# Extremally Disconnected Groups
of Measurable Cardinality

Ol’ga Sipacheva  Department of General Topology and Geometry, Faculty of
Mechanics and Mathematics, M. V. Lomonosov Moscow State University, Leninskie
Gory 1, Moscow, 199991 Russia<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Given an arbitrary measurable cardinal $\kappa$, a nondiscrete Hausdorff
extremally disconnected group of cardinality $\kappa$ is constructed.

###### Key words and phrases:

Topological group, extremally disconnected, measurable cardinal

###### 2020 Mathematics Subject Classification:

54H11, 54G05, 03E35

This note is concerned with Arhangel’skii’s old problem of the existence in
ZFC of a nondiscrete Hausdorff extremally disconnected group [1]. The
nonexistence of a countable group with these properties in consistent with ZFC
[2]. Here we show that, for any measurable cardinal $\kappa$, there exists a
nondiscrete Hausdorff extremally disconnected group of cardinality $\kappa$.
We begin with basic definitions; for more details on measurable cardinals and
related ultrafilters, see, e.g., [3].

######  Definition 1.

A topological space is _extremally disconnected_ if the closure of any open
set is open in this space.

######  Definition 2.

Let $\kappa$ be an uncountable cardinal. A filter $\mathscr{F}$ is _$\kappa$
-complete_ if $\bigcap_{\alpha<\lambda}A_{\alpha}\in\mathscr{F}$ for any
cardinal $\lambda<\kappa$ and any $A_{\alpha}\in\mathscr{F}$,
$\alpha<\lambda$.

######  Definition 3.

An uncountable cardinal $\kappa$ is said to be _measurable_ if there exists a
$\kappa$-complete nonprincipal ultrafilter on $\kappa$.

The nonexistence of measurable cardinals is consistent with ZFC, while the
consistency of their existence cannot be proved within ZFC (see [3, Lemma 10.4
and Theorem 12.12]).

######  Definition 4.

Let $\delta$ be an ordinal, and let $X_{\alpha}\subset\delta$ for
$\alpha<\delta$. The _diagonal intersection_ $\Delta_{\alpha<\delta}$ of the
sequence $(X_{\alpha})_{\alpha<\delta}$ is
$\\{\alpha<\delta:\alpha\in\bigcap_{\beta<\alpha}X_{\alpha}$.

######  Definition 5.

A filter $\mathscr{F}$ on a cardinal $\kappa$ is _normal_ if it is closed
under diagonal intersections.

Obviously, a nonprincipal normal ultrafilter on $\kappa$ is $\kappa$-complete
if and only if it is _uniform_ (that is, contains no elements of cardinality
less than $\kappa$).

######  Fact (see [3, Theorem 10.20]).

Every measurable cardinal $\kappa$ carries a normal $\kappa$-complete
nonprincipal ultrafilter.

Given a set $X$ and a cardinal $\kappa$, we use the standard notation

$[X]^{\kappa}=\\{Y\subset X:|Y|=\kappa\\},\qquad[X]^{<\kappa}=\\{Y\subset
X:|Y|<\kappa\\}.$

For any cardinal $\kappa$, the set $[\kappa]^{<\omega}$ of all finite subsets
of $\kappa$ is the Boolean group freely generated by $\kappa$ with respect to
the operation $\triangle$ of symmetric difference. For any filter
$\mathscr{F}$ on $\kappa$, the subgroups $\langle A\rangle$ generated by
$A\in\mathscr{F}$ form a base of neighborhoods of zero in a group topology on
$[\kappa]^{<\omega}$. We denote the group $[\kappa]^{<\omega}$ with this
topology by $B(\kappa_{\mathscr{F}})$ and the zero element of this group by
${\mathbf{0}}$. Obviously, if the filter $\mathscr{F}$ is free, then
$B(\kappa_{\mathscr{F}})$ is a nondiscrete Hausdorff group.

Our purpose is to prove the following assertion.

######  Theorem.

Given any normal $\kappa$-complete nonprincipal ultrafilter $\mathscr{U}$ on a
cardinal $\kappa$, $B(\kappa_{\mathscr{U}})$ is a nondiscrete Hausdorff
extremally disconnected topological group. Moreover, all subsets of
cardinality less than $\kappa$ in $B(\kappa_{\mathscr{U}})$ are closed and
discrete, and $B(\kappa_{\mathscr{U}})$ is a $P_{\kappa}$-space, i.e., the
intersection of any family of fewer than $\kappa$ open sets in
$B(\kappa_{\mathscr{U}})$ is open.

The proof of this theorem is based on a lemma about symmetric products of
ultrafilters, which are defined by analogy with the usual (Fubini, or tensor)
products of ultrafilters as follows.

######  Definition 6.

Let $\mathscr{F}$ and $\mathscr{G}$ be two filters on sets $X$ and $Y$,
respectively. The family

$\mathscr{F}\otimes\mathscr{G}=\\{A\subset X\times X:\\{x\in X:\\{y\in
Y:(x,y)\in A\\}\in\mathscr{G}\\}\in\mathscr{F}\\}$

is called the _product_ of $\mathscr{F}$ and $\mathscr{G}$.111In the
literature, the terms _tensor product_ and _Fubini product_ and notations
$\mathscr{F}\cdot\mathscr{G}$ and $\mathscr{F}\times\mathscr{G}$ are also
used.

######  Definition 7.

Let $k$ be a positive integer, and let $\mathscr{F}_{1}$, …, $\mathscr{F}_{k}$
be filters on a cardinal $\kappa$. We define the _symmetric product_
$[\mathscr{F}_{1}\otimes\dots\otimes\mathscr{F}_{k}$ of $\mathscr{F}_{1}]$, …,
$\mathscr{F}_{k}$ recursively. The symmetric product of a single factor is set
equal to this factor, and for $k>1$,

$[\mathscr{F}_{1}\otimes\dots\otimes\mathscr{F}_{k}]\\\
=\\{A\subset[\kappa]^{k}:\\{F\in[\kappa]^{k-1}:\\{\alpha\in\kappa\setminus(\max
F+1):F\cup\\{\alpha\\}\in A\\}\in\mathscr{F}_{n}\\}\\\
\in[\mathscr{F}_{1}\otimes\dots\otimes\mathscr{F}_{k-1}]\\}.$

In particular, for two filters $\mathscr{F}$ and $\mathscr{G}$ on $\kappa$,

$[\mathscr{F}\otimes\mathscr{G}]=\\{A\subset[\kappa]^{2}:\\{\alpha\in\kappa:\\{\beta\in\kappa\setminus\alpha:\\{\alpha,\beta\\}\in
A\\}\in\mathscr{G}\\}\in\mathscr{F}\\}.$

For the symmetric product of $k$ copies of a filter $\mathscr{F}$, we use the
notation $\bigl{[}\bigotimes^{k}{\mathscr{F}}\bigr{]}$.

It is well known that the product of any two filters (ultrafilters) is a
filter (ultrafilter); see, e.g., [4, p. 156]. It easily follows by induction
that the symmetric product of any $k$ uniform ultrafilters on an infinite
cardinal $\kappa$ is an ultrafilter (it suffices to note that the diagonal
$\Delta=\\{(\alpha,\dots,\alpha):\alpha\in\kappa\\}$ of $\kappa^{k}$ is not in
$[\mathscr{F}_{1}\otimes\dots\otimes\mathscr{F}_{k}]$ and none of the initial
intervals of $\kappa$ is not in $\mathscr{F}_{k}$ and consider the direct
image [4, p. 155] of the ultrafilter
$[\mathscr{F}_{1}\otimes\dots\otimes\mathscr{F}_{k-1}]\otimes\mathscr{F}_{k}$
restricted to

$X=\\{(F,\alpha):F\in[\kappa]^{k-1},\ \alpha\in\kappa\setminus(\max
F+1)\\}\subset[\kappa]^{k-1}\times\kappa$

under the map $f\colon X\to[\kappa]^{k}$ defined by
$f((F,\alpha))=F\cup\\{\alpha\\}$). Note also that
$[\mathscr{F}\otimes\mathscr{G}]$ is nonprincipal if so is $\mathscr{F}$ or
$\mathscr{G}$.

######  Lemma.

For any positive integer $k$, any cardinal $\kappa$, and any normal
$\kappa$-complete ultrafilter $\mathscr{U}$ on $\kappa$, the sets $[A]^{k}$,
$A\in\mathscr{U}$, form a base of the ultrafilter
$\bigl{[}\bigotimes^{k}{\mathscr{U}}\bigr{]}$.

###### Proof.

We prove the lemma by induction on $k$. For $k=1$, there is nothing to prove.
Suppose that $k>1$ and the assertion holds for all smaller $k$.

We identify each $F\in[\kappa]^{k-1}$ with the increasing $(k-1)$-tuple of the
elements of $F$. Being endowed with the corresponding colexicographic order
$\preccurlyeq$, $[\kappa]^{k-1}$ is a well-ordered set of order type $\kappa$
(because each $F\in[\kappa]^{k-1}$ has less than $\kappa$ predecessors). For
each $\alpha<\kappa$ and every positive integer $m$, let $\gamma_{m}(\alpha)$
denote the order type of $([\alpha+1]^{m},\preccurlyeq)$. Then
$\kappa\setminus(\gamma_{k-1}(\alpha)+1)\in\mathscr{U}$ (because $\mathscr{U}$
is $\kappa$-complete and, therefore, uniform), whence

$None$
$D=\Delta_{\alpha\in\kappa}(\kappa\setminus(\gamma_{k-1}(\alpha)+1)\in\mathscr{U}\quad\text{and}\quad\alpha>\gamma_{k-1}(\beta)\text{
for any }\alpha,\beta\in D,\ \alpha>\beta.$

Take any $\widetilde{A}\in\bigl{[}\bigotimes^{k}{\mathscr{U}}\bigr{]}$. By the
induction hypothesis (and by the definition of the symmetric product of
filters), there exists an $A^{\prime}\in\mathscr{U}$ and a family
$\\{A_{F}:F\in[A^{\prime}]^{k-1}\\}$, where $A_{F}\in\mathscr{U}$, such that
$A^{\prime}\subset D$ and
$\widetilde{D}=\\{F\cup\\{\alpha\\}:F\in[A^{\prime}]^{k-1},\alpha\in
A_{F}\setminus(\max F+1)\\}\subset\widetilde{A}$. Let us number the elements
of $[A^{\prime}]^{k-1}$ by ordinals in $\preccurlyeq$-increasing order:

$[A^{\prime}]^{k-1}=\\{F_{\alpha}:\alpha<\kappa\\},\qquad F_{\alpha}\prec
F_{\beta}\quad\text{for}\quad\alpha<\beta.$

Clearly, the ordinal number $\alpha$ of any
$F_{\alpha}\in([A^{\prime}]^{k-1},\preccurlyeq)$ does not exceed the ordinal
number of $F_{\alpha}$ in $([\kappa]^{k},\preccurlyeq)$. We set
$A_{\alpha}=A_{F_{\alpha}}\setminus(\max F_{\alpha}+1)$ and
$A=\Delta_{\alpha\in\kappa}A_{\alpha}\cap A^{\prime}$. If
$\alpha_{1}<\dots<\alpha_{k}$, $\alpha_{i}\in A$, then
$\\{\alpha_{1},\dots,\alpha_{k-1}\\}=F_{\alpha}$ for some $\alpha\in\kappa$.
Since $\alpha_{k}\in A\subset D$, we have
$\alpha_{k}>\gamma_{k-1}(\alpha_{k-1})$. Therefore, the ordinal number of the
set $F_{\alpha}$ in $([\kappa]^{k-1},\preccurlyeq)$ is less than $\alpha_{k}$
(because $F_{\alpha}\in[\alpha_{k-1}+1]^{k-1}$), and hence
$\alpha<\alpha_{k}$. Thus, it follows from
$\alpha_{k}\in\Delta_{\beta\in\kappa}A_{\beta}$ that $\alpha_{k}\in
A_{\alpha}\subset A_{F_{\alpha}}$. By the definition of $A_{F_{\alpha}}$, we
have $\\{\alpha_{1},\dots,\alpha)_{k}\\}\in\widetilde{A}$, and the
arbitrariness of $\alpha_{1},\dots,\alpha_{k}\in A$ implies
$[A]^{k}\subset\widetilde{A}$. ∎

Now we can prove the theorem.

###### Proof of the theorem.

Let $U\subset B(\kappa_{\mathscr{U}})$ be an open set such that
${\mathbf{0}}\in\overline{U}\setminus U$. To prove the extremal
disconnectedness of $B(\kappa_{\mathscr{U}})$, we must find an
$A\in\mathscr{U}$ for which $\langle A\rangle\subset\overline{U}$.

Since $\mathscr{U}$ is $\sigma$-complete, it follows that there exists a
positive integer $k$ for which ${\mathbf{0}}\in\overline{U\cap[\kappa]^{k}}$
(otherwise, for each $k$, there is an $A_{k}\in\mathscr{U}$ such that $\langle
A_{k}\rangle\cap U\cap[\kappa]^{k}$, and for $A=\bigcap A_{k}$, we have
$A\in\mathscr{U}$ and $\langle A\rangle\cap U=\varnothing$, which contradicts
the assumption ${\mathbf{0}}\in\overline{U}$).

Let $D=\Delta_{\alpha\in\kappa}(\kappa\setminus(\gamma_{k}(\alpha)+1)$ (recall
that $\gamma_{k}(\alpha)$ is the order type of
$([\alpha+1]^{k},\preccurlyeq)$). Then $D\in\mathscr{U}$, and according to the
lemma, there exists a $B\in\mathscr{U}$, $B\subset D$, for which
$[B]^{k}\subset U\cap[\kappa]^{k}$ (otherwise,
$U\cap[\kappa]^{k}\notin\bigl{[}\bigotimes^{k}{\mathscr{U}}\bigr{]}$ and there
exists a $B\in\mathscr{U}$ such that $[B]^{k}\cap
U\cap[\kappa]^{k}=\varnothing$ and hence $\langle B\rangle\cap
U\cap[\kappa]^{k}$, which contradicts the assumption
${\mathbf{0}}\in\overline{U\cap[\kappa]^{k}}$).

Since $U$ is open, it follows that each $F\in[B]^{k}$ is contained in $U$
together with its neighborhood; in other words, for each $F\in[B]^{k}$, there
exists an $A_{F}\in\mathscr{U}$ such that $F+\langle
A_{F}\rangle=\\{F\triangle G:G\in\langle A_{F}\rangle\\}\subset U$. As in the
proof of the lemma, we number the elements of $[B]^{k}$ by ordinals in
$\preccurlyeq$-increasing order:

$[B]^{k}=\\{F_{\alpha}:\alpha<\kappa\\},\qquad F_{\alpha}\prec
F_{\beta}\quad\text{for}\quad\alpha<\beta;$

then we set $A_{\alpha}=A_{F_{\alpha}}\setminus(\max F_{\alpha}+1)$ and
$A=\Delta_{\alpha\in\kappa}A_{\alpha}\cap B$.

Take $\alpha_{1},\dots,\alpha_{n}\in A$, where $n\in\omega$, $n\geq k$, and
$\alpha_{1}<\dots<\alpha_{n}$. We have
$\\{\alpha_{1},\dots,\alpha_{k}\\}=F_{\alpha}$ for some $\alpha\in\kappa$.
Since $A\subset B\subset D$, $\beta>\gamma_{k}(\delta)$ for any
$\beta,\delta\in D$ such that $\beta>\delta$, and
$F_{\alpha}\in[\alpha_{k}+1]^{k}$, it follows that $\alpha<\alpha_{k+i}$ for
$i\geq 1$. Therefore, $\alpha_{k+i}\in A_{\alpha}\subset A_{F_{\alpha}}$
(because $\alpha_{k+i}\in\Delta_{\beta\in\kappa}A_{\beta}$). Thus,
$\\{\alpha_{1},\dots,\alpha_{n}\\}=F_{\alpha}\triangle\\{\alpha_{k+1},\dots,\alpha_{n}\\}\in
F_{\alpha}+\langle A_{F_{\alpha}}\rangle\subset U$.

Now take $\alpha_{1},\dots,\alpha_{m}\in A$, where $m\in\omega$, $0<m<k$, and
$\alpha_{1}<\dots<\alpha_{m}$, and let $A^{\prime}$ be any element of
$\mathscr{U}$. We must show that $\\{\alpha_{1},\dots,\alpha_{m}\\}+\langle
A^{\prime}\rangle\cap U\neq\varnothing$. We set
$A^{\prime\prime}=A^{\prime}\cap A\setminus\alpha_{m}$. For any
$\alpha_{m+1},\dots,\alpha_{k}\in A^{\prime\prime}$,
$\alpha_{m+1}<\dots<\alpha_{k}$, we have $\alpha_{1},\dots,\alpha_{k}\in A$.
According to what was shown above, $\\{\alpha_{1},\dots,\alpha_{k}\\}\in U$.
On the other hand,
$\\{\alpha_{1},\dots,\alpha_{k}\\}\in\\{\alpha_{1},\dots,\alpha_{n}\\}+\langle
A^{\prime\prime}\rangle\subset\\{\alpha_{1},\dots,\alpha_{n}\\}+\langle
A^{\prime}\rangle$.

Thus, $\langle A\rangle\subset\overline{U}$, as required.

It follows from the $\kappa$-completeness of $\mathscr{U}$ and the definition
of the topology of $B(\kappa_{\mathscr{U}})$ that the intersection of fewer
than $\kappa$ neighborhoods of ${\mathbf{0}}$ is again a neighborhood of
${\mathbf{0}}$. Hence the intersection of fewer that $\kappa$ neighborhoods of
any point in $B(\kappa_{\mathscr{U}})$ is a neighborhood of this point, and
the intersection (union) of fewer than $\kappa$ open (closed) sets is open
(closed). Thus, any set of cardinality less than $\kappa$ in
$B(\kappa_{\mathscr{U}})$ is closed (and discrete). ∎

## References

  * [1] A. Arhangelski, “Groupes topologiques extrémalement discontinus,” C. R. Acad. Sci. Paris Sér. A-B 265, A822–A825 (1967).
  * [2] E. Reznichenko and O. Sipacheva, “Discrete subsets in topological groups and countable extremally disconnected groups,” Proc. Amer. Math. Soc. 149, 2655–2668 (2021).
  * [3] T. J. Jech, _Set Theory: The Third Millennium Edition, Revised and Expanded_ (Springer, Berlin, 2002).
  * [4] W. W. Comfort and S. Negrepontis, _The Theory of Ultrafilters_ (Springer-Verlag, Berlin, 1974).


# Observational tests of a black hole in effective loop quantum gravity

Yun-Long Liu  Department of Physics, South China University of Technology,
Guangzhou 510641, China  Zhao-Qing Feng  Department of Physics, South China
University of Technology, Guangzhou 510641, China  Xiang-Dong
Zhang111Corresponding author<EMAIL_ADDRESS>Department of Physics,
South China University of Technology, Guangzhou 510641, China

###### Abstract

A new polymer black hole solution in loop quantum gravity was proposed
recently. The difference between the polymer black hole and Schwarzschild
black hole is captured by a quantum parameter $A$. In order to get the
constraints on parameter $A$, we consider the observational constraints
imposed on $A$ by using the Solar System experiments and calculate the
deflection of light, Shapiro time delay, perihelion precession and obtain the
effects associated with parameter $A$. Moreover, the parameterized post-
Newtonian approach of this loop quantum gravity black hole was also carried
out. It turns out the tightest constraint on $A$ can be improved to
$0<A<4.0\times 10^{-6}$.

##  I INTRODUCTION

Einstein’s general relativity(GR) has been proposed for more than a century
and still being an exciting and prosperous field. It has undergone more and
more sophisticated tests with the development of tools like atomic clocks,
radio telescopes. currently, the gravity test projects such as the deflection
of light, the perihelion advance, and Shapiro time Delay have been carried out
in the weak field zone especially in solar system Confrontation_will_2014 .
While In the strong field regime, the gravity tests include the observation of
binary pulsars Binary_taylor_1994 ; Doublebinarypulsar_yunes_2009 ;
Testing_seymour_2018 , the direct detections of gravitational waves
Tests_ligo_2016  and the image of the black hole shadow First_EHT_2019 ;
Gravitational_ehtcollaboration_2020 . Remarkably, the predictions of GR are
all consistent with these observations up to now.

However, despite all of these successes, GR is still far from being perfect.
From theoretical consideration, unifying GR with quantum mechanics into a
consistent quantum gravity theory remains the biggest theoretical challenge to
fundamental physics. Among various approaches to quantum gravity, loop quantum
gravity (LQG) is notable with its background independence and non-perturbative
features Ro04 ; Th07 ; As04 ; Ma07 .

Recently, an attractive polymerized black hole solution is constructed by
solving the effective equations of LQG type_bodendorfer_2021 ;
Mass_bodendorfer_2021 . This solution results in quantum extensions of the
Schwarzschild black hole, the classical singularity of the Schwarzschild black
hole has been resolved by connecting the black hole region with the white hole
region through a bounce. Some aspects of this Schwarzschild-like metric have
already been studied. For example, the thermodynamic properties of the
effective polymer black hole and the corresponding quantum corrections as
functions of black and white hole masses have been investigated in Ref.
Quantum_mele_2021 . In Ref. Gravitational_fu_2021 , the influences of the
quantum effects on the weak and strong bending angles of light rays have been
studied. More investigations on the polymerized black holes can be found in
Ref. consistent_bouhmadi-lopez_2020 ; Properties_gan_2020 .

Moreover, the rotating black holes are typically found in nature, In recent
Ref.Testing_brahma_2021 , the authors found a rotational solution of loop
quantum gravity black hole (LQGBH) by the Newman-Janis algorithm. Then they
investigate the test of loop quantum gravity by the shadow cast of the
rotating black hole as well as other experiments. And they obtain a constraint
on the quantum correction parameter $A$ as $0<A<7.7\times 10^{-5}$. Note that
the more precise experiment constraints on the quantum parameter may help us
to get a deeper understanding of LQG and consider that the gravitational
experiment in the Solar system provides very high precision
Observational_zhu_2020 . In this paper, we will analyze the influence of the
polymerized spacetime solution of the classical observations in the Solar
system such as the light deflection, Shapiro time delay, perihelion advance,
and the Parameterized Post-Newtonian(PPN) approach. In addition, in this
paper, we ignore the effects of the angular momentum of the spacetime, due to
the rotation of the Sun or Earth being expected to be very small.

The structure of this paper is organized as follows. In Sec.II, we give a
brief introduction to review the metric of LQGBH. Then in Sec.III, we obtain
the conserved quantities and the motion equations of a test body in the
nonrotating loop quantum gravity spacetime. We study the observational tests
including deflection of light, Shapiro time delay and perihelion precession as
well as the PPN approach in Sec.IV, and we obtain the constraints on the
dimensionless parameter $A$. The main conclusions and some discussions are
given in Sec.V.

##  II Nonrotating loop quantum gravity Black hole

Starting with the non-rotating loop quantum gravity black hole (LQGBH)
type_bodendorfer_2021 ; Mass_bodendorfer_2021 ; Testing_brahma_2021 , the
Schwarzschild-like metric in the loop quantum gravity reads

$\displaystyle
ds^{2}=-8AM_{b}^{2}\mathcal{A}(r)dt^{2}+\frac{dr^{2}}{8AM_{b}^{2}\mathcal{A}(r)}+\mathcal{B}(r)(d\theta^{2}+\sin^{2}(\theta)d\phi^{2}).$
(1)

The metric functions $\mathcal{A}(r)$ and $\mathcal{B}(r)$ are given by

$\displaystyle\mathcal{A}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{B}(r)}\left(\frac{r^{2}}{8AM_{b}^{2}}+1\right)\left(1-\frac{2M_{b}}{\sqrt{8AM_{b}^{2}+r^{2}}}\right),$
(2) $\displaystyle\mathcal{B}(r)$ $\displaystyle=$
$\displaystyle\frac{512A^{3}M_{b}^{4}M_{w}^{2}+\left(\sqrt{8AM_{b}^{2}+r^{2}}+r\right)^{6}}{8\sqrt{8AM_{b}^{2}+r^{2}}\left(\sqrt{8AM_{b}^{2}+r^{2}}+r\right)^{3}},$
(3)

where $M_{b}$ and $M_{w}$ is the mass of asymptotically Schwarzschild black
hole and white hole respectively, and the dimensionless parameter $A$ is
defined by $A=(\lambda_{k}/M_{b}M_{w})^{2/3}/2$ with $\lambda_{k}$ being the
parameter coming from the holonomy correction type_bodendorfer_2021 ;
Mass_bodendorfer_2021 .

Without loss of generality, we consider the most interesting and meaningful
scheme that $M_{b}=M_{w}=M$ type_bodendorfer_2021 ; Mass_bodendorfer_2021 ;
Testing_brahma_2021 . It displays a symmetric spacetime reflecting at the
transition surface ($r=0$). In this scheme, the metric functions
$\mathcal{A}(r)$ and $\mathcal{B}(r)$ reduce to

$\displaystyle\mathcal{A}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{\mathcal{B}(r)}\left(\frac{r^{2}}{8AM^{2}}+1\right)\left(1-\frac{2M}{\sqrt{8AM^{2}+r^{2}}}\right),$
(4) $\displaystyle\mathcal{B}(r)$ $\displaystyle=$ $\displaystyle
2AM^{2}+r^{2}.$ (5)

This metric solves the interior singularity of black hole. While the positive
solutions of $\mathcal{A}(r)=0$ corresponds to the black hole horizon of LQGBH
as

$\displaystyle r_{+}=2\sqrt{M^{2}-2AM^{2}}.$ (6)

It should be noted that horizons will disappear when $A>1/2$. In the $A\to 0$
limit, the expressions of $8AM^{2}\mathcal{A}(r)$ and $\mathcal{B}(r)$ can be
simplified to

$\displaystyle 8AM^{2}\mathcal{A}(r)$ $\displaystyle\to$ $\displaystyle
1-\frac{2M}{r},$ (7) $\displaystyle\mathcal{B}(r)$ $\displaystyle\to$
$\displaystyle r^{2}.$ (8)

Hence, the metric will go back to Schwarzschild case in the classical limit.

##  III motion Equations of test particles

Let us consider the evolution of a test particle in the LQGBH spacetime. Note
that we have two Killing vectors $K_{t}^{\mu}=(\frac{\partial}{\partial
t})^{\mu}$ and $K_{\phi}^{\mu}=(\frac{\partial}{\partial\phi})^{\mu}$. Then
the corresponding conserved quantities can be obtained as

$\displaystyle E$ $\displaystyle=$ $\displaystyle-mg_{tt}\dot{t}=p_{t},$ (9)
$\displaystyle J$ $\displaystyle=$ $\displaystyle
mg_{\phi\phi}\dot{\phi}=p_{\phi},$ (10)

where $E$ and $J$ denote the conserved energy and the angular momentum of the
test body, respectively.

In the following part of paper, we focus on the case that test body moving in
the equatorial plane ($\theta=\pi/2$ and $\dot{\theta}=0$). Considering the
normalization condition for 4-velocity $v_{a}$ as
$v_{a}v^{a}=g_{\mu\nu}\dot{x^{\mu}}\dot{x^{\nu}}=k$ ($k=-1$ for massive
particles while $k=0$ for massless one), we have

$\displaystyle
g_{rr}\dot{r}^{2}=-g_{tt}\dot{t}^{2}-g_{\phi\phi}\dot{\phi}^{2}+k.$ (11)

Taking into account the Eq. (9), (10), (11), and the variables with unity
mass($\bar{E}=E/m$, $\bar{J}=J/m$), equations of motion for test body can be
re-expression as

$\displaystyle\frac{dt}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{\bar{E}}{8AM^{2}\mathcal{A}(r)},$ (12)
$\displaystyle\frac{dr}{d\tau}$ $\displaystyle=$
$\displaystyle\sqrt{\bar{E}^{2}-V_{eff}},$ (13)
$\displaystyle\frac{d\phi}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{\bar{J}}{\mathcal{B}(r)},$ (14)

where the effective potential $V_{eff}$ is defined as follows

$\displaystyle
V_{eff}=\frac{8AM^{2}\mathcal{A}(r)\bar{J}^{2}}{\mathcal{B}(r)}+8AM^{2}\mathcal{A}(r)k.$
(15)

In the following, for simplicity, the $\bar{\ }$ is eliminated, and we
directly use the $E$ and $J$ to denote $\bar{E}$ and $\bar{J}$ respectively.

##  IV Observational tests in the loop quantum gravity

Using the motion of the test body that we have discussed above, the classical
tests including the deflection of light, Shapiro time delay, perihelion
advance, and geodesic precession of the spinning object will be studied in
this section.

###  IV.1 Deflection of Light

Starting from the motion equation of massless particle ($k=0$), Eqs.(13)-(15)
implies

$\displaystyle\frac{dr}{d\phi}=\left(\frac{dr}{d\tau}\right)/\left(\frac{d\phi}{d\tau}\right)=\sigma\sqrt{\mathcal{B}(r)\left(\frac{\mathcal{B}(r)}{b^{2}}-8AM^{2}\mathcal{A}(r)\right)},$
(16)

where an ”impact” parameter $b$ represents the ratio of $J$ and $E$ ($b=J/E$)
is introduced. Moreover, the parameter $\sigma=\pm 1$ correspond to the
outgoing and ingoing trajectories, respectively.

The light deflection angle can be expressed as

$\displaystyle\Delta\phi=2\int_{r_{0}}^{\infty}\frac{1}{\sqrt{\mathcal{B}(r)\left(\frac{\mathcal{B}(r)}{b^{2}}-8AM^{2}\mathcal{A}(r)\right)}}\,dr-\pi,$
(17)

where $r_{0}$ is the turning point of the trajectories which is given by
$\frac{dr}{d\phi}|_{r=r_{0}}=0$. Combine this condition with Eq.(16), the
relation between impact parameter $b$ and the closest approach $r_{0}$ reads

$\displaystyle
b=\sqrt{\frac{\mathcal{B}\left(r_{0}\right)}{8AM^{2}\mathcal{A}\left(r_{0}\right)}}.$
(18)

To simplify the integration in Eq.(17), we do the transformation

$\displaystyle r=\frac{r_{0}}{u}.$ (19)

Then the light deflection angle formula (17) can be reexpressed as

$\displaystyle\Delta\phi=2\int_{0}^{1}\frac{r_{0}}{u^{2}\sqrt{\frac{1}{b^{2}}\mathcal{B}\left(r_{0}/u\right)\left(\mathcal{B}\left(r_{0}/u\right)-8AM^{2}\mathcal{A}\left(r_{0}/u\right)b^{2}\right)}}\,du-\pi.$
(20)

In the weak field limit, the magnitude of $\epsilon=M/r_{0}$ is tiny. By
replacing $r_{0}$ in Eq.(20) with $\epsilon$ and expanding the equation, the
approximations of the integration in terms of $\epsilon$ is given by

$\displaystyle\Delta\phi=$ $\displaystyle
2\int_{0}^{1}\left(\frac{1}{\sqrt{1-u^{2}}}+\frac{\epsilon\left(u^{2}+u+1\right)}{(u+1)\sqrt{1-u^{2}}}+\frac{\epsilon^{2}\left(3\left(u^{2}+u+1\right)^{2}-4A(u+1)^{2}\left(2u^{2}+1\right)\right)}{2(u+1)^{2}\sqrt{1-u^{2}}}\right)\,du$
(21) $\displaystyle-\pi+O\left(\epsilon^{3}\right).$

Then one obtains the deflection angle of the light as

$\displaystyle\Delta\phi\approx\Delta\phi_{GR}-4\pi\left(\frac{M}{r_{0}}\right)^{2}A\approx\Delta\phi_{GR}\left(1-\pi
A\frac{M_{0}}{r_{0}}\right),$ (22)

where $\Delta\phi_{GR}$ is the light deflection in general relativity.

In the solar system, we assume that the closest distance approaching the Sun
$r_{0}$ of the electromagnetic wave signal is approximately $R_{S}$, where
$R_{S}$ is the radius of the Sun. Hence, the effect of parameter $A$ on light
deflection can be described as $\Delta\phi_{\text{GR}}\pi AM_{S}/R_{S}$ with
$M_{S}$ being the mass of the Sun. Using the measurement data of the radio
waves deflection among four quasars sources with the Very Long Baseline Array
(VLBA) PROGRESS_fomalont_2009  in the solar system and considering $A>0$, the
constraint of the quantum parameter $A$ can be obtained for

$\displaystyle 0<A<74.9836\quad(68\%\,C.L.).$ (23)

###  IV.2 Shapiro time Delay

In this section, we still consider the case of a massless test body moving in
curved spacetime. Considering Eq.(12), (13) and (15) when the parameter $k$
equals to zero ($k=0$), the differential equation of massless particles
between $t$ and $r$ reads

$\displaystyle\frac{dt}{dr}=\left(\frac{dt}{d\tau}\right)/\left(\frac{dr}{d\tau}\right)=\pm\sqrt{\frac{\mathcal{B}(r)}{64A^{2}M^{4}\mathcal{A}(r)^{2}\left(\mathcal{B}(r)-8Ab^{2}M^{2}\mathcal{A}(r)\right)}}.$
(24)

Then the time difference of the electromagnetic wave signal moving from the
closest approach point $P_{0}$ of the Sun to the point $X$ of satellite or
planet as

$\displaystyle\Delta
t(r_{X})=\int_{r_{0}}^{r_{X}}\sqrt{\frac{\mathcal{B}(r)}{64A^{2}M^{4}\mathcal{A}(r)^{2}\left(\mathcal{B}(r)-8Ab^{2}M^{2}\mathcal{A}(r)\right)}}\,dr.$
(25)

where $r_{0}$ is the radius of the closest approach point $P_{0}$ and $r_{X}$
is the radius of the point $X$. Doing the same transformation as in Eq.(19)
and expanding the expression of time $\Delta t(r_{X})$, with the perturbation
quantity $\epsilon=M/r_{0}$ in the weak field, we found that

$\displaystyle\Delta t(r_{X})$ $\displaystyle=$ $\displaystyle
M\int_{\frac{r_{0}}{r_{X}}}^{1}\sqrt{\frac{1}{1-u^{2}}}\left(\frac{1}{u^{2}\epsilon}+\frac{(3u+2)}{u(u+1)}+\frac{\left(-16A(u+1)^{2}+3u(5u+8)+12\right)\epsilon}{2(u+1)^{2}}\right)\,du$
(26) $\displaystyle+O\left(\epsilon^{2}\right).$

Then the time $\Delta t$ up to the sub-leading order can be expressed as

$\displaystyle\Delta t\left(r_{X}\right)$ $\displaystyle\approx$
$\displaystyle\sqrt{r_{X}^{2}-r_{0}^{2}}+M\left(\sqrt{\frac{r_{X}-r_{0}}{r_{X}+r_{0}}}+2\cosh^{-1}\left(\frac{r_{X}}{r_{0}}\right)\right)+\frac{1}{2}M\epsilon\left(\frac{r_{X}}{r_{X}+r_{0}}\right)^{3/2}$
(27)
$\displaystyle\times\left(\frac{\sqrt{r_{X}-r_{0}}\left(4r_{X}+5r_{0}\right)}{r_{X}^{3/2}}+2(16A-15)\left(\frac{r_{X}+r_{0}}{r_{X}}\right){}^{3/2}\sin^{-1}\left(\frac{\sqrt{r_{X}-r_{0}}}{\sqrt{2r_{X}}}\right)\right).$

Now, we consider the Shapiro time delay by sending an electromagnetic wave
signal out from a source (satellite or Earth) $X$, and receiving the signal
reflected by another reflection body (satellite or planet) $Y$. Whether the
electromagnetic wave signal passes perihelion (the turning point of the
trajectories $P_{0}$) or not, the calculation of time delay can be divided
into two categories, the inferior conjunction case and superior one.

In the inferior conjunction case, in which the object $Y$ that reflects the
radar signal is located between Earth (or spacecraft, denoted by $X$) and the
Sun. The time delay with the effect of the parameter $A$ reads

$\displaystyle\delta t_{I}$ $\displaystyle\approx$ $\displaystyle
4\log\left(\frac{r_{X}}{r_{Y}}\right)M+\frac{2r_{0}\left(r_{X}-r_{Y}\right)}{r_{X}r_{Y}}M+\frac{2(8A-6)\left(r_{X}-r_{Y}\right)}{r_{X}r_{Y}}M^{2}$
(28) $\displaystyle=$ $\displaystyle\delta
t^{GR}_{I}+\frac{16A\left(r_{X}-r_{Y}\right)}{r_{X}r_{Y}}M^{2},$

where $\delta t^{GR}_{I}$ referred to as the Shapiro time delay in Einstein’s
general relativity.

In the superior conjunction case the object $Y$ reflects the radar signal and
the object $X$ is on opposite sides of the Sun. By taking the similar
procedures that in Eq (28). The time delay in this case is then given by

$\displaystyle\delta t_{S}$ $\displaystyle\approx$ $\displaystyle
4M+4\log\left(\frac{4r_{X}r_{Y}}{r_{0}^{2}}\right)M-\frac{2\left(r_{0}\left(r_{X}+r_{Y}\right)\right)}{r_{X}r_{Y}}M$
(29) $\displaystyle+\left(\frac{16\pi
A-15\pi+8}{r_{0}}-\frac{4(4A-3)\left(r_{X}+r_{Y}\right)}{r_{X}r_{Y}}\right)M^{2}$
$\displaystyle=$ $\displaystyle\delta
t^{GR}_{S}+16A\left(\frac{\pi}{r_{0}}-\frac{r_{X}+r_{Y}}{r_{X}r_{Y}}\right)M^{2}.$

In astronomical measurements of the Cassini experiment, the researchers always
measure the relative change of the radar signal frequency in the superior
conjunction case, rather than measuring the detail time delays directly.
Combining with the time delay expression as Eq.(29) in the superior
conjunction case, the relative change of the frequency of the radar signal
reads

$\displaystyle\delta\nu$ $\displaystyle=$ $\displaystyle\frac{d}{dt}\delta
t_{S}=\frac{d}{dt}\left(\delta
t^{GR}_{S}+16A\left(\frac{\pi}{r_{0}}-\frac{r_{X}+r_{Y}}{r_{X}r_{Y}}\right)M^{2}\right)$
(30) $\displaystyle\approx$ $\displaystyle\delta\nu^{GR}_{S}-\frac{16AM^{2}\pi
r_{0}^{\prime}(t)}{r_{0}^{2}}.$

According to the Cassini experimenttest_bertotti_2003 , the frequency shift
caused by quantum gravity effect is

$\displaystyle\delta\nu_{A}\approx\frac{4096\pi}{729}\left(\frac{M_{S}}{R_{S}}\right)^{2}v_{E}A<10^{-14},$
(31)

where $v_{E}=r_{0}^{\prime}(t)$ is the Earth’s average orbit velocity, $M_{S}$
and $R_{S}$ are the mass and the radius of the Sun respectively. By
transferring geometric units($c=G=1$) to SI one and use the experiment data,
the constraint on parameter $A$ reads

$\displaystyle 0<A<1.27.$ (32)

###  IV.3 Perihelion Advance

Considering the motion equation for massive particle ($k=-1$). Eqs. (13)-(15)
imply a differential equation of massive particles between $r$ and $\phi$ as

$\displaystyle\frac{dr}{d\phi}=\left(\frac{dr}{d\tau}\right)/\left(\frac{d\phi}{d\tau}\right)=\sigma\sqrt{\frac{\mathcal{B}(r)\left(E^{2}\mathcal{B}(r)-8AM^{2}\mathcal{A}(r)\left(J^{2}+\mathcal{B}(r)\right)\right)}{J^{2}}},$
(33)

where the parameter $\sigma=\pm 1$ corresponds to the outgoing and ingoing
moving, respectively. By doing the standard transformation $u=r_{0}/r$,
Eq.(33) can be restructured to

$\displaystyle\left(\frac{du}{d\phi}\right)^{2}=\frac{u^{4}\mathcal{B}\left(r_{0}/u\right)}{r_{0}^{2}}\left(\frac{E^{2}\mathcal{B}\left(r_{0}/u\right)}{J^{2}}-8AM^{2}\mathcal{A}\left(r_{0}/u\right)\left(\frac{\mathcal{B}\left(r_{0}/u\right)}{J^{2}}+1\right)\right).$
(34)

By introducing the small parameter $\epsilon=M/r_{0}$, the above equation can
be expressed as:

$\displaystyle u^{\prime\prime}(\phi)+u(\phi)=f,$ (35)

where $f$ is the function of $u(\phi)$ as

$\displaystyle f$ $\displaystyle=$
$\displaystyle\frac{1}{J^{2}\mathcal{I}\epsilon}\Big{(}M^{2}+2\left(2E^{2}-5\right)AM^{2}\mathcal{I}\epsilon
u(\phi)+\left(22AM^{2}+3J^{2}\right)\epsilon^{2}u(\phi)^{2}$ (36)
$\displaystyle+8A\mathcal{I}\left(\left(E^{2}-4\right)AM^{2}-2J^{2}\right)\epsilon^{3}u(\phi)^{3}+32A\left(2AM^{2}+J^{2}\right)\epsilon^{4}u(\phi)^{4}\Big{)},$

and

$\displaystyle\mathcal{I}=\sqrt{8A\epsilon^{2}u(\phi)^{2}+1}.$ (37)

Next, we use the perturbation method to obtain the approximate solution of the
above equation. Hence, we need to consider the approximations of the
differential equation in terms of $\epsilon$ in the weak field where the
magnitude of $\epsilon=M/r_{0}$ is small, and the above differential equation
can be truncated as

$\displaystyle u^{\prime\prime}(\phi)+u(\phi)-\frac{Mr_{0}}{J^{2}}$
$\displaystyle=$
$\displaystyle\frac{2\left(2E^{2}-5\right)AM^{2}u(\phi)}{J^{2}}+\frac{3\left(6AM^{2}+J^{2}\right)u(\phi)^{2}\epsilon}{4J^{2}}$
(38)
$\displaystyle+\frac{A\left(\left(E^{2}-4\right)AM^{2}-2J^{2}\right)u(\phi)^{3}\epsilon^{2}}{2J^{2}}+O\left(\epsilon^{3}\right).$

The unperturbed solution of Eq.(38) can be obtained by solving the equation

$\displaystyle u_{0}^{\prime\prime}(\phi)+u_{0}(\phi)-\frac{Mr_{0}}{J^{2}}=0,$
(39)

the solution of Eq.(39) reads

$\displaystyle u_{0}(\phi)=\frac{Mr_{0}}{J^{2}}(1+e\cos(\phi)).$ (40)

Second, consider the first-order correction $u_{1}$ of the unperturbed orbit
$u_{0}$. The approximate solution can be expanded as $u\approx u_{0}+u_{1}$,
where the $u_{1}$ should satisfy

$\displaystyle u_{1}^{\prime\prime}(\phi)+u_{1}(\phi)-\frac{Mr_{0}}{J^{2}}$
$\displaystyle\approx$
$\displaystyle\frac{2\left(2E^{2}-5\right)AM^{2}u_{0}(\phi)}{J^{2}}+\frac{3\left(6AM^{2}+J^{2}\right)u_{0}(\phi)^{2}\epsilon}{4J^{2}}$
(41)
$\displaystyle+\frac{A\left(\left(E^{2}-4\right)AM^{2}-2J^{2}\right)u_{0}(\phi)^{3}\epsilon^{2}}{2J^{2}}.$

Plugging the Eq.(40) into Eq.(41), we can find that

$\displaystyle u_{1}^{\prime\prime}(\phi)+u_{1}(\phi)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{3}A_{n}\cos^{n}(\phi),$ (42)

where

$\displaystyle\mathcal{A}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{J^{8}\epsilon}\left(8\left(E^{2}-4\right)A^{2}M^{8}+2AJ^{2}M^{6}+\left(2\left(2E^{2}-5\right)A+3\right)J^{4}M^{4}\right),$
(43) $\displaystyle\mathcal{A}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{J^{8}\epsilon}\left(2e\left(12\left(E^{2}-4\right)A^{2}M^{8}-6AJ^{2}M^{6}+\left(\left(2E^{2}-5\right)A+3\right)J^{4}M^{4}\right)\right),$
(44) $\displaystyle\mathcal{A}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{J^{8}\epsilon}\left(3e^{2}M^{4}\left(8\left(E^{2}-4\right)A^{2}M^{4}-10AJ^{2}M^{2}+J^{4}\right)\right),$
(45) $\displaystyle\mathcal{A}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{J^{8}\epsilon}\left(8Ae^{3}M^{6}\left(\left(E^{2}-4\right)AM^{2}-2J^{2}\right)\right).$
(46)

Considering the initial conditions

$\displaystyle u_{1}(0)=0,\quad u_{1}^{\prime}(0)=0.$ (47)

The solution $u_{1}$ reads

$\displaystyle
u_{1}(\phi)=\sum_{n=0}^{3}C_{n}\cos(n\phi)+S_{1}\phi\sin(\phi),$ (48)

where

$\displaystyle C_{0}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{0}+\frac{\mathcal{A}_{2}}{2},$ (49) $\displaystyle
C_{1}$ $\displaystyle=$
$\displaystyle-\left(\mathcal{A}_{0}+\frac{\mathcal{A}_{2}}{3}-\frac{\mathcal{A}_{3}}{32}\right),$
(50) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{6}\mathcal{A}_{2},$ (51) $\displaystyle C_{3}$
$\displaystyle=$ $\displaystyle-\frac{1}{32}\mathcal{A}_{3},$ (52)
$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}_{1}\phi}{2}+\frac{3\mathcal{A}_{3}\phi}{8}.$
(53)

It is easy to see that the perihelion advance only depends on the
$S_{1}\phi\sin(\phi)$ term in $u_{1}$. Hence, the
$\sum_{n=0}^{3}C_{n}\cos(n\phi)$ term can be ignored, and the approximation
solution of $u$ reads

$\displaystyle u(\phi)$ $\displaystyle\approx$
$\displaystyle\frac{Mr_{0}}{J^{2}}+\frac{eMr_{0}}{J^{2}}\left(\mathcal{P}\sin(\phi)+\cos(\phi)\right)$
(54) $\displaystyle=$
$\displaystyle\frac{Mr_{0}}{J^{2}}\left(1+e\sqrt{1+\mathcal{P}^{2}}\cos(\phi-\phi_{0})\right),$

where

$\displaystyle\mathcal{P}$ $\displaystyle=$
$\displaystyle\left(2E^{2}A-5A+3\right)\frac{M^{2}\phi}{J^{2}}-6A\left(e^{2}+1\right)\frac{M^{4}\phi}{J^{4}}+3\left(E^{2}-4\right)A^{2}\left(e^{2}+4\right)\frac{M^{6}\phi}{J^{6}},$
(55) $\displaystyle\phi_{0}$ $\displaystyle=$
$\displaystyle\frac{\delta\phi_{0}}{2\pi}\phi=\arctan(P).$ (56)

Now, considering the orbit of the solution (54), the perihelion radius $r_{-}$
and aphelion radius $r_{+}$ of the orbit read respectively

$\displaystyle\frac{r_{0}}{r_{+}}=\frac{(1-e)r_{0}}{M}\frac{M^{2}}{J^{2}},$
(57)
$\displaystyle\frac{r_{0}}{r_{-}}=\frac{(1+e)r_{0}}{M}\frac{M^{2}}{J^{2}}.$
(58)

Combining the Eqs.(57) and (58), we found that

$\displaystyle\frac{M^{2}}{J^{2}}=\frac{2M}{\left(1-e^{2}\right)\left(r_{-}+r_{+}\right)}=\frac{M}{\left(1-e^{2}\right)\kappa}\sim\frac{M}{r_{0}}=\epsilon,$
(59)

where $\kappa:=(r_{+}+r_{-})/2$ is the semi-major axis of the orbit. Therefore
$M^{2}/J^{2}$ and $\epsilon=M/r_{0}$ have the same order of magnitude. Hence,
the angular $\phi_{0}$ in Eq.(56) can be expanded in terms of $M^{2}/J^{2}$,
and the angular shift of the perihelia per orbit $\delta\phi_{0}$ reads

$\displaystyle\delta\phi_{0}\approx
2\pi\left(3+\left(2E^{2}-5\right)A\right)\frac{M^{2}}{J^{2}}+O\left(\left(\frac{M^{2}}{J^{2}}\right)^{2}\right).$
(60)

By solving $dr/d\phi|_{r_{0}}=0$ with Eq.(33), the relation between the energy
$E$ and the closest approach $r_{0}$ reads

$\displaystyle
E^{2}=8AM^{2}\mathcal{A}(r_{0})\left(\frac{J^{2}}{\mathcal{B}(r_{0})}+1\right)=1-2\epsilon+O(\epsilon^{2}).$
(61)

Therefore, in the weak field approximation, the Eq.(60) can be simplified as

$\displaystyle\Delta\phi=\delta\phi_{0}\approx\frac{6\pi
M}{\left(1-e^{2}\right)\kappa}(1-A+O(\epsilon^{1}))\approx\Delta\phi^{GR}(1-A).$
(62)

Now, we could get an upper bound of the parameter $A$ by using the
observational data. For the experimental data of the anomalous Mercury
perihelion advance from the MESSENGER mission Precession_park_2017 , the
precession rate of perihelion caused by the gravitoelectric effect reads

$\displaystyle\Delta\phi=(42.9799\pm 0.0009)^{\prime\prime}/\text{century}.$
(63)

For the motion of Mercury around the Sun, the observed error of anomalous
perihelion advance is $0.0009^{\prime\prime}/\text{century}$. The contribution
of LQG is expected to be less than the observational error. Therefore, the
constraint range of parameters $A$ turns out to be

$\displaystyle 0<A<2.09\times 10^{-5}.$ (64)

Analogically, using the perihelion advance experimental data of the LAGEOS
satellites that move around the Earth Accurate_lucchesi_2010

$\displaystyle\Delta\phi=\Delta\phi^{GR}(1+(0.28\pm 2.14)\times 10^{-3}),$
(65)

we can obtain the constraint range of parameters $A$ as

$\displaystyle 0<A<1.86\times 10^{-3}.$ (66)

###  IV.4  Parameterized Post-Newtonian(PPN) approach

Now we are going to calculate the PPN parameters of the Schwarzschild like
metric (1) in the loop quantum gravity and obtain the relation between the PPN
parameter and the parameter $A$ that contains the LQG effect. First, we
perform the following transformations New_rezzolla_2014

$\displaystyle r$ $\displaystyle=$ $\displaystyle\sqrt{\bar{r}^{2}-2AM^{2}}.$
(67)

The metric (1) then can be reformulated as

$\displaystyle
ds^{2}=-\mathcal{N}^{2}(\bar{r})dt^{2}+\frac{\mathcal{B}^{2}(\bar{r})}{\mathcal{N}^{2}(\bar{r})}d\bar{r}^{2}+\bar{r}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (68)

where

$\displaystyle\mathcal{N}^{2}(\bar{r})$ $\displaystyle=$ $\displaystyle
1-\frac{2M^{2}}{\bar{r}^{2}}(-3A+\sqrt{A}\sqrt{6+\frac{\bar{r}^{2}}{AM^{2}}}),$
(69) $\displaystyle\mathcal{B}^{2}(\bar{r})$ $\displaystyle=$
$\displaystyle\frac{\bar{r}^{2}}{-2AM^{2}+\bar{r}^{2}}.$ (70)

From the transformation (67), we know that $\bar{r}>\sqrt{2A}M$ or
$\bar{r}<-\sqrt{2A}M$ must be satisfied to insure that
$\mathcal{B}^{2}(\bar{r})>0$. On the one hand, the metric component of (68)
can be expanded in terms of $M/\bar{r}$ as Metric_johannsen_2011 ;
New_rezzolla_2014

$\displaystyle\mathcal{N}^{2}(\bar{r})$ $\displaystyle=$ $\displaystyle
1-2\frac{M}{\bar{r}}+6A\frac{M^{2}}{\bar{r}^{2}}+O\left(\left(M/\bar{r}\right)^{3}\right),$
(71) $\displaystyle\frac{\mathcal{B}^{2}(\bar{r})}{\mathcal{N}^{2}(\bar{r})}$
$\displaystyle=$ $\displaystyle
1+2\frac{M}{\bar{r}}+O\left(\left(M/\bar{r}\right)^{2}\right).$ (72)

On the other hand, the PPN approximation of the asymptotic spacetime can be
described as

$\displaystyle
ds^{2}=-\mathcal{G}^{2}(\bar{r})dt^{2}+\mathcal{F}^{2}(\bar{r})d\bar{r}^{2}+\bar{r}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (73)

where

$\displaystyle\mathcal{G}^{2}(\bar{r})$ $\displaystyle=$ $\displaystyle
1-2\frac{M}{\bar{r}}+2\left(\beta-\gamma\right)\frac{M^{2}}{\bar{r}^{2}},$
(74) $\displaystyle\mathcal{F}^{2}(\bar{r})$ $\displaystyle=$ $\displaystyle
1+2\gamma\frac{M}{\bar{r}}.$ (75)

Here $\gamma$ and $\beta$ are the PPN parameters. Comparing equations
(71)-(72) and (74)-(75), we immediately obtain the corresponding equations as
follows

$\displaystyle\beta-\gamma=3A,\quad\gamma=1.$ (76)

or equivalently

$\displaystyle\gamma=1,\quad\beta=3A+1.$ (77)

Next, we consider observational constraints that imposed to PPN parameter
$\beta$ by the MESSENGER mission Precession_park_2017  which reads

$\displaystyle-6.6\times 10^{-5}<\beta-1<1.2\times 10^{-5},$ (78)

and this immediately in turn implies

$\displaystyle 0<A<4\times 10^{-6}.$ (79)

Table 1:  Summary of estimates for upper bounds of the parameter $A$. Experiments/Observations | $A$ | Datasets
---|---|---
Light deflection | 74.9836 | VLBI observation of quasarsPROGRESS_fomalont_2009
Time delay | 1.27 | Cassini experimenttest_bertotti_2003
Perihelion advance | $2.09\times 10^{-5}$ | MESSENGER missionPrecession_park_2017
| $1.86\times 10^{-3}$ | LAGEOS satellitesAccurate_lucchesi_2010
PPN approach($\beta=3A+1$) | $4.0\times 10^{-6}$ | MESSENGER missionPrecession_park_2017
Shadow of black holeTesting_brahma_2021  | 0.24 | Shadow of M87*First_EHT_2019
Tests strong equivalence principleTesting_brahma_2021  | $7.7\times 10^{-5}$ | Lunar laser ranging data Progress_williams_2004

##  V CONCLUSIONS

LQG provides an elegant resolution of both the black hole and big bang
singularities. Recently, a new polymer black hole model has been proposed in
LQG type_bodendorfer_2021 ; Mass_bodendorfer_2021 ; Testing_brahma_2021 . In
this paper, we study the classical tests of polymerized black hole in the
effective loop quantum gravity including the light deflection, Shapiro time
delay, perihelion advance, and PPN methods. Based on these classical
observations, we calculate the influences of the parameter $A$ and then obtain
the constraints on the parameter $A$ using the latest astronomical
observations in the Solar System.

The upper bounds of the parameter $A$ are summarized in Table 1. We
interestingly observed that the MESSENGER mission gives a nice constraint on
the parameter $A$ through perihelion advance as $0<A<2.09\times 10^{-5}$.
Moreover, the best constraint comes from the PPN method which gives rise to
$0<A<4.0\times 10^{-6}$. On the contrary, the observations such as the light
deflection and Shapiro time delay do not give tight constraints on parameter
$A$. The reason is that parameter $A$ has more effect on the ”nonlinearity” in
the superposition of gravity than the spatial curvature produced by unit rest
mass.

It is worth noting that in 2018, the joint European-Japanese BepiColombo
project launched two spacecraft that will explore Mercury
BepiColombo_benkhoff_2010 ; New_will_2018 . Through the BepiColombo
spacecraft, the accuracy of Mercury’s perihelion advance measurements will be
further improved by an order of magnitude compared to the MESSENGER mission
and thus more tighter constraint on the parameter $A$ will be obtained
accordingly. We would like to leave this for future study.

###### Acknowledgements.

This work is supported by NSFC with No.11775082.

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