[table]capposition=top 11institutetext: Indian Institute of Technology
Kharagpur, West Bengal, India

# Verifiable and Energy Efficient Medical Image Analysis with Quantised Self-
attentive Deep Neural Networks

Rakshith Sathish 11 Swanand Khare 11 Debdoot Sheet 11

###### Abstract

Convolutional Neural Networks have played a significant role in various
medical imaging tasks like classification and segmentation. They provide
state-of-the-art performance compared to classical image processing
algorithms. However, the major downside of these methods is the high
computational complexity, reliance on high-performance hardware like GPUs and
the inherent black-box nature of the model. In this paper, we propose
quantised stand-alone self-attention based models as an alternative to
traditional CNNs. In the proposed class of networks, convolutional layers are
replaced with stand-alone self-attention layers, and the network parameters
are quantised after training. We experimentally validate the performance of
our method on classification and segmentation tasks. We observe $50-80\%$
reduction in model size, $60-80\%$ lesser number of parameters, $40-85\%$
fewer FLOPs and $65-80\%$ more energy efficiency during inference on CPUs. The
code will be available at https://github.com/Rakshith2597/Quantised-Self-
Attentive-Deep-Neural-Network.

###### Keywords:

Self-attention Quantisation Medical image analysis.

##  1 Introduction

Deep neural networks have played a significant role in medical image analysis.
Since the advent of UNet[18] to UNetr[5], the performance of neural networks
on various tasks like classification, segmentation, and restoration has
improved considerably. Deeper and broader convolutional neural networks
generally show an improvement in performance at the cost of an increase in the
number of learnable parameters, model size and total floating-point operations
performed during a single forward pass of the data through the network.
Moreover, these models require specialised high-performance hardware even
during inference. This reliance on larger models and high-performance hardware
hinders the last-mile delivery of AI solutions to improve the existing
healthcare system, especially in resource constrained developing and under-
developed countries.

Challenges: The performance and trustability of deep neural network-based
methods are of utmost importance, especially in the medical domain. The
performance of these methods decreases as we try to reduce the number of
learnable parameters in the model. As an example, in the case of image
classification, deeper networks have been shown to be superior to shallow
networks with fewer parameters [6, 8]. Despite the good performance measured
in terms of quantitative evaluation metrics, deep neural network (DNN) are
known to make the right decision for the wrong reasons [4]. This limits the
trustability of DNN-based frameworks in practical application. Additionally,
the black box nature of the convolutional neural networks makes them
unreliable for clinical applications. Developing a method that relies on fewer
parameters and is clinically verifiable is a challenging task. Also, an
efficient model is expected to replicate the performance during inference at a
resonable execution speed even in the absence of GPUs.

Attention-based networks were proposed to augment DNNs with explainability in
the case of natural images. However, due to the inherent differences in the
nature of images, we cannot assume an equivalent performance in the medical
images. As an example, in detecting objects in natural images, the objects of
interest often have a well-defined shape and structure, which are absent in
the case of medical images. In the case of medical image classification, the
biomarkers are usually unstructured pathologies with variable appearance. In
this work, we try to verify the effectiveness of replacing convolutions with
attention in neural networks for medical images.

Related works: Transformers [21], based solely on attention mechanisms has
revolutionised the way models are designed for natural language tasks.
Motivated by their success, [27], [26], [17] and [25] explored the possibility
of using self-attention to solve various vision tasks. Among these, the stand-
alone self-attention proposed by [17] established that self-attention could
potentially replace convolutional layers altogether. Even though it is
efficient compared to other DNNs, such models can be further improved by
quantising the weights and activations of the networks [15]. The quantisation
of deep neural networks has shown significant progress in recent years
[24][1]. The ability to quantise the neural network trained in high precision
without substantial loss in performance during inference simplifies the
process.

Our Approach: Inspired by the success of [17] in natural image classification
tasks, we propose the design of a new class of networks for medical image
classification and segmentation, in which we replace the convolution layers
with self-attention layers. Furthermore, we optimise the networks for
inference by quantising the parameters thereby decreasing energy consumption.
To the best of our knowledge, a quantised fully self-attentive network for
classification and segmentation of medical images and comparison with its
convolutional counterparts has not been attempted so far. Schematic overview
of the proposed method is illustrated in Fig. 1.

Figure 1: Overview of the proposed method. Convolutional layers in deep neural
network architectures are replaced with self-attention layers and networks
with parameters at FP32 precision are trained till convergence. To optimise
the model for storage and faster inference, the network parameters are
quantised without loss in performance.

##  2 Method

###  2.1 Stand-alone self-attention

Attention was introduced by [3] for a neural machine translation model.
Attention modules can learn to focus on essential regions within a context,
making it an important component of neural networks. Self-attention [21] is
defined as attention applied to a single context instead of across multiple
contexts; that is, Key, Query and Values are derived from the same context.
[17] introduced the stand-alone self-attention layer, which can replace
convolutions to construct a fully attentional model. Motivated by the initial
success of [17] in natural images, we explore the feasibility of using such
modules in the proposed class of networks for medical image analysis.

To compute attention for each pixel
$\mathbf{x}_{i,j}\in\mathbb{R}^{C_{in}\times 1\times 1}$ in an image or an
activation map, local regions with spatial extent $h\times w$ around
$\mathbf{x}_{i,j}$ are used to derive the keys and values. Learned linear
transformations are performed on $\mathbf{x}_{i,j}$ and its local regions to
obtain query ($\mathbf{Q}$), keys ($\mathbf{K}$) and values ($\mathbf{V}$) as

$\mathbf{Q}=\mathbf{W_{Q}}\mathit{\mathbf{x}_{i,j}}$ (1)
$\mathbf{K}=\mathbf{W_{K}}\mathit{\mathbf{x}_{h,w}}$ (2)
$\mathbf{V}=\mathbf{W_{V}}\mathit{\mathbf{x}_{h,w}}$ (3)

where $\mathbf{W_{Q}}\in\mathbb{R}^{C_{out}\times C_{in}}$,
$\mathbf{W_{K}}\in\mathbb{R}^{C_{out}\times C_{in}}$ and
$\mathbf{W_{V}}\in\mathbb{R}^{C_{out}\times C_{in}}$ are learnable
transformation matrices and $\mathbf{x}_{h,w}\in\mathbb{R}^{C_{in}\times
h\times w}$ is the local region centered at $\mathbf{x}_{i,j}$.

Self-attention on its own does not encode any positional information, which
makes it permutation equivariant. Relative positional embedding [19] as used
in [17] are incorporated into the attention module. The keys
$\mathbf{K}\in\mathbb{\rm I\\!R^{C_{out}}}\times h\times w$ are split into
$\mathbf{K}_{1},\mathbf{K}_{2}\in\mathbb{\rm I\\!R^{C_{out/2}\times h\times
w}}$ each and column offset $\mathbf{E}_{col}$ and row offset
$\mathbf{E}_{row}$ of the positional embedding are added to these separately.
After this, we concatenate $\mathbf{K}_{1},\mathbf{K}_{2}$ to obtain a new key
($\mathbf{K}\textquoteright\in\mathbb{\rm I\\!R^{C_{out}\times h\times w}}$)
which contains the relative spatial information of pixels in the local region
of size $h\times w$. Thus, the relative spatial attention for a pixel $x_{ij}$
is mathematically defined as in Eq. 4 and is graphically illustrated in Figure
2.

$\mathbf{y_{i,j}}=\sum_{\\{u,v\\}\in
N_{h,w}(i,j)}\mathit{softmax}_{u,v}(\mathbf{Q}_{i,j}^{\top}\mathbf{K}_{u,v})\mathbf{V}_{u,v}$
(4)

where $N_{h,w}(i,j)$ is the neighbourhood of size $h\times w$ centered at
$(i,j)$.

Figure 2: Self-attention mechanism with local context. Operations are
performed on a per-pixel basis to compute attention as shown in the figure.
Linear transformations for obtaining query, keys and values are implemented
using 2D convolution (Conv2d) operation. The learnt relative positional
embedding are added to the keys to incorporate the inter-pixel relationships
within the local context.

We use these attention blocks instead of 2D convolutional blocks in our
networks. During training, all the weights and activations are represented and
stored with a precision of FP32. The parameters are quantised to INT8
precision for inference.

###  2.2 Quantisation of network parameters

We perform quantisation using the FBGEMM (FaceBook GEneral Matrix
Multiplication) [10] backend of PyTorch for x86 CPUs, which is based on the
quantisation scheme proposed by [9]. In order to be able to perform all the
arithmetic operations using integer arithmetic operations on quantised values,
we require the quantisation scheme to be an affine mapping of integers $q$ to
real numbers $r$ as

$r=S(q-Z)$ (5)

where $S$ and $Z$ are quantisation parameters. We have employed a post-
training $8$-bit quantisation of all the weights and operations for our
proposed model.

###  2.3 Network architecture

Classification: The architecture of the proposed classification network is
illustrated in Fig. 3(a) with the details of the constituent modules in Fig.
3(c). The network consists of a series of alternating attention blocks and
attention down blocks followed by fully-connected linear layers. The feature
maps are downsampled using the max-pooling operation. The size of the output
linear layer is equal to the number of target classes. The network is trained
to perform multi-label classification using a binary cross-entropy loss.

(a) Classification network (SaDNN-cls)

(b) Segmentation network (SaDNN-seg)

(c) Network components

Figure 3: Architecture of the proposed Self-attentive Deep Neural Networks
(SaDNN). Detailed architecture of the networks for classification and
segmentation are shown in (a) and (b) respectively. Components of the various
blocks in these networks are detailed in (c).

Segmentation: The proposed segmentation network has a fully attention-based
encoder-decoder architecture as shown in Fig. 3(b). The encoder unit consists
of stand-alone self-attention blocks with ReLU activation and max-pooling
operations with the number of feature maps increasing progressively with each
attention block. The decoder consists of attention blocks and max-unpooling
operations. The size of activation maps of the decoder matches with the
corresponding layer in the encoder. The unpooling operations are performed
using the indices transferred from the pooling layers in the encoder. To
prevent the loss of subtle information, we employ activation concatenation in
the decoder, similar to UNet [18]. The network is trained using soft dice loss
[12].

##  3 Experiments

###  3.1 Datasets

Classification: To evaluate the performance of the fully self-attentive
network (SaDNN-cls) on classification tasks, we have used the NIH Chest X-ray
dataset of $14$ Common Thorax Disease [22]. The dataset comprises $112,120$
frontal-view X-ray images of $30,805$ patients with fourteen disease labels.
These disease classes can co-occur in an image; therefore, the classification
problem is formulated as multi-label classification. The train, validation and
test split provided in the dataset was used for the experiments.

Segmentation: A subset of the medical segmentation decathlon dataset [2] is
used to evaluate the performance of the proposed fully-attentive network
(SaDNN-seg) for liver segmentation. Out of the $131$ ground truth paired $3$D
CT volumes-Ground truth pairs available in the dataset, $80$ per cent were
randomly chosen for training, and the remaining $20$ per cent were used for
testing.

###  3.2 Implementation Details

Training: The proposed models were trained using an Adam Optimiser [11] with a
learning rate of $1\times 10^{-4}$. The models for classification task were
trained for $15$ epochs and the models for segmentation were trained for $25$
epochs.

Baselines: Performance of the proposed quantised self-attention network for
the classification task is compared with ResNet-18, ResNet-50 and their
$8-$bit quantised versions q-ResNet-18, and q-ResNet-50. To assess the
performance of the segmentation network, we chose a modified UNet[18] (UNet-
small) and SUMNet[13] architecture trained on the same dataset split and their
quantised versions q-UNet-small and q-SUMNet as baselines.

System specifications: All networks were trained on a high-performance server
with a NVIDIA $V100$ GPU, $x86_{6}4$ Intel(R) Xeon(R) Silver $4110$ CPU @
$2.10GHz$, $96$ GB RAM and $1$ TB HDD running on Ubuntu $18.01.1$ LTS OS. The
inference of quantised models was also performed on the same class of CPUs.

##  4 Results and Discussions

###  4.1 Qualitative Analysis

visualisation of predictions of the proposed q-SaDNN-seg network and its
unquantised version SaDNN-seg are presented in Fig. 4. Over-segmented regions
in the predicted segmentation maps are marked in green, under-segmented
regions are marked in red and correctly segmented region is shown in white. We
observe that the tendency of the original unquantised network SaDNN-seg to
over-segment is significantly reduced post quantisation. However, the
quantisation of network parameters causes the q-SaDNN-seg to under-segment the
target organ. This is reflected in the slightly lower Dice coefficient (DSC)
of the proposed model as seen in Table 6.

(a) Sample 1

(b) Ground Truth

(c) SaDNN-seg

(d) q-SaDNN-seg

(e) Sample 2

(f) Ground Truth

(g) SaDNN-seg

(h) q-SaDNN-seg

Figure 4: Comparison of segmentation predictions. Figure shows sample input CT
images in (a) and (e) with the corresponding ground truths of liver in (b) and
(f) respectively. Segmentation map as predicted by SaDNN-seg, with the over-
segmented region marked in green and under-segmented region marked in red are
presented in (c) and (g) for the two sample images. Similar visualisation of
segmentation by the proposed q-SaDNN-seg are presented in (d) and (h).

###  4.2 Quantitative Analysis

The performance of the proposed quantised fully self-attentive network and
baselines for multi-label classification task is reported in terms of accuracy
in Table 6. It can be observed that the proposed network can achieve
performance slightly better than the existing deep residual convolutional
neural networks. Table 6 shows the comparison of the proposed segmentation
network with the baselines in terms of DSC. The proposed quantised network
performs almost as good as the quantised versions of the baseline
convolutional neural networks.

Model | Accuracy
---|---
ResNet-18 | 0.89
q-ResNet-18 | 0.88
ResNet-50 | 0.84
q-ResNet-50 | 0.83
SaDNN-cls (ours) | 0.90
q-SaDNN-cls (ours) | 0.89

Figure 5: Evaluation of classification

Model | DSC
---|---
UNet-small | 0.88
q-UNet-small | 0.88
SUMNet | 0.89
q-SUMNet | 0.89
SaDNN-seg (ours) | 0.88
q-SaDNN-seg (ours) | 0.85

Figure 6: Evaluation of segmentation

###  4.3 Computational Analysis

The DNNs used for the experiments exhibited superior classification and
segmentation performance in terms of quantitative metrics, but they require a
considerable amount of computations and memory access operations to be
performed. Deploying a framework which needs excessive computations to be
performed results in large energy consumption, which is not feasible in
diverse resource-constrained scenarios. Therefore, it is key to have an
energy-efficient model without degradation in performance. A rough estimate of
energy cost per operation in $45nm$ $0.9V$ IC design can be calculated using
Table 1 presented in [7, 23, 14].

Table 1: Approximate energy cost in $45nm$ $0.9V$ for different multiplication and addition operations Operation | Energy (pJ)
---|---
MUL | ADD
8-bit INT | 0.2 pJ | 0.03 pJ
16-bit FP | 1.1 pJ | 0.40 pJ
32-bit FP | 3.7 pJ | 0.90 pJ

The number of multiplication and addition operations in a standalone self-
attention layer [20] can be calculated as

$Ops_{mul}=Ops_{add}=2b^{2}c$ (6)

where $b$ is the block (local region) size and $c$ is the number of channels.

The total number of parameters, MACs, energy consumed during forward pass and
model size of the proposed q-SaDNN-cls and q-SaDNN-seg networks are reported
in Table 2 and Table 3 with graphical comparisons in Fig. 7. Models with the
least area in the radar charts are more efficient. The proposed q-SaDNN-cls
network is $58.59\%$ smaller than quantised ResNet-18 and $80.75\%$ smaller
than quantised ResNet-50 in terms of model size. In terms of total MAC units,
the the propsed networks have $65.93\%$ fewer MACs than ResNet-18, $85.32\%$
fewer than ResNet-50. Similarly, in terms of the total trainable parameters,
the proposed networks have $59.17\%$ lesser parameters than ResNet-18 and
$80.62\%$ lesser than ResNet-50.

Table 2: Comparison of classification networks Model | #Params | MACs | Model size | Energy
---|---|---|---|---
ResNet-18 | 11.17 M | 9.10 G | 44.79 MB | 20.93 J
q-ResNet-18 | 11.17 M | 9.10 G | 11.40 MB | 1.04 J
ResNet-50 | 23.53 M | 21.11 G | 94.45 MB | 48.53 J
q-ResNet-50 | 23.53 M | 21.11 G | 24.52 MB | 2.41 J
SaDNN-cls | 4.56 M | 3.10 G | 18.30 MB | 7.13 J
q-SaDNN-cls | 4.56 M | 3.10 G | 4.72 MB | 0.35 J

(a) Classification

(b) Segmentation

Figure 7: Graphical comparison of proposed networks. Figure shows radar chart
based comparison of proposed (a) classification network and (b) segmentation
network in terms of number of parameters, MACs, model size and energy. The
model with the least area within the plot is the best one.

Similar improvement in efficiency of computing can be observed in the case of
segmentation as well. The segmentation network q-SaDNN-seg is $73.06\%$
smaller than q-UNet-small and $64.94\%$ smaller than q-SUMNet in terms of
model size. In terms of total MAC units, the q-SaDNN-seg has $34.94\%$ fewer
than SUMNet. In terms of the trainable parameters, q-SaDNN-seg has $74.37\%$
lesser parameters than UNet-small and $66.21\%$ lesser than SUMNet. It is to
be noted that the proposed models are superior in terms energy consumption as
well.

Table 3: Comparison of segmentation networks Model | #Params | MACs | Model size | Energy
---|---|---|---|---
UNet-small | 31.03 M | 218.60 G | 118.48 MB | 502.78 J
q-UNet-small | 31.03 M | 218.60 G | 29.77 MB | 25.13 J
SUMNet | 23.53 M | 425.98 G | 91.07 MB | 979.75 J
q-SUMNet | 23.53 M | 425.98 G | 22.88 MB | 48.97 J
SaDNN-seg | 7.95 M | 277.15 G | 30.47 MB | 637 J
q-SaDNN-seg | 7.95 M | 277.15 G | 8.02 MB | 31.87 J

(a) Test image

(b) Saliency map

Figure 8: Figure shows (a) a sample image from the test set used in our
experiments with the clinically relevant region as provided in the dataset
marked in green and (b) saliency map of q-SaDNN-cls. Regions shown in red in
the saliency map are perceived as most important and those in blue to be least
important by the network during prediction.

###  4.4 Analysis of clinical relevance

Validating the results of the model with respect to clinically relevant
information to provide some explanations for the decision made by the model is
an important factor that determines trustability. The clinically relevant
region provided in the NIH Chest X-ray dataset as marked by a radiologist and
the saliency map based explanation generated using RISE[16] for the proposed
quantised self-attention deep neural network for classification are shown in
Fig. 8. It can be observed that the proposed model focuses on the clinically
relevant region while making the decision.

##  5 Conclusion

We proposed a class of quantised self-attentive neural networks which can be
used for medical image classification and segmentation. In these networks,
convolutional layers are replaced with attention layers which have fewer
learnable parameters. Computation of attention while considering a small local
region surrounding a pixel prevents degradation of performance despite the
absence of local feature extraction which is typically performed in a CNN. We
show that our energy efficient method achieves performance at par with the
commonly used CNNs with fewer number of parameters and model size. These
attributes make our proposed models affordable and easy to adopt in resource
constrained settings.

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# Revisiting quantum black holes from effective loop quantum gravity

Geeth Ongolea<EMAIL_ADDRESS>Parampreet Singhb<EMAIL_ADDRESS>Anzhong Wanga<EMAIL_ADDRESS>a GCAP-CASPER, Physics Department,
Baylor University, Waco, Texas, 76798-7316, USA
b Department of Physics and Astronomy, Louisiana State University, Baton
Rouge, LA 70803, USA

###### Abstract

We systematically study a family of loop quantizations for the classical
Kruskal spacetimes using the effective description motivated from loop quantum
gravity for four generic parameters, $c_{o},m,\delta_{b}$ and $\delta_{c}$,
where the latter two denote the polymerization parameters which capture the
underlying quantum geometry. We focus on the family where polymerization
parameters are constant on dynamical trajectories, and of which the Ashtekar-
Olmedo-Singh (AOS) and Corichi-Singh (CS) models appear as special cases. We
study general features of singularity resolution in all these models due to
quantum gravity effects and analytically extend the solutions across the white
hole (WH) and black hole (BH) horizons to the exterior. We find that the
leading term in the asymptotic expansion of the Kretschmann scalar is
$r^{-4}$. However, for AOS and CS models black holes with masses greater than
solar mass the dominant term behaves as $r^{-6}$ for the size of the
observable universe and our analysis can be used to phenomenologically
constrain the choice of parameters for other models. In addition, one can
uniquely fix the parameter $c_{o}$ by requiring that the Hawking temperature
at the BH horizon to the leading order be consistent with its classical value
for a macroscopic BH. Assuming that the BH and WH masses are of the same
order, we are able to identify a family of choices of $\delta_{b}$ and
$\delta_{c}$ which share all the desired properties of the AOS model.

##  I Introduction

Black holes (BHs) initially were only discovered as solutions to Einstein’s
General Relativity (GR) [1] but now observations have confirmed their
existence [2, 3]. However to understand these objects in entirety, GR is
insufficient as it predicts its own demise near the singularities present
inside BHs [4]. Spacetimes near these singularities enter the Planckian regime
and it is widely believed that a quantum theory of gravity is necessary to
understand these esoteric objects.

Loop quantum gravity (LQG) is a non-perturbative theory of quantum gravity
[5]. Its techniques have been used to resolve the cosmological singularities,
which is now a well-established research field, usually referred to as loop
quantum cosmology (LQC) [6, 7, 8]. Since the interior of a static BH is
isometric to the vacuum Kantowski-Sachs (KS) cosmological spacetime,
techniques from LQC have been borrowed to study loop quantum black holes
(LQBHs) [9]. A majority of LQBH models describe the spacetimes in an effective
or semi-classical fashion, as such a description has been proved very
successful in LQC [10, 11, 12], and captures the leading-order quantum
corrections with the introduction of polymerization parameters. In particular,
it shows explicitly that the big bang singularity is replaced by a quantum
bounce (see for eg. [13, 14, 15]) and the subsequent cosmological
perturbations are consistent with current observations (see for eg. [16, 8]).
It is the general belief in the community that such an effective description
should be also applicable to LQBHs, although a detailed derivation of the
effective description from the full quantum dynamics of LQBHs is still absent.
In this manuscript our analysis will assume the validity of this effective
spacetime description.

In the quantum theory of LQBHs, the elementary variables are holonomies of
connection variables which contribute to trigonometric functions of the
connection at the level of the Hamiltonian. Different quantizations of LQBH
models differ in their choices of the two polymerization parameters,
$\delta_{b}$ and $\delta_{c}$, which capture the edge length of loop in the
quantization procedure. At an effective level this amounts to the following
replacements of the two symmetry reduced Ashtekar-Barbero connection variables
$b$ and $c$ [17, 9]

$\displaystyle b\rightarrow\frac{\sin(\delta_{b}b)}{\delta_{b}},\quad
c\rightarrow\frac{\sin(\delta_{c}c)}{\delta_{c}},$ (1.1)

in the classical Hamiltonian. It is clear that when $\delta_{i}\rightarrow
0\;(i=b,c)$, the classical Hamiltonian is recovered and quantum effects are
expected to be negligibly small. On the other hand, such effects will become
large when they are large. Generally speaking, $\delta_{b}$ and $\delta_{c}$
are functions of the phase space variables $b$, $c$ and their corresponding
canonical conjugates $p_{b}$ and $p_{c}$, which are connected with the
spacetime metric through Eq.(2.1). Different choices of $\delta_{b}$ and
$\delta_{c}$ are frequently referred to as different schemes, and various ones
are available in the literature. Among them, three major schemes can be
identified:

  * •

$\mu_{o}$-(like) schemes: Inspired by the $\mu_{o}$ scheme in LQC [18, 19], in
this scheme the polymerization parameters $\delta_{b}$ and $\delta_{c}$ are
fixed as constants, see, for example, [20, 21, 22, 23, 24, 25, 26, 27, 28,
29], and references therein. In these studies, the dependence of $\delta_{i}$
on some physical parameters involved in the models was considered. However, in
all of them, either some inconsistencies or unexpected properties were
identified. In particular, it was found that in some models physical
quantities depend on the choice of the fiducial cell size, while in others
singularity resolutions cannot be consistently identiﬁed with a curvature
scale. For details, we refer the reader to [30]. Note that of the above
choices, Corichi-Singh (CS) model overcome various problems [24], but results
in a highly asymmetric mass of the white hole (WH) formed after the bounce.
Later different types of models which result in symmetric value of masses of
BH and WH have been developed [31, 30].

  * •

$\bar{\mu}$-(like) schemes: Inspired by the $\bar{\mu}$ scheme in LQC [14]
here the polymerization parameters are chosen to be functions of the phase
space variables. This was first implemented for LQBHs in [32] and adapted
later in [33]. Although this scheme has alleviated all the shortcomings of the
$\mu_{o}$-scheme in cosmology [6], it fails when applied to LQBHs [30]. In
particular, large quantum corrections appear near BH horizons, in which the
gravitational fields are already very weak classically. Basically, this
quantization does not distinguish between the coordinate and the physical
singularity which is illustrated in Kasner spacetimes too [34]. Furthermore,
recent work has shown that these effects are, in fact, so strong that BH/WH
horizons do not exist at all [35, 36]. Instead, only transition surfaces
exist, and the number of such surfaces are infinite. As a result, the
corresponding spacetime is geodesically complete. This result turns out to be
consistent with earlier studies on $\bar{\mu}$ quantization of Kantowski-Sachs
cosmological models _with matter_ where coordinate singularity does not exist,
central singularity is eliminated and the effective loop quantum spacetime is
geodesically complete [37].

It should be noted that such large quantum effects are closely related to the
use of the Kantowski-Sachs metric as describing the internal spacetimes of
BHs. In particular, in the Kantowski-Sachs spacetime the physical length
perpendicular to the two-spheres of the classical Schwarzschild BH vanishes
not only at the BH singularity but also at its event horizon. Then, the
polymerization will lead to significant quantum effects at both locations. To
cure the singularity, such large effects are essential. However, these large
effects at BH horizons shall lead to inconsistency physically, especially when
massive BHs are considered. This is because near such massive BH horizons, the
gravitational fields are classically very weak, and quantum gravitational
effects are expected to be negligible.

  * •

Dirac observables: In this scheme the polymerization parameters are particular
functions of the phase space variables such that they are constant along the
trajectories of the dynamical equations. To counter the problems faced by the
$\mu_{o}$ and $\bar{\mu}$ schemes, Dirac-observable scheme in the extended
phase space was first introduced in [38, 30, 39]. In these works the two
polymerization parameters were uniquely fixed at the transition surface. For
massive BHs, they are given by [30]

$\displaystyle\delta^{\text{(AOS)}}_{b}$ $\displaystyle=$
$\displaystyle\left(\frac{\sqrt{\Delta}}{\sqrt{2\pi}\gamma^{2}m}\right)^{1/3},$
$\displaystyle\delta^{\text{(AOS)}}_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{2L_{o}}\left(\frac{\gamma\Delta^{2}}{4\pi^{2}m}\right)^{1/3},$
(1.2)

where $\gamma$ and $m$ denote, respectively, the Barbero-Immirzi and mass
parameters, $\Delta\left(\equiv 4\sqrt{3}\pi\gamma\ell_{p}^{2}\right)$ denotes
the minimal area gap and $L_{o}$ the size of the fiducial cell, while
$\ell_{p}$ is the Planck length. With the above choice of polymerization
parameters, it was found that the corresponding LQBH has the following
desirable properties: 1) The physical quantities and properties are
independent of $L_{o}$. 2) The spacetime near the transition surface is
symmetric. 3) There exists a universal mass-independent upper bound on the
curvature invariants of the effective spacetime at the transition surface. 4)
No mass amplification of the WH exists. 5) There exist negligible quantum
corrections at classical scales, including the locations of massive BH/WH
horizons.

In addition, working within the four-dimensional phase space, spanned by the
four Ashtekar phase variables ($b,p_{b};c,p_{c}$), Dirac observables at the
Hamiltonian level were also considered in [40, 41, 42, 43, 44, 45].

It is interesting to note that the Ashtekar-Olmedo-Singh (AOS) scheme can be
considered as a mixture of the $\mu_{o}$ and $\bar{\mu}$ like schemes. Then, a
natural question arises: Are there other choices of polymerization parameters
$\delta_{b}$ and $\delta_{c}$ for which similar desirable properties as those
of the AOS solution exist? The goal of this manuscript will be to explore this
avenue in detail. But before we go further, it is important to note the
following remark.

Remark: It is to be emphasized that the above comparison between different
schemes is restricted to Kruskal spacetimes in GR. Whether the noted
advantages and disadvantages extend to other black hole spacetimes and
gravitational collapse scenarios is far from clear. In particular, a recent
study showed that a $\mu_{o}$ like scheme would not permit formation of
trapped surfaces in gravitational collapse which $\bar{\mu}$ like scheme
permits generically [46]. In addition, the $\bar{\mu}$-scheme was also used in
the quantization of the Lemaitre-Tolman-Bondi metric in [47, 48, 49]. Yet,
another different approach was explored in [50, 51], and the model was further
studied in [52].

To generalize the AOS model, following the line of previous studies of LQBHs
[53, 28], we first make a general survey over the most general solutions of
the effective Hamiltonian which contains five free parameters, including
$\delta_{b}$ and $\delta_{c}$ which are treated as constant on dynamical
trajectories. Then, using the gauge freedom, we find that one of the five free
parameters can be gauged away simply by the shift of the timelike coordinate,
$T\rightarrow T+T_{0}$. Hence, only four of the five free parameters are
physically essential, denoted, respectively, by $c_{o}$, $m$, $\delta_{b}$ and
$\delta_{c}$. It should be noted that the physical meaning of $c_{o}$ was
studied in detail in [29]. However, unlike [29], in our current analysis we do
not adopt the AOS choice of $\delta_{b}$ and $\delta_{c}$ given by Eq.(• ‣ I),
and instead consider them as arbitrary constants. Among these four parameters,
$m$ is related to the mass parameter of the solution, and $c_{o}$
characterizes not only the position of the transition surface [29], but also
the Hawking temperature at the BH horizon. It should be noted that the four-
parameter solutions are valid only inside the BH and WH horizons,
$T_{\text{WH}}\leq T\leq T_{\text{BH}}$, where $T_{\text{WH}}$ and
$T_{\text{BH}}$ denote the locations of the WH and BH horizon, respectively.
To study the asymptotical behavior of the LQBH spacetime, we first extend the
solutions analytically beyond these surfaces [cf. Section II.C]. Since the
extension is analytical, it is also unique.

After analytically extending the solutions across the WH and BH horizons, we
find the leading term in the asymptotic expansion of the Kretschmann scalar is
$r^{-4}$. This result is well known for the AOS model and we find it to hold
for a family of models we study. In addition to $r^{-4}$ term there also
exists an $r^{-5}$ correction along with the $r^{-6}$ term that captures the
classical behavior. We find that the $r^{-4}$ term becomes dominant over the
$r^{-6}$ term only when $r>r_{c}$, where $r_{c}$ is the critical radius. In
addition, we also calculate the value $r_{c_{4}}$, the critical radius beyond
which the $r^{-4}$ term starts to dominate over the $r^{-5}$ term. Similarly,
we calculate $r_{c_{5}}$ to find the dominant term among $r^{-5}$ and $r^{-6}$
term. For both the AOS and CS models, we find that
$r_{c},r_{c_{4}},r_{c_{5}}>L_{\text{obs}}$ for solar mass BHs, where
$L_{\text{obs}}$ denotes the size of our observational Universe. This suggests
that the asymptotic behavior can still be well described by its classical
limit within our observable universe. In particular, the Kretschmann scalar
can be still considered as falling off like $r^{-6}$ in our observable
universe.

On the other hand, requiring that the deviation of the Hawking temperature at
the horizon of a massive BH from its classical counterpart be negligible, we
find that $c_{o}$ is uniquely fixed [cf. Eqs. (3.22) and (3.23)] to

$\displaystyle c_{o}=\frac{L_{o}\delta_{c}\gamma}{8m},$ (1.3)

which is precisely the choice made in [24, 38, 30, 39]. Once $c_{o}$ is fixed,
we find that the ratio of the BH and WH masses and the dependence of the
curvatures on $m$ at the transition surface are crucially dependent on the
choice of the two polymerization parameters $\delta_{b}$ and $\delta_{c}$. In
particular, we find that as long as they are all inversely proportional to
$m^{1/3}$, that is

$\displaystyle\delta_{b}=\alpha_{b}\left(\frac{\ell_{p}}{m}\right)^{1/3},\;\;\;L_{o}\delta_{c}=\alpha_{c}\left(\frac{\ell_{p}}{m}\right)^{1/3},$
(1.4)

where $\alpha_{b}$ and $\alpha_{c}$ are two $m$-independent otherwise
arbitrary dimensionless constants, the Kretschmann scalar is always
independent of $m$ [cf. Fig. 1], no matter what the values of $\alpha_{b}$ and
$\alpha_{c}$ are, although the amplitude of the Kretschmann scalar indeed
depends on specific values of $\alpha_{b}$ and $\alpha_{c}$. On the other
hand, when any of the dependence of these two parameters on $m$ is different
from $m^{-1/3}$, the amplitude of the Kretschmann scalar at the transition
surface will sensitively depend on the values of $m$, as shown explicitly in
Figs. 2 \- 4. In addition, simply requiring the ratio of the BH and WH masses
be one imposes a relation between $\alpha_{b}$ and $\alpha_{c}$ [cf.
Eq.(3.34)]. In particular, assuming that $\delta_{b}$ takes the form (1.4), we
find that $M_{\text{BH}}=M_{\text{WH}}$ leads to

$\displaystyle\alpha_{c}=\left(\frac{\gamma^{3}\ell_{p}}{2}\right)\alpha_{b}^{4},\;\left(M_{\text{BH}}=M_{\text{WH}}\right).$
(1.5)

Therefore, we identify a class of parameters, described by Eqs.(1.3) - (1.5)
with $\alpha_{b}$ being the free parameter, which share the same properties as
the AOS solution. The latter corresponds to the particular choice

$\displaystyle\alpha_{b}^{\text{AOS}}=\left(\frac{\sqrt{\Delta}}{\sqrt{2\pi}\gamma^{2}\ell_{p}}\right)^{1/3}.$
(1.6)

The rest of the paper is organized as follows: In Sec.II, starting with the KS
spacetime and its classical Hamiltonian, we first obtain the effective
Hamiltonian by using the replacements (1.1), and re-derive the five-parameter
solutions, where $B_{o},\;p_{c}^{o},\;c_{o}$ together with $\delta_{b}$ and
$\delta_{c}$ appearing in Eqs.(II.1) and (2.15) are the five parameters. But,
as shown there, $B_{o}$ can be eliminated by the simple replacement,
$T\rightarrow T+\hat{T}_{o}$, so only four of them are physical ones, where
$p_{c}^{o}$ is related to the mass parameter $m$ via the relation
$p_{c}^{o}=4m^{2}$. In the same section, we consider the marginally trapped
surfaces, and found that they can happen at $\dot{p}_{c}=0$ as well as at
$N\rightarrow\infty$. The former corresponds to the transition surface, and
the metric crosses it smoothly, while the latter represents BH and WH
horizons, as shown explicitly in Sec.II.2 and II.3. In Sec.II.3, the metric is
also analytically extended beyond these two horizons. In Sec.III.1, spacetimes
outside the horizons are studied, including the asymptotic and near horizon
regions, while in Sec.III.2, spacetimes across the transition surface and near
the WH horizon are investigated. The paper concludes in Sec.IV, where we
present our main findings.

##  II loop quantum black holes with constant polymerization parameters

The spacetime inside a classical spherically symmetric black hole can always
be written in the KS form [18]

$ds^{2}=-N^{2}dT^{2}+\frac{p_{b}^{2}}{L_{o}^{2}\left|p_{c}\right|}dx^{2}+\left|p_{c}\right|d\Omega^{2},$
(2.1)

where $N,\;p_{b},\;p_{c}$ are all functions of $T$ only, and
$d\Omega^{2}\equiv d\theta^{2}+\sin^{2}\theta d\phi^{2}$ with
$-\infty<T,\;x<\infty$, $\theta\in[0,\pi]$ and $\phi\in[0,2\pi]$. However, due
to the independence of the metric on $x$, the corresponding Hamiltonian is not
well defined, as it is involved in integration over $x$. Then, one usually
first introduce a fiducial cell with length $L_{o}$, so that $x\in[0,L_{o}]$.
Physics should not depend on the choice of $L_{o}$, so at the end of the day
we can always take the limit $L_{o}\rightarrow\infty$, without loss of the
generality. The functions $p_{b}$ and $p_{c}$ are the dynamical variables,
which satisfy the Poisson brackets

$\displaystyle\\{c,p_{c}\\}=2G\gamma,\quad\\{b,p_{b}\\}=G\gamma,$ (2.2)

where $G$ is the Newton gravitational constant, and $b$ and $c$ are the
corresponding phase space conjugate momenta of $p_{b}$ and $p_{c}$,
respectively.

It should be noted that the KS metric (2.1) is invariant under the gauge
transformations

$T=f(\hat{T}),\quad x=\alpha\hat{x}+x_{o},$ (2.3)

via the redefinition of the lapse function and the length of the fiducial
cell,

$\hat{N}=Nf_{,\hat{T}},\quad\hat{L}_{o}=\frac{L_{o}}{\alpha},$ (2.4)

where $f(\hat{T})$ is an arbitrary function of $\hat{T}$, $\alpha$ and $x_{o}$
are constants, and $f_{,\hat{T}}\equiv df/d\hat{T}$. Using the above gauge
freedom, we can always choose the lapse function as

$\displaystyle
N^{\text{GR}}=\frac{\gamma\;{\text{sgn}}(p_{c})\left|p_{c}\right|^{1/2}}{b}.$
(2.5)

Then, the corresponding classical Hamiltonian in GR is given by [30]

$\displaystyle H^{\text{GR}}[N^{\text{GR}}]$ $\displaystyle\equiv$
$\displaystyle N^{\text{GR}}\mathcal{H}^{\text{GR}}$ (2.6) $\displaystyle=$
$\displaystyle-\frac{1}{2G\gamma}\left(2c\;p_{c}+\left(b+\frac{\gamma^{2}}{b}\right)p_{b}\right).~{}~{}~{}$

###  II.1 General Spacetimes

To the leading order, it is expected that, as in LQC [6], the quantum effects
can be well captured by replacing the two canonical phase space variables $b$
and $c$ via the relations given by Eq.(1.1) in the classical lapse function
and Hamiltonian [5, 6], where the two polymerization parameters $\delta_{b}$
and $\delta_{c}$ are in general functions of the phase space variables
($b,p_{b};c,p_{c}$). However, in the current paper we shall focus ourselves
only to the cases in which they are constants. Inserting the above
replacements into Eqs.(2.5) and (2.6) we find that the effective Hamiltonian
for LQBHs is given by

$\displaystyle H_{\mathrm{eff}}$ $\displaystyle\equiv$ $\displaystyle
N\mathcal{H}=\frac{L_{o}}{G}\left(O_{b}-O_{c}\right),$ (2.7)

with

$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{\gamma\delta_{b}\;{\text{sgn}}\left(p_{c}\right)\sqrt{|p_{c}|}}{\sin{\left(\delta_{b}b\right)}},$
$\displaystyle O_{b}$ $\displaystyle\equiv$
$\displaystyle-\frac{p_{b}}{2\gamma
L_{o}}\left(\frac{\sin\left(\delta_{b}b\right)}{\delta_{b}}+\frac{\gamma^{2}\delta_{b}}{\sin\left(\delta_{b}b\right)}\right),$
$\displaystyle O_{c}$ $\displaystyle\equiv$
$\displaystyle\frac{\left|p_{c}\right|}{\gamma
L_{o}}\frac{\sin\left(\delta_{c}c\right)}{\delta_{c}}.$ (2.8)

It can be shown that $O_{b}$ and $O_{c}$ are the only two independent Dirac
observables that can be constructed in the current case. Note that
$\sqrt{|p_{c}|}$ is the geometric radius of the two-spheres with $T,x=$
Constant. Thus, without loss of the generality (as far as the effective semi-
classical approximations are concerned), we can always assume that $p_{c}\geq
0$. Then, the corresponding Hamiltonian equations for the four phase variables
$(b,p_{b};c,p_{c})$ are given respectively by

$\displaystyle\dot{c}$ $\displaystyle=$
$\displaystyle-2\frac{\sin\left(\delta_{c}c\right)}{\delta_{c}},$ (2.9)
$\displaystyle\dot{p}_{c}$ $\displaystyle=$
$\displaystyle{2}p_{c}\cos\left(\delta_{c}c\right),$ (2.10)
$\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\frac{\sin\left(\delta_{b}b\right)}{\delta_{b}}+\frac{\gamma^{2}\delta_{b}}{\sin\left(\delta_{b}b\right)}\right),$
(2.11) $\displaystyle\dot{p}_{b}$ $\displaystyle=$
$\displaystyle\frac{1}{2}p_{b}\cos\left(\delta_{b}b\right)\left(1-\frac{\gamma^{2}\delta_{b}^{2}}{\sin^{2}\left(\delta_{b}b\right)}\right).$
(2.12)

The integration of the first three equations yields [29]

$\displaystyle\sin\left(\delta_{c}c\right)$ $\displaystyle=$
$\displaystyle\frac{2c_{o}e^{-2T}}{1+c_{o}^{2}e^{-4T}},$ $\displaystyle p_{c}$
$\displaystyle=$ $\displaystyle
p_{c}^{o}\left(c_{o}^{2}e^{-2T}+e^{2T}\right),$
$\displaystyle\cos\left(\delta_{b}b\right)$ $\displaystyle=$ $\displaystyle
b_{o}\tanh\left(\frac{b_{o}T}{2}+B_{o}\right),$ (2.13)

where $c_{o},\;p_{c}^{o},\;B_{o}$ are integration constants, and

$\displaystyle b_{o}\equiv\sqrt{1+\gamma^{2}\delta_{b}^{2}}.$ (2.14)

Substituting the above solutions to the effective Hamiltonian (2.7), we find
that

$\displaystyle p_{b}$ $\displaystyle=$ $\displaystyle-
p_{b}^{o}\cosh^{2}\left(\frac{b_{o}T}{2}+B_{o}\right)$ (2.15)
$\displaystyle~{}~{}~{}~{}\times\Bigg{[}1-b_{o}^{2}\tanh^{2}\left(\frac{b_{o}T}{2}+B_{o}\right)\Bigg{]}^{1/2},~{}~{}~{}$

where $p_{b}^{o}\equiv\frac{4c_{o}p_{c}^{o}\delta_{b}}{b_{o}^{2}\delta_{c}}$.
However, since we assume $p_{c}\geq 0$, we must require $p_{c}^{o}>0$.
Corresponding to the above solution, it can be shown that the two Dirac
observables are given by

$\displaystyle O_{b}=O_{c}=\frac{2c_{o}p_{c}^{o}}{\gamma L_{o}\delta_{c}}.$
(2.16)

It can be also shown that

$\displaystyle N^{2}$ $\displaystyle=$ $\displaystyle
p_{c}^{o}\gamma^{2}\delta_{b}^{2}\frac{c_{o}^{2}e^{-2T}+e^{2T}}{1-b_{o}^{2}\tanh^{2}\left(\frac{b_{o}T}{2}+B_{o}\right)},$
$\displaystyle g_{xx}$ $\displaystyle=$
$\displaystyle\alpha^{2}\cosh^{4}\left(\frac{b_{o}T}{2}+B_{o}\right)$ (2.17)
$\displaystyle\times\frac{1-b_{o}^{2}\tanh^{2}\left(\frac{b_{o}T}{2}+B_{o}\right)}{c_{o}^{2}e^{-2T}+e^{2T}},$

where
$\alpha^{2}\equiv\frac{16p_{c}^{o}c_{o}^{2}\delta_{b}^{2}}{b_{o}^{4}L_{o}^{2}\delta_{c}^{2}}$.
It is clear that the above solutions contain five free parameters
($c_{o},p_{c}^{o},B_{o},\delta_{b},\delta_{c}$). By using the following
arguments we show that only four are physical.

  * •

From Eq.(II.1) we can see that $p_{c}$ is always positive and non-zero. In
fact, at $T_{\cal{T}}\equiv(1/4)\ln c_{o}^{2}$ it reaches its minimal value

$\displaystyle p_{c}^{\cal{T}}\left(T_{\cal{T}}\right)\equiv
2p_{c}^{o}|c_{o}|.$ (2.18)

Apart from it, $p_{c}$ is increasing in both directions, $T>T_{\cal{T}}$ and
$T<T_{\cal{T}}$. Thus, the surface $T=T_{\cal{T}}$ acts as a transition
surface, which will be denoted as surface ${\cal{T}}$.

  * •

The lapse function becomes unbounded at

$\displaystyle T_{\pm}=\frac{1}{b_{o}}\left(\ln\frac{b_{o}\pm 1}{b_{o}\mp
1}-2B_{o}\right).$ (2.19)

Clearly, these two singularities restrict the above solutions to the region
$T_{-}<T<T_{+}$. Then, extensions are needed beyond these two surfaces in
order to obtain a geodesically complete spacetime, provided that the spacetime
is not singular at these two points. As a matter of fact, these surfaces
represent the black and white hole horizons, respectively [29], and the
extensions can be easily carried out as in the classical case. In particular,
requiring the extension be analytical, it is also unique, as in the classical
case.

Before showing our above claims, let us first simplify the above solutions by
using the gauge residuals left from the gauge freedom (2.3). First, by the
shift symmetry $T\rightarrow\hat{T}=T+\hat{T}_{o}$, we find that the metric
(2.1) takes the form
$ds^{2}=-\hat{N}^{2}d\hat{T}^{2}+\hat{g}_{xx}dx^{2}+\hat{p}_{c}d\Omega^{2}$,
where $\left(\hat{N}^{2},\;\hat{g}_{xx},\;\hat{p}_{c}\right)$ are given by
Eqs.(II.1) and (II.1) with the replacements

$\displaystyle\left(T,B_{o},p_{c}^{o},c_{o},L_{o},\alpha\right)\rightarrow\left(\hat{T},\hat{B}_{o},\hat{p}_{c}^{o},\hat{c}_{o},\hat{L}_{o},\hat{\alpha}\right),$
(2.20)

where

$\displaystyle\hat{B}_{o}$ $\displaystyle=$ $\displaystyle
B_{o}-\frac{b_{o}}{2}\hat{T}_{o},\quad\hat{p}_{c}^{o}=e^{-2\hat{T}_{o}}p_{c}^{o},$
$\displaystyle\hat{c}_{o}$ $\displaystyle=$ $\displaystyle
c_{o}e^{2\hat{T}_{o}},\quad\hat{L}_{o}=L_{o},\quad\hat{\alpha}=\alpha
e^{\hat{T}_{o}}.$ (2.21)

Since $\hat{T}_{o}$ is an arbitrary constant, without loss of the generality,
we can always set it to

$\displaystyle\hat{T}_{o}=\frac{2}{b_{o}}\left(B_{o}-\tanh^{-1}\left(\frac{1}{b_{o}}\right)\right).$
(2.22)

Then, we find that in terms of $\hat{T}$, the two horizons given by Eq.(2.19)
now are located, respectively, at

$\displaystyle\hat{T}_{\text{BH}}\equiv\hat{T}_{+}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\hat{T}_{\text{WH}}\equiv\hat{T}_{-}$
$\displaystyle=$ $\displaystyle-\frac{2}{b_{o}}\ln\frac{b_{o}+1}{b_{0}-1}<0.$
(2.23)

As indicated by their subscriptions, $\hat{T}_{\text{BH}}$ and
$\hat{T}_{\text{WH}}$ will correspond to the locations of the black and white
hole horizons, respectively. The above choice is also consistent with that
adopted in [24, 30], so that the coordinate $\hat{T}$ is strictly negative
between the black and white hole horizons. It is interesting to note that
$T_{\text{WH}}\rightarrow-\infty$ as $b_{o}\rightarrow 1$ (or
$\delta_{b}\rightarrow 0$), which corresponds to the classical limit, and the
WH horizon turns into the spacetime singularity, at which now we have
$\left.p_{c}(T=-\infty)\right|_{\delta_{b}=0}=0$.

On the other hand, considering the rescaling freedom of Eq.(2.3) for the
$x$-coordinate, the metric can be finally cast in the form

$\displaystyle
ds^{2}=-\hat{N}^{2}d\hat{T}^{2}+\hat{g}_{xx}d\hat{x}^{2}+\hat{p}_{c}d\Omega^{2},$
(2.24)

with

$\displaystyle\hat{N}^{2}$ $\displaystyle=$
$\displaystyle\hat{p}_{c}^{o}\gamma^{2}\delta_{b}^{2}\frac{\hat{c}_{o}^{2}e^{-2\hat{T}}+e^{2\hat{T}}}{1-b_{o}^{2}\tanh^{2}\left(\frac{b_{o}\hat{T}}{2}+\hat{B}_{o}\right)},$
$\displaystyle\hat{g}_{xx}$ $\displaystyle=$
$\displaystyle\gamma^{2}\delta_{b}^{2}\cosh^{4}\left(\frac{b_{o}\hat{T}}{2}+\hat{B}_{o}\right)$
$\displaystyle\times\frac{1-b_{o}^{2}\tanh^{2}\left(\frac{b_{o}\hat{T}}{2}+\hat{B}_{o}\right)}{\hat{c}_{o}^{2}e^{-2\hat{T}}+e^{2\hat{T}}},$
$\displaystyle\hat{p}_{c}$ $\displaystyle=$
$\displaystyle\hat{p}_{c}^{o}\left(\hat{c}_{o}^{2}e^{-2\hat{T}}+e^{2\hat{T}}\right),$
$\displaystyle\hat{B}_{o}$ $\displaystyle=$
$\displaystyle\tanh^{-1}\left(\frac{1}{b_{o}}\right).$ (2.25)

Note that in the expression of $\hat{g}_{xx}$ the factor
$\gamma^{2}\delta_{b}$ is kept, as this will allow us to take the classical
limit $\delta_{b}\rightarrow 0$, considering the fact,

$\displaystyle\cosh^{2}\left(y+\hat{B}_{o}\right)=\frac{\cosh^{2}y}{\gamma^{2}\delta_{b}^{2}}\left(b_{o}+\tanh
y\right)^{2},$ $\displaystyle
1-b_{o}^{2}\tanh^{2}\left(y+\hat{B}_{o}\right)=-\frac{\gamma^{2}\delta_{b}^{2}}{\left(b_{o}+\tanh
y\right)^{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\tanh
y\left[2b_{o}+\left(b_{o}^{2}+1\right)\tanh y\right].~{}~{}~{}~{}~{}~{}$
(2.26)

Therefore, from Eq.(2.14) we find that in the current case there are only four
essential parameters, $\hat{p}_{c}^{o},\;\delta_{b},\;\delta_{c}$ and
$\hat{c}_{o}$, which uniquely determine the properties of the spacetimes. In
addition, if we require that the above solutions will reduce to the classical
one as $\delta_{b},\;\delta_{c}\rightarrow 0$, we must require
$\hat{c}_{o}=\hat{c}_{o}\left(\delta_{c}\right)$ and

$\displaystyle\lim_{\delta_{c}\rightarrow
0}\hat{c}_{o}\left(\delta_{c}\right)=0,$ (2.27)

as can be seen from Eqs.(2.9), (2.10) and (II.1). Moreover, in the classical
limit the BH horizon is located at $\hat{p}^{\text{GR}}_{c}(0)=(2m)^{2}$,
which corresponds to $\hat{p}_{c}^{o}=4m^{2}$, and shall be adopted in the
rest of this paper.

It is interesting to note that in [24, 30] $\hat{c}_{o}$ was chosen as

$\displaystyle\hat{c}_{o}^{\text{(CS, AOS)}}=\frac{\gamma
L_{o}\delta_{c}}{8m},$ (2.28)

which clearly satisfies Eq.(2.27). With this choice, Eq.(2.16) yields

$\displaystyle\hat{O}^{\text{(CS, AOS)}}_{b}=\hat{O}^{\text{(CS,
AOS)}}_{c}=m.$ (2.29)

However, in the following we shall leave this possibility open, and only
impose the condition (2.27). Moreover, for massive BHs, in [30] $\delta_{b}$
and $\delta_{c}$ were chosen as that given in Eq.(• ‣ I). In addition, in [24]
the following choice was adopted

$\displaystyle\delta^{\text{(CS)}}_{b}=\frac{\sqrt{\Delta}}{r_{o}},\quad\delta^{\text{(CS)}}_{c}=\frac{\sqrt{\Delta}}{L_{o}},$
(2.30)

where $r_{o}$ denotes the geometric radius of the fiducial metric
$ds_{o}^{2}=dx^{2}+r_{o}^{2}d^{2}\Omega$ considered in [24]. Again, in this
paper we shall also leave these choices open, in order to have a general
survey of the 4-dimensional (4D) phase space.

In addition, without causing any confusion, in the rest of this paper we shall
drop all superscript hats in Eqs. (2.24) - (2.27), so the spacetimes to be
considered in the rest of this paper are described by the metric

$\displaystyle ds^{2}=-{N}^{2}d{T}^{2}+{g}_{xx}dx^{2}+{p}_{c}d\Omega^{2},$
(2.31)

with

$\displaystyle N^{2}$ $\displaystyle=$
$\displaystyle\frac{p_{c}{\cal{D}}^{2}}{\cal{A}},$ $\displaystyle g_{xx}$
$\displaystyle=$
$\displaystyle\frac{4m^{2}\left[b_{o}\cosh\left(\frac{b_{o}T}{2}\right)+\sinh\left(\frac{b_{o}T}{2}\right)\right]^{4}}{p_{c}{\cal{D}}^{2}}{\cal{A}},$
$\displaystyle p_{c}$ $\displaystyle=$ $\displaystyle
4m^{2}\left(c_{o}^{2}e^{-2T}+e^{2T}\right),$ (2.32)

where

$\displaystyle{\cal{A}}$ $\displaystyle\equiv$ $\displaystyle
2\left(b_{o}^{2}+1\right)-\left(b_{o}+1\right)^{2}e^{b_{o}T}-\left(b_{o}-1\right)^{2}e^{-b_{o}T},$
$\displaystyle{\cal{D}}$ $\displaystyle\equiv$
$\displaystyle\left(b_{o}+1\right)e^{b_{o}T/2}+\left(b_{o}-1\right)e^{-b_{o}T/2}.$
(2.33)

Now let us turn to our previous claims regarding to the existence of
transition surface and BH and WH horizons. To this goal, let us first consider
the existence of a marginally trapped surface. The latter can be found by
calculating the expansions of the in-going and out-going radially moving light
rays [54, 4, 30, 55, 56]. Introducing the unit vectors, $u_{\mu}\equiv
N\delta^{T}_{\mu}$ and $s_{\mu}\equiv\sqrt{g_{xx}}\delta^{x}_{\mu}$, we
construct two null vectors $\ell_{\mu}^{\pm}=\left(u_{\mu}\pm
s_{\mu}\right)/\sqrt{2}$, which define, respectively, the in-going and out-
going radially-moving light rays. Then, the expansions of these light rays are
given by

$\Theta_{\pm}\equiv
m^{\mu\nu}\nabla_{\mu}\ell_{\nu}^{\pm}=-\frac{\dot{p}_{c}}{\sqrt{2}Np_{c}},$
(2.34)

where $m_{\mu\nu}\equiv g_{\mu\nu}+u_{\mu}u_{\nu}-s_{\mu}s_{\nu}$. A
marginally trapped surface is defined as the location at which
$\Theta_{+}\Theta_{-}=0$ [4, 30, 55, 56, 57]. Clearly, in the current case
this is possible only when: a) $\dot{p}_{c}=0$, or b) $N=\infty$. In the
following, let us consider them separately.

###  II.2 Transition Surface

From Eq.(II.1) we find that

$\displaystyle\dot{p}_{c}$ $\displaystyle=$
$\displaystyle\frac{8m^{2}}{e^{2T}}\left(e^{4T}-c_{o}^{2}\right)=\begin{cases}>0,&T>T_{\cal{T}},\cr=0,&T=T_{\cal{T}},\cr<0,&T<T_{\cal{T}},\cr\end{cases}~{}~{}~{}~{}~{}$
(2.35)

where

$\displaystyle T_{\cal{T}}=\frac{1}{2}\ln c_{o}.$ (2.36)

It is clear that in the region $T>T_{\cal{T}}$ both of $\Theta_{\pm}$ are
negative, so the spacetime in this region is trapped. On the other hand, in
the region $T<T_{\cal{T}}$ both of them are positive, and the corresponding
spacetime now becomes anti-trapped. In addition, the metric coefficients are
smooth across $T=T_{\cal{T}}$. Therefore, this marginally trapped surface is a
transition surface that separates a trapped region from an anti-trapped one.
The geometric radius $\sqrt{p_{c}}$ of the two-spheres is increasing apart
from this surface in both directions. At this transition surface, the area of
the two-spheres reaches its minimal value,

$\displaystyle A^{\text{min}}\equiv 4\pi p_{c}\left(T_{\cal{T}}\right)=32\pi
m^{2}c_{o}.$ (2.37)

For the choice of Eq.(2.28), we have

$\displaystyle A^{\text{min}}_{\text{(AOS)}}$ $\displaystyle=$ $\displaystyle
4\pi m\gamma L_{o}\delta_{c}$ (2.38) $\displaystyle=$ $\displaystyle
4\pi\gamma^{2}\left(\frac{3}{2}\right)^{1/3}\left(\frac{m}{\ell_{pl}}\right)^{2/3}\;\ell_{pl}^{2}.$

Note that in the last step of the above equation, we used the value of
$L_{o}\delta_{c}$ given by Eq.(• ‣ I) for massive black holes [30]. Thus, for
solar mass black holes, we have

$\displaystyle\left.A^{\text{min}}_{\text{(AOS)}}\right|_{m\gtrsim
m_{\bigodot}}\gtrsim 10^{25}\ell_{pl}^{2}.$ (2.39)

###  II.3 Black and White Hole Horizons and Analytical Extensions Beyond Them

As noted above, the lapse function $N$ becomes unbounded at $T=T_{\text{BH}}$
and $T=T_{\text{WH}}$, where $T_{\text{BH}}$ and $T_{\text{WH}}$ are given by
Eq.(II.1). Clearly, on these surfaces, $\Theta_{\pm}=0$. So, they also
represent marginally trapped surfaces. To see the nature of these surfaces,
let us first note that ${\cal{D}}$ defined above is strictly positive
${\cal{D}}>0$, while ${\cal{A}}$ vanish at the two points defined by
Eq.(II.1). In fact, ${\cal{A}}(T)$ can be written as

$\displaystyle{\cal{A}}(T)=\frac{\left(b_{o}+1\right)^{2}}{e^{b_{o}T}}\left(1-e^{b_{o}T}\right)\left(e^{b_{o}T}-e^{b_{o}T_{\text{WH}}}\right).$
(2.40)

It can be shown that the singularities at $T=0$ and $T=T_{\text{WH}}$ are
coordinate ones, similar to the classical Schwarzschild solution at $r=2m$. In
particular, near these singularities we have
$N^{2}\propto\left(e^{b_{o}T}-e^{b_{o}T_{\text{A}}}\right)^{-1}$ and
$g_{xx}\propto\left(e^{b_{o}T}-e^{b_{o}T_{\text{A}}}\right)$, where
$T_{\text{A}}$ denotes one of the locations of the two horizons. To make
extensions across each of them, it is sufficient to consider the neighborhood
of these horizons, at which we find that

$\displaystyle N^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{b_{o}^{2}f_{1}(T_{A})e^{2b_{o}T}}{e^{b_{o}T}-e^{b_{o}T_{\text{A}}}},$
$\displaystyle g_{xx}$ $\displaystyle\simeq$ $\displaystyle
f_{2}(T_{A})\left(e^{b_{o}T}-e^{b_{o}T_{\text{A}}}\right),$ (2.41)

where

$\displaystyle f_{1}(T_{BH})$
$\displaystyle=4b_{o}\left(c_{o}^{2}+1\right)m^{2},$ $\displaystyle
f_{1}(T_{WH})$
$\displaystyle=4m^{2}b_{o}\left(\frac{b_{o}-1}{b_{o}+1}\right)^{\frac{4}{b_{o}}-2}$
$\displaystyle~{}~{}~{}\times\left(\left(\frac{b_{o}+1}{b_{o}-1}\right)^{\frac{8}{b_{o}}}c_{o}^{2}+1\right),$
$\displaystyle f_{2}(T_{BH})$
$\displaystyle=\frac{b_{o}^{3}}{\left(c_{o}^{2}+1\right)},$ $\displaystyle
f_{2}(T_{WH})$
$\displaystyle=\frac{b_{o}^{3}\left(\frac{b_{o}+1}{b_{o}-1}\right)^{\frac{4}{b_{o}}+2}}{\left(\frac{b_{o}+1}{b_{o}-1}\right)^{\frac{8}{b_{o}}}c_{o}^{2}+1}.$
(2.42)

It is interesting to note that, when $b_{o}=1$, $f_{1,2}(T_{\text{BH}})$
reduce to those of the classical BH, while $f_{1,2}(T_{\text{WH}})$ diverge
under this limit. This is because now the WH horizon turns into the classical
spacetime singularity, as noted previously.

Then, we find that the 2D metric in the ($T,x)$-plane takes the form

$\displaystyle d\Sigma_{2}\simeq
f_{2}(T_{A})\left(e^{b_{o}T}-e^{b_{o}T_{\text{A}}}\right)\left(-dt^{2}+dx^{2}\right),$
(2.43)

where

$\displaystyle t\simeq
f(T_{A})\ln\left(e^{b_{o}T}-e^{b_{o}T_{\text{A}}}\right)+t_{A},$ (2.44)

with $f(T_{A})\equiv\sqrt{f_{1}/f_{2}}$, and $t_{A}$ being a constant. Setting
$v=t+x,w=t-x$, we find that

$\displaystyle d\Sigma_{2}\simeq-
f_{2}(T_{A})\left(e^{b_{o}T}-e^{b_{o}T_{\text{A}}}\right)dvdw.$ (2.45)

To eliminate the coordinate singularity, we further introduce $\hat{v}$ and
$\hat{w}$ via the relations $\hat{v}=e^{av},\;\hat{w}=e^{aw}$, so that

$\displaystyle
d\Sigma_{2}\simeq-\frac{f_{2}(T_{A})}{a^{2}e^{2at_{A}}}\left(e^{b_{o}T}-e^{b_{o}T_{\text{A}}}\right)^{1-2af(T_{A})}d\hat{v}d\hat{w}.~{}~{}~{}$
(2.46)

Clearly, choosing

$\displaystyle a=\frac{1}{2f(T_{A})},$ (2.47)

we find that the coordinate singularity disappears in the
($\hat{v},\hat{w}$)-coordinates. On the other hand, since the functions
${\cal{D}},\;{\cal{A}},\;p_{c}$ and $\cosh(b_{o}T/2+B_{o})$ are all analytical
functions of $T$, it is not hard to be convinced that such extensions are
analytical across each of these two singular surfaces. As a result, these
extensions are also unique, and the extended spacetimes take precisely the
form of Eqs.(2.24) and (II.1) in the ($T,x,\theta,\phi$) coordinates in each
of the three regions: $-\infty<T<T_{\text{WH}}$,
$T_{\text{WH}}<T<T_{\text{BH}}$ and $T_{\text{BH}}<T<\infty$. Across the two
horizons, the spacetimes are smoothly connected by the
($\hat{v},\hat{w}$)-coordinates.

##  III Local and Global Properties of the extended spacetimes

In this section we shall study the local and global properties of the extended
spacetimes in details. The extensions introduced in the last section allow us
to study the spacetimes in each of the three regions separately.

###  III.1 Spacetime Outside of the BH Horizon

Due to the choice of the free parameter $\hat{T}_{o}$ of Eq.(2.22), the black
hole horizon now is located at $T_{\text{BH}}=0$. Then, from Eq.(2.40) we find
that

$\displaystyle{\cal{A}}_{+}(T)$ $\displaystyle\equiv$
$\displaystyle-{\cal{A}}(T)=\frac{\left(b_{o}+1\right)^{2}}{e^{b_{o}T}}$ (3.1)
$\displaystyle\times\left(e^{b_{o}T}-1\right)\left(e^{b_{o}T}-e^{b_{o}T_{\text{WH}}}\right)>0,~{}~{}~{}$

for $T>0$. Thus, the normal vector $N^{+}_{\mu}\equiv\delta^{T}_{\mu}$ to the
hypersurface $T=$ constant becomes spacelike, as now we have

$\displaystyle
g^{\mu\nu}N^{+}_{\mu}N^{+}_{\nu}=\frac{{\cal{A}}_{+}}{p_{c}{\cal{D}}^{2}}>0.$
(3.2)

As a result, the metric in this region takes the form

$\displaystyle
ds^{2}_{+}=-N^{2}_{+}dx^{2}+\frac{p_{c}{\cal{D}}^{2}}{{\cal{A}}_{+}}dT^{2}+{p}_{c}d\Omega^{2},$
(3.3)

with

$\displaystyle N_{+}$ $\displaystyle\equiv$
$\displaystyle\frac{2m\sqrt{{\cal{A}}_{+}}}{\sqrt{p_{c}}{\cal{D}}}\cosh^{2}\left(\frac{b_{o}T}{2}\right)$
(3.4)
$\displaystyle\times\left[b_{o}+\tanh\left(\frac{b_{o}T}{2}\right)\right]^{2}.~{}~{}~{}~{}~{}$

Introducing the two unit vectors, $u^{+}_{\mu}\equiv N_{+}\delta_{\mu}^{x}$
and $s^{+}_{\mu}\equiv\sqrt{g_{TT}}\delta_{\mu}^{T}$, we can construct two
null vectors, $\ell_{\mu}^{(+,\pm)}\equiv\left(u^{+}_{\mu}\pm
s^{+}_{\mu}\right)/\sqrt{2}$, which define respectively the in-going and out-
going radially moving light rays. Then, the expansions of these light rays are
given by

$\Theta^{(+)}_{\pm}\equiv
m^{\mu\nu}\nabla_{\mu}\ell_{\nu}^{(+,\pm)}=\pm\sqrt{\frac{{\cal{A}}_{+}}{2p_{c}{\cal{D}}^{2}}}\;\frac{\dot{p}_{c}}{p_{c}},$
(3.5)

with

$\dot{p}_{c}=8m^{2}e^{-2T}\left(e^{4T}-c_{o}^{2}\right)>0,$ (3.6)

for $T>0$, provided that $c_{o}^{2}<1$. This is always the case, as long as
the temperature of massive BHs does not deviate significantly from its
classical value, as to be shown below. Therefore, in this region the expansion
of the in-going null rays is always negative, while the expansion of the out-
going null rays is always positive, so the spacetimes in this region are
untrapped [4, 55, 56]. Thus, the horizon located at $T=0$ separates the
untrapped region ($T>0$) from the trapped one ($T_{\text{WH}}<T<0$), and acts
like a BH horizon, although in the trapped region the spacetime is free of any
kind of spacetime singularities. The analytic extension can be carried out by
introducing the ($\hat{v},\hat{w}$)-coordinates, presented in the last
section.

To study further the properties of the spacetimes in this region, let us
consider the asymptotic behaviors ($T\rightarrow\infty$) of the spacetimes and
their near-horizon properties ($T\gtrsim 0$), separately. We majorly use the
Kretschmann scalar for analyzing the behavior of the spacetime and this has
been calculated using the xAct package in Mathematica.

####  III.1.1 Asymptotic Behavior of the Spacetime

As $T\rightarrow\infty$, we find that the corresponding Kretschmann scalar is
given by

$\displaystyle K\equiv
R_{\alpha\beta\delta\gamma}R^{\alpha\beta\delta\gamma}=\frac{a_{0}}{r^{4}}+\frac{a_{1}}{r^{4+b_{o}}}+\frac{a_{2}}{r^{4+2b_{o}}}+\frac{a_{3}}{r^{4+3b_{o}}}+\frac{a_{4}}{r^{8}}+\mathcal{O}\left(\frac{1}{r^{4(1+b_{o})}}\right),$
(3.7)

where $r\equiv 2me^{T}$ and

$\displaystyle a_{0}$ $\displaystyle\equiv
4\left(b_{o}-1\right)^{2}\left(b_{o}^{2}-4b_{o}+6\right),$ $\displaystyle
a_{1}$
$\displaystyle\equiv-\frac{16(2m)^{b_{o}}b_{o}\left(b_{o}-1\right)\left(b_{o}^{4}-7b_{o}^{3}+17b_{o}^{2}-7b_{o}-8\right)}{\left(b_{o}+1\right)^{2}},$
$\displaystyle a_{2}$
$\displaystyle\equiv\frac{16(2m)^{2b_{o}}b_{o}\left(b_{o}^{7}-21b_{o}^{6}+90b_{o}^{5}-84b_{o}^{4}-57b_{o}^{3}+89b_{o}^{2}+2b_{o}-8\right)}{\left(b_{o}+1\right)^{4}},$
$\displaystyle a_{3}$
$\displaystyle\equiv\frac{16(2m)^{3b_{o}}\left(b_{o}-1\right)b_{o}\left(5b_{o}^{7}+28b_{o}^{6}-265b_{o}^{5}+284b_{o}^{4}+103b_{o}^{3}-184b_{o}^{2}-3b_{o}+8\right)}{\left(b_{o}+1\right)^{5}},$
$\displaystyle a_{4}$
$\displaystyle\equiv-128m^{4}\left(b_{o}-1\right)\left(b_{o}^{3}-11b_{o}^{2}+42b_{o}-46\right)c_{o}^{2}.$
(3.8)

It is interesting to note that the above expressions are valid for any choice
of $(\delta_{b},\delta_{c},c_{o},m)$. In particular, they reduce precisely to
its classical limit, $K_{\text{GR}}=48m^{2}/r^{6}$, when $\delta_{b}=0$,
independent of the choices of $\delta_{c},c_{o}$ and $m$. In fact, when
$\delta_{b}=0$ we have $b_{o}=1$, and then from the above expressions we find
that $a_{2}=48m^{2}$, while all other terms identically vanish.

On the other hand, $a_{0}\not=0$, as long as $\delta_{b}\not=0$. That is, the
Kretschmann scalar is asymptotically vanishing as $r^{-4}$, instead of
$r^{-6}$, as that in the classical case. This is true not only in the
$\mu_{0}$-scheme, but also true in the general case, in which $\delta_{b}$ and
$\delta_{c}$ are the Dirac observables of the 4-dimensional phase spacetime
($b,p_{b};c,p_{c}$) [44].

However, it must be noted that the $r^{-4}$ term becomes dominant only when
$r>r_{c}$, where $r_{c}$ is defined by

$\displaystyle\frac{a_{0}}{r^{4}_{c}}=\frac{a_{2}}{r^{4+2b_{o}}_{c}}.$ (3.9)

In particular, for the AOS choice in Eq.(• ‣ I), we find that in the large
mass limit

$\displaystyle r_{c}^{AOS}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{8\left(2/3\right)^{1/3}}{\gamma^{2}}\right)^{1/2}\frac{m^{5/3}}{\ell_{p}^{2/3}}+\mathcal{O}\left(m^{4/3}\right).~{}~{}~{}~{}$
(3.10)

Thus, for solar mass BHs, we have

$\displaystyle r_{c,M_{\odot}}^{AOS}$ $\displaystyle\simeq$ $\displaystyle
2.06\times 10^{31}\;{\text{cm}}>L_{\text{obs}},~{}~{}~{}~{}$ (3.11)

where $L_{\text{obs}}\;(\simeq 4.4\times 10^{28}\;{\text{cm}})$ denotes the
size of our observational Universe.

Similarly, we can define the critical value $r_{c_{4}}$, beyond which the
$r^{-4}$ starts to dominate over the $r^{-5}$ term. And $r_{c_{5}}$ as the
critical radius beyond which the $r^{-5}$ term becomes dominant over the
$r^{-6}$ term. These are defined by

$\displaystyle\frac{a_{0}}{r^{4}_{c_{4}}}$
$\displaystyle=\frac{a_{1}}{r^{4+b_{o}}_{c_{4}}}$
$\displaystyle\frac{a_{1}}{r^{4+b_{o}}_{c_{5}}}$
$\displaystyle=\frac{a_{2}}{r^{4+2b_{o}}_{c_{5}}}$ (3.12)

This expressions in the large mass limit for the AOS choice in Eq.(• ‣ I) are
as follows

$\displaystyle r_{c_{4}}^{AOS}$
$\displaystyle\simeq\frac{2^{11/3}\;m^{5/3}}{3^{7/6}\;\gamma\;\ell_{p}^{2/3}}+\mathcal{O}\left(m\right),$
$\displaystyle r_{c_{5}}^{AOS}$
$\displaystyle\simeq\frac{3^{5/6}m^{5/3}}{{2}^{1/3}\gamma{\ell_{p}^{2/3}}}+\mathcal{O}\left(m\right).$
(3.13)

And for a solar mass BH, we have

$\displaystyle r_{c_{4},M_{\odot}}^{AOS}$ $\displaystyle\simeq 2.75\times
10^{31}\;{\text{cm}}>L_{\text{obs}},$ $\displaystyle
r_{c_{5},M_{\odot}}^{AOS}$ $\displaystyle\simeq 1.54\times
10^{31}\;{\text{cm}}>L_{\text{obs}}.$ (3.14)

It is clear from the above calculations that $r^{-4}$ and $r^{-5}$ terms will
be dominant at distances much beyond our observational universe. Hence, for
BHs with solar mass and greater, the asymptotic behavior of the AOS spacetime
can be well described by its classical limit ($K\simeq r^{-6}$) within our
observational Universe.

On the other hand, for the CS choice (2.30), we find

$\displaystyle r_{c}^{CS}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{16}{3\pi^{2}\gamma^{6}}\right)^{1/2}\frac{m^{3}}{\ell_{p}^{2}}+\mathcal{O}\left(m^{2}\right),$
$\displaystyle r_{c_{4}}^{CS}$ $\displaystyle\simeq$
$\displaystyle\frac{2^{4}m^{3}}{3^{3/2}\pi\gamma^{3}\ell_{p}^{2}}+\mathcal{O}\left(m\right),$
$\displaystyle r_{c_{5}}^{CS}$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{3}m^{3}}{\pi\gamma^{3}\ell_{p}^{2}}+\mathcal{O}\left(m\right),$
(3.15)

and for a solar mass BH we obtain $r_{c,M_{\odot}}^{CS}\simeq 4.18\times
10^{82}\;{\text{cm}}$, $r_{c_{4},M_{\odot}}^{CS}\simeq 5.57\times
10^{82}\;{\text{cm}}$ and $r_{c_{5},M_{\odot}}^{CS}\simeq 3.13\times
10^{82}\;{\text{cm}}\gg L_{\text{obs}}$. Therefore, in both models the
asymptotic behavior of the spacetime can be well described by its classical
limit ($K\simeq r^{-6}$) within our observational Universe.

To understand further the asymptotic behavior of the spacetimes, let us
calculate the effective energy momentum tensor, given by

$\displaystyle\kappa T_{\mu\nu}\equiv G_{\mu\nu}=\rho
u_{\mu}u_{\nu}+p_{r}r_{\mu}r_{\nu}+p_{\bot}\left(\theta_{\mu}\theta_{\nu}+\phi_{\mu}\phi_{\nu}\right),$
(3.16)

where $u_{\mu}$ denotes the unit timelike vector along $T$ direction, and
$r_{\mu}$, $\theta_{\mu}$ and $\phi_{\mu}$ are the spacelike unit vectors
along $r$, $\theta$, and $\phi$ directions respectively. In addition $\rho$,
$p_{r}$ and $p_{\bot}$ are the energy density and pressures along the radial
and tangential directions and $\kappa=8\pi G$. To the leading order, they are
given by

$\displaystyle\rho$
$\displaystyle\simeq-\frac{4(2m)^{b_{o}}\left(b_{o}-1\right)b_{o}^{2}}{r^{2+b_{o}}\left(b_{o}+1\right)^{2}}+\mathcal{O}\left(\frac{1}{r^{2(1+b_{o})}}\right),$
(3.17) $\displaystyle p_{r}$
$\displaystyle\simeq\frac{2(b_{o}-1)}{r^{2}}+\mathcal{O}\left(\frac{1}{r^{2+b_{o}}}\right),$
(3.18) $\displaystyle p_{\bot}$
$\displaystyle\simeq\frac{\left(b_{o}-1\right)^{2}}{r^{2}}+\mathcal{O}\left(\frac{1}{r^{2+b_{o}}}\right).$
(3.19)

From the above expressions one can see that when $(b_{o}=1)$ the spacetime is
vacuum, and when $b_{o}\not=1$, an effective matter field exists, which always
violates the weak energy condition [4].

####  III.1.2 Spacetime Properties Near the BH Horizon

To study the properties of the spacetime near the BH horizon, an important
quantity is the Hawking temperature. Following [39], for a spherically
symmetric static spacetime of the form

$\displaystyle ds^{2}=-g_{tt}dt^{2}+g_{xx}dx^{2}+p_{c}d\Omega^{2},$ (3.20)

the Hawking temperature is given by

$\displaystyle
T_{H}=\frac{\hbar}{k_{B}\mathcal{P}},\quad\mathcal{P}=\lim_{t\to
0}\frac{4\pi\left(g_{tt}g_{xx}\right)^{\frac{1}{2}}}{\partial_{t}g_{xx}},$
(3.21)

where $k_{B}$ is the Boltzmann constant. We find that for the metric (3.3),
the Hawking temperature is given by

$\displaystyle T_{H}=\frac{T_{H}^{\text{GR}}}{\cal{P}},$ (3.22)

where $T_{H}^{\text{GR}}\left(\equiv\hbar/(8\pi k_{b}m)\right)$ denotes the
corresponding Hawking temperature of the classical BH and

$\displaystyle{\cal{P}}\equiv\left(1+c_{o}^{2}\right)\frac{\gamma
L_{o}\delta_{c}}{8mc_{o}}.$ (3.23)

It is clear that we must choose

$\displaystyle c_{o}=\frac{\gamma L_{o}\delta_{c}}{8m},$ (3.24)

in order to make sure that the temperatures of massive BHs do not deviate
significantly from their classical counterparts. It is remarkable to note that
the above choice is precisely the one adopted by CS and AOS [24, 30], given by
Eq.(2.28), for which we have

$\displaystyle{\cal{P}}^{\text{(CS, AOS)}}=1+c_{o}^{2}.$ (3.25)

###  III.2 Spacetime Inside the BH/WH Horizons

Recall that the metric inside the BH/WH horizons for the BH takes the form

$\displaystyle ds^{2}=-{N}^{2}d{T}^{2}+{g}_{xx}dx^{2}+{p}_{c}d\Omega^{2},$
(3.26)

where $N$, ${g}_{xx}$ and $p_{c}$ are given by Eq.(II.1). In the rest of this
paper, we shall adopt the value of $c_{o}$ given by Eq.(3.24), which will
ensure the quantum corrections to the Hawking temperature are negligible for
macroscopic BHs. This choice reduces the 4-dimensional phase space to three,
now spanned by ($m,\delta_{b},\delta_{c}$). However, unlike previous studies,
we shall keep the choices of $\delta_{b}$ and $\delta_{c}$ open.

To gain a deeper understanding of this 3-dimensional phase space, let us first
consider the spacetime near the transition surface.

####  III.2.1 Properties of the spacetime near the Transition surface

To our goals, let us focus ourselves on the behavior of the the Kretschmann
scalar near the transition surface. Since the expression of the Kretschmann
scalar is humongous, its explicit form will not be written out explicitly in
this paper.

To begin with, let us first note that the Kretschmann scalar is independent of
the mass parameter $m$ at the transition surface for the AOS choice of
($\delta_{b},\delta_{c}$), given by Eq.(• ‣ I) [30]. Therefore, we would first
consider the case in which $\delta_{b}$ and $\delta_{c}$ take the forms

$\displaystyle\delta_{b}=\alpha_{b}\left(\frac{\ell_{p}}{m}\right)^{1/3},\quad
L_{o}\delta_{c}=\alpha_{c}\left(\frac{\ell_{p}}{m}\right)^{1/3},$ (3.27)

where $\alpha_{b}$ and $\alpha_{c}$ are two positive and otherwise arbitrary
constants. For the AOS solution, we have

$\displaystyle\alpha^{\text{(AOS)}}_{b}$ $\displaystyle=$
$\displaystyle\left(\frac{\sqrt{\Delta}}{\sqrt{2\pi}\gamma^{2}\ell_{p}}\right)^{1/3},$
$\displaystyle\alpha^{\text{(AOS)}}_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{\gamma\Delta^{2}}{4\pi^{2}\ell_{p}}\right)^{1/3}.$
(3.28)

However, we find that for any non-zero and positive values of
($\alpha_{b},\alpha_{c}$), the the Kretschmann scalar $K_{\cal{T}}$ evaluated
at the transition surface is always independent of the mass parameter $m$. In
Fig. 1, we plot three different choices,
$\left(\alpha_{b},\alpha_{c}\right)=\left\\{(1.97,3),\;(2,0.5),\;(1,1)\right\\}$,
denoted by Lines a, b and c, respectively, from which one can see clearly that
$K_{\cal{T}}$ is independent of mass. To compare them with the case considered
in [38, 30], the AOS choice of Eq.(III.2.1) is also plot out and denoted by
the Line AOS. It should be also noted that in each of these cases
$K_{\cal{T}}$ does not depend on $m$, but it does depend on the specific
values of ($\alpha_{b},\alpha_{c}$).

Figure 1:  The Kretschmann scalar $K_{\cal{T}}$ evaluated at the transition
surface vs the mass parameter $m$ for different choices of
($\delta_{b},\delta_{c}$). The Lines a, b and c correspond, respectively, to
the choices $(\alpha_{b},\alpha_{c})=\\{(1.97,3);\;(2,0.5);\;(1,1)\\}$. The
Line AOS corresponds to the choice of Eq.(III.2.1).

To see how $K_{\cal{T}}$ depends on the powers of $m$ appearing in
$\delta_{b}$ and $\delta_{c}$, let us consider the case

$\displaystyle\delta_{b}=\alpha_{b}\left(\frac{\ell_{p}}{m}\right)^{\beta_{b}},\quad
L_{o}\delta_{c}=\alpha_{c}\left(\frac{\ell_{p}}{m}\right)^{\beta_{c}},$ (3.29)

where $\beta_{b}$ and $\beta_{c}$ are other two arbitrary constants. In
particular, in Fig. 2 we choose

$\displaystyle\left(\delta_{b},\delta_{c}\right)=\left(\delta^{\text{(AOS)}}_{b},\frac{1}{{2L_{o}}}{\left(\frac{\gamma\Delta^{3/2}}{4\pi^{2}m}\right)^{1/2}}\right).$
(3.30)

From this figure we can see that now $K_{\cal{T}}$ increases as $m$ is
increasing. On the other hand, in Fig. 3, we consider the case

$\displaystyle\left(\delta_{b},\delta_{c}\right)=\left(\left(\frac{\sqrt{\Delta}}{\sqrt{2\pi}\gamma^{2}m}\right)^{1/2},\delta^{\text{(AOS)}}_{c}\right),$
(3.31)

from which we can see that $K_{\cal{T}}$ is also increasing as $m$ increases.

In addition, if $\beta_{b,c}<1/3$, then $K_{\cal{T}}$ will be decreasing as
$m$ increases and when $\beta_{b,c}>1/3$, $K_{\cal{T}}$ increases as $m$
increases. To show these properties, in Fig. 4, we plot the case

$\displaystyle\left(\delta_{b},\delta_{c}\right)=\left(\left(\frac{\sqrt{\Delta}}{\sqrt{2\pi}\gamma^{2}m}\right)^{1/2},\frac{1}{{2L_{o}}}{\left(\frac{\gamma\Delta^{3/2}}{4\pi^{2}m}\right)^{1/2}}\right),~{}~{}~{}~{}~{}$
(3.32)

denoted by Line f, and the one

$\displaystyle\left(\delta_{b},\delta_{c}\right)=\left(\left(\frac{\sqrt{\Delta}}{\sqrt{2\pi}\gamma^{2}m}\right)^{1/4},\frac{1}{{2L_{o}}}{\left(\frac{\gamma\Delta^{5/2}}{4\pi^{2}m}\right)^{1/4}}\right),~{}~{}~{}~{}~{}$
(3.33)

denoted by Line g.

Figure 2: Plot of $K_{\cal{T}}$ vs $m$ for the choice of
($\delta_{b},\delta_{c}$) given by Eq.(3.30) with
$\left(\beta_{b},\beta_{c}\right)=\left(1/3,1/2\right)$. To compare with the
AOS choice, Eq.(III.2.1), the corresponding line is also plotted out, denoted
by AOS. Figure 3: Plot of $K_{\cal{T}}$ vs $m$ for the choice of Eq.(3.31)
with $\left(\beta_{b},\beta_{c}\right)=\left(1/2,1/3\right)$. The AOS choice,
Eq.(III.2.1), corresponds to the Line AOS. Figure 4: Plot of $K_{\cal{T}}$ vs
$m$. Line f corresponds to the choice of Eq.(3.32) with
$\left(\beta_{b},\beta_{c}\right)=\left(1/2,1/2\right)$, and Line g to the
choice of Eq.(3.33) with
$\left(\beta_{b},\beta_{c}\right)=\left(1/4,1/4\right)$.

From these figures it is clear that $K_{\cal{T}}$ will be independent of the
mass parameter $m$, as long as $\delta_{b,c}$’s are all proportional to
$m^{-1/3}$, no matter what the coefficients $\alpha_{b,c}$’s will be. The AOS
choice is only a point on the 2-dimensional plane spanned by
$\left(\alpha_{b},\alpha_{c}\right)$, given by Eq.(III.2.1). At any point of
this plane, $K_{\cal{T}}$ will not depend on the choice of $m$. On the other
hand, outside of this plane, that is, as long as $\beta_{b}\not=1/3$ and/or
$\beta_{c}\not=1/3$, $K_{\cal{T}}$ will depend on $m$.

In [38, 30], $\delta_{b}$ and $\delta_{c}$ were uniquely determined by
requiring that the physical areas of the plaquettes $\Box(\theta,\phi)$ and
$\Box(x,\phi)$ at the transition surface be equal to the area gap $\Delta$. A
natural question to ask is: do there exist other conditions that can uniquely
determine $\delta_{b}$ and $\delta_{c}$? Our above considerations show that
requiring that $K_{\cal{T}}$ be independent of the mass parameter $m$ only
uniquely fixed $\beta_{b}$ and $\beta_{c}$, but not their amplitudes
$\alpha_{b}$ and $\alpha_{c}$. If we further require that the masses of the BH
and WH be equal, can we uniquely determine the amplitudes $\alpha_{b}$ and
$\alpha_{c}$? To answer these questions, let us turn to consider the radii of
the BH and WH horizons.

####  III.2.2 Properties of Spacetime near WH/BH horizons

The geometric radii of the WH and BH horizons are given by
$r_{A}=\sqrt{p_{c}(T_{A})}$, where $T_{A}$ denotes the location of the horizon
$A$. In Fig. 5, we plot the ratio between the two radii for the choices of
$\delta_{b}$ and $\delta_{c}$, given by Eqs.(3.32) and Eqs.(3.33), denoted
respectively by Line $x$ and $y$. Line AOS corresponds to the AOS choice.

Figure 5: Plots of the ratio $r_{\text{WH}}/r_{\text{BH}}$ vs the mass
parameter $m$. The lines $x$, AOS and $y$ correspond to the choices of
($\delta_{b},\delta_{c}$), given respectively by Eqs.(3.32), (• ‣ I) and
(3.33).

On the other hand, from Eqs.(II.1) and (II.1), we find that
$r_{\text{BH}}=r_{\text{WH}}$ implies that

$\displaystyle L_{o}\delta_{c}$
$\displaystyle=\frac{8m}{\gamma}\left(\frac{\sqrt{\gamma^{2}\delta_{b}^{2}+1}-1}{\sqrt{\gamma^{2}\delta_{b}^{2}+1}+1}\right)^{{2}/{\sqrt{\gamma^{2}\delta_{b}^{2}+1}}}.$
(3.34)

Substituting Eq.(3.27) into the above equation, we find that for massive BHs

$\displaystyle\alpha_{c}=\left(\frac{\gamma^{3}\ell_{p}}{2}\right)\alpha_{b}^{4}.$
(3.35)

Therefore, imposing the conditions: a) $K_{\cal{T}}$ be independent of $m$,
and b) $r_{\text{BH}}=r_{\text{WH}}$, the two parameters $\delta_{b}$ and
$\delta_{c}$ can be uniquely determined by Eqs.(3.27) and (3.35), modulated a
free parameter $\alpha_{b}$. In other words, there exits a family of choices
of $\delta_{b}$ and $\delta_{c}$, for which the above kinds of properties are
true.

##  IV Conclusions

In this paper, we have systematically studied the most general solutions of
LQBHs of the effective Hamiltonian obtained from its classical counterpart by
the replacements (1.1), with $\delta_{b}$ and $\delta_{c}$ being arbitrary
constants. These solutions usually contain five free parameters, three are the
integration constants of the dynamical equations [29], and two are the
polymerization parameters $\delta_{b}$ and $\delta_{c}$. However, using the
gauge freedom (2.3), one of them can be gauged away simply by the replacement,
$T\rightarrow T+\hat{T}_{o}$, as shown explicitly by Eqs.(2.20) - (2.22).
Therefore, there are generically only four free parameters, $c_{o}$, $m$,
$\delta_{b}$ and $\delta_{c}$. These solutions were already studied by several
authors from different aspects, including Refs. [29, 24, 30].

However, our current studies are different from those previous ones in the
sense that we have also taken $\delta_{b}$ and $\delta_{c}$ as free parameters
and explored their effects on various properties of the LQBH spacetimes. On
the other hand, in [24] Corichi and Singh studied the solutions with the
choice (2.30), while in [30, 29] the choice (• ‣ I) was adopted.

The four-parameter solutions are usually obtained in the KS spacetime, as
shown explicitly by Eqs.(2.24) and (II.1), which becomes singular at both of
the WH and BH horizons. Therefore, extensions beyond these horizons are
needed, in order to study the asymptotic properties of the spacetimes. We have
carried out such analytical extensions in Sec. II.D. Since the extensions are
analytical, they are also unique. Once these are done, we are allowed to study
the spacetimes in each of the three regions separately, the two asymptotic
regions outside of the WH and BH horizons, and the internal region in between
them.

Since the two asymptotic regions have quite similar properties, it is
sufficient to study only one of them, which we choose the one in the BH side
with $T>0$ [cf. Sec. III.A.1]. For any given four free parameters
($c_{o},m,\delta_{b},\delta_{c}$), the Kretschmann scalar in the asymptotic
limit always takes the form of Eqs.(3.7) and (III.1.1). From these expressions
it can be seen that the leading order is $1/r^{4}$, as long as
$\delta_{b}\not=0$, irrespective to any specific values of the four
parameters. On the other hand, when $\delta_{b}=0$, for which we have
$b_{o}=1$, it reduces to its classical value, $K_{\text{GR}}=48m^{2}/r^{6}$.
However, more careful analysis showed that the $r^{-4}$ terms becomes dominant
over the $r^{-6}$ term only when $r>r_{c}$, where $r_{c}$ is defined by
Eq.(3.9). In addition, we also show that the $r^{-4}$ term dominates over the
$r^{-5}$ term only when $r>r_{c_{4}}$ given in Eq.(III.1.1). And we further
show that $r^{-5}$ is dominant over $r^{-6}$ term only when $r>r_{c_{5}}$ also
given in Eq.(III.1.1). For both the AOS and CS choices, we found that $r_{c}$,
$r_{c_{4}}$ and $r_{c_{5}}$ are greater than $L_{\text{obs}}$ for solar mass
BHs, where $L_{\text{obs}}\;(\simeq 4.4\times 10^{28}\;{\text{cm}})$ denotes
the size of our observational Universe. Therefore, in both cases the
asymptotic behavior of the spacetime can be well described by its classical
limit ($K\simeq r^{-6}$) within our observational Universe.

We have further calculated the Hawking temperature of the BH and found that it
is given by Eqs.(3.22) and (3.23). For macroscopic BHs, the gravitational
fields near the BH horizons are very weak, and we expect that the temperature
should be very close to that of classical BH. With this requirement, it is
clear that the free parameter $c_{o}$ must be chosen so that Eq.(3.24) holds,
i.e.

$\displaystyle c_{o}=\frac{\gamma L_{o}\delta_{c}}{8m}.$ (4.1)

It is remarkable to note that this was precisely the choice adopted in [24,
30].

To explore the effects of the polymerization parameters $\delta_{b}$ and
$\delta_{c}$, we have first studied the Kretschmann scalar $K_{\cal{T}}$ at
the transition surface $T=T_{\cal{T}}$, and found that it is independent of
the mass parameter $m$ as long as they take the forms given by Eq.(3.27) for
any chosen values of $\alpha_{b}$ and $\alpha_{c}$, as shown explicitly in
Fig. 1. Clearly, the AOS choice of Eq.(III.2.1) is only a point in the plane
spanned by ($\alpha_{b},\alpha_{c}$). To see how $K_{\cal{T}}$ depends on the
powers of $m$ in the expressions of $\delta_{b}$ and $\delta_{c}$, we have
studied the more general forms Eq.(3.29), and found that
$\beta_{b}=\beta_{c}=1/3$ is the unique choice that leads $K_{\cal{T}}$ to be
independent of $m$, as shown explicitly by Figs. 2 \- 4.

Finally, we have also explored the condition at which the WH and BH masses are
equal, and found it is given by Eq.(3.34). If we further require that the
Kretschmann scalar at the transition surface be independent of $m$, we find
that $\delta_{b}$ and $\delta_{c}$ must take the form (3.27) with $\alpha_{c}$
being given by Eq.(3.35). Therefore, it is concluded that as long as
$\delta_{b}$ and $\delta_{c}$ are chosen as

$\displaystyle\delta_{b}=\alpha_{b}\left(\frac{\ell_{p}}{m}\right)^{1/3},\;\;L_{o}\delta_{c}=\alpha_{b}^{4}\left(\frac{\gamma^{3}\ell_{p}}{2}\right)\left(\frac{\ell_{p}}{m}\right)^{1/3},~{}~{}~{}~{}$
(4.2)

the corresponding LQBHs shall have the same desired properties as those of the
AOS solution [30], for any given non-zero and positive $\alpha_{b}$.

With the choice of $\delta_{b}$ and $\delta_{c}$ given by Eq.(4.2), an
interesting question is: is it possible to impose any constraint on the choice
of the free parameter $\alpha_{b}$ from observations? As the Event Horizon
Telescope (EHT) just observed two supermassive BHs, one is the $M87^{*}$ BH
located at the center of the more distant Messier 87 galaxy [3], and the other
is the Sagittarius $A^{*}$ BH located at the center of our own Milky Way
galaxy [58], it would be very interesting to see the effects of $\alpha_{b}$
on the shadows of supermassive BHs. In Fig. 6, we plot the dependence of
$r_{c}$ on $\alpha_{b}$ for a solar mass, Sagittarius $A^{*}$ and $M87^{*}$
BHs, respectively, from which it can be seen that $r_{c}$ becomes very small,
so that the quantum effects are expected to be large, only when $\alpha_{b}$
is very large. For more details, we would like to come back to this issue in
another occasion, as such considerations clearly are out of the scope of the
current paper.

Figure 6: Plots of the critical radius $r_{c}$ vs $\alpha_{b}$ for a solar
mass, Sagittarius $A^{*}$ and $M87^{*}$ BHs, respectively.

###### Acknowledgements.

G.O. is supported through Baylor Physics graduate program, and P.S. is
supported by NSF grant PHY-2110207, while A.W. is partly supported by the NSF
grant, PHY-2308845.

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