text:
\section{Introduction}

Dark energy composes about 69\% of the total energy density of the present universe.
It is almost 15 times larger than all the visible matter we see in our universe.
But so far the existing evidences of dark energy are mostly indirect \cite{DEexp}.
The mystery of dark energy poses a great challenge to modern science.
Since the cosmological constant $\Lambda$ was introduced by Einstein into his field equation of general relativity (GR),
it has long been viewed as representing dark energy.
Up to now, various models of dark energy (such as scalar fields) have been proposed to interpret the current astronomy observations \cite{DEmodel}.

NASA has already proposed a developed mission on the gravity observation and dark energy detection explorer in the Solar System (GODDESS)
by flying a constellation of long-baseline atom-interferometer gravity gradiometers
and measuring the trace of the force field gradient tensor in the Solar System \cite{Nan18}.
It will possibly be able to isolate the new force field signal from overwhelmingly stronger gravity effects and achieve a direct detection of dark energy as a scalar field \cite{Nan18}.

Perhaps, dark energy has already been detected through non-gravitational effects \cite{Vagnozzi21}.
The coupling of dark energy to photons leads to its production in the strong magnetic field of the solar tachocline via a mechanism analogous to the Primakoff process for axions.
This allows for detectable signals on Earth. In fact, the electron recoil excess recently reported by the XENON1T \cite{XENON-1T} (deep underground at the INFN laboratory, Italy) collaboration has been explained by chameleon-screened dark energy, indicating the first direct detection of dark energy \cite{Vagnozzi21}.
Future detectors such as XENONnT \cite{XENONnT}, PandaX-4T \cite{Zhang4T}, and LUX-ZEPLIN \cite{LUXZEPLIN} are on the way.

Early in 2017, He and Zhang proposed to probe dark energy directly through gravitational effects \cite{HZ2017}.
In general, dark energy is characterized by an equation-of-state parameter $w$, which may evolve with the cosmological redshift $z$, namely $w=w\(z\)$ \cite{DEmodel}.
Different models of dark energy can be described by the parameter $w$ when it takes the corresponding model values,
such as the cosmological constant model ($\,w=-1\,$), the quintessence model ($-1<w <-\frac{1}{3}\,$), and
the phantom model ($\,w<-1$) \cite{DEmodel,HZ2017}. However, $w$ can be treated as a constant on astrophysical scales, like our Solar System \cite{HZ2017}.
Based on this, it was proved model-~and~state-independently in He~\&~Zhang~$\(2017\)$ that the local repulsion from dark energy can be described well by the SdS$_{w}$ metric \cite{HZ2017},
\begin{eqnarray}
\label{eq:dSS2}
\nonumber
\text{d} S_{w}^2 \,=\,\!&-& \bigg[1\!-2\,\frac{\,M\,}{r}-2\left( \frac{\,r_{\!o}^{}\,}{r} \right)^{\!3w+1}\bigg]\,\text{d} t^{2}\\[2mm]
&+&\frac{1}{\, \bigg[1-2\,\frac{\,M\,}{r}-2\( \frac{\,r_{\!o}^{}\,}{r} \)^{\!3w+1}\bigg] \,}\,\text{d} r^2\!+ r^{2}\left(\text{d} \theta^2+\sin^{2}\theta\,\text{d}\phi^2\right),
\end{eqnarray}
where the mass parameter $M$ is determined by both visible and dark matter of the gravitational system,
$\,r_o^{}\,$ is a model-parameter characterizing the size of the present universe, and $(t,~r,~\theta,~\phi)$ are the spherical (spacetime) coordinates.
Here we adopt the geometrized unit system ($G=c=1$).
At large $r$, the SdS$_{w}$ spacetime can be conformally and isometrically embedded into the Friedmann-Robertson-Walker spacetime \cite{HZ2017}.
Besides, the scale factor $\,r_o^{}\,$ can be given in each dark energy model when comparing with the cosmological data;
it takes the value $\,r_{\!o}^{} = \sqrt{6/\Lambda}\,$ for the cosmological constant model only and not for general dark energy models \cite{HZ2017}.
By definition, the pressure $p_{i}$, the energy density $\rho_{i}$, and the equation-of-state parameter $w$ are connected by
\begin{eqnarray}
\begin{array}{rcl}
\displaystyle
w &=& \sum\limits_{i}\,p_{i}\,\bigg/\sum\limits_{i}\,\rho_{i} <-\frac{1}{3},
\end{array}
\end{eqnarray}
where $i$ represents the $i$th contribution described by a different dark energy model \cite{DEmodel}. Note that there is only one $w$-term responsible for all these dark energy contributions in the SdS$_{w}$ metric. When $w=-1$, the metric \eqref{eq:dSS2} reduces to the well-known Schwarzschild-de Sitter (SdS) one.

For the SdS case, various kinds of studies on $\Lambda$ have been carried out previously.
In the early 1980s, Islam \cite{Islam83} showed that the light orbital equation is independent of $\Lambda$ in the SdS spacetime.
Since then it was generally believed that dark energy plays no role on the bending of light. By 2007, Rindler~and~Ishak
had concluded that $\Lambda$ contributes to the deflection of light after considering the measurements done by observers \cite{RI2007} .
However, the conclusion has led to a debate of more than ten years \cite{Ishak2008,Arakida12,Italy,Sereno08,Bhadra10,other1,Ishak2010,Ishak-Rev}.
The difficulty of understanding the influence of dark energy on light bending is due to the fact that the SdS spacetime is not flat at spatial infinity.
To overcome this difficulty, the Gauss-Bonnet theorem have been applied to a special case of the static spacetime of spherical symmetry in the literature \cite{Gibbons08,Ishihara16,Ishihara17,Arakida18}.
However, the difficulty has still not been resolved completely in the SdS spacetime; some concepts and definitions remain to be clarified \cite{Arakida18}.
Ten years later in 2017, He and Zhang \cite{HZ2017} extended the debate to the general SdS$_{w}$ spacetime.

In this work, we attempt to geometrize the bending of light using the Gauss-Bonnet theorem and propose a method to overcome the difficulty of measuring local dark energy effects.
In order to illustrate our basic ideas in a clear way, we just focus on the ideal SdS$_{w}$ case in which a point-like mass is surrounded by dark energy with a generic state parameter $w<-\frac{1}{3}$.
In fact, when $w\geq-\frac{1}{3}$, the metric \eqref{eq:dSS2} is still a solution to the Einstein equation,
which can be verified straightforwardly by following the derivations presented in He~\&~Zhang~$\(2017\)$ \cite{HZ2017}.
So the SdS$_{w}$ metric is also applicable to the case of $w\geq-\frac{1}{3}$.
However, in this case, the metric becomes asymptotically flat, and thus it is quite trivial to understand the role of the $w$-term in the bending of light.

This work is organized as follows. In section~\ref{sec:2}, we introduce the projection tensors. In section~\ref{sec:3},
we analyze the dark energy effect on massive particles, and demonstrate that it acts as a kind of gravitational force in the SdS$_{w}$ spacetime. In section~\ref{sec:4}, by presenting new techniques,
we perform the geometrization for the bending of light, and extend the concept of light deflection to any curved static spacetime.
In section~\ref{sec:5}, we give a strict solution to the problem of the influence of dark energy on light bending in the SdS$_{w}$ spacetime, and propose a method to overcome the difficulty of measuring the local dark energy effect. We conclude in section~\ref{sec:6}. In appendices \ref{app:A}, \ref{app:B}, \ref{app:C}, \ref{app:D} and \ref{app:E}, we present additional results and explanations of our theories that clarify and support the results in the main text.

\vspace*{3mm}
\section{\hspace*{+0.0mm}Projection tensors}
\label{sec:2}
\vspace*{1mm}

For any observer, let $U^{a}$ be their four-velocity with $U^{a}U_{a}=-1$, where $a$ marks the abstract index notation. For convenience, we will use the notation through this work, which was widely used in Wald's book\,\cite{Wald84}.
Let $h_{ab}=g_{ab}+U_{a}U_{b}$ and $\pi_{ab}=-\,U_{a}U_{b}$, where $g_{ab}$ is the metric tensor used to describe the entire spacetime. Then, we have
\begin{eqnarray}
\begin{array}{rcl}
\displaystyle
\nonumber
h_{ab}\,h^{b}_{\,\,\,c} &=& h_{ac},~~ h_{ab}\,\pi^{b}_{\,\,\,c} = 0,~~\pi_{ab}\,\pi^{b}_{\,\,\,c} = \pi_{ac}.
\end{array}
\end{eqnarray}
Indeed, $h_{ab}$ and $\pi_{ab}$ are space-like and time-like projection tensors, respectively.
And we have
\begin{eqnarray}
\begin{array}{rcl}
\displaystyle
\nonumber
h^{a}_{\,\,\,b}\,U^{b}=0,\,\,h_{ab}\,U^{b}=0,\,\,\pi^{a}_{\,\,\,b}\,U^{b}=U^{a},~\pi_{ab}\,U^{b}=U_{a}.
\end{array}
\end{eqnarray}
For a given four-vector $V^{a}$, let $\,V^{\,\,\,a}_{\parallel}\,$ and $\,V^{\,\,\,a}_{\perp}\,$ be the components which are parallel and perpendicular to $\,U^{a}$, respectively. Then we obtain
\begin{eqnarray}
\label{eq:Vperp}
\begin{array}{rcl}
\displaystyle
V^{\,\,\,a}_{\perp} &=& h^{a}_{\,\,\,b}\,V^{b},~~V^{\,\,\,a}_{\parallel} = \pi^{a}_{\,\,\,b}\,V^{b},\\[3mm]
V_{\perp \,a} &=& h_{ab}\,V^{b},~~ V_{\parallel \,a} =\pi_{ab}\,V^{b},
\end{array}
\end{eqnarray}
where $V_{a}$ is the corresponding dual vector. Usually, one has $V_{a}=g_{ab}\,V^{b}$. Clearly, $\,h^{a}_{\,\,\,b}\,$ projects the four-vector onto the local space of the observer.
It is noteworthy that $h_{ab}$ is actually the metric tensor of some subspace of the entire spacetime.
Denote $\,\Xi\,$ as this subspace, and parametrize it by a set of coordinates $z^{\mu}$, $\mu=0, 1, 2, 4$. The tensors $\,g_{ab}\,$ and $\,h_{ab}\,$ can be expressed in general as
\begin{eqnarray}
\label{eq:gab}
g_{ab} &=& g_{\mu\nu}\,\(\text{d} z^{\mu}\)_{a}\,\(\text{d} z^{\nu}\)_{b},\\[1mm]
\pi_{ab} &=& \pi_{\mu\nu}\,\(\text{d} z^{\mu}\)_{a}\,\(\text{d} z^{\nu}\)_{b},\\[1mm]
\label{eq:hab}
h_{ab} &=& h_{\mu\nu}\,\(\text{d} z^{\mu}\)_{a}\,\(\text{d} z^{\nu}\)_{b},
\end{eqnarray}
respectively, where $\pi_{\mu\nu}= -\,U_{\mu}\,U_{\nu}$ and $h_{\mu\nu}=g_{\mu\nu}+U_{\mu}\,U_{\nu}$.  Thus, $h_{ab}$ can be induced directly from $g_{ab}$, namely $h_{ab}=g_{ab}\big|_{\Xi}$.
Accordingly, it can be described well in the same coordinate system as $\,g_{ab}$.
In a similar way, $V^{a}$ and $V_{a}$ can be written as
\begin{eqnarray}
\label{eq:Va}
V^{a} = V^{\mu}\,\,\partial_{\mu}^{\,\,\,a},~~~~\textit{V}_{\textit{a}} = \textit{V}_{\mu}\,\(\text{d} \textit{z}^{\mu}\)_{\textit{a}},
\end{eqnarray}
with $\partial_{\mu}^{\,\,\,a}=\(\partial/\partial z^{\mu}\)^{a}$.
Note that, for $h_{ab}$, all its Lorentz indices are contracted.
So the subspace described by $h_{ab}$ is actually a physical space in which measurements can be made by the observer.
Let $V^{\mu}_{\perp}=h^{\mu}_{\,\,\,\nu} \,V^{\nu}$, and $V_{\perp \,\mu}=h_{\mu\nu} \,V^{\nu}$.
Then we have
\begin{eqnarray}
\label{eq:Vperp1}
V_{\perp}^{\,\,\,a} &=& V^{\mu}_{\perp}\,\,\partial_{\mu}^{\,\,\,a},~~~V_{\perp \,a} = V_{\perp \,\mu}\,\(\text{d} z^{\mu}\)_{a}.
\end{eqnarray}
Clearly, any four-vector $V^{a}$ can be projected into the local space of the observer.
For the projected $\,V_{\perp}^{\,\,\,a}\,$ by $\,h^{a}_{\,\,\,b}\,$ and $\,V_{\perp \,a}\,$ by $\,h_{ab}$, they are both three-vectors with Lorentz indices contracted.
From the viewpoint of physics, they are both physical quantities and can be measured by the observer.
So those quantities derived directly from them, such as the intersection angle given by equation\,\eqref{eq:thetaM} of appendix \ref{app:A},
are all independent of the choice of coordinates and thus physically measurable, especially in our notations and conventions.

\vspace*{3mm}
\section{\hspace*{+0.0mm}The Newtonian analogy}
\label{sec:del}
\label{sec:3}
\vspace*{1mm}

In GR, the Einstein equation determining the motion of matter allows a Newtonian interpretation, and thus a degree of intuitive understanding, which is often unavailable from the formalism alone.
Let us consider the Newtonian motion of a massive test-particle in the gravitational field including both contributions of the mass source and dark energy.
In Newtonian gravity, the test-particle experiences an acceleration in a gravitational field, and thus can be used to help us
obtain the exact form of the attractive Newtonian force of the mass source (including both visible and dark matter).
The force induced by dark energy can be interpreted in a quite similar way to the attractive Newtonian force.
In principle, we can obtain the exact form of the \textit{total gravitational force} including both the Newtonian attraction and the dark energy contribution from GR, starting from the metric tensor.

To begin with, we define the Newtonian limit by three requirements: the gravitational field is weak, and it is static
as well as the particles are moving very slowly compared to the speed of light \cite{Carroll14}.
Next, let us see how to introduce the total gravitational force including contributions from both the mass source and dark energy.
Firstly, its form should be intrinsically determined by the metric tensor $g_{ab}$.
Secondly, it should allow to reproduce the conventional results of Newtonian gravity in the Newtonian limit.
Thirdly, it should not include any information about the motion of the observer or the test-particle so that the total gravitational force is intrinsic to the spacetime itself.
When taking all these requirements into consideration,
we will find that only as an instantaneous observer who is static relative to the gravitational field,
one can feel the gravitational field directly and thus obtain the exact form of the total gravitational force correctly in the Newtonian limit.

In theories of stationary spacetime, for any observer, their four-acceleration is usually nonzero and given by $\hat{A}^{a}=U^{b}\nabla_{b}U^{a}$,
where $\nabla_{b}$ is the covariant derivative operator associated with the metric tensor $g_{ab}$ \cite{Wald84}.
Let $V^{a}$ be the four-velocity of a test-particle moving on a geodesic and passing through the local space of the observer, then the observed three-acceleration by this observer
would be $\hat{a}^{a}=-\hat{A}^{a}+\frac{2}{(V^{c}U_{c})^2}\,(V^{b}\hat{A}_{b})\,V_{\perp}^{\,\,\,a}$.
Setting $V^{a}=U^{a}$, we have $\hat{a}^{a}=-\hat{A}^{a}$.
In this case, the three-acceleration $\hat{a}^{a}$ is measured in the rest frame of the test-particle, usually named as the \textit{proper acceleration} \cite{Rindler06}.
However, the proper acceleration is still observer dependent.
If the instantaneous observer is static relative to the gravitational field,
the measured proper acceleration $\hat{a}^{a}$ by this observer is just the gravitational three-force $\vec{g}$ on per unit mass of the particle, especially in the Newtonian limit \cite{Wald84,Carroll14,Rindler06}.
The specific $\vec{g}$-form obtained directly from this proper acceleration will be referred to as the \textit{proper form} of the gravitational force.
Clearly, the measured proper acceleration $\hat{a}^{a}$ by the observer is uniquely determined by the metric tensor,
and, therefore, so is the proper form of the gravitational three-force $\vec{g}$.

In the case of some small physical separation, $\delta^{a}$, between the test-particle and observer,
we have the three-acceleration, $\hat{a}^{a}=- (1+\delta^{b}\nabla_{b})\, \hat{A}^{a} +R^{a}_{\,\,\,bcd}\,\,U^{b}U^{c}\delta^{d}$, where $R^{a}_{\,\,\,bcd}$ is the Riemann tensor of $\(1, 3\)$-type.
Thus, the separation in this case will lead to some additional $\delta$-terms.
However, in GR, the measurements have to be made by the instantaneous observer at the point of the test-particle or a sufficiently small region around the test-particle.
So $\delta^{a}$ is very small in an actual measurement, and thus these $\delta$-terms can be ignored directly.
Alternatively, the proper acceleration can be obtained by making $\delta$-corrections to the measured acceleration by the observer located at a small separation of $\delta^{a}$ to the test-particle.

As shown in equation\,\eqref{eq:dSS2}, the SdS$_{w}$ metric is static. So there exists a time-like Killing vector $\cal{K}^{\textit{a}}\!=\!(\partial/\partial\,\!\textit{t})^{\textit{a}}$. Let $\chi\!=\!\sqrt{-\cal{K}_{\textit{a}}\cal{K}^{\textit{a}}}$. Here the range that we are interested in is $1\!-\!2\frac{\,M\,}{r}\!-\!2\left(\! \frac{\,r_{\!o}^{}\,}{r} \!\right)^{\!3w\!+\!1}\!>\!0$. Then, $\chi\!=\!\sqrt{-g_{00}}=\sqrt{1\!-\!2\frac{\,M\,}{r}\!-\!2\left(\! \frac{\,r_{\!o}^{}\,}{r} \!\right)^{\!3w\!+\!1}}$.
By definition, we have $\hat{A}^{a}=\nabla^{a}\rm{ln} \,\chi$ \cite{Wald84}.
Correspondingly, $\hat{A}_{a}=\nabla_{a}\rm{ln} \,\chi=\({\it d}\,\rm{ln} \,\chi\)_{\it{a}}=\frac{1}{\sqrt{-\it{g_{{\rm 00}}}}}\,\frac{\partial\,\!\sqrt{-\it{g_{{\rm 00}}}}}{\partial\,\!{\it r}}\,\(\text{d}\,\!\it{r}\)_{\it{a}}$,
where $\({\it d}\,\rm{ln} \,\chi\)_{a}$ is the total differential of $\rm{ln}\,\chi$.
We therefore obtain the proper form of the three-acceleration of the massive test-particle as follows,
\begin{eqnarray}
\label{eq:4Aa}
\nonumber
\hat{a}^{a}\!=-\hat{A}^{a}&=&-g^{ab}\,\hat{A}_{b}=-\(g^{\mu\nu}\,\partial_{\mu}^{\,\,\,a}\,\partial_{\nu}^{\,\,\,b}\)\,\left[\frac{\,\,1}{\sqrt{-g_{00}}}\,\frac{\partial\!\,\sqrt{-g_{00}}}{\partial\,\!r}\,\(\text{d}\,\!r\)_{b}\right]\\[3mm]
\nonumber
&=&-\frac{\,\,g^{rr}}{\sqrt{-g_{00}}}\,\frac{\partial\!\,\sqrt{-g_{00}}}{\partial\,\!r}\,\(\frac{\partial}{\partial\,\!r}\)^{\!a}\\[3mm]
\nonumber
&=&-\frac{1}{r}\left[\frac{\,M\,}{r}\!+\!(3w\!+\!1)\left(\! \frac{\,r_{\!o}^{}\,}{r} \!\right)^{\!3w\!+\!1}\right]\,\(\frac{\partial}{\partial\,\!r}\)^{\!a}\\[3mm]
&=&-\frac{1}{r}\frac{1}{\sqrt{1\!-\!2\frac{\,M\,}{r}\!-\!2\left(\! \frac{\,r_{\!o}^{}\,}{r} \!\right)^{\!3w\!+\!1}}}\,\left[\frac{\,M\,}{r}\!+\!(3w\!+\!1)\left(\! \frac{\,r_{\!o}^{}\,}{r} \!\right)^{\!3w\!+\!1}\right]\textit{e}_{r}^{\,a},
\hspace*{10mm}
\end{eqnarray}
where  $\textit{e}_{r}^{\,a}=\frac{1}{\sqrt{g_{rr}}}\(\frac{\partial}{\partial\,\!r}\)^{\!a}$ is the unit vector along the radial direction.
In our notations, we have $g^{ab}=g^{\mu\nu}\,\partial_{\mu}^{\,\,\,a}\,\partial_{\nu}^{\,\,\,b}$ and $\(\text{d} z^{\nu}\)_{b}\,\partial_{\mu}^{\,\,\,b}\,=\delta^{\,\,\nu}_{\mu}$,
which have been used in the first and second lines, respectively.
From equation\eqref{eq:4Aa}, we see that besides the attractive Newtonian force, there exits a {\it dark force} (generated by dark energy) on the massive test-particle.
Evidently, the dark force and the Newtonian force are on equal footing.
The observational data requires $\,3w+2 < 0\,$ at $\,6\,\sigma$\, level\,\cite{2015BAO}.
Thus, the $w-$term leads to a repulsive dark force.
For the special case of the cosmological constant $\Lambda$, we have $\,r_{\!o}^{} = \sqrt{6/\Lambda}\,$.
In this case, we can rewrite equation\,\eqref{eq:4Aa} by setting $w=-1$ in a more familiar form
\begin{eqnarray}
\label{eq:g3force}
\nonumber
\vec{g}= -\frac{1}{r}\frac{1}{\sqrt{1\!-\!2\frac{\,M\,}{r}\!-2\frac{\Lambda}{\,6\,}r^2}}\(\frac{\,M\,}{\,\,r\,}\!-\!\frac{\Lambda}{3}r^2\)\,\boldsymbol{\hat{r}},
\hspace*{10mm}
\end{eqnarray}
with $|\boldsymbol{\hat{r}}|=1$.
Under the weak field approximation, one gets ~$\vec{g}\,\simeq-\(\frac{\,M\,}{\,\,r^2\,}\!-\!\frac{\Lambda}{3}r\)\,\boldsymbol{\hat{r}}$,
which agrees with the dark force shown by Ho and Hsu in \,\cite{Ho15}.
Now we successfully obtain the proper form of the total gravitational force via equation\,\eqref{eq:4Aa}, independently of specific dark energy models.

As seen from the static observer with respect to the gravitational field,
the strength of the three-force $\,\vec{g}\,$ on the test-particle, also known as the gravitational field strength,
equals the magnitude of the three-acceleration \eqref{eq:4Aa}, namely $g=\left|\vec{g}\right|=\left|\hat{a}^{a}\right|$.
Exactly, the gravitational field strength $g$ can be written as
\begin{eqnarray}
\label{eq:gforce}
\nonumber
g \,&=&\sqrt{\hat{a}^{a}\,\hat{a}_{a}}=\sqrt{\hat{A}^{a}\,\hat{A}_{a}}\\[3mm]
\nonumber
&=&\sqrt{\left[\frac{\,\,g^{rr}}{\sqrt{-g_{00}}}\,\frac{\partial\!\,\sqrt{-g_{00}}}{\partial\,\!r}\,\(\frac{\partial}{\partial\,\!r}\)^{\!a}\right] \left[\frac{1}{\sqrt{-g_{00}}}\,\frac{\partial\,\!\sqrt{-g_{00}}}{\partial\,\!r}\,\(\text{d}\,\!r\)_{a}\right]}\\[3mm]
\nonumber
&=&\!\bigg|\frac{1}{\sqrt{\(-g_{00}\)g_{rr}}\,\,}\,\(\frac{\partial\,\!\sqrt{-g_{00}}}{\partial\,\!r}\)\!\bigg|\\[3mm]
&=&\frac{1}{\,r\,}\frac{\,1\,}{\sqrt{1\!-\!2\frac{\,M\,}{r}\!-\!2\left(\! \frac{\,r_{\!o}^{}\,}{r} \!\right)^{\!3w\!+\!1}}}\,\bigg|\!\frac{\,M\,}{r}\!+\!(3w\!+\!1)\left(\! \frac{\,r_{\!o}^{}\,}{r} \!\right)^{\!3w\!+\!1}\!\bigg|,
\hspace*{10mm}
\end{eqnarray}
where we have used $\(\text{d}\,\!r\)_{a}\(\frac{\partial}{\partial\,\!r}\)^{\!a}=1$ in the third line.
When taking $r_{\!o}^{}\to\infty$, one has
\begin{eqnarray}
\label{eq:CCforce}
\nonumber
g\simeq\frac{\,M\,}{r^2}\frac{1}{\sqrt{1\!-\!2\frac{\,M\,}{r}}},
\hspace*{10mm}
\end{eqnarray}
which is just the Rindler's form of the field strength shown by equation\,$\(11.15\)$ in \,\cite{Rindler06} for the special case of the Schwarzschild spacetime.
In this case, we see that, for large $r$, the field strength recovers the Newtonian inverse-square law in the Newtonian limit, namely $g\!\simeq\!\frac{\,\,M\,\,\,}{\,r^2\,}$.
It is important to mention that the SdS$_{w}$ metric is true not only for the cosmological constant but also for other possible forms of dark energy (such as a single effective scalar field).
In general, there is no explicit cosmological constant $\Lambda$ involved in the SdS$_{w}$ metric;
that is, the parameter $r_{\!o}^{}$ can not be expressed in terms of $\Lambda$ for general dark energy models.
Additionally, we do not have the explicit $r_{\!o}^{}$ value for each dark energy model at present. Hence, the Newtonian analogue cannot be achieved directly from arbitrary state parameter $w$.
One has to first set it to $w=-1$, i.e., the cosmological constant case. Thereafter one should set $\Lambda=0$ (to achieve $r_{\!o}^{}=\infty$), and finally for $r\gg\,M$, one recovers the Newtonian analogy.
Anyway, we have shown that the three-force $\vec{g}$ (or, equivalently, $\hat{a}^{a}$) given by equation\eqref{eq:4Aa} satisfies all the requirements mentioned earlier,
and we therefore successfully obtain the proper form of the total gravitational force including dark energy.

The dark force acts on astrophysical scales (such as the Solar-System) that can be tens of orders of magnitudes lower than the cosmological scale.
Requiring $\vec{g}=0$ or $g=0$, we derive the critical radius $r_{\text{cri}}^{}$ without any further approximations,
\begin{eqnarray}
r_{\text{cri}}^{} \,=\, r_{\!o}^{}\(\!\frac{M}{\,\left|3w\!+\!1\right|\,r_{\!o}^{}\,}\!\)^{\!\!-\frac{1}{3w}}
\label{Arc}
\label{eq:Arc}
\end{eqnarray}
where the repulsive dark force can balance the attractive Newtonian force.
This coincides with Ho\,\,\&\,\,Hsu~$\(2015\)$ for the cosmological constant model\,\cite{Ho15} and with He\,\,\&\,\,Zhang~$\(2017\)$ for general dark energy models \cite{HZ2017}.
However, the post-Newtonian approximation adopted in He~\&~Zhang~$\(2017\)$ begins to break down when $r\simr_{\text{cri}}^{}$.
Here the formula \eqref{eq:4Aa} remains valid in the outer region with $r\gtrsimr_{\text{cri}}^{}$.
So we have extended the formula for the total gravitational force to the far-away region.
Now it is reasonable to say that the repulsive dark force will dominate over the Newtonian attraction in the outer region.

One major goal of the GODDESS mission is to detect any possible deviation from the inverse square law behavior (mentioned above) \cite{Nan18}.
It can be expected that the dark force, also referred to as `{\it the fifth force}' by the GODDESS team \cite{Nan18}, will lead to the deviation.
It means that the GODDESS mission will have a chance of successful detection of dark energy \cite{Nan18}.
The measured force (or field) strength by GODDESS depends on its distance to the mass source.
The relative changes in the force (or field) strength has huge impact to GODDESS on differentiating the dark force from the purely Newtonian force \cite{Nan18}.
So the exact form of the total gravitational force \eqref{eq:4Aa} or the gravitational field strength \eqref{eq:gforce} can help the GODDESS team to optimize schemes
and develop strategies for the detection of dark energy.
For example, the critical radius $r_{\text{cri}}^{}$ \eqref{eq:Arc} derived from equation\,\eqref{eq:4Aa} or \eqref{eq:gforce} can tell us how far the explorer should be from the Sun, the Earth or other planets, which is important to the choice of mission trajectories.

\vspace*{3mm}
\section{\hspace*{+0.0mm}Geometrization of light bending}
\label{sec:del}
\label{sec:4}
\vspace*{1mm}

In any static spacetime, there always exists a local space $\Xi_{p}$ perpendicular to the four-velocity $U^{a}$ of the static observer at each point $p$.
The local space $\Xi_{p}$ is actually determined by a local metric tensor, denoted as $h_{ab}|_{\Xi_{p}}$, which is independent of the choice of local coordinates,
and, as seen from the local observer at point $p$, it represents a physical space.
We can paste these local spaces $\{\Xi_{p}\}$ at different points $\{p\}$ together to get a three-dimensional manifold $\Xi$, namely $\Xi=\substack{\bigcup\\p}\,\Xi_{p}$.
In current theories of gravity, the entire spacetime is uniquely determined by one metric tensor $g_{ab}$.
Consequently, all the metric tensors $\{h_{ab}|_{\Xi_{p}}\}$ at different points $\{p\}$ can be written in a unified form. Denote it by $h_{ab}$.
Clearly, $h_{ab}$ is the induced metric tensor from $g_{ab}$, namely $h_{ab}=g_{ab}|_{\Xi}$ (see section~\ref{sec:2} for details).
For the static observer, as the global extension of $h_{ab}|_{\Xi_{p}}$, $h_{ab}$ is uniquely given by $g_{ab}$.
In GR, the metric tensor $g_{ab}$ can be parametrized by a set of global coordinates, like $\{z^{\mu}\}$, as shown by equation\,\eqref{eq:gab}.
Therefore, we can describe the metric tensor $h_{ab}$ on the hypersurface $\Xi$ in the same global coordinates, exactly as demonstrated by equation\,\eqref{eq:hab} (see section~\ref{sec:2} for details).

At each point $p$, consider a locally embedded surface $\Sigma_{p}\subset\Xi_{p}$, representing a slice of $\Xi_{p}$.
In analogy to what we have done for $\Xi$, we can paste these local surfaces $\{\Sigma_{p}\}$ at different points $\{p\}$ together
and obtain a two-dimensional manifold $\Sigma=\substack{\bigcup\\p}\,\Sigma_{p}$.
In differential geometry, $\Xi$ is a (three-dimensional) global space.
As see from the observer at each point $p$, $\Sigma$ is a global surface embedded into the global space $\Xi$,
with each $\Sigma_{p}$ determined by the newly induced tensor $\hat{h}_{ab}|_{\Sigma_{p}}\equiv\,\!(h_{ab}|_{\Xi_{p}})|_{\Sigma_{p}}$~$=h_{ab}|_{\Sigma_{p}}$.
Indeed, there always exists such a global surface in any curved static spacetime.
For instance, in the SdS$_{w}$ spacetime, we can construct such a surface by setting $\theta =\frac{\pi}{2}$, without loss of generality (see section~\ref{sec:5} for details).
For any surface $\Sigma$, we denote $\hat{h}_{ab}$ to be its metric tensor.
Generally, we have $\hat{h}_{ab}=h_{ab}|_{\Sigma}$, which is observer dependent.
In math lingo, $(\Sigma,\,\hat{h}_{ab})$ is a two-dimensional Riemannian manifold.
Now it can be viewed as a physical surface on which measurements may take place.

For each static observer, the metric tensor $\hat{h}_{ab}$, as the global extension of $\hat{h}_{ab}|_{\Sigma_{p}}$,
is not uniquely determined by $h_{ab}$.
Actually, the tensor $\hat{h}_{ab}|_{\Sigma_{p}}$ is dependent on the choice of the local slice $\Sigma_{p}~(\subset\Xi_{p})$.
Besides, we can always get the static observer at each point $p$ to choose a desired slice $\Sigma_{p}$ by certain rules, and then paste these local slices $\{\Sigma_{p}\}$ together to construct a global surface $\Sigma$.
To carry it out, we can define a global surface $\Sigma$ in advance.
Then, we could have the local observer at each point $p$ choose a neighborhood of $p$ on the surface $\Sigma$ and define it as the local slice $\Sigma_{p}$.
Clearly, the choice of $\Sigma_{p}$ is flexible, depending on how to define or choose the global surface $\Sigma$.
In mathematics, $\hat{h}_{ab}|_{\Sigma_{p}}$ is closely related with $\Sigma_{p}$.
Thus, it also shows a dependence on the choice or the definition of $\Sigma$.
As the global extension of $\hat{h}_{ab}|_{\Sigma_{p}}$, the metric tensor $\hat{h}_{ab}$ can not be uniquely determined by $h_{ab}$; more exactly, it also depends on the choice of the global surface $\Sigma$.
Taking into account all these factors, we come to the conclusion that the physical surface $(\Sigma,\,\hat{h}_{ab})$ can be chosen at our convenience,
which lays a solid foundation for the geometrization of light bending.

\begin{figure}[t]
\centerline{
\includegraphics[width=1.03\columnwidth]{PTransport.pdf}}
\vspace*{-2mm}
\caption{Parallel transport. The vector field $V^{a}\!=\!V^{a}\(\lambda\)$ denotes the parallel transport of $V^{a}\(0\)$ along the positively oriented path $\cal{S\!\to\!_{A}\!\to\!_{B}\!\to\!O\!\to_{M}\to\!S}$ at point $\lambda$, with a polar angle $\varphi\(\lambda\)$ from the $x$-axis.}
\label{fig:PTrans}
\end{figure}

Now we choose a global surface $\Sigma$. As figure~\ref{fig:PTrans} illustrates, there is a simple, closed, positively oriented piecewise regular curve on $\Sigma$,
and it is composed of two oriented paths $L_{0}\doteq\cal{S\!\to\!_{A}\!\to\!_{B}\!\to\!O}$ and $\bar{L}\doteq\cal{O\!\to_{M}\to\!S}$.
Let $\gamma: \!I\!=\![0,l]\to\Sigma$ be a map of a closed interval $I$ to the closed curve,  with $\gamma\(\lambda_{0}\)\!=\!\cal{O}$ and $\gamma\(0\)\!=\!\gamma\(l\)\!=\!\cal{S}$, where $\lambda_{0}$ is the arc length at point $\cal{O}$, and $l$ is the total arc length of the closed curve. Here $\gamma$ has already been parametrized by arc length $\lambda$.
Now choose a Cartesian coordinate system $\rm{O}\it{xy}$ with origin at $\rm{O}=\(0, 0\)$.
For any vector $V^{a}\(0\)$ tangent to $\Sigma$,
let $V^{a}=V^{a}\(\lambda\)$ be the parallel transport of $V^{a}\(0\)$ along the oriented curve $\gamma$, with $\varphi\!=\!\varphi\(\lambda\)$ being the angle from the $x$-axis to $V^{a}$ at point $\lambda$. Thus, we obtain $\varphi\!=\!\varphi\(\lambda_{0}\)$ at $\lambda=\lambda_{0}$ and $\varphi\!=\!\varphi\(l\)$ at $\lambda=l$.
Along the same track as $\bar{L}$, the oriented path $L\doteq\cal{S\!\to_{M}\to\!O}$ is traced in the opposite direction. Denote $\bar{V}^{a}=\bar{V}^{a}(\bar{\lambda})$ as the parallel transport of $V^{a}\(0\)$ along $L$ at $\bar{\lambda}=l\!-\!\lambda$ and $\bar{\varphi}\!=\!\bar{\varphi}(\bar{\lambda})$ as the corresponding intersection angle between $\bar{V}^{a}$ and the $x$-axis. In this notation, we have $\bar{\varphi}\!=\!\bar{\varphi}\(\bar{\lambda}_{0}\)$ at $\bar{\lambda}_{0}=l\!-\!\lambda_{0}$. Similarly, the parallelly transported $V^{a}\(l\)$ along $L$ and the corresponding intersection angle are denoted by $\bar{V}_{*}^{a}=\bar{V}_{*}^{a}(\bar{\lambda})$ and $\bar{\varphi}_{*}\!=\!\bar{\varphi}_{*}(\bar{\lambda})$ at point $\bar{\lambda}$, respectively. In particular, $\bar{\varphi}_{*}\!=\!\bar{\varphi}_{*}\(\bar{\lambda}_{0}\)$ at $\bar{\lambda}_{0}=l\!-\!\lambda_{0}$.
According to the theorem of uniqueness of parallel transport, we have $\bar{V}_{*}^{a}(\bar{\lambda_{0}})=V^{a}\(\lambda_{0}\)$ and $\bar{\varphi}_{*}\(\bar{\lambda}_{0}\)=\varphi\(\lambda_{0}\)$,
which are an immediate consequence of the theorem of existence and uniqueness of differential equations  \cite{Carmo16,Chern00}.
When any two vectors are parallelly transported together along the same track and in the same direction, their intersection angle keeps to be unchanged \cite{Rindler06}.
Thus, $\bar{\varphi}_{*}\(\bar{\lambda}_{0}\)-\bar{\varphi}\(\bar{\lambda}_{0}\)=\varphi\(l\)-\varphi\(0\)$.
Then we have $\varphi\(\lambda_{0}\)-\bar{\varphi}\(\bar{\lambda}_{0}\)=\varphi\(l\)-\varphi\(0\)$.
It is noteworthy that these results are all independent of the choice of $V^{a}\(0\)$.
Following the same steps as above, we can derive $\varphi\(\lambda\)-\bar{\varphi}\(\bar{\lambda}\)=\varphi\(l\)-\varphi\(0\)$ for any point $\lambda$.
Hence, we finally find the following relationship,
\begin{eqnarray}
\label{eq:varphi0}
\begin{array}{rcl}
\displaystyle
\bar{\varphi}\(\bar{\lambda}\)-\varphi\(\lambda\)=\bar{\varphi}\(\bar{\lambda}_{0}\)-\varphi\(\lambda_{0}\),
\hspace*{0mm}
\end{array}
\end{eqnarray}
which holds well for any $\gamma$-like curve.
Basing on this newly discovered relationship, we can extend the concept of light deflection to any curved static spacetime.

To do this, we need to perform the geometrization of light bending in advance.
First of all, we define a generalized deflection angle exactly as
\begin{eqnarray}
\label{eq:GaussianCurvature0}
\begin{array}{rcl}
\displaystyle
\alpha_{M}=\bar{\varphi}\(\bar{\lambda}_{0}\)-\varphi\(\lambda_{0}\),
\hspace*{0mm}
\end{array}
\end{eqnarray}
which is applicable to any curved static spacetime. Hereafter, we refer to this angle as the \textit{Gaussian deflection angle}.
Then, denote by $K$ the Gaussian curvature. By the Gauss-Bonnet theorem, we have (see appendix \ref{app:E} for details)
\begin{eqnarray}
\label{eq:GBtoPT}
\begin{array}{rcl}
\displaystyle
\varphi\(l\)-\varphi\(0\)=\int\!\!\!\int_{D}\,K\text{d} \sigma
\hspace*{0mm}
\end{array}
\end{eqnarray}
where $\text{d} \sigma$ is the element of area and $\it{D} \(\subseteq\rm{\Sigma}\)$ denotes the simple, connected region bounded by the closed curve $\gamma$.
Combining with the above derivations, the Gaussian deflection angle can be further derived as follows,
\begin{eqnarray}
\label{eq:GaussianCurvature}
\begin{array}{rcl}
\displaystyle
\alpha_{M}=\bar{\varphi}\(\bar{\lambda}\)-\varphi\(\lambda\)=-\int\!\!\!\int_{D}\,K\text{d} \sigma,
\hspace*{0mm}
\end{array}
\end{eqnarray}
which is independent of specific spacetime models.
It clearly indicates that the nature of light bending is the curvature of spacetime.
Besides, the left hand side of the equation is invariant under coordinate transformations, and so is the right hand side.
Now we have geometrized the deflection of light successfully, and extended the definition of the deflection angle to the most general static spacetime.

The mathematics involved in equation\,\eqref{eq:GaussianCurvature} is fairly simple. Denote $\partial{\rm D}$ as the closed boundary of the region $D$, namely $\gamma=\partial{\rm D}$.
Choose ${\it D}$ to be a geodesic polygon (that is, polygon whose sides are arcs of geodesics). Hereafter, ${\it D}$ is referred to as the \textit{lensing patch}, on which the measurement can be made by local observers.
In the case of gravitational lensing, light follows null geodesics.
Thus, the geodesic curvature of $\partial{\rm D}$ is zero, namely, $k_{g}\equiv0$ \cite{Carmo16,Chern00}.
Let $\theta_{i}$ be the $i$th external angle of $\partial{\rm D}$, which is actually a measurable intersection angle (see appendix \ref{app:A} for details) by the static observer at the $i$th vertex $\gamma\(\lambda_{i}\)$.
Here $\lambda_{i}$ is the arc length at the $i$th vertex.
Then, according to the Gauss-Bonnet theorem, we also have (see appendix \ref{app:E} for details)
\begin{eqnarray}
\label{eq:polygon}
\begin{array}{rcl}
\displaystyle
\alpha_{M}=\sum_{i}\theta_{i}-2\pi.
\hspace*{0mm}
\end{array}
\end{eqnarray}
Generally speaking, it asserts that the Gaussian deflection angle is equal to the excess over $2{\rm \pi}$ of the sum of the external angles of the geodesic polygon.
Take a geodesic triangle for example. Denote the $i$th interior angle as $\psi_{i}$. When dealing with the geodesic triangle, we have $\alpha_{M}=\pi-\sum_{i=1}^{3}\psi_{i}$. If the triangle is on a flat surface, we have $\alpha_{M}=0$. Also, on a sphere-like surface, $\alpha_{M}<0$ \cite{Carmo16}. Similarly, on a pseudosphere-like surface, $\alpha_{M}>0$ \cite{Carmo16}.
Obviously, the formula \eqref{eq:polygon} provides a remarkable relation between the geodesic polygon and the deflection of light .

The Gaussian deflection angle $\alpha_{M}$ \eqref{eq:GaussianCurvature0} is actually a generalized deflection angle. In some special cases, it can reduce to the \textit{usual deflection angle}.
To show this, we need to choose a spacetime region where the metric is flat (or conformally flat).
A typical example of this is the region of spatial infinity in the Schwarzschild spacetime.
Hereafter, we name this kind of region as the \textit{laboratory area}.  Its properties are presented detailedly in appendix~\ref{app:A};
in the laboratory area, the light ray travels along a physically straight line, and we can not probe its bending effect.
As illustrated in figure~\ref{fig:PTrans}, let $L_{0}$ be a simple, oriented piecewise regular curve in the laboratory area.
It contains three segments of straight lines: $\,\overline{\cal{SA}}\,$, $\,\overline{\cal{AB}}$, and $\,\overline{\cal{BO}}$,
satisfying $\,\overline{\cal{SA}}\,\bot\,\(\partial/\partial\it{x}\)^{a}$, $\,\overline{\cal{AB}}\,\bot\,\(\partial/\partial\it{y}\)^{a}$, and $\,\overline{\cal{BO}}\,\bot\,\(\partial/\partial\it{x}\)^{a}$.
These segments can be parts of light trajectories.
If so, the intersection angle between any two segments is actually an measurable angle (see appendix \ref{app:E} for details).
Due to the flatness or the conformal flatness of the laboratory area, one has $\varphi\(\lambda_{0}\)=\varphi\(0\)$ when taking the parallel transport of $V^{a}\(0\)$ along the path $L_{0}$.
Then, $\alpha_{M}=\bar{\varphi}\(\bar{\lambda}_{0}\)-\varphi\(0\)$, which agrees with the definition of the \textit{usual deflection angle}.
Thus, the Gaussian deflection angle $\alpha_{M}$ recovers the usual deflection angle.
We therefore via equations\,\eqref{eq:GaussianCurvature0} and \eqref{eq:GaussianCurvature} give a reasonable and natural generalization of the usual deflection angle.

Turn now to the application of the Gaussian deflection angle to measuring the local spacetime effect on the bending of light. To obtain the Gaussian deflection angle,
one needs to measure the external angle $\theta_{i}$ or the interior angle $\psi_{i}$ at each vertex $\gamma\(\lambda_{i}\)$ of $\partial{\rm D}$.
These external (or interior) angles are actually the intersection angles between any two light trajectories (as null geodesics),
and each angle can be directly measured by the static observer at each vertex.
In reality, measurements may be performed by moving observers passing by each vertex.
In this case, the measured values by the moving observer for $\,\theta_{i}\,$ or $\,\psi_{i}\,$ need to be made relativistic corrections.
In fact, the finally measured $\,\theta_{i}\,$ or $\,\psi_{i}\,$ by the static observer at each vertex $\gamma\(\lambda_{i}\)$ can be obtained by using the general relativistic aberration relationships \cite{Ishak-Rev}.
Anyway, we can always obtain the Gaussian deflection angle from these locally measurable intersection angles $\,\{\theta_{i}\}\,$ or $\,\{\psi_{i}\}\,$ via the formula\,\eqref{eq:polygon}.
In addition, the Gaussian deflection angle, as a global quantity, is fully determined by the integral of the Gaussian curvature $K$ over the lensing patch $D$, namely the total curvature (see appendix \ref{app:E} for details).
Thus, using the Gauss-Bonnet theorem, we establish a relation between the global properties of the lensing patch $D$, like the total curvature,
and the local properties of the curve $\partial{\rm D}$ composed of null geodesics, such as $\,\theta_{i}\,$ or $\,\psi_{i}\,$ at each vertex.
By definition, $K$ quantifies properties intrinsic to the spacetime surface $\Sigma$.
Therefore, from this perspective, the Gaussian deflection angle can be used as a potentially interesting probe of the intrinsic properties of spacetime.

\vspace*{3mm}
\section{\hspace*{+0.0mm}Local SdS$_{w}$ spacetime effect on the bending of light}
\label{sec:5}
\vspace*{1mm}

Come back to the case of the SdS$_{w}$ spacetime. When $M=0$, the SdS$_{w}$ metric is conformally flat \cite{HZ2017}.
In this case, the light ray travels along a physically straight line, which is the same as the Minkowski vacuum case (cf~appendix~\ref{app:A}).
It means that in this case, we can not probe the local spacetime effect of dark energy on the bending of light.
However, for the case of $M\neq0$, it becomes rather difficult to deal with the SdS$_{w}$ metric in a similar way.
In fact, it is almost impossible to demonstrate what role dark energy plays in the bending of light based on the traditional theories\footnote{In traditional theories for light deflection, the usual deflection angle plays a central role, and various approaches have been proposed accordingly to probe the bending of light. Hereafter, this kind of approaches are referred to as the traditional approaches.} only
(see~appendices~\ref{app:C} and \ref{app:D} for details).
In this section, we attempt to understand the role of dark energy in light bending by introducing the concepts and techniques presented in the previous section.
As already mentioned earlier, in the SdS$_{w}$ spacetime, there always exists a subspace $\Xi$ locally perpendicular to the four-velocity $U^{a}$ of the static observer at each point.
The subspace $\Xi$ can be globally described by the induced metric tensor,
\begin{eqnarray}
\label{eq:Xi_hab}
\nonumber
h_{ab}&=&\frac{1}{1\!-2\frac{\,M\,}{r}-\!2\left(\! \frac{\,r_{\!o}^{}\,}{r} \!\right)^{\!3w+1}}\(\text{d} r\)_{a}\(\text{d} r\)_{b}+ r^{2}\(\text{d} \theta\)_{a}\(\text{d} \theta\)_{b}+r^{2}\sin^{2}\!\theta\,\(\text{d}\phi\)_{a}\(\text{d}\phi\)_{b}.
\end{eqnarray}
We then take $\,\theta =\frac{\pi}{2}$, without lossing generality, and therefore obtain the following metric tensor,
\begin{eqnarray}
\label{eq:Sigma_hab}
\nonumber
\hat{h}_{ab}=\dfrac{1}{1-2\frac{\,M\,}{r}-2\left(\! \frac{\,r_{\!o}^{}\,}{r} \!\right)^{\!3w+1}}\(\text{d} r\)_{a}\(\text{d} r\)_{b}+ r^{2}\(\text{d}\phi\)_{a}\(\text{d}\phi\)_{b}.
\end{eqnarray}
Thus the subspace $\Xi$ reduces to the $\(\it{r},\phi\)$-plane determined by $\hat{h}_{ab}$.
We therefore construct a physical surface $\Sigma$, which is characterized by the following metric,
\begin{eqnarray}
\label{eq:Sigma_SdS}
\nonumber
\text{d} s^{2}&=&E\,\text{d}\,\!u^{2}+2 F\,\text{d}\,\!u\,\text{d}\upsilon+G\,\text{d}\upsilon^{2}=\frac{1}{1\!-2\frac{\,M\,}{r}-2\left(\! \frac{\,r_{\!o}^{}\,}{r} \!\right)^{\!3w+1}}\, \,\text{d} r^2\,+r^{2}\,\text{d}\phi^2.
\end{eqnarray}
This metric is orthogonally parametrized by $\(u, \upsilon\)\!=\!\(r, \phi\)$ (see appendix \ref{app:E} for details). In this case,
\begin{eqnarray}
\label{eq:EFG}
E=\frac{1}{1\!-2\frac{\,M\,}{r}-\!2\left(\! \frac{\,r_{\!o}^{}\,}{r} \!\right)^{\!3w+1}},~F=0,~G=r^2.
\end{eqnarray}
With these, we further derive the Gaussian curvature by using equation\,\eqref{eq:GaussK}, exactly as follows,
\begin{eqnarray}
\label{eq:K_SdSw}
K=\frac{\partial}{\partial\,\!r}\(\sqrt{1\!-2\frac{\,M\,}{r}-\!2\left(\! \frac{\,r_{\!o}^{}\,}{r} \!\right)^{\!3w+1}}\).
\end{eqnarray}
As shown in appendix \ref{app:E}, the total curvature $K_{\rm{tot}}$ can be written as
\begin{eqnarray}
\label{eq:totalK}
\nonumber
K_{\rm{tot}}&=&\int\!\!\!\int_{D}\,K\text{d} \sigma
=-\frac{1}{2}\oint_{\partial\rm{D}}\(-\frac{(\sqrt{E})_{\!\upsilon}}{\sqrt{G}}\,\rm{d}\it{u}+\frac{(\sqrt{G})_{\!u}}{\sqrt{E}}\,\rm{d}\it{\upsilon}\)
\hspace*{0mm}
\end{eqnarray}
Combining this with equation\,\eqref{eq:GaussianCurvature} and equation\,\eqref{eq:EFG}, the Gaussian deflection angle can be finally reexpressed as
\begin{eqnarray}
\label{eq:ralphaw}
\alpha_{M}
&=&\frac{1}{2}\,\mathlarger{\oint}_{\partial{\rm D}}\sqrt{1\!-2\frac{\,M\,}{r}-\!2\(\! \frac{\,r_{\!o}^{}\,}{r} \!\)^{\!3w+1}}\text{d} \phi,
\vspace*{6mm}
\end{eqnarray}
where $\phi=\phi\(r\)$ is uniquely determined by the boundary curve $\partial{\rm D}$. Note that the formula \eqref{eq:ralphaw} is derived without any of further assumptions or approximations.
Obviously, the Gaussian deflection angle $\alpha_{M}$ takes a model-independent form, with various dark energy models described by different $w$ values \cite{DEmodel}.
From the formula\,\eqref{eq:ralphaw}, we can further conclude that dark energy does contribute the bending of light via the $w$-dependent term.
So it is possible to extract the information about $\(w, r_{\!o}^{}\)$ by making precise measurements via equation\,\eqref{eq:ralphaw} for the bending of light.
Clearly, this will have many applications.
For instance, as expected from the model predictions of the cosmological constant as the dark energy candidate, we have $w=-1$ and $\Lambda=6/{r_{\!o}^{}}^{\!\!2}$,
which can be tested directly through measuring the bending of light on astrophysical scales, independently of current cosmological observations.

\begin{figure}[!htb]
  \centerline{
      \includegraphics[width=0.71\columnwidth]{PathInt.pdf}
      }
  \caption[PTV]{An example of the lensing patch. It is enclosed by a specific boundary curve, which comprises four segments of null geodesics: $\overline{\cal{SA}}$,~\,$\widetilde{\cal{AB}}$,~\,$\overline{\cal{BO}}$,~\,and~\,$\widetilde{\cal{OS}}$, with impact parameters $b=$ 0, $b_{0}+\delta_{0}$, 0, and $b_{0}-\delta_{0}$, respectively.}
  \label{fig:BC}
\end{figure}

The boundary curve $\partial{\rm D}$ involved in the formula\,\eqref{eq:ralphaw} can be chosen at our convenience,
which is one of the significant advantages of the Gaussian deflection angle over the usual deflection angle\footnote{
This arises from the fact that the Gaussian deflection angle \eqref{eq:GaussianCurvature0} can be rewritten as a surface integral \eqref{eq:GaussianCurvature} and thus it actually has one more degree of freedom in the definition than the usual deflection angle.}.
For certain choices, we can estimate the contribution of dark energy to the Gaussian deflection angle in an analytic way.
Now, consider a closed, positively oriented curve $\partial{\rm D}$ such that
it consists of four segments of light trajectories: $\overline{\cal{SA}}$,~\,$\widetilde{\cal{AB}}$,~\,$\overline{\cal{BO}}$,~\,and~\,$\widetilde{\cal{OS}}$, as illustrated in figure \ref{fig:BC}.
Their impact parameters are: $b=$ 0, $b_{0}+\delta_{0}$, 0 and $b_{0}-\delta_{0}$, respectively.
According to the light orbit equation (LOE)\,\eqref{eq:SdSwlightLC} (see He\,\,\&\,\,Zhang~$\(2017\)$ \cite{HZ2017} for details), these segments can be described by
\begin{eqnarray}
\label{eq:4lightLC}
\nonumber
\frac{\text{d} \phi}{\text{d} r}&=&0~~~(\overline{\cal{SA}}),\\[3mm]
\nonumber
\(\!\frac{1}{r^2}\frac{\text{d} r}{\text{d} \phi}\!\)^{\!\!2}&=&\frac{1}{\(b_{0}\!+\!\delta_{0}\)^2}-\frac{1}{\,r^2\,}\!\left[1\!-2\frac{M}{r}-2\!\left(\! \frac{\,r_{\!o}^{}\,}{r}\!\right)^{\!3w+1}\right]~(\widetilde{\cal{AB}}),\\[3mm]
\nonumber
\frac{\text{d} \phi}{\text{d} r}&=&0~~~(\overline{\cal{BO}}), \\[3mm]
\nonumber
\(\!\frac{1}{r^2}\frac{\text{d} r}{\text{d} \phi}\!\)^{\!\!2}&=&\frac{1}{\(b_{0}\!-\!\delta_{0}\)^2}-\frac{1}{\,r^2\,}\!\left[1\!-2\frac{M}{r}-2\!\left(\! \frac{\,r_{\!o}^{}\,}{r}\!\right)^{\!3w+1}\right]~(\widetilde{\cal{OS}}),
\nonumber
\vspace*{1mm}
\end{eqnarray}
respectively. Among these segments, the ones with $b=0$ are actually parts of radial null geodesics, respectively.
To calculate the Gaussian deflection angle, we need to perform a transformation, $\text{d}\phi\to(\frac{\text{d}\,\!r}{\text{d}\phi})^{-1}\,\text{d}\,\!r$.
With this transformation, by substituting the four LOEs into the formula\,\eqref{eq:ralphaw} and integrating it along the oriented curve $\partial{\rm D}$ as well as making some necessary approximations,
one can further provide an order-of-magnitude estimate for the Gaussian deflection angle.

To do this, we first set $\,R_{\cal{A}}<R_{\cal{B}}\,$ and $\,R_{\cal{S}}<R_{\cal{O}}\,$, exactly as portrayed in figure~\ref{fig:BC},
where $R_{\cal{P}}$ denotes the radius at which the point $\cal{P}$ is located.
Then we use $\Delta$ to characterize half the absolute size of the change in the distance to the mass center $M$ from the photon traveling along a given path,
such as \,$\widetilde{\cal{AB}}$\, or \,$\widetilde{\cal{OS}}$.
By setting $\delta_{0} \ll b_{0}$, and $\Delta \ll b_{0}$, we have $r\simeq\,\!b_{0}$ for the closed curve $\partial{\rm D}$.
Thus the parameter $b_{0}$ can be used to characterize the distance of the lensing patch $D$ to the mass center $M$.
Now we consider the gravitational potential to be fairly weak, $\frac{M}{r}\ll1$ and $\left(\! \frac{\,r_{\!o}^{}\,}{r}\!\right)^{\!3w+1}\ll1$.
More precisely, we further set $\Delta\simeq\frac{\,R_{\cal{B}}-R_{\cal{A}}\,}{2}\sqrt{1\!-2\frac{\,M\,}{b_{0}}-\!2\left(\! \frac{\,r_{\!o}^{}\,}{b_{0}} \!\right)^{\!3w+1}}\approx\frac{\,R_{\cal{B}}-R_{\cal{A}}\,}{2}$ for the path $\,\widetilde{\cal{AB}}\,$,
and $\Delta\simeq\frac{\,R_{\cal{O}}-R_{\cal{S}}\,}{2}\sqrt{1\!-2\frac{\,M\,}{b_{0}}-\!2\left(\! \frac{\,r_{\!o}^{}\,}{b_{0}} \!\right)^{\!3w+1}}\approx\frac{\,R_{\cal{O}}-R_{\cal{S}}\,}{2}$ for the path $\,\widetilde{\cal{OS}}\,$.
Let the two paths, $\,\widetilde{\cal{AB}}\,$ and $\,\widetilde{\cal{OS}}\,$, have nearly the same $\Delta$.
As a result, the lensing patch $D$ may be a long, narrow belt. However, its length can be much smaller than $b_{0}$ in our scheme for the direct probe of dark energy.
Then we assume $\( \frac{\,r_{\!o}^{}\,}{b_{0}}\)^{\!3w+1}\ll\frac{M}{b_{0}}$.
Under these assumptions and approximations, we can estimate the Gaussian deflection angle for the case of $w\simeq-1$ as follows,
\begin{eqnarray}
\label{eq:ApproxGDA}
\begin{array}{rcl}
\displaystyle
\alpha_{M}~=&&\frac{1}{2}\,\mathlarger{\oint}_{\partial{\rm D}}\frac{\sqrt{1\!-2\frac{\,M\,}{r}-\!2\(\! \frac{\,r_{\!o}^{}\,}{r} \!\)^{\!3w+1}} }{ \(\frac{\text{d} r}{\text{d} \phi}\)} \text{d} r\\[4mm]
=&&\left[~...~\right]_{\overline{\cal{SA}}}+\left[~...~\right]_{\widetilde{\cal{AB}}}+\left[~...~\right]_{\overline{\cal{BO}}}+\left[~...~\right]_{\widetilde{\cal{OS}}}\\[4mm]
\sim&&0+\frac{\frac{\Delta}{b_{0}\!+\!\delta_{0}}}{\sqrt{\frac{M}{b_{0}\!+\!\delta_{0}}+\(\frac{b_{0}\!+\!\delta_{0}}{r_{\!o}^{}}\)^2}}
+0-\frac{\frac{\Delta}{b_{0}\!-\!\delta_{0}}}{\sqrt{\frac{M}{b_{0}\!-\!\delta_{0}}+\(\frac{b_{0}\!-\!\delta_{0}}{r_{\!o}^{}}\)^2}}\\[6mm]
\sim&&0+\frac{\Delta}{b_{0}}\sqrt{\frac{b_{0}}{M}}\left[1-\frac{1}{2}\frac{b_{0}}{M}\(\frac{b_{0}}{r_{\!o}^{}}\)^2\right]\(1-\frac{1}{2}\frac{\delta_{0}}{b_{0}}\)\\[4mm]
+&&0-\frac{\Delta}{b_{0}}\sqrt{\frac{b_{0}}{M}}\left[1-\frac{1}{2}\frac{b_{0}}{M}\(\frac{b_{0}}{r_{\!o}^{}}\)^2\right]\(1+\frac{1}{2}\frac{\delta_{0}}{b_{0}}\)\\[4mm]
=&&\(\frac{\Delta}{b_{0}}\)\(\frac{\delta_{0}}{b_{0}}\)\sqrt{\frac{b_{0}}{M}}\left[1-\frac{1}{2}\frac{b_{0}}{M}\(\frac{b_{0}}{r_{\!o}^{}}\)^{2}\right],
\end{array}
\end{eqnarray}
where $\left[~...~\right]_{\rm{part}}$ represents the line integral~\eqref{eq:ralphaw} along a part of the boundary curve $\partial{\rm D}$.
Let's think about this line by line. The first line comes directly from equation\,\eqref{eq:ralphaw}.
In the second line, the four LOEs have been substituted into $\frac{\text{d}\,\!r}{\,\,\text{d}\phi\,\,}$, respectively.
As expected, this line allows simply setting $M=0$ and finding that the Gaussian deflection angle is zero to a first approximation.
The third line uses the two approximate formulae of $\Delta$ (mentioned above) for the two paths, $\,\widetilde{\cal{AB}}\,$ and $\,\widetilde{\cal{OS}}$, respectively.
The fourth line has been derived by using $\delta_{0} \ll b_{0}$ and $(\! \frac{\,r_{\!o}^{}\,}{b_{0}}\!)^{\!3w+1}\ll\frac{M}{b_{0}}$.
The last line gives an approximate formula quite roughly, but this will suffice for an order-of-magnitude estimate.
In the last line, the Gaussian deflection angle takes a simple form, providing a straightforward way to understand the role of dark energy in the bending of light.

For comparison, we also calculate the \textit{usual deflection angle} by using some techniques developed for the Gaussian deflection angle (see appendix \ref{app:D} for details).
For the special case of the cosmological constant ($w=-1$), the \textit{usual deflection angle} can be approximated as
\begin{eqnarray}
\label{eq:UDA4w=-1}
\alpha_{M}\sim\,\frac{4M}{b} \left[1+\(\frac{b}{r_{o}}\)^2\right]=\frac{4M}{b}+\frac{2M\Lambda}{3}\,b,
\end{eqnarray}
where the $\Lambda$ term is consistent with the result presented in the literature \cite{Italy,Sereno08,Bhadra10,Ishak-Rev,Arakida12}.
When setting $\Lambda=0$, we find that the usual deflection angle recovers the conventional result of light bending, $\alpha_{M}=\frac{4M}{b}$.
We can derive the dark energy correction $\Delta\alpha_{M}$ to the usual deflection angle,
\begin{eqnarray}
\label{eq:approxSDA}
\frac{\Delta\alpha_{M}}{\alpha_{M}}\sim\(\frac{b}{r_{o}}\)^{2}.
\end{eqnarray}
Accordingly, we find that effects of $\Lambda$ on the bending of light are quite small for a real astrophysical system.
For example, let $b$ equal the size of our Solar System, like $b=$ 0.1 $r_{\text{cri}}^{}$. Then we have $\frac{\Delta\alpha_{M}}{\alpha_{M}}\sim10^{-18}$ from equation\,\eqref{eq:approxSDA}.
Similar to Sereno (2019)\,\cite{Italy,Sereno08}, we can conclude from the point of view of the traditional (gravitational lensing) theories that effects due to $\Lambda$ are too small to be detected,
although the $\Lambda$ term does contributes to the usual deflection angle.

From the formula \eqref{eq:ApproxGDA},
we also get the dark energy correction to the Gaussian deflection angle,
\begin{eqnarray}
\label{eq:relativeDA}
\frac{\Delta\alpha_{M}}{\alpha_{M}}\sim\frac{1}{2}\frac{\,\,b_{0}\,\,}{\,\,M\,\,}\(\frac{b_{0}}{r_{\!o}^{}}\)^{\!2}.
\end{eqnarray}
Clearly, it is about $\sim\frac{1}{2}\(\!\frac{\,\,b_{0}\,\,}{\,\,M\,\,}\!\)$ times larger than that to the usual deflection angle.
It is worth rephrasing that we can further enhance the local effect of dark energy via the proper choice of the boundary curve $\partial{\rm D}$.

In our Solar System, we have $M\sim1.0 \,M_{\odot}$, where $M_{\odot}$ is the mass of the Sun.
Set $b_{0}=$ 0.1 $r_{\text{cri}}^{}$ and $\delta_{0}\sim\Delta\sim1.0~\textrm{AU}$ (astronomical unit). In this case, it can be verified that all the assumptions and approximations hold well throughout the derivations of the formula\,\eqref{eq:ApproxGDA}.
Then, $\frac{1}{2}\(\!\frac{b_{0}}{\,\,M\,\,}\!\)\sim2.2\times10^{14}$.
It clearly indicates that the influence of dark energy on light bending can be enhanced by 14 orders of magnitude.
Notice that in our scheme, the impact parameter is fixed to a design value, and it is not an observable.
From equations\,\eqref{eq:ApproxGDA} and \eqref{eq:relativeDA}, we further obtain $\alpha_{M}\sim0.63''$ and $\frac{\Delta\alpha_{M}}{\alpha_{M}}\sim2.5\times10^{-4}$.
It means that we can directly probe the existence of dark energy and measure the equation-of-state parameter $w$ on a much shorter length scale than the Solar-System's
at the distance of $\sim$ 0.1 $r_{\text{cri}}^{}$ from the Sun once a spatial resolution of $\sim\!10^{-5}$ arcseconds can be reached by the current lensing experiments.
In fact, this spatial resolution has been achieved by GRAVITY \cite{GRAVITY18},
which is often used to detect the gravitational microlensing events.

The universe is presently dominated by not only dark energy, but also the pressure-less (baryonic and dark) matter.
However, when considering the fact that the Sun contains more than 99\% of the mass of the Solar System,
as well as the SdS$_{w}$ metric holds well for a point-like mass $M$ or for regions outside a spherically symmetric mass-distribution \cite{HZ2017},
it can be found that our estimate remains valid, especially as an order-of-magnitude estimate.
So we can safely draw the conclusion that the current lensing experiments are already sensitive to probing dark energy via the method presented above.

Nevertheless, these lensing experiments are all designed based on the traditional approaches where
it is the usual deflection angle that plays a central role rather than the Gaussian deflection angle \eqref{eq:GaussianCurvature}.
Within the framework of traditional theories, the local dark energy effect is too weak to be detected only by using traditional approaches (as we discussed just below equation\,\eqref{eq:approxSDA} or shown by Sereno (2019) \cite{Italy,Sereno08} for the $\Lambda$ effects, for example);
in other words, we have no chance to amplify the local effect of dark energy via the choice of the path of integration $\partial{\rm D}$
in analogy to what we have done for the Gaussian deflection angle \eqref{eq:ralphaw}.
This explains why despite intense effort no experiment on the bending of light shows any deviation from traditional theories, till today.
Anyway, we have successfully proposed a method to overcome the difficulty of measuring local dark energy effects.

\vspace*{4mm}
\section{\hspace*{+0.0mm}Conclusions}
\label{sec:6}
\vspace*{1.5mm}

Recently, the XENON1T reported a non-gravitational signature of dark energy \cite{Vagnozzi21}.
If it is confirmed by future experiments,
we have to face the scenario in which dark energy couples to photons directly.
Taking dark energy as a scalar field for example, its quanta interact with photons, as assumed in \cite{Vagnozzi21}.
If so, these quanta may be produced by the magnetized objects like the Sun and then propagate outward to the observer, forming local fluctuations in the pressure and energy density of dark energy
while still keeping $w$ nearly constant on astrophysical scales so as not to contradict with the existing observational data.
These fluctuations may manifest as tiny noises or perturbations when measuring the local dark energy effect via the bending of light.
On the other hand, in principle this scenario can only be dealt with in quantum gravity.
As was done in \cite{QGbending15,Bai17}, new scalar fields were introduced into the standard model of particle physics,
and their quanta interact with photons at the quantum level.
It turns out that the final result of quantum gravity agrees with the classical GR result to a first-order approximation \cite{Bai17}.
However, when returning back to the study of dark energy, it might become rather different to calculate the deflection angle,
and the computations need to be accomplished strictly using modern field theory techniques, which is far beyond the scope of our work.

In this work, we focus on the classical GR, and showed that the bending of light can serve as an important tool for the direct probe of dark energy on the Solar System scales.
By using the famous Gauss-Bonnet theorem, we geometrized light deflection, and demonstrated explicitly that in any curved static spacetime the Gaussian deflection angle is equivalent to the total curvature.
For the general case of the SdS$_{w}$ spacetime, we concluded that dark energy does affect the deflection of light. Measuring such effect can directly probe the existence and nature of dark energy.

In section~\ref{sec:2}, we introduced the projection tensor $h^{a}_{\,\,\,b}$ and the induced metric tensor $h_{ab}$.
Both of them are observer dependent.
Generally, $h^{a}_{\,\,\,b}$ projects any four-vector onto the observer's local space that is described by the metric tensor $h_{ab}$.
In physics, the projection of the four-vector by $h^{a}_{\,\,\,b}$ is physically measurable.
For example, the proper three-acceleration $\hat{a}^{a}$ measured by the static observer can be regarded as the projection of the four-acceleration $\hat{A}^{a}$. More exactly, we have $\hat{a}^{a}=- h^{a}_{\,\,\,b}\hat{A}^{b}$~ $(=-\hat{A}^{a})$, with $\pi^{a}_{\,\,\,b}\,\hat{A}^{b}\!=\!0$. It is independent of the choice of coordinates, since all the Lorentz indices are contracted. This is indeed the reason why the gravitational three-force $\vec{g}$ can be directly measured by the static observer in the SdS$_{w}$ spacetime.

In section~\ref{sec:3}, we derived the proper form for the dark force induced by dark energy, in analogy to what we have done for the traditional Newtonian gravity.
Then we further showed that the dark force is repulsive and its strength grows with $r$. This is in contrast to the attractive Newtonian force caused by both visible and dark matter.
We also derived the critical radius $r_{\text{cri}}^{}$, at which the dark force balances the Newtonian attraction.
$r_{\text{cri}}^{}$ plays an important role in helping us to design experiments and develop strategies for the direct probe of dark energy through gravitational effects.
Taking the gravitational lensing effect for example, the lensing patch $D$ cannot be too far away from the region $r\simr_{\text{cri}}^{}$, otherwise it will become very difficult for the direct probe of dark energy.
Another example is about the GODDESS mission, which has been proposed by NASA to detect the dark force (also referred to as the fifth force).
Using the explicit form of the dark force or its strength, we can maximally optimize detection schemes and quantitatively develop measurement strategies for the GODDESS mission.

Section~\ref{sec:4} was built on the Gauss-Bonnet theorem.
We first defined the Gaussian deflection angle by adopting new techniques, like taking parallel transport.
Using the Gauss-Bonnet theorem, we then proved strictly that the Gaussian deflection angle is identical to the integral of the Gaussian curvature over the lensing patch $D$,
which is applicable in any curved static spacetime.
We therefore geometrized the bending of light after a sequence of strict mathematical derivations.
We also showed that the Gaussian deflection angle is intrinsically a generalized deflection angle;
in some special cases, the Gaussian deflection angle reduces to the usual deflection angle.
From the Gauss-Bonnet theorem, we also derived a relationship between the Gaussian deflection angle and the measured external (or interior) angles
by the static observers at every vertex of the closed boundary curve $\partial{\rm D}$.
From this relationship, the Gaussian deflection angle can be obtained directly.
Note that all these results are independent of the choice of coordinates, forming a strict mathematical basis for the theories of direct probe of dark energy through the bending of light.

In section~\ref{sec:5}, we demonstrated that the light ray is deflected as it travels in the SdS$_{w}$ spacetime,
and gained a clear understanding of the role of dark energy in the bending of light.
For the Gaussian deflection angle, we did explicit calculations in the general SdS$_{w}$ case (cf appendix \ref{app:E}), and presented its analytical form in equation\,\eqref{eq:ralphaw} strictly;
the deflection angle takes a general form, where the equation-of-state parameter $w$ can describe different forms of dark energy when it takes their corresponding values.
Thus we model-independently concluded that dark energy does affect the deflection of light.
We also showed in equation\,\eqref{eq:ralphaw} that the dark energy contribution is fully determined by the two generic parameters $\(w, r_{\!o}^{}\)$.
Therefore, it is possible to obtain $\(w, r_{\!o}^{}\)$ from measuring the Gaussian deflection angle.
With extracted information about the two parameters, we could further discriminate between different dark energy models and thus identify the right model for describing dark energy.

In addition, we proposed a method to overcome the difficulty of measuring the local effect of dark energy on the bending of light.
Under the weak field approximation, we calculated the direct contribution of dark energy to the Gaussian deflection angle,
trying to maximize the effect of dark energy by choosing the lensing patch $D$.
We also estimated the usual deflection angle by using the techniques developed in section~\ref{sec:4} (See appendices \ref{app:A}, \ref{app:B}, \ref{app:C}, and \ref{app:D} for details).
By comparisons, we concluded that the direct contribution of dark energy to light deflection can be enhanced by 14 orders of magnitude if we choose the boundary curve $\partial{\rm D}$ properly.
When applying this enhancement to detect dark energy in our Solar System, we found that the direct contribution of dark energy is already sensitive to the current lensing measurements.

Finally, we conclude that in general, it is important to make precise measurements on the bending of light via the Gaussian deflection angle \eqref{eq:GaussianCurvature},
and this will allow us to probe the existence of dark energy directly and discriminate different dark energy models with $w=-1$ versus $w\neq-1$.
By the theories established in this work, it can be expected that the direct probe of dark energy can been achieved at much shorter scales than our Solar-System's in the near future.

\acknowledgments

The author thanks Prof. SL Xiong for his support and Prof. HJ He for useful discussions as well as the two anonymous reviewers for their valuable comments and constructive suggestions.
This work is supported by the National Program on Key Research and Development Project (Grant No. 2016YFA0400800, 2016YFA0400802, 2017YFA0402600) from the Minister of Science and Technology of China (MOST), the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences  (Grant No. QYZDY-SSW-SLH008), the Strategic Priority Research Program on Space Science of the Chinese Academy of Sciences (Grant No. XDB23040400), and the National SKA Program of China (Grant No. 2020SKA0120300). The authors thank supports from the National Natural Science Foundation of China under Grants U1838202, 11833003, U1838201, U2031205, 11673023, 11733009, U1838111, U1838113, U1838105, 11503027, and U1838104.

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text:
Q: recaptcha doesn't work in Vue component.give grecaptcha.render is not a function when grecaptcha not loaded how to add grecaptcha onload method in Vue component.when not loaded grecaptcha.render is not a function
const script = document.createElement("script");
 script.src = `${URL}?render=explicit&hl=TR`;
 script.async = true;
 script.defer = true;
 document.body.appendChild(script);
 this.initReCaptcha();

initReCaptcha: function () {
    setTimeout(function () {
       if (typeof grecaptcha === 'undefined') {
           this.initReCaptcha();
       } else {
          grecaptcha.render('recaptcha', {
          sitekey: 'XXXX',
          callback: this.submit,
          'expired-callback': this.expired
                        });
                    }
                }.bind(this), 100);
            }

is working when i set the timeout 1000.but it's too late.

A: I found that out that since two days ago a lot of people got the error:
 grecaptcha.render is not a function

Well, this works for me. Maybe it helps someone in the
I break it into peaces so you can read it easily:
mounted(){
    if (typeof grecaptcha === "undefined") {
        var script = document.createElement("script");
        script.src = "https://www.google.com/recaptcha/api.js?render=explicit";
        script.onload = this.renderWait;
        document.head.appendChild(script);
    } else this.render();
}

If grecaptcha is not loaded I will create a script and wait for it to be loaded.
If so renderWait is called
renderWait() {
        setTimeout(() => {
            if (typeof grecaptcha !== "undefined" && typeof grecaptcha.render !== "undefined")  this.render();
            else this.renderWait();
        }, 200);
},

When we have finally grecaptcha render it:
render() {
        const recaptchaDiv = document.createElement('div');
        recaptchaDiv.className = 'outside-badge';
        document.body.appendChild(recaptchaDiv);

        this.widgetId = grecaptcha.render(recaptchaDiv, {
            sitekey: this.sitekey,
            size: "invisible",
            badge: this.badge || "bottomright",
            theme: this.theme || "light",
            callback: token => {
                this.callback(token);
                grecaptcha.reset(this.widgetId);
            }
        });
        this.loaded = true;
 },

Maybe it helps you :)

A: The recaptcha script was updated last May 4, 2018 9:07:30.349 PM GMT (based on the timestamp of the recaptcha script). The grecaptcha is being loaded, however the render function has a delay that is why the recaptcha is not being displayed in the UI (grecaptcha.render is not a function). The possible fix is to validate the render function before attempting to load the recaptcha.
Here is the fix:
initReCaptcha: function () {
    setTimeout(function () {
        if (typeof grecaptcha === 'undefined' || typeof grecaptcha.render ==='undefined') {
            this.initReCaptcha();
        } else {
            grecaptcha.render('recaptcha', {
                sitekey: 'XXXX',
                callback: this.submit,
                'expired-callback': this.expired
            });
        }
    }.bind(this), 100);
}

I hope it will help you. Cheers
** Fixed condition as qRoC wrote in comments, thanks.

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An airline captain was breaking in a new blonde stewardess. The route they were flying had a layover in another city.
He knew which room she was in at the hotel and called her up wondering what happened. She answered the phone, crying, and said she couldn't get out of her room.


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The new Mistral+ promises to be our best optical sorting solution for the waste and recycling market.


possibility to register new accounts for xmpp.jit.si.
xmpp.jit.si is noticeably suffering from the lack of attention.
to use another one or deploy their own.
there and we will fully close the server on June 1st.


16/3 10-12 P1311 Spare time ?, ?
For each round, group 2-5 present at seminar a, group 6-9 at seminar b. Note the different order of the seminars.
The spare time on March 16 is reserved for those who due to extraordinary reasons have not been able to complete all their parts of the course within the normal schedule. Only those who have something left to do in order to pass the course have to attend that lecture.
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Join us for Mingei Fusion!
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text:
AGATHA CHRISTIE BY THE PRICKING OF MY THUMBS PDF
admin August 5, 2019 August 5, 2019 No Comments on AGATHA CHRISTIE BY THE PRICKING OF MY THUMBS PDF
By The Pricking Of My Thumbs. An old woman in a nursing home speaks of a child buried behind the fireplace When Tommy and Tuppence visited an elderly. By the Pricking of My Thumbs has ratings and reviews. Laurel said: I' m going to go ahead and give this five stars, because it is fairly rare f. When Tommy and Tuppence visit an elderly aunt in her nursing home, Tuppence is concerned by the odd behavior of some staff and residents.
Author: Tezilkree Tujin
Whole neighbourhood was out searching for her.
Youthful in two Christie books written in the s, middle-aged in a World-War II spy novel, Tommy and Tuppence were unusual in that chrsitie aged according to real time, unlike Hercule Poirot or Miss Marplewhose age remained more or less the same from their first novels in the s, to their last novels in the s.
The manner in which Christie writes about older people too, I suppose indicative of her own era, implies that old old people are at least slightly mad, just by virtue of being old and no one can expect them to be normal. During this marriage, Agatha published six novels, a collection of short stories, and a number of short stories in magazines.
Causes appear to be natural but Tuppence's suspicions are aroused when a note from Ada mentions that fellow-nursing home dweller Mrs Lancaster is not safe. Her disappearance caused an outcry from the public, tumbs of whom were admirers of her novels. Paperbackpages.
Picture Tuppence sitting infront of a fireplace at Sunny Ridge, as Tommy is upstairs visiting his old aunt. He finds the painting was done by an artist called Boscowan. The artist has died but Mrs.
During the Second World War, Christie worked in the pharmacy at University College Hospital of University College, London, where she acquired a knowledge of poisons that she put to good use in her post-war crime novels. I think I need to rhumbs this one someday, just in one sitting.
HISTORY OF PUKHTOON PDF
It's terribly disappointing, especially because Tommy and Tuppence are usually so delightful.
Agatha Christie – Marple: By the Pricking of My Thumbs () – Rotten Tomatoes
Masterful Agatha Christie classic. So it's terrible to place Christie's name on it. Was this review helpful to you? Tuppence has the impression that Sir Phillip knows more about the whole affair. Keep track of everything you watch; tell your friends.
So, five stars for atmosphere, but also for another reason: Add the first question. And it was not so much about the prospect of being poisoned or having overdosed at the premises that was worrying but neglect and abuse in the hands of the staff. Sometimes, a quiet, slow paced story with a gradual build up can be just as eerie. Still, I have about forty more of her novels to go, and I'm going to read them all.
Eventually she finds it in a small village called Sutton Chancellor.
The dynamic and dialogue between them is always a joy to read, and whilst part of me misses the snappier exchanges they had in their youth, another part of me likes that we get to see them age throughout the series – and change in sgatha ability and temperament accordingly.
Mar 18, Cchristie Mouloudj rated it it was ok Shelves: During the First World War, she worked at a hospital as a nurse; later working at a hospital pharmacy, a job that influenced her work, as many of the murders in her books are carried out with poison. Have your say Cancel reply Enter your comment here What are they doing now? Tuppence is saved just in time. It was strictly a Tommy and Tuppence Beresford mystery.
The main one that struck me agata that, despite both Beresfords, Tommy AND Tuppence, having done sterling work for the various defense agencies and persons of England, in BOTH wars, now only Tommy is employed by whichever mysterious agencies which employ him.
EL CUCHILLO EN LA MANO PATRICK NESS PDF
By the Pricking of My Thumbs agatua this time.
"Marple" By the Pricking of My Thumbs (TV Episode ) – IMDb
The next year, she became the President of the Detection Club. A Major Waters, he asked if by any possibility a child had been buried here. Robert Barnard said that this novel "Begins rather well, with a vicious old aunt of Tommy's in a genteel old people's home, but declines rapidly into a welter of half-realised plots and a plethora of those conversations, all too familiar in late Christie, which meander on through irrelevancies, repetitions and inconsequentialities to end nowhere as if she had sat at the feet of Samuel Beckett.
Aug 03, Christine rated it it was amazing Shelves: A cache of precious stones. Jun 20, Hannah rated it really liked it. There is something he wants to discuss but not over the phone. Maurice Richardson in The Observer of 17 November said, "Not her best though it has patches of her cosy euphoria and aura of the sinister.
Makes one appreciate the economy of dialogue — all point, or at least possible point, in early Christie.
christke Season 4 The Walking Dead: Emma Boscowan a sculptor and the ex-wife of Bosowan, an artist who painted the House on the Canal. I like that this one gets really dark in subject matter, and the tension and sinister atmosphere in the climactic scene are excellent; I just wish it had ended with a little more punch, rather than fizzling out just as the narrative reached its fever pitch.
The surprise conclusion makes this a fun book, not counting all the killings! Tuppence meets with the people of Sutton Chancellor. Because when Chistie is great, no one else comes close.
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AGATHA CHRISTIE LISTERDALE MYSTERY PDF
HARTILE PREVESTITOARE PDF
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FUNDAMENTOS DE LA TEORIA ELECTROMAGNETICA REITZ MILFORD SOLUCIONARIO PDF
EUROREGION BESKIDY PDF
DESCARGAR DIEZ NEGRITOS AGATHA CHRISTIE PDF

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Social networking sites have become ubiquitous and have permeated every aspect of modern-day lives. We use them to communicate with people we know, to network and build personal and professional relationships, for entertainment, business, advertising and a host of other purposes.
It goes without saying that educational institutions do not want to be left behind in the internet race and medical schools in Australia too have got into web-mode. They look for ways and means of intertwining the application process with social networks. So, if colleges are doing things differently, it goes without saying that applicants will also have to move in sync with them and understand what role social media plays in the medical school application process.
Admissions updates– Many medical schools use different social networking sites to communicate with the applicants. Application statuses and updates are sent via popular sites. It is important that you check these updates. Some may be scheduling changes while there might be some modifications in procedures that are being communicated via the social network. All of these are important for you.
Maintaining cohesiveness– Some schools have been adopting novel methods of using social media to enhance and increase class cohesion. Some create incoming class-orientation groups on networks like Facebook and many schools also agree that these measures are very successful in meeting their objectives.
Medical colleges conduct standard team-building activities on campus, which can be pretty expensive endeavours. Using social media helps in saving those costs. Another advantage is that the applicants are encouraged to organise the pre-orientation activities on their own. Expensive school-staff coordination is out and interesting class-cohesion is in.
Ideally, if you are an active user on a social network site that has groups from various medical schools, it will give you a view of different opinions. There are online forums that you can register on too. All of these can give you a pretty good idea of what that particular school is really like and can help you decide which one will match your needs the best.
Though social media wears a cloak of informality, it is important to maintain a professional face in your communication on the site. If you have your sights on getting into medical school, it's best to resist the temptation of posting pictures of parties or social events on your accounts on these sites.
Be responsible with the manner in which you use social media. If you have an account that is linked to any medical school, ensure that the information is not public. Set your privacy settings appropriately. Do not post any information about the application or blog about your experiences at the interview. Your personal information should also not be easily accessible.
Medical schools understand the importance of the role of social media and have incorporated it into their standard workings as well as into the application process. Take advantage of the media, leverage its benefits and use it to make a good impression. After all, medical school is much more than just college; it's the start of your career as a doctor.
Does dyslexia affect admission into medical school?


text:
Ortem
Orakçı Machinery
Plaskar
Ormold
Orakçı Architecture
Novada Edremit
Orakçı Group continues to carry out its activities in its 40 year journey on the framework of its goal of "stable and sustainable growth".
Establishment Years
Hikmet Orakçı, 16 years after doing mathematics teaching in various provinces of Anatolia goes into business life. In 1978 with the name Hikmet Orakçı Trade started to provide in the fields of service infrastructure, environmental planning and service buildings construction works to municipalities. He founded his first limited company Hoca Construction Co. Ltd. In 1995.
As the municipalities began to purchase services from the private sector in cleaning services at the beginning of the 1990s, Orakçı Group takes the steps of garbage collection and cleaning services and is among the first companies in the sector. In 1998, these services were collected under the Ortem roof. Together with the year 1999, it adds services for the creation and maintenance of parks and gardens to its services that are given to the municipalities.
First Branch in Homeland
In 1996, Bandırma Municipality garbage collection and cleaning service work is taken and first service site outside Bursa is established. In the following years, new sites will be established in many provinces of Turkey and a sound basis of being a national company will be laid.
Second Generation at Work
After completing their education, Özgür Orakçı, Betül Orakçı and Barış Orakçı start to work in various departments of the group.
Hikmet Orakçı, who believes in institutionalization, transfers the administration to his children over time and continues to work as the founding president himself.
First Industrial Enterprise
Towards the years 2000, direction to manufacturing starts. In 2002, Orakçı Machine is founded which is the first industrial enterprise of Orakçı Group and On-vehicle equipment starts to be manufactured.
Orakçı Group started to manufacture plastic injection parts in 2004 by stepping into the automotive sector, and join Plaskar Inc. Group.
Expansion of Activity Areas
The Orakçı Group significantly expands the service issues it provides in the sectors it is in. In the services provided to the municipalities, the group which adopts the concept of "Integrated Waste Management" starts to work on "Designing and Operating Solid Waste Regular Storage Areas" in 2001 as well as garbage collection and cleaning services. In 2005, signed the "Separate Collection and Separation Project of Packaging Waste at the Source" and established the first licensed separation facility. Then it starts to provide the operating service of the sanitary landfills. Now Ortem is involved in all aspects of waste management, from collection transportation to recycling and disposal. Along with the year 2002, besides the contracting works in the construction sector, "Stone Mill Operation, Aggregate and Ready Mix Concrete Dealership" started. Along with the year 2006, it has started to produce ready-mix concrete, transfer and pumping services with its plants which it set up by taking a different step in the construction sector.
New Structure - Change
Orakçı Group is restructured in 2007 in accordance with sectors, regions and division of labor in order to achieve the targeted growth and development under a more appropriate organization and a more institutional structure.
Orakçı Group Headquarter Is In Its New Place
The Orakçı Group builds its Headquarter itself in 2009 and gathers its companies and management in one spot.
The Orakçı Group headquarter is committed to working as a management center responsible for coordinating finance, human resources, strategy and business development functions, as well as for identifying the common vision and strategies of group and increasing the value provided to its stakeholders through synergies created within the group.
Growth and export
Orakci Group extends its activities by continuing its development without slowing down; leans to exports and establishes its foreign trade company. The group which started exporting in the machine manufacturing sector is currently exporting to Azerbaijan, UAE, Bulgaria, Algeria, Italy, Kuwait, Kosovo, Libya, Uzbekistan, Russia, Saudi Arabia, Syria, Jordan, Venezuela, Malaysia, Dubai, Philippines, Sri lanka, Singapore, Chili, South Africa and Yemen.
Along with the year 2010, construction projects in the construction sector starts with Orakçı Construction. In 2011, in ready-mixed concrete production sector, a second power plant for the construction of Hydroelectric Power Plant was established and operated. The mold maintenance workshop, which was launched in 2008 within the scope of Plaskar Inc., has become a structure that produces plastic injection molds for different sectors, mainly automotive. This production was restructured under Ormold along with the year 2014. The Orakçı Group has begun preparations for establishing international partnerships along with the year 2012. New Investment and Partnership, after 2012 particularly in machinery, automotive and service sectors about the issues on establishing cooperation in the international arena, investing, participating in tenders continue rapidly and results are began to received.
Within this time period, partnerships are established in technology fields, new dealerships are given and new service markets are preluded. Parallel to this, the production areas and the organization are restructured in order to meet the increase in sales figures.
Orakçı Machinery and Ormold in Their New Places
Orakçı Machinery completes its new factory building in order to meet the capacity increases in on-vehicle equipment production and starts production on site in its new place in 2016. Ormold is moved to its factory in Gebze in line with its big targets in plastic injection production.
Housing Projects & Shopping Centers
The new excitement was the Çorlu Trio Park project while Orakçı Building was in the process of completing housing projects in Bursa. With a new partnership, Novada Shopping Center, which actualized in Edremit, has taken its place in minds as the biggest living and shopping center of the bay.
Orakçı Group is growing every year with new investments and moving towards its targets with firm steps.
© Copyright 2021 Orakçı Şirketler Grubu


Heroin overdoses have been rising sharply in the United States over the last several years. In 2014, over 10,500 people died of heroin overdoses in the U.S. Heroin is sold illegally, so there is no control over the quality or strength of the drug. Also, it is sometimes mixed with other poisonous substances.
Street names for heroin include "junk", "smack", dope, and "skag".
People use heroin to get "high." But if they overdose on it, they get extremely sleepy or may become unconsciousness and stop breathing.
Bardsley CH. Opioids. In: Marx JA, Hockberger RS, Walls RM, eds. Rosen's Emergency Medicine: Concepts and Clinical Practice . 8th ed. Philadelphia, PA: Elsevier Saunders; 2013:chap 162.
Centers for Disease Control and Prevention. Injury prevention & control: opioid overdose. www.cdc.gov/drugoverdose/opioids/heroin.html . Updated March 14, 2016. Accessed March 30, 2016.
Doyon S. Opioids. In: Tintinalli JE, Kelen GD, Stapczynski JS, Ma OJ, Cline DM, eds. Emergency Medicine: A Comprehensive Study Guide . 6th ed. New York, NY: McGraw-Hill; 2004:chap 167.
Levine DP, Brown PD. Infections in injection drug users. In: Bennett JE, Dolin R, Blaser MJ, eds. Mandell, Douglas, and Bennett's Principles and Practice of Infectious Diseases . 8th ed. Philadelphia, PA: Elsevier Saunders; 2015:chap 317.
National Institute on Drug Abuse. Heroin. www.drugabuse.gov/drugs-abuse/heroin . Updated March 2015. Accessed September 15, 2015.
National Institute on Drug Abuse Research Report Series. Heroin. www.drugabuse.gov/publications/research-reports/heroin . Updated November 2014. Accessed September 15, 2015.


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