0$ the right-hand side can be well approximated by
$t_{c}$. Now, expanding the $t_{b}$ part of the above expression around the
horizon, we find

$t_{b}+\nu_{2}t_{b}^{2}+\nu_{3}t_{b}^{3}+{\cal{O}}\left(t_{b}^{4}\right),$
(4.25)

where

$\displaystyle\nu_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{6}m\gamma^{2}\delta_{b}\Omega_{b},$
$\displaystyle\nu_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{60}m\gamma^{2}\delta_{b}\Omega_{b}\left(10-\gamma^{2}\delta_{b}^{2}\right).$
(4.26)

The above coefficients $\nu_{i}$ are negligibly small for large black holes.
For example, for a BH of mass $10^{6}$, they are of the order $\sim 10^{-9}$.
Hence, for macroscopic black holes Eq.(4.24) can also be well approximated by

$t_{b}\simeq t_{c},$ (4.27)

near the black hole horizon, similar to the case $\alpha_{1}=0$. This linear
relation can be confirmed by the plot of Eq.(4.24) for various values, as seen
in Fig. 5. For plotting the curves b, c, and d corresponding to positive
$\Omega_{c}$, the parameters are chosen respectively as,

$\displaystyle\left(\frac{m}{m_{p}}\right)$
$\displaystyle=\left(10^{6},10^{8},10^{10}\right),$
$\displaystyle\left(\omega_{cc},\omega_{cb},\Omega_{c}\right)$
$\displaystyle=\left(\frac{\delta_{c}}{3m},0,\frac{\delta_{c}}{3m}\right),$
$\displaystyle\left(\omega_{bb},\omega_{bc},\Omega_{b}\right)$
$\displaystyle=\left(\frac{\delta_{b}}{3m},0,\frac{\delta_{b}}{3m}\right),$
(4.28)

where $\delta_{i}$’s are given by Eq.(2.26).

Since in the current case ($\alpha_{1}\not=0$) $t_{b}\simeq t_{c}$ also holds
near the horizon for macroscopic black holes, the thermodynamics of the black
hole horizon is quite similar to the case $\alpha_{1}=0$. In particular, its
temperature is also given by Eqs.(4.22) and (4.23), and the difference to that
of the Schwarzschild black hole calculated in GR is negligibly small for
macroscopic black holes.

Again, a plot of the relative difference between the Kretschmann scalar K and
$K_{GR}$ is given in Fig. 6 for the $\alpha_{1}\not=0$ case, which also shows
the negligible quantum effects near the horizons for massive LQBHs.

###  IV.2 Spacetimes near Transition Surfaces

It is evident from Figs. 3 and 5 that the above approximation, $t_{b}\simeq
t_{c}$, is no longer valid once we start to probe the spacetime near and to
the other side of the transition surface. We break this analysis again into
two cases, $\alpha_{1}=0$ and $\alpha_{1}\neq 0$.

####  IV.2.1 $\alpha_{1}=0$

In this case, the relation between $t_{b}$ and $t_{c}$ is given by Eq.(4.14),
which is valid everywhere in the interior. Combining this equation with the
metric (4.8) we can calculate the curvature invariants to analyze the
spacetime near the transition surface. We find that this can be done by
xActxact , a package for tensor computations in Mathematica, although the
exact expressions are too complicated to be written down here. For this
reason, we only plot out the Kretschmann scalar here for illustration, as
other scalars like the Ricci scalar, Ricci tensor squared, have similar
features. In particular, in Figs. 7 and 8 we plot the Kretschmann scalar
respectively for $\Omega_{c}<0$ and $\Omega_{c}>0$, but all with
$\Omega_{b}=0$. In addition, we also provide Table 1, in which we show the
explicit dependence of the maximal amplitude $K_{m}$ of the Kretschmann scalar
on the mass $m$, the location of the maximal amplitude of the Kretschmann
scalar, denoted by $\tau_{m}$, and the location of the transition surface
denoted by $\tau_{\text{ts}}$. To compare with the AOS solution, we also give
the maximal amplitudes of the Kretschmann scalar for the AOS solution.

Figure 7: The Kretschmann scalar near the transition surface $\tau_{ts}\simeq-11.6201$ denoted by the vertical line for the case $\alpha_{1}=0$. Here $m=10^{6}$, $(\omega_{cc},\omega_{cb})=(-{\delta_{c}}/{3m},0)$, and $(\omega_{bb},\omega_{bc})=(-{\delta_{b}}/{3m},{\delta_{b}}/{3m})$, so that $(\Omega_{b},\Omega_{c})=(0,-{\delta_{c}}/{3m}<0)$, where $\delta_{i}$’s are given by Eq.(2.26). Figure 8: The Kretschmann scalar near the transition surface denoted by the vertical line for the case $\alpha_{1}=0$. Here $m=10^{6}$, $(\omega_{cc},\omega_{cb})=({\delta_{c}}/{3m},0)$, and $(\omega_{bb},\omega_{bc})=({\delta_{b}}/{3m},-{\delta_{b}}/{3m})$, so that $(\Omega_{b},\Omega_{c})=(0,{\delta_{c}}/{3m}>0)$, where $\delta_{i}$’s are given by Eq.(2.26). Table 1: The maximal amplitude $K_{m}$ of the Kretschmann scalar $K$ for the case $\alpha_{1}=0$ with different choices of the mass parameter $m$. Here $\tau_{m}$ denotes the location of the maximal point of $K$, and $\tau_{ts}$ the location of the corresponding transition surface (throat). To compare it with that given by the AOS solution, we also give the maximal values of $K_{m}^{AOS}$. Here we choose $\omega_{bb}=-{\delta_{b}}/{3m}$, $\omega_{bc}={\delta_{b}}/{3m}$, $\omega_{cc}=-{\delta_{c}}/{3m}$, and $\omega_{cb}=0$, so that $(\Omega_{b},\Omega_{c})=(0,-{\delta_{c}}/{3m}<0)$, where $\delta_{i}$’s are given by Eq.(2.26). $m/m_{p}$ | $\tau_{m}$ | $K_{m}$ | $\tau_{ts}$ | $K_{m}^{AOS}$
---|---|---|---|---
$10^{6}$ | -12.0147 | $2.46\times 10^{48}$ | -11.6201 | 82188.3628
$10^{8}$ | -15.0848 | $1.56\times 10^{52}$ | -14.6902 | 82188.3642
$10^{10}$ | -18.1549 | $3.60\times 10^{75}$ | -17.7603 | 82188.3642
$10^{12}$ | -21.225 | $2.21\times 10^{75}$ | -20.8304 | 82188.3642
$10^{14}$ | -24.2951 | $2.87\times 10^{70}$ | -23.9005 | 82188.3642
$10^{16}$ | -27.3653 | $2.82\times 10^{69}$ | -26.9706 | 82188.3641
$10^{18}$ | -30.4354 | $9.59\times 10^{77}$ | -30.0408 | 82188.3618

From Figs. 7 and 8 and Table 1 we can see that the Kretschmann scalar remains
finite across the transition surfaces, but the maximal amplitude of the
Kretschmann scalar sensitively depends on the mass $m$, which is in sharp
contrast to the AOS solution in which the maximal amplitude $K_{m}^{AOS}$ of
the Kretschmann scalar remains the same AOS18a ; AOS18b ; AO20 .

Another unexpected feature is that the maximal point of the Kretschmann scalar
usually is not precisely at the transition surface, $\tau_{m}\not=\tau_{ts}$.
Although this looks strange, a closer examination shows that this is due to
two main facts: (1) the appearance of the factor $1/C_{bc}$ in the lapse
function of the metric (4.8), and (2) the dependence of $t_{b}$ on $t_{c}$,
which will lead to the modifications of $g_{xx}(t_{b},t_{c})$, in comparison
to the corresponding AOS component $g^{\text{AOS}}_{xx}(t_{b},t_{c})$ in which
we have $t_{b}=t_{c}=T$.

Figure 9: The function ${\cal{D}}^{2}$ defined by Eq.(IV.1) for the case
$\alpha_{1}=0$, for which we have $C_{bc}=1/{\cal{D}}$. The vertical (green)
line marks the position of the transition surface. When plotting this curve,
we have chosen the relevant parameters exactly as those given in Fig. 7.

In particular, when $\alpha_{1}=0$, we have $1/C_{bc}={\cal{D}}$, as can be
seen from Eq.(IV.1), where ${\cal{D}}$ is defined by Eq.(IV.1). In Fig. 9 we
plot out the function ${\cal{D}}^{2}$ for the same choices of the parameters
as given in Fig. 7, from which we can see that it changes dramatically near
the maximal point $\tau_{m}\simeq-12.0147$ of the Kretschmann scalar. In Figs.
10 and 11, we plot out the metric components $g_{t_{c}t_{c}}$ and $g_{xx}$
given in Eq.(4.8) vs $t_{c}$, where $g_{tt}\equiv\left|g_{t_{c}t_{c}}\right|$.
From these figures we can see clearly that both of these components change
dramatically near the maximal point $\tau_{m}$ of the Kretschmann scalar. To
compare it with the AOS solution, in each of these two figures, we also plot
the corresponding quantities for the AOS solution, from which it can be seen
that no such behavior appears in the AOS solution.

We also study the location of the white horizon and find that it is very near
the transition surface. In particular, the ratio between WH and BH horizon
radii now is much smaller than $1$ and sensitively depends on the mass
parameter $m$, as shown explicitly in Table.2. Whereas in the AOS model this
ratio is very close to $1$.

Figure 10: The metric component $g_{t_{c}t_{c}}$ given in Eq.(4.8), where $g_{tt}\equiv|g_{t_{c}t_{c}}|$. The inserting is the plot of the same quantity for the AOS solution. When plotting this curve, we have chosen the relevant parameters exactly as those given in Fig. 7. Figure 11: The metric component $g_{xx}$ given in Eq.(4.8). The inserting is the plot of the same quantity for the AOS solution. When plotting this curve, we have chosen the relevant parameters exactly as those given in Fig. 7. Table 2: The ratio of the WH and BH horizon radii for the case $\alpha_{1}=0$ with different choices of the mass parameter $m$. Here we use the same choices as those in 8, except for $m$. $m/m_{p}$ | $\frac{r_{\scaleto{WH}{2pt}}}{r_{\scaleto{BH}{2pt}}}$
---|---
$10^{6}$ | $5.5872\times 10^{-5}$
$10^{8}$ | $2.9462\times 10^{-6}$
$10^{10}$ | $1.5148\times 10^{-7}$
$10^{12}$ | $7.6577\times 10^{-9}$
$10^{14}$ | $3.8242\times 10^{-10}$
$10^{16}$ | $1.8923\times 10^{-11}$
$10^{18}$ | $9.2972\times 10^{-13}$

####  IV.2.2 $\alpha_{1}\not=0$

In this case, the explicit relation between $t_{b}$ and $t_{c}$ is given by
Eq.(4.24). This relation allows us to write down the metric and calculate the
curvature invariants. Similar to the $\alpha_{1}=0$ case, the exact
expressions of them are too complicated to be written down explicitly here,
and instead we find that it sufficient to simply plot them out. Since they all
have similar behavior, we plot out only the Kretschmann scalar. In particular,
we plot it for $\Omega_{c}<0$ and $\Omega_{c}>0$, respectively in Figs. 12 and
13. The vertical line in each of these figures represents the location of the
transition surface, and is usually different from the maximal point of the
Kretschmann scalar, quite similar to the case $\alpha_{1}=0$ and for similar
reasons.

Figure 12: The Kretschmann scalar for the case $\alpha_{1}\not=0$ with
$\Omega_{c}<0$. In particular, the parameters are chosen as $m=10^{6}$,
$(\omega_{cc},\omega_{cb},\Omega_{c})=(-\frac{\delta_{c}}{3m},0,-\frac{\delta_{c}}{3m})$,
$(\omega_{bb},\omega_{bc},\Omega_{b})=(-\frac{\delta_{b}}{3m},0,-\frac{\delta_{b}}{3m})$,
where $\delta_{i}$’s are given by Eq.(2.26). Figure 13: The Kretschmann scalar
for the case $\alpha_{1}\not=0$ with $\Omega_{c}>0$. In particular, the
parameters are chosen as $m=10^{6}$,
$(\omega_{cc},\omega_{cb},\Omega_{c})=(\frac{\delta_{c}}{3m},\frac{\delta_{c}}{3m},\frac{2\delta_{c}}{3m})$,
$(\omega_{bb},\omega_{bc},\Omega_{b})=(\frac{\delta_{b}}{3m},\frac{\delta_{b}}{3m},\frac{2\delta_{b}}{3m})$,
where $\delta_{i}$’s are given by Eq.(2.26).

##  V Conclusions

In this paper, we studied the 4-dimensional canonical phase space approach,
explored respectively by BMM BMM19  and GM GQ21  recently, in which the two
parameters $\delta_{i}\;(i=b,c)$ appearing in the polymerization quantization
Ashtekar20

$b\rightarrow\frac{\sin(\delta_{b}b)}{\delta_{b}},\quad
c\rightarrow\frac{\sin(\delta_{c}c)}{\delta_{c}},$ (5.1)

are considered as functions of the two Dirac variables $O_{b}$ and $O_{c}$
GQ21

$\delta_{i}=f_{i}\left(O_{b},O_{c}\right),\;(i,=b,c),$ (5.2)

where $O_{b}$ and $O_{c}$ are given by Eqs.(2.10) and (2.11). Note that BMM
only considered the particular case $\delta_{i}=f_{i}\left(O_{i}\right)$ BMM19
, the same as the AOS choice given in Eq.(2.14), although AOS considered them
in the extended 8-dimensional phase space $\Gamma_{\text{ext}}$. The
corresponding dynamical equations are given by Eqs.(3.7) and (3.8), which
allow analytical solutions in terms of $t_{b}$ and $t_{c}$, where $t_{b}$ and
$t_{c}$ are all functions of $T$ only, given by Eq.(3.6).

To compare the AOS and BMM/GM approaches, in Section II we first presented the
AOS model, and discuss how to uniquely fix the two Dirac observables
$\delta_{i}$’s [cf. Eqs.(2.15) and (2.16)] in the extended phase space. In the
large mass limit, these conditions lead to $\delta_{i}$’s given explicitly by
Eq.(2.26).

In the BMM/GM model, the black and white horizons, in general, all exist, and
naturally divide the whole spacetime into the external and internal regions,
where $T$ is timelike in the internal region and spacelike in the external
region. In Section III, we briefly introduce the BMM/GM approach and focused
on studies of the external region of the spacetime. We found that the
asymptotical flatness condition of the spacetime requires

$\Omega_{b}\geq 0,$ (5.3)

where $\Omega_{b}$ is defined in Eq.(3.19), which excludes the BMM choice
$\delta_{i}=f_{i}\left(O_{i}\right)$ BMM19 , for which we always have
$\Omega^{\text{BMM}}_{b}<0$, as shown explicitly by Eq.(III.1). Despite the
significant difference of the metrics of the AOS and BMM/GM models, we found
that, to the leading order, the asymptotical behavior of the spacetime in the
two models is universal and independent of the mass parameter $m$ for the
curvature invariants [cf. Eqs.(III.2) and (III.3)]. But, to the next leading
order, they are different. In particular, the Kretschmann scalar behaves as

$K\simeq\frac{A_{0}}{r^{4}}+{\cal{O}}\left(\frac{1}{r^{4}\xi}\right),$ (5.4)

as $r\rightarrow\infty$, where $A_{0}$ is a constant and independent of $m$,
and $r$ the geometric radius of the two-spheres. For the case
$\alpha_{1}\not=0$, we have
$\xi=\frac{2}{\alpha_{0}}\ln\left(\frac{r}{2m}\right)$, and for
$\alpha_{1}=0$, we have $\xi=\left(\frac{r}{2m}\right)^{b_{o}}$. Here
$\alpha_{1}$ is defined in Eq.(III.1). The differences from the next leading
order can be understood more clearly from the metric and the effective energy-
momentum tensor, given, respectively, by Eqs.(3.35), (III.2), (3.48) and
(III.3). On the other hand, asymptotically the AOS solution takes the global
monopole form (3.36), found previously in a completely different content BV89
. Nevertheless, the leading behavior of the Kretschmann scalar in both cases
is in sharp contrast to the classical case BBCCY20 ; AO20 , for which we have
$K_{\text{GR}}=48m^{2}/r^{6}$.

In Section IV, we conducted our studies on the internal region of the
spacetime. We first showed that the quantum gravitational effects near the
black hole horizon are negligible for massive black holes, and both the
Kretschmann scalar and Hawking temperature are indistinguishable from those of
GR, as shown explicitly by Figs. 4 and 6, and Eq.(4.22). However, despite the
fact that all the physical quantities are finite, and the Schwarzschild black
hole singularity is replaced by a transition surface whose radius is always
finite and non-zero, the internal region near the transition surface is
dramatically different from that of the AOS model in several respects: (1)
First, the location of the maxima of the curvature invariants, such as the
Kretschmann scalar is displaced from the transition surface as shown
explicitly by Figs.7, 8, 12 and 13. Detailed investigations of the metric
components reveal that this is because of the dependence of the two Dirac
observables $\delta_{i}$’s on the 4D phase space of the Ashtekar variables
($b,c,p_{b},p_{c}$), which considerably modifies the structure of the
spacetime. In particular, we plotted the metric components $g_{t_{c}t_{c}}$
and $g_{xx}$, respectively in Figs. 10 and 11, and then compared them with
those given in the AOS model, where the maxima are always located at the
transition surface AOS18a ; AOS18b ; AO20 . (2) The maxima of these curvature
invariants depend on the choice of the mass parameter $m$. In particular,
Table 1 shows such dependence for the Kretschmann scalar, which also shows
that such dependence is absent in the AOS model. (3) The location of the white
hole horizon is very near to the transition surface, and the ratio of the two
horizon radii is much smaller than $1$, and depends sensitively on $m$ as
shown in Table.2. All these results are significantly different from those
obtained in the AOS model.

In review of the results presented in this paper, it is clear that further
investigations are highly demanded for LQBH models, in which the two
polymerization parameters $\delta_{b}$ and $\delta_{c}$ appearing in Eq.(5.1)
are considered as Dirac observables of the 4-dimensional phase space, spanned
by ($b,p_{b};c,p_{c}$), before accepting them as viable LQBH models in LQG. In
particular, in Giesel:2021rky  the consistent gauge-fixing conditions in
polymerized gravitational systems were studied, and it would be very
interesting to check how these conditions affect the results presented in this
paper as well as results obtained in other LQBH models.

Notes-in-addition: When we were finalizing our manuscript, we came across
three very interesting and relevant articles NGM22a ; GM22 ; NGM22b . We will
briefly comment on them here. First, in NGM22a  the authors studied the
physical meaning of the three integration constants, $C_{1},C_{2}$ and
$\bar{p}^{0}_{c}$, obtained from the integration of the three dynamical
equations for the variables $c,b$ and $p_{c}$, respectively, and found that
$C_{1}$ is related to the location of the transition surface, $C_{2}$ can be
gauged away by the redefinition of time $t\rightarrow t+t_{0}$, where $t_{0}$
is a constant, while $\bar{p}^{0}_{c}$ is related to the mass parameter. A
similar consideration was also carried out in GSSW2020  but for the BMM
polymer black hole solution BMM19a . Second, in GM22  the authors studied the
integrability of
$G_{b}(t_{b})\equiv\int^{0}_{t_{b}}F_{cb}(t^{\prime}_{b})dt^{\prime}_{b}=\int^{0}_{t_{c}}F_{bc}(t^{\prime}_{c})dt^{\prime}_{c}\equiv
G_{c}(t_{c})$, the invertibility of $t_{i}=G_{i}^{-1}[G_{j}(t_{j})]$, and the
overlap of the images of $G_{i}$’s. It was shown that $F_{ij}$’s are always
integrable so that $G_{i}$’s always exist. The images of $G_{i}$’s can be
always made overlapped by using the redefinitions of the two time-variables
$t_{i}^{\prime}=t_{i}+t_{i}^{0}$. In addition, $t_{i}=G_{i}^{-1}$ is always
invertible except at the zero points $G_{i}(t_{i}^{\text{ext}})=0$. Moreover,
these zero points never correspond to the same moment $T$, so at least one of
the two $G_{i}$’s is invertible at any given moment $T$. It must be noted that
all the studies carried out in GM22  were restricted to the internal region.
When restricting our studies to this region, our results are consistent with
theirs, whenever the problems of the integrability, invertibility and overlap
of the images, all studied in GM22 , are concerned. Finally, in NGM22b , the
authors considered the quantization of the AOS extended phase space model, and
found the conditions that guarantee the existence of physical states in the
regime of large black hole masses, among other interesting results.

## Acknowledgements

We would like to thank G.A. Mena Marugán and P. Singh for their valuable
discussions, comments and suggestions. We also acknowledge the support
provided by the Baylor University High Performance and Research Computing
Services for carrying out the numerical computations of this paper. This work
is supported in part by the Zhejiang Provincial Natural Science Foundation of
China under Grant No. LR21A050001 and LY20A050002, the National Key Research
and Development Program of China Grant No.2020YFC2201503, the National Natural
Science Foundation of China under Grant No. 11675143 and No. 11975203, and the
Fundamental Research Funds for the Provincial Universities of Zhejiang in
China under Grant No. RF-A2019015.

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