text:
\section{Introduction}

Despite the enormous successes of the Einstein's theory of gravity, the latter appears to be about ``twenty five percent wrong''. So far the scientists proposed two possible solutions of the problem that are known under the name of ``dark matter'' or ``dark gravity", and both are extensions of the Einstein's field equations. The first proposal consists on modifying the right side of the Einstein's equations, while according to the second proposal it is modified the left hand side.
Indeed, in order to take into account of all the observational evidences: galactic rotation curves, structure formation in the universe, CMB spectrum, bullet cluster, and gravitational lensing, it seems needed to somehow modify  the Einstein's field equations.
However, in this paper we propose the following different approach, namely: ``understand gravity instead of modifying it''.

In this document we do not pretend to provide a definitive answer to the ``mystery of missing mass'' or ``missing gravity in the universe'', but we only focus on the galactic rotation curves. Nevertheless, we believe our result to be quite astonishing on both the theoretical and observational side.

The analysis here reported, which follows the previous paper \cite{Li:2019ksm}\footnote{The previous seminal paper \cite{Li:2019ksm} did not address the issue of conformally coupled matter that completely changes the geometrical interpretation of our proposal underlining the crucial role of the asymptotic but harmless spacetime singularity.
Notice that in \cite{Li:2019ksm} the massive particles break explicitly the conformal invariance, even if slightly, making the solution no longer exact.
Moreover, we will show in this paper that in presence of conformally coupled matter we do not need to resort to the global structure of space-time and to invoke the small inhomogeneities on the cosmological scale or the presence of the cosmological constant, which will turn out to be too small to affect on the rotation curves on a galactic scale: ``everything will be limited to the single galaxies''.}, is universal and apply to any conformally invariant theory, nonlocal \cite{Krasnikov,kuzmin,Modesto:2011kw,Modesto:2014lga,Modesto:2017sdr}, or local \cite{LM-Sh, LWQG}, that has the Schwarzschild-metric as an exact \cite{Li:2015bqa} and stable solution \cite{Briscese:2018oyx,Briscese:2021mob,Briscese:2018bny,Briscese:2019rii,Modesto:2021okr,Modesto:2021ief,spallucci}.

However, for the sake of simplicity we will focus on Einstein's conformal gravity, whose  general covariant action functional \cite{Deser:1970hs} reads
\begin{eqnarray}
 S=\int \! d^4x \sqrt{-\hat{g}} \, \left(\phi^2\hat{R}+6\hat{g}^{\mu\nu}\partial_\mu\phi \partial_\nu \phi- 2 h  \phi^4 \right)
 \label{CEG}  ,
\end{eqnarray}
which is defined on a pseudo-Riemannian spacetime Manifold $\mathcal{M}$ equipped with a metric tensor field $\hat{g}_{\mu\nu}$, a scalar field $\phi$ (the dilaton), and it is invariant under the following Weyl conformal transformation:
\begin{eqnarray}
\hat{g}^\prime_{\mu\nu}=\Omega^2\hat{g}_{\mu\nu} \, ,  \quad \phi^\prime=\Omega^{-1}\phi \, ,
\label{ConfInv}
\end{eqnarray}
where $\Omega(x)$ is a general local function.
In (\ref{CEG}) $h$ is a dimensionless constant that has to be selected extremely small in order to have a cosmological constant compatible with the observed value. However, we here assume $h=0$ because the presence of a tiny cosmological constant will not affect our result (see section (\ref{effectCosmoConstant}) for more details).
For completeness and in order to show the exactness of the solutions that we will expand later on, we here remind the equations of motion for the theory (\ref{CEG}) for $h=0$,
\begin{eqnarray}
&& \phi^2 \hat{G}_{\mu\nu}  =
   \nabla_\nu \partial_\mu \phi^2 - \hat{g}_{\mu\nu} \hat{\Box} \phi^2
 -  6 \left( \partial_\mu \phi \partial_\nu \phi - \frac{1}{2} \hat{g}_{\mu\nu} g^{\alpha \beta}
 \partial_\alpha \phi \partial_\beta \phi \right) \, ,  \nonumber \\
 &&
 \hat{\Box} \phi = \frac{1}{6} \hat{R} \phi \, .
 \label{ECGEoM}
 \end{eqnarray}

The Einstein-Hilbert action for gravity is recovered whether the Weyl conformal invariance is broken spontaneously in exact analogy with the Higgs mechanism in the standard model of particle physics (for more details we refer the reader to \cite{Bambi:2016wdn}.)
One possible vacuum of the theory (\ref{CEG}) (exact solution of the equations of motion (\ref{ECGEoM})) is
$\phi = {\rm const.} = \kappa^{-1}_4 = 1/\sqrt{16 \pi G}$, together with the metric satisfying $R_{\mu\nu} \propto \hat{g}_{\mu\nu}$.
Therefore, replacing $\phi = 1/\sqrt{16 \pi G} + \varphi$ in the action (\ref{CEG}) and using the conformal invariance to eliminate the gauge dependent Goldstone degree of freedom $\varphi
, we
finally end up with the Einstein-Hilbert action in presence of the cosmological constant,
\begin{eqnarray}
S_{\rm EH} = \frac{1}{16 \pi G} \int d^4 x \sqrt{-\hat{g}} \, \left(\hat{R} - 2 \Lambda \right) ,
\label{EH}
\end{eqnarray}
where $\Lambda$ is consistent with the observed value for a proper choice of the dimensionless parameter $h$ in the action (\ref{CEG}).
 Ergo, Einstein's gravity is simply the theory (\ref{CEG}) in the spontaneously broken phase of Weyl conformal invariance \cite{NK,Bambi:2016wdn}.

Let us now expand about the exact solutions in conformal gravity. Given the conformal invariance (\ref{ConfInv}), any rescaling of the metric $\hat{g}_{\mu\nu}$ accompanied by a non trivial profile for the dilaton field $\phi$, is also an exact solution, namely
 \begin{eqnarray}
 \hat{g}_{\mu\nu}^*=Q^2(x) \, \hat{g}_{\mu\nu} \, \quad \phi^* = Q(x)^{-1} \, \phi ,
 \label{rescaling}
 \end{eqnarray}
 solve the EoM obtained varying the action (\ref{CEG}) respect to $\hat{g}_{\mu\nu}$ and $\phi$.

 So far the rescaling (\ref{rescaling}) has been used to show how the singularity issue disappearance in conformal gravity
 \cite{NK, Bambi:2016wdn, Bambi:2017yoz, Chakrabarty:2017ysw}.

 However, and contrary to the previous papers, we here focus on a {\em not-asymptotically flat} rescaling of the Schwarzschild metric as a workaround to the non-Newtonian galactic rotation curves. Moreover, the logic in this project is literally opposite to the one implemented in the past works and it is somehow anti-intuitive. In fact, here, instead of removing the spacetime's singularities, we apparently deliberately introduce an unreachable asymptotic singularity. However, as it will be proved later on, the spacetime stays geodetically complete. Indeed, the proper time to reach the singularity at the edge of the Universe will turn out to be infinite.

 Notice that in order to give a physical meaning to the metric (\ref{rescaling}), conformal symmetry has to be broken spontaneously to a particular vacuum specified by the function $Q(x)$. The uniqueness of such rescaling will be discussed in section \ref{Unique}. In the spontaneously broken phase of conformal symmetry observables are still invariant under diffeomorphisms.

\section{The spherically symmetric solution in conformal gravity}
As explained in the introduction, given an exact solution of Einstein's conformal gravity, any rescaled metric is an exact solution too, if the metric is accompanied by a non-trivial profile for the dilaton. Therefore, we here consider the following conformal rescaling of the Schwarzschild spacetime,
\begin{eqnarray}
&& d \hat{s}^{ * 2 } = Q^2(x) \left[ -  \left(1-\frac{2GM }{ x}  \right)  dt^2+\frac{dx^2}{1-\frac{2GM}{ x} }+x^2(d\theta^2+\sin^2\theta d\varphi^2) \right] \, , \\
&& Q(x) = \frac{1}{1- \frac{\gamma*}{2} x } \, , \nonumber \\
&& \phi^* = Q(x)^{-1} \, \kappa_4^{-1} \,  ,
\label{Qmetric}
\end{eqnarray}
where we identified $x$ with the radial coordinate. The reason of the particular rescaling $Q(x)$ will be clarify shortly making use of a more suitable radial coordinate. Notice that $Q(x)$ is singular for $x= 2/\gamma^*$, and, therefore, the metric is defined for  $x <  2/\gamma^*$. However, we will prove in the next section that the asymptotic singularity is unattainable, namely it requires an infinte amount of proper time to be reached. As a remnant of the previous work As a remnant of the previous work \cite{Li:2019ksm}, we named $\gamma^*/2$ the free inverse length scale present in the solution. However, in order to manifestly identify the effect of the conformal symmetry it would be useful and more suitable to define:
$\gamma^*/2 \equiv \ell_{\rm c}$, which we will refer to as the characteristic scale of the system.

In order to show that the scaling factor $Q(x)$ in (\ref{Qmetric}) is the only one compatible with (i) $g_{00} = -1/g_{11}$
(we will expand on the uniqueness of the metric in the section \ref{Unique}), we make a coordinate transformation to the usual radial Schwarzschild coordinate ``$r$'', which identifies the physical radius of the two-sphere (ii).
The new radial coordinate $r$ is related to $x$ as follows,
\begin{eqnarray}
&& x=\frac{r}{1 + \frac{\gamma^*}{2} r } \, ,
\label{xTOr}
\\
&& \frac{ \partial x(r) }{\partial r} = \frac{1}{Q(r)^2} \,  ,
\label{xTOrder}
\end{eqnarray}
and the metric turns into:
 \begin{eqnarray}
&& d \hat{s}^{ * 2 }  = -  Q^2(r) \left( 1-\frac{2GM Q(r)}{ r}  \right)  dt^2+\frac{dr^2}{Q^2(r) \left( 1-\frac{2GM Q(r)}{ r}  \right)} + r^2 (d\theta^2+\sin^2\theta d\varphi^2)  \, ,
\nonumber \\
&& Q(r) = 1+ \frac{\gamma^*}{2} r  \qquad \left( \mbox{notice that} \,\, x = \frac{r}{Q(r)} \right) \, , \nonumber \\
&& \phi^*(r) = Q(r)^{-1} \, \kappa_4^{-1}\, .
\label{Qmetricr}
\end{eqnarray}
It deserves to be notice that any rescaling that differs from the one in (\ref{Qmetric}) is not consistent with the two requirements above, namely (i) and (ii). Therefore, in the infinite class of exact solutions conformally equivalent to the Schwarzschild metric, there is only one geometry non-asymptotically flat consistent with $g_{00} = -1/g_{11}$ and two-dimensional transverse area $4 \pi r^2$. Notice that $Q(r)$ in (\ref{Qmetricr}) is only linear in $r$, which is the minimal modification of the metric compatible with analyticity.
As mentioned above, we will expand further on the uniqueness of the metric in section (\ref{Unique}).

\subsection{Regularity of the Kretschmann and Weyl square invariants}
As a first check of the regularity, we look at the spacetime in $x=2/\gamma^*$.
Since the Schwarzschild spacetime is Ricci flat, before the rescaling the first non-trivial curvature invariant is the Kretschmann scalar, which reads:
\begin{eqnarray}
     \hat{K} := \hat{R}_{\alpha \beta \gamma \delta} \hat{R}^{\alpha \beta \gamma \delta} = \hat{C}_{\alpha \beta \gamma \delta} \hat{C}^{\alpha \beta \gamma \delta} = \frac{ 48 G^2 M^2}{x^6} \, ,
     \label{Kre}
\end{eqnarray}
where in the last equality we used that $\hat{R}_{\alpha \beta}=0$ and introduced the Weyl tensor $\hat{C}_{\alpha \beta \gamma \delta}$. Under the Weyl rescaling (\ref{ConfInv}) the Weyl tensor, for the following position of the indexes, is invariant, namely
\begin{eqnarray}
\hat{C}^{* \alpha}\,_{ \beta \gamma \delta} = \hat{C}^{\alpha}\,_{ \beta \gamma \delta}\, .
\end{eqnarray}
Hence, the Kretschmann scalar (\ref{Kre}) for the metric (\ref{Qmetric}) turns into:
\begin{eqnarray}
 \hat{C}^{*2} = \hat{C}^{* \alpha}\,_{\beta \gamma \delta} \, \hat{C}^{* \mu}\,_{\nu\rho\sigma} \,  \hat{g}^*_{\alpha \mu} \hat{g}^{* \beta \nu} \hat{g}^{* \gamma \rho} \hat{g}^{* \delta \sigma}
 = \hat{C}^{\alpha}\,_{\beta \gamma \delta} \, \hat{C}^{\mu}\,_{\nu\rho\sigma} \,  \hat{g}_{\alpha \mu} \hat{g}^{\beta \nu} \hat{g}^{\gamma \rho} \hat{g}^{\delta \sigma} Q^2(x) \, Q^{-2}(x) \, Q^{-2}(x)\,  Q^{-2}(x) = \frac{\hat{C}^2}{Q^4(x)} \, .
 \label{C}
\end{eqnarray}
Finally, for the metric (\ref{Qmetric}) we find:
\begin{eqnarray}
\hat{C}^{*2} = \frac{\hat{K}}{Q^4(x)} = \frac{ 48 G^2 M^2}{x^6} \, \left( 1- \frac{\gamma^*}{2} x \right)^4
\, ,
\end{eqnarray}
which is zero in the limit $x \rightarrow  \gamma^*/2$. The letter point, as we will show explicitly in the next subsection, represents the spatial infinity for the metric (\ref{Qmetric}) because nothing can reach such point in finite proper time. Therefore, the curvature invariant approaches asymptotically zero.

Using the radial coordinate $r$ the curvature invariant $\hat{C}^{*2}$ turns into ($x = r/Q(r)$):
 \begin{eqnarray}
\hat{C}^{*2}(r) : = \hat{C}^{*2}(x(r)) = \frac{ \hat{C}^{2}  (x(r))}{Q^4(x(r))} = \frac{ 48 G^2 M^2}{r^6} \, Q^6(r) \, Q^{-4}(r) =
 \frac{ 48 G^2 M^2}{r^6} \,
\left( 1+ \frac{\gamma^*}{2} r \right)^2 \, ,
\end{eqnarray}
which is now zero for $r \rightarrow + \infty$ according to the inverse coordinate transformation from $x$ to $r$, namely
\begin{eqnarray}
 r= \frac{x}{ 1 - \frac{\gamma^*}{2} x} \, ,
 \label{rTOx}
\end{eqnarray}
which diverges to infinity for $x \rightarrow  \gamma^*/2$.

On the other hand, the Kretschmann scalar for the metric (\ref{Qmetricr}) is:
\begin{eqnarray}
&& \hat{K}^*=\frac{1}{2r^6} \left[-2\gamma^{*2}GMr^3(16+12\gamma^*r+3\gamma^{*2}r^2)+\gamma^{*2}r^4(16+12\gamma^*r+3\gamma^{*2}r^2) \right. \nonumber\\
&&
\hspace{1cm}
\left. +4G^2M^2(24+16\gamma^*r+8\gamma^{*2}r^2+4\gamma^{*3}r^3+\gamma^{*4}r^4) \right]  .
\end{eqnarray}
At large distance the Kretschmann invariant for the metric (\ref{Qmetricr}) tends to a constant $\hat{K}^*\rightarrow 3\gamma^{*4}/2$, which means that the metric (\ref{Qmetricr}) descries asymptotically a spacetime of constant curvature. Indeed, at large scales the metric (\ref{Qmetricr}) approaches the Anti de-Sitter spacetime with
 scalar curvature $R\rightarrow - 3\gamma^{*2}$ in the limit $r\rightarrow+\infty$.

Therefore, the two curvature invariants computed above, namely $\hat{C}^{*2}$ and $\hat{K}^*$, are asymptotically finite, and $x=2/\gamma^*$ is not a curvature singularity. The letter point, as we will show explicitly in the next subsection, represents the spatial infinity for the metric (\ref{Qmetric}) because nothing can reach such point in finite proper time.

Although in this paper we are concerned with the spacetime far outside the event horizon (indeed all the probes in the galaxy are stars and not black holes), the reader may worry about the singularity at $x = 0$ or $r = 0$. However, the resolution of singularities has been rigorously dealt with in several previous articles \cite{NK, Bambi:2016wdn} and the results found there can be exported directly to the metric (\ref{Qmetric}). Indeed, it is sufficient to rescale the latter metric as explicitly done in \cite{Bambi:2016wdn}. For completeness, starting from the metric (\ref{Qmetricr}), we here provide an explicit example of geodetically complete spacetime from short to large distances, namely
 \begin{eqnarray}
&& d \hat{s}^{ * 2 }  = S(r) \left[ -  Q^2(r) \left( 1-\frac{2GM Q(r)}{r}  \right)  dt^2+\frac{dr^2}{Q^2(r) \left( 1-\frac{2GM Q(r)}{r}  \right)} + r^2 d \Omega^{(2)}  \right] \, ,
\nonumber \\
&& Q(r) = 1+ \frac{\gamma^*}{2} r  \qquad \left( \mbox{notice that} \,\, x = \frac{r}{Q(r)} \right) \, , \nonumber \\
&& \phi^*(r) = S(r)^{-1/2}  Q(r)^{-1} \, \kappa_4^{-1}\, , \nonumber \\
&& S(r) = 1+ \frac{L^4}{r^4} \, ,
\label{QmetricrSF}
\end{eqnarray}
where $L$ is a parameter with dimension of length (for more details and observational constraints on $L$ see \cite{Bambi:2016wdn, FiniteConformal, Bambi:2017yoz}).

\subsection{Geodetic completion: conformally coupled particles}
For the sake of simplicity from now on in the paper we will remove the label ``$^*$" to the metric and to the dilaton field.
Let us start with a conformally coupled particle whose action reads:
\begin{eqnarray}
S_{\rm cp} = - \int \sqrt{ - f^2 \phi^2 \hat{g}_{\mu\nu} d x^\mu d x^\nu}
=  - \int \sqrt{ - f^2 \phi^2 \hat{g}_{\mu\nu} \frac{d x^\mu}{d \lambda} \frac{d x^\nu}{d \lambda} }
\, d \lambda \, ,
\label{Scp}
\end{eqnarray}
where $f$ is a positive constant coupling strength, $\lambda$ is the world-line parameter, and $x^\mu(\lambda)$ is the trajectory of the particle\footnote{Notice that assuming the conformal symmetry to be spontaneously broken to

 $\phi = \kappa_4^{-1}$ and taking the unitary gauge, the action~(\ref{Scp}) turns into the usual one for a particle with mass $m = f \kappa_4^{-1}$ ($f>0$). Different values for $f$ provide different mass scales.}. From (\ref{Scp}), the Lagrangian reads:
\begin{eqnarray}
L_{\rm cp} = - \sqrt{ - f^2 \phi^2 \hat{g}_{\mu\nu} \dot{x}^\mu \dot{x}^\nu } \, ,
\label{Lcp}
\end{eqnarray}
and the translation invariance in the time-like coordinate $t$ implies:
\begin{eqnarray}
\frac{\partial L_{\rm cp}}{\partial \dot{t} } = - \frac{f^2 \phi^2 \hat{g}_{tt} \dot{t}}{L_{\rm cp}} = {\rm const.} = - E \quad \Longrightarrow \quad \dot{t} = \frac{L_{\rm cp} E }{f^2 \phi^2 \hat{g}_{tt}} .
\label{ConstE}
\end{eqnarray}
Since we are interested in evaluating the proper time for the particle to reach the singularity of the universe located in $x= 2/\gamma^*$, we choose the proper time gauge, namely $\lambda = \tau$. Therefore, $E$ can be formally interpreted like the energy of the test-particle and
\begin{eqnarray}
\frac{d \hat{s}^2}{d \tau^2} = - 1
\quad \Longrightarrow \quad
\Bigg\{\hat{g}_{\mu\nu} \dot{x}^\mu \dot{x}^\nu = - 1 \quad {\rm and}
\quad
L_{\rm cp}= - f \phi
\quad \Longrightarrow \quad  \dot{t} = - \frac{ E }{f \phi \, \hat{g}_{tt}} \Bigg\} \, .
\label{PTG}
\end{eqnarray}
Replacing $\dot{t}$ from (\ref{PTG}) in
\begin{eqnarray}
\hat{g}_{\mu\nu} \dot{x}^\mu \dot{x}^\nu = - 1 \, ,
\end{eqnarray}
and using the solution of the EOM for $\phi$, namely $\phi = Q^{-1} \kappa^{-1}_4$, we end up with the following first order differential equation for $x(\tau)$,
\begin{eqnarray}
&& g_{tt} \dot{t}^2 + g_{xx} \dot{x}^2 = -1 \quad \Longrightarrow \quad
g_{tt} \frac{E^2}{f^2 \phi^2 g_{tt}^2 }  + g_{xx} \dot{x}^2 = -1
\quad \Longrightarrow \quad
\frac{E^2}{f^2 \phi^2 }  + g_{xx} g_{tt} \dot{x}^2 = - g_{tt} \, ,
\\
&& Q(x)^4 \dot{x}^2 + Q(x)^2 \left( 1 - \frac{2 G M}{x} \right) - \frac{E^2 \kappa_4^2}{f^2} Q(x)^2 = 0 \, ,
\label{Geox}
\end{eqnarray}
or, introducing the dimensionless parameter $e^2 \equiv \frac{E^2 \kappa_4^2}{f^2}$,
\begin{eqnarray}
 Q(x)^2 \dot{x}^2 =  \frac{2 G M}{x} + e^2 -1  \, .
\label{GeoX}
\end{eqnarray}
Since we are interested to investigate the asymptotic completeness of the spacetime for large $x$, we can assume $x \gg 2 G M$ and (\ref{GeoX}) simplifies to
\begin{eqnarray}
 Q(x)^2 \dot{x}^2 \simeq e^2 -1  \, ,
\label{GeoX2}
\end{eqnarray}
which must be positive because $Q(x)^2 \dot{x}^2$ is surely positive.
Replacing $Q(x)$ from (\ref{Qmetric}) into (\ref{GeoX2}) we get:
\begin{eqnarray}
  \frac{|\dot{x}|}{| 1 - \frac{\gamma^*}{2} x |}  \simeq \sqrt{e^2 -1} > 0  \, .
\label{GeoX3}
\end{eqnarray}
We here would like to study a particle moving from smaller to larger values of $x$, then $\dot{x} >0 $, moreover, $x < \gamma^*/2$, therefore (\ref{GeoX3}) simplifies to
\begin{eqnarray}
  \frac{\dot{x}}{1 - \frac{\gamma^*}{2} x}  \simeq \sqrt{e^2 -1} > 0  \, ,
\label{GeoX4}
\end{eqnarray}
and the solution is:
\begin{eqnarray}
\tau = \frac{2}{\gamma^* \, \sqrt{e^2 -1}} \log\left( \frac{ 1 - \frac{\gamma^*}{2} x_0}{1 - \frac{\gamma^*}{2} x} \right) \qquad {\rm or} \qquad
x(\tau) = \frac{2}{\gamma^*}  \left[ 1-e^{-\frac{\gamma  \tau }{2}} \left( 1 - \frac{\gamma^*}{2} x_0 \right) \right] \, .
\label{SolGeo}
\end{eqnarray}
According to the solution (\ref{SolGeo}), the proper time to reach the edge of the universe located in $x= 2/\gamma^* = \ell_{\rm c}$ is infinity. Moving to the radial coordinate $r$ defined in (\ref{xTOr}),
\begin{eqnarray}
\lim_{x \rightarrow \frac{\gamma^*}{2}} r = \lim_{x \rightarrow \frac{\gamma^*}{2}} \frac{x}{ 1 - \frac{\gamma^*}{2} x} = + \infty \, .
\end{eqnarray}
Therefore, a massive particle will reach $r=+ \infty$ in an infinite amount of proper time.
Indeed, in the coordinate $r$, the radial geodesic equation (\ref{GeoX}) turns into:
\begin{eqnarray}
&& Q(r)^2 \dot{r}^2 \left(\frac{\partial x[ r(\tau)] }{ \partial r}\right)^2=  \frac{2 G M Q(r) }{r} + e^2 -1  \, ,
\nonumber  \\
&& \frac{\dot{r}^2}{Q(r)^2} =  \frac{2 G M Q(r) }{r} + e^2 -1
\quad (\mbox{where we used} \,\, (\ref{xTOr}) \,\, \rm{and} \,\, (\ref{xTOrder})) \, ,
\label{confrar}  \\
&& \frac{\dot{r}}{Q(r)} \simeq \sqrt{G M \gamma^* + e^2 -1} : = {\rm c}
\quad {\mbox{for} }  \quad  r \gg 2 G M, \quad \dot{r} > 0 \, , \quad {\rm and} \quad   G M \gamma^* + e^2 -1 >0 \, ,\nonumber  \\
&& \tau = \frac{2 \log \left(\frac{1 + \frac{\gamma^*}{2} r}{ 1 + \frac{\gamma^*}{2}  r_0}\right)}{{\rm c} \, \gamma^* }
\quad \Longrightarrow \quad
r= \frac{2 e^{\frac{ {\rm c} \gamma^*  \tau }{2}}+ \gamma^*  r_0 e^{\frac{ {\rm c} \gamma  \tau }{2}}-2}{\gamma^* } \, .
\end{eqnarray}
So far we found that the proper time for a particle (conformally coupled) to reach the edge of the Universe is infinite in both $x$ and $r$ coordinates, in the former case the boundary is located at the finite value $x= \gamma^*/2$, in the latter case it is located in $r= +\infty$.
In the next section we will study the geodesic motion of massless particles.

\subsection{Geodetic completion: massless particles}
For massless particles the correct action, which is invariant under reparametrizations of the world line, $p^\prime= f(p)$, is:
\begin{eqnarray}
S_{\gamma} = \int \mathcal{L}_{ \gamma } d \lambda = \int e(p)^{-1} \phi^2\hat{g}_{\mu\nu}  \frac{d x^\mu}{d p} \frac{d x^\nu}{d p} d p \, ,
\label{Lm0}
\end{eqnarray}
where $e(p)$ is an auxiliary field that transforms as $e^\prime(p^\prime)^{-1} = e(p)^{-1} (d p^\prime/d p)$ in order to guarantee the invariance of the action. The action (\ref{Lm0}) is not only invariant under general coordinate transformations, but also under the Weyl conformal rescaling (\ref{ConfInv}).

The variation respect to $e$ gives:
\begin{eqnarray}
\frac{ \delta S_{\gamma}}{\delta e} = 0 \quad \Longrightarrow \quad  - \int d p  \, \frac{\delta e}{e^2} \phi^2 \, \hat{g}_{\mu\nu} \, \dot{x}^{\mu} \dot{x}^\nu = 0 \quad \Longrightarrow \quad   d \hat{s}^2 =  \hat{g}_{\mu\nu} dx^\mu d x^\nu = 0 \, ,
\label{ds0}
\end{eqnarray}
which is equivalent to say that massless particles travel along the light cone.

The variation respect to $x^\mu$ gives the geodesic equation in presence of the dilaton field, namely (in the gauge $e(p)={\rm const.}$)
\begin{eqnarray}
\frac{D^2(g = \phi^2 \hat{g}) x^\lambda}{d p^2} =
\frac{D^2(\hat{g}) x^\lambda}{d p^2} + 2 \frac{\partial_\mu \phi}{\phi} \frac{ d x^\mu}{d p} \frac{ d x^\lambda }{d p}
-  \frac{\partial^\lambda \phi }{\phi} \frac{ d x^\mu}{d p} \frac{ d x_\mu }{d p}
= 0 \, ,
\label{geom0}
\end{eqnarray}
where $D^2(\hat{g})$ is the covariant derivative respect to the metric $\hat{g}_{\mu\nu}$.

However, when we contract equation (\ref{geom0}) with the velocity $d x_\lambda/d p$ and we use $d\hat{s}^2 =0$ obtained in (\ref{ds0}), we get the following on-shell condition,
\begin{eqnarray}
\frac{d x_\lambda}{d p} \, \frac{D^2(\hat{g}) x^\lambda}{d p^2} + 2  \frac{d x_\lambda}{d p} \, \frac{\partial_\mu \phi}{\phi} \frac{ d x^\mu}{d p} \frac{ d x^\lambda }{d p}
-  \frac{d x_\lambda}{d p} \, \frac{\partial^\lambda \phi }{\phi} \frac{ d x^\mu}{d p} \frac{ d x_\mu }{d p}
= 0 \quad \Longrightarrow \quad \frac{d x_\lambda}{d p} \, \frac{D^2(\hat{g}) x^\lambda}{d p^2} = 0 \, .
\end{eqnarray}
Therefore, the the covariant derivative $\frac{D^2(\hat{g}) x^\lambda}{d p^2}$ must be proportional to the velocity, namely
\begin{eqnarray}
\frac{D^2(\hat{g}) x^\lambda}{d p^2} = f \, \frac{d x^\lambda}{d p}  \quad (f = {\rm const.})
\label{constre}
\end{eqnarray}
because the velocity is null on the light cone.
Under a reparametrization of the world line $q = q(p)$ eq.(\ref{constre}) becomes
\begin{eqnarray}
\frac{d^2 x^{\lambda} }{dq^2} + \Gamma^\lambda_{\mu\nu}  \frac{d x^\mu }{d p}  \frac{d x^\nu }{d p}
=  \frac{d x^\lambda}{d p} \left( \frac{dp}{dq} \right) \left(f \frac{dq}{dp} - \frac{d^2 q}{dp^2} \right) .
\label{parame}
\end{eqnarray}
Choosing the dependence of $q$ on $p$ such us to make vanish the right-hand side of (\ref{parame}), we end up we the geodesic equation in the affine parametrization. Hence, we can redefine $q \rightarrow \lambda$ and, finally, we get the affinely parametrized geodesic equation for photons in the metric $\hat{g}_{\mu\nu}$,
\begin{eqnarray}
\frac{D^2(\hat{g}) x^\lambda}{d \lambda^2} = 0 \, .
\label{affine}
\end{eqnarray}
We can now investigate the conservations laws based on the symmetries of the metric $\hat{g}_{\mu\nu}$. Let us consider the following scalar,
\begin{eqnarray}
\hat{\alpha} = \hat{g}_{\mu\nu} v^\mu \frac{d x^\nu}{d \lambda} =
\hat{g}_{\mu\nu} v^\mu u^\nu
\, .
\label{alpha}
\end{eqnarray}
where $v^\mu$ is a general vector and $u^\mu$ the four velocity. Taking the derivative of (\ref{alpha}) respect to $\lambda$ and using the geodesic equation (\ref{affine}) we get:
\begin{eqnarray}
\frac{d}{d \lambda} \hat{\alpha} = \frac{1}{2}  v^\mu \partial_\mu \hat{g}_{\rho \nu} \frac{d x^\rho}{d \lambda}
\frac{d x^\nu}{d \lambda} + \hat{g}_{\mu\nu} \partial_\rho v^\mu \frac{d x^\nu}{d \lambda} \frac{d x^\rho}{d \lambda}
= \frac{1}{2} [\mathcal{L}_v \hat{g} ]_{\rho \nu }  \frac{d x^\rho}{d \lambda} \frac{d x^\nu}{d \lambda} \, ,
\end{eqnarray}
where $[\mathcal{L}_v \hat{g}]$ is the Lie derivative of $\hat{g}_{\mu\nu}$ by a vector field $v^\mu$.
Thus, if $v^\mu$ is a Killing vector field, namely $[\mathcal{L}_v \hat{g}]=0$, $\hat{\alpha}$ is conserved:
\begin{eqnarray}
\frac{d}{d \lambda} \left[ \hat{g}_{\mu\nu} v^\mu \frac{d x^\nu}{d \lambda} \right] = 0
\, .
\label{CL}
\end{eqnarray}
The metric~(\ref{Qmetric}) is time-independent and spherically symmetric (in particular it is invariant under $t \rightarrow t + \delta t$ and $\varphi \rightarrow \varphi + \delta \varphi$). Therefore, we have the following Killing vectors associated with the above symmetries
\begin{eqnarray}
\xi^{\alpha} = (1, 0, 0, 0) \, , \quad \eta^{\alpha} = (0, 0, 0, 1) \, .
\end{eqnarray}
Since the metric is independent of the $t$- and $\varphi$-coordinates, according to (\ref{alpha}) we can construct the following conserved quantities
\begin{eqnarray}
&& e = - \xi \cdot u = - \xi^\alpha u^\beta \hat{g}_{\alpha \beta} = - \hat{g}_{ t \beta} u^\beta = - \hat{g}_{ t t } u^t =
Q^2(x) \left(  1 - \frac{2 M}{x} \right) \frac{d t}{d \lambda} = Q^2(x) \left(  1 - \frac{2 M}{x} \right) \dot{t} \, ,
\label{Ldott} \\
&& \ell = \eta \cdot u = \eta^\alpha u^\beta \hat{g}_{\alpha \beta} =  \hat{g}_{ \phi \beta} u^\beta =  \hat{g}_{ \phi \phi } u^\phi =
Q^2(x) x^2 \sin^2 \theta \, \dot{\varphi} \, ,
\label{Ldotphi}
\end{eqnarray}
where the null vector
\begin{eqnarray}
u^{\alpha} = \frac{d x^\alpha}{d \lambda}
\end{eqnarray}
satisfies
\begin{eqnarray}
u \cdot u = \hat{g}_{\alpha \beta}  \frac{d x^\alpha}{d \lambda}  \frac{d x^\beta}{d \lambda} = 0 \, ,
\label{NullV}
\end{eqnarray}
as a consequence of (\ref{ds0}).

From~(\ref{NullV}) in the equatorial plane (i.e. $\theta = \pi/2$), we get the following equation
\begin{eqnarray}
- \left(  1 - \frac{2 G M}{x} \right) \dot{t}^2 + \frac{\dot{x}^2}{\left(  1 - \frac{2 G M}{x} \right)}
+ x^2  \, \dot{\varphi}^2 = 0 \, .
\label{uu}
\end{eqnarray}
Note that the rescaling of the metric cancels out in the above equation (\ref{uu}) for null geodesics, but $Q^2(x)$ will appear again when the conserved quantities (\ref{Ldott}) and (\ref{Ldotphi}) are taken into account. Let us solve (\ref{Ldott}) for $\dot{t}$ and (\ref{Ldotphi}) for $\dot{\varphi}$ and, afterwards, replace the results in (\ref{uu}). The outcome is:
\begin{eqnarray}
-  \frac{e^2}{Q(x)^4 \left(1 - \frac{2GM}{x} \right)}
+ \frac{\dot{x}^2}{  1 - \frac{2 G M}{x} }
+ \frac{\ell^2}{ Q(x)^4 x^2 } = 0 \, .
\label{uu2}
\end{eqnarray}
Let us focus on the radial geodesics (i.e. $\ell=0$), which will be sufficient to verify the geodesic completeness. Equation~(\ref{uu2}) simplifies to:
\begin{eqnarray}
-  \frac{e^2}{Q(x)^4 }
+\dot{x}^2 =0 \, \quad \Longrightarrow \quad Q^2(x) | \dot{x} | = e \, .
\label{Geom0}
\end{eqnarray}
The above first order differential equation can be easily integrated for a photon traveling towards the boundary $x=2/\gamma^*$, namely for $\dot{x} > 0$. The result of the integration is:
\begin{eqnarray}
x(\lambda) = \frac{4 \lambda - 2 \gamma^*  \lambda  x_0  +  4 x_0  }{2 \gamma^*  \lambda - \gamma^{* 2} \lambda  - x_0 + 4} \, ,
\end{eqnarray}
where $x_0$ is the initial position from which the photon is emitted, and
\begin{eqnarray}
\lim_{\lambda \rightarrow +\infty} x(\lambda) = \frac{2}{\gamma^*} \, .
\end{eqnarray}
It turns out that photons cannot reach $x=2/\gamma^*$ for any finite value of the affine parameter $\lambda$.

In the coordinate $r$ the geodesic equation (\ref{Geom0}) turns into:
\begin{eqnarray}
{Q(x[r])^4 }
 \left( \frac{\partial x[r(\tau)]}{\partial r} \right)^2
\dot{r}^2 = e^2
 \, \quad \Longrightarrow \quad | \dot{r} | = e \, ,
\label{Geom2}
\end{eqnarray}
and a massless particle can reach $r= + \infty$ only for $\lambda = \infty$. The above equation (\ref{Geom2}) has been derived in the appendix (\ref{RadialGeor0}) also directly starting from the metric (\ref{Qmetricr}).

\section{Uniqueness of the solution}\label{Unique}

In the first part of this paper the rescaling of the metric $Q(x)$ was chosen compatibly with the relation $g_{00} = -1/g_{11}$, as evident in the coordinate $r$. In this section we would like to provide three fundamental reasons
to support such choice.
(i) The first one is related to the null energy condition, which asserts that $p + \rho \geqslant 0$ \cite{Pons:2014oya}. Indeed, in order to preserve the null energy condition we must impose $g_{00} = -1/g_{11}$.

(ii) The second one is related to the acceleration of the light in the Newtonian regime. Indeed, if the velocity of light has to remain constant in empty space surrounding a point-like mass, then photons should experience zero acceleration \cite{Dadhich:2012pda}. Using the last result in the previous subsection, namely $| \dot{r} | = e$ we get $\ddot{r} = 0$, which is true only if the relation $g_{00} = -1/g_{11}$ for the components of the metric tensor is satisfied. Let us expand on this point.
For a general spherically symmetric metric,
\begin{eqnarray}
ds^2 = - A(r) dt^2 + B(r) dr^2 + r^2 d \Omega^2 ,
\end{eqnarray}
making use again of (\ref{Ldott}), namely
\begin{eqnarray}
e = A(r) \dot{t} \, ,
\end{eqnarray}
 and $ds^2 = 0$, the radial geodesic equation reads
 \begin{eqnarray}
 \hspace{-0.5cm}
- A(r) \dot{t}^2  + B(r) \dot{r}^2 = 0  \quad \Longrightarrow \quad
-  \frac{e^2}{A(r) }  + B(r) \dot{r}^2 = 0
  \quad \Longrightarrow \quad   \dot{r}^2 = \frac{e^2}{A(r)  B(r)}
   \quad \Longrightarrow \quad 2 \, \dot{r} \, \ddot{r} = \left(\frac{e^2}{A(r)  B(r)}  \right)^{\prime} \dot{r} \, ,
 \end{eqnarray}
 where $^\prime$ means derivative respect to $r$. Finally,
 \begin{eqnarray}
 \ddot{r} = \left(\frac{e^2}{ 2 A(r)  B(r)}  \right)^{\prime} \, .
 \end{eqnarray}
 %
 Therefore, in order to do not experience acceleration in the radial coordinate we must have:
 $A(r) B(r) = {\rm const.}$. Notice that here the radial coordinate is not the physical radial distance because the spacetime is not asymptotically flat.  However, according to the Taylor expansion of (\ref{ellrr}) in the Newtonian intermedium regime $\ell_r \approx r$ and the acceleration above vanishes.

(iii) Last but not least we should consider the impact of the large distance modification of the Schwarzschild metric on the homogeneity and isotropy of the Universe.

Let us start considering the following coordinate transformation from the radial coordinate $r$ to $\rho$,
\begin{eqnarray}
&& \rho = \frac{4 r }{2 (1 + \alpha r + \beta r^2)^{1/2} + 2 + \alpha r}  \, , \\
&& \tau =  \int dt R(t) \, ,
\label{rrho}
\end{eqnarray}
in the following general not asymptotically flat metric,
 \begin{eqnarray}
 d \hat{s}^{ * 2 } = -   \left( 1 + \alpha r +   \beta r^2 \right)  dt^2+\frac{dr^2}{\left( 1 + \alpha r + \beta r^2 \right) } + r^2 d \Omega^{(2)}  \, .
\label{QmetricrLargerG}
\end{eqnarray}
The above metric (\ref{QmetricrLargerG}) in the new coordinates reads:
\begin{eqnarray}
 d \hat{s}^{ * 2 } = \frac{1}{R^2(\tau)}
 \left[ \frac{ 1 - \frac{\alpha^2 \rho^2}{16} + \frac{\beta \rho^2}{4} }{ \left(1 - \frac{\alpha \rho}{4} \right)^2 - \frac{\beta \rho^2}{4} }\right]^2
 \left\{
 - d \tau^2
 + \frac{R(\tau)^2}{\left[ 1 - \left(  \frac{\alpha^2}{16} - \frac{\beta}{4} \right) \rho^2 \right]^2 } \left(d \rho^2 + \rho^2 d \Omega^{(2)} \right)
\right\} \, .
\label{QmetricrLargerG1}
\end{eqnarray}
Now, in a geometry which is both homogeneous and isotropic about all points, any observer can serve as the origin of the radial coordinate $\rho$; thus in his own local rest frame each observer is able to make the above general coordinate transformation using his own particular $\rho$. Moreover, in conformal gravity we can make an overall rescaling of the metric to finally end up with a comoving Robertson-Walker (RW) spacetime written in spatially isotropic coordinates with spatial curvature $K = \beta - \alpha^2/4$,
\begin{eqnarray}
 d \hat{s}^{ * 2 } =
F(\tau, \rho)
 \left[
 - d \tau^2
 + \frac{R(\tau)^2}{ \left(1 + K \rho^2/4 \right)^2 } \left(d \rho^2 + \rho^2 d \Omega^{(2)} \right)
\right] \, .
\label{QmetricrLargerG2}
\end{eqnarray}
For the case of the metric (\ref{Qmetricr}), taking $r \gg 2 G M$ and $GM \gamma^* \ll 1$,
 \begin{eqnarray}
 d \hat{s}^{ * 2 }  \approx -   \left( 1 + \gamma^* r +   \frac{\gamma^{* 2}}{4} r^2 \right)  dt^2+\frac{dr^2}{\left( 1 + \gamma^* r +   \frac{\gamma^{* 2}}{4} r^2 \right) } + r^2 d \Omega^{(2)}  \, .
\label{QmetricrLarger}
\end{eqnarray}
 we can identify the constants $\alpha = \gamma^*$ and $\beta = \gamma^{* 2}/{4}$, and in the new coordinates $(\tau, \rho)$ the metric (\ref{QmetricrLarger}) takes the following RW form,
 \begin{eqnarray}
  d \hat{s}^{ * 2 } = \frac{1}{R^2(\tau)}
  \frac{ 1 }{ \left(1 -  \frac{\gamma^*}{2} \rho\right)^2 }
 \left[
 - d \tau^2
 + R(\tau)^2 \left( d \rho^2 + \rho^2 d \Omega^{(2)} \right)
\right] \, ,
\label{QmetricrCosmo}
\end{eqnarray}
which coincides with the metric (\ref{Qmetric}) for $x\gg 2 GM$ upon reintroducing the time coordinate $t$ defined in (\ref{rrho}).

Therefore, the metric proposed in this paper is the {\em only one} that does not affect the homogeneity of the Universe at large scales. Finally, we notice that the metric (\ref{QmetricrLarger}) is asymptotically (for large $r$) Anti-de Sitter, whose stability is guarantee from the fact that it comes form a rescaling of the Schwarzschild metric, which is known to be stable.

\section{The cosmological constant is not an issue in our model} \label{effectCosmoConstant}
It is commonly accepted that the value of the cosmological constant is non zero ($\Lambda\sim10^{-56}$cm$^{-2}$).
Therefore, we will have to more correctly consider the rescaling of the Schwarzschild-de Sitter spacetime instead of (\ref{Qmetric}) or (\ref{Qmetricr}), namely
\begin{eqnarray}
&& d \hat{s}^{ * 2 }  = Q^2(x) \left[ -  \left(1-\frac{2GM}{c^2 x} - \frac{\Lambda}{3} x^2 \right) dt^2+\frac{dx^2}{1-\frac{2GM}{ x} - \frac{\Lambda}{3} x^2}+x^2 \Omega^{(2)} \right] \, , \\
&& Q(x) = \frac{1}{1- \frac{\gamma*}{2} x } \, ,
\label{QmetricdS}
\end{eqnarray}
or in the radial coordinate $r$,
 \begin{eqnarray}
&& d \hat{s}^{ * 2 }  = -  Q^2(r) \left( 1-\frac{2GM Q(r)}{r} - \frac{\Lambda}{3} \frac{r^2}{Q^2(r)} \right) c^2 dt^2+\frac{dr^2}{Q^2(r) \left( 1-\frac{2GM Q(r)}{ r} - \frac{\Lambda}{3} \frac{r^2}{Q^2(r)} \right)} + r^2 d\Omega^{(2)}  \, ,
\nonumber \\
&& Q(r) = 1+ \frac{\gamma*}{2} r   \, .
\label{QmetricrdS}
\end{eqnarray}
Notice that the metric is till in the form $g_{00}(r) = - 1/g_{11}(r)$.
If we focus on (\ref{QmetricrdS}) and we consider the limit $r \gg 2 G M$ together with the approximation $G M \gamma^* \ll 1$, the metric (\ref{QmetricrdS}) simplifies to:
 \begin{eqnarray}
 d \hat{s}^{ * 2 }  &\approx&  - \left(Q^2(r)-\frac{\Lambda}{3}r^2\right)dt^2+\frac{dr^2}{\left(Q^2(r)-\frac{\Lambda}{3}r^2\right)}+r^2d\Omega^{(2)} \nonumber \\
&&  = - \left( 1 + \gamma^* r + \frac{\gamma^{* 2} }{4} r^2 - \frac{\Lambda}{3}r^2\right)dt^2+\frac{dr^2}{\left( 1 + \gamma^* r + \frac{\gamma^{* 2}}{4} r^2 -\frac{\Lambda}{3}r^2\right)}+r^2d \Omega^{(2)}
\, .
\label{QmetricrdSlimit}
\end{eqnarray}
However, since $\gamma^{*2}\gg\Lambda$ (we will see later that $\gamma^* \sim 10^{-21} {\rm m}^{-1}$) then the presence of the cosmological constant will not affect our analysi
\footnote{Looking at the Mannheim's paper \cite{Mannheim:2010xw}, in the appendix A5 the potential is defined as usually like $-(g_{00} + 1)/2$ (see the paragraph before formula A43 and formula A45). However, this is inconsistent with the physical velocity that we get from the metric (\ref{QmetricrLargerG}). Indeed, the usual derivation of the potential, which one can find for example in Landau's book ``Classical Field Theory'', does not work for spacetimes not asymptotically flat, which is the case of (\ref{QmetricrLargerG}). The correctness of Mannheim's paper lies in the fact that the scales in his model are much larger than the galactic extension.
It deserves to be notice that for special values of $\gamma_0$ and $k$ in Mannheim's paper, namely $\gamma_0 = \gamma^*$ and $k^2 = - \gamma^{*2}/4$, the exact solution (\ref{QmetricrLargerG}) (it is (5) in Mannheim's paper) of Weyl conformal gravity turns out to be a conformal rescaling of the Minkowski spacetime.
}.

Finally we want to make the following speculative comment.
It deserves to be notice that the value of the radius of de Sitter's spacetime (proportional to the inverse of the square root of the cosmological constant) is about the radius of the Universe. Therefore, for $\ell_c$ comparable to the radius of the Universe the two contributions quadratic in $r$ in (\ref{QmetricrdSlimit}) can cancel each other.

\section{The orbital velocity}
In this section we compute the orbital velocity
of a conformally coupled probe particle on the equatorial plane in the geometry (\ref{Qmetricr}) and (\ref{Qmetric}) respectively assuming zero radial velocity.
For completeness let us remind here the Action for a conformally coupled particle (\ref{Scp}),
\begin{eqnarray}
S_{\rm cp} = - \int \sqrt{ - f^2 \phi^2 \hat{g}_{\mu\nu} d x^\mu d x^\nu}
=  - \int \sqrt{ - f^2 \phi^2 \hat{g}_{\mu\nu} \frac{d x^\mu}{d \lambda} \frac{d x^\nu}{d \lambda} }
\, d \lambda \, ,
\label{Scp2}
\end{eqnarray}
from which the Lagrangian reads:
\begin{eqnarray}
L_{\rm cp} = - \sqrt{ - f^2 \phi^2 \hat{g}_{\mu\nu} \dot{x}^\mu \dot{x}^\nu } \, .
\label{Lcp2}
\end{eqnarray}
Since both the metrics (\ref{Qmetricr}) and (\ref{Qmetric}) are nvariant whether we make the replacements $t \rightarrow t + {\rm const}.$ and $\varphi \rightarrow \varphi + {\rm const}.$. Therefore, from the Lagrangian (\ref{Lcp2}) we obtain the following conserved quantities (for $\theta=\pi/2$),
\begin{gather}
\frac{\partial{L_{\rm cp}}}{\partial \dot{t}} = - \frac{ f^2 \phi^2\hat{g}_{tt}\dot{t}}{{L_{\rm cp}}}
 = - E \, , \qquad
	\frac{\partial{L_{\rm cp}}}{\partial \dot{\varphi}} = - \frac{ f^2 \phi^2\hat{g}_{\varphi\varphi}\dot{\varphi}}{{L_{\rm cp}}}=  \ell \,  .
	\label{conserva}
\end{gather}
In the proper time gauge $\lambda \equiv \tau$, $d \hat{s}^2/d \lambda^2 =-1$ and $L_{\rm cp} = - f \phi$. Hence, from
(\ref{conserva}),
\begin{eqnarray}
\dot{t}= \frac{E}{f \, \phi \, \hat{g}_{tt}} \, , \qquad
	\dot{\varphi} = - \frac{\ell}{f \, \phi \, \hat{g}_{\varphi\varphi}} \, .
	\label{dotti}
\end{eqnarray}

\subsubsection{The orbital velocity in the metric (\ref{Qmetricr})}
 %
 Let us in this section focus on the metric (\ref{Qmetricr}). Again in the proper time gauge and for $\theta = \pi/2$,
 \begin{eqnarray}
 \frac{d \hat{s}^2}{d \lambda^2} = - 1 \quad \Longrightarrow \quad \hat{g}_{tt} \dot{t}^2 + \hat{g}_{rr} \dot{r}^2 + \hat{g}_{\varphi \varphi} \dot{\varphi}^2 = -1 \, ,
 \label{orbitalprop}
 \end{eqnarray}
 and replacing (\ref{dotti}) in (\ref{orbitalprop}), we get:
 \begin{eqnarray}
 \hat{g}_{tt} \left( \frac{E}{f \, \phi \, \hat{g}_{tt}} \right)^2 + \hat{g}_{rr} \dot{r}^2
 + \hat{g}_{\varphi \varphi} \left(  - \frac{\ell}{f \, \phi \, \hat{g}_{\varphi\varphi}}  \right)^2 = -1
 \quad \Longrightarrow \quad
    \frac{E^2}{f^2 \, \phi^2 \, \hat{g}_{tt}}  + \hat{g}_{rr} \dot{r}^2
 +  \frac{\ell^2}{f^2 \, \phi^2 \, \hat{g}_{\varphi\varphi}}   = -1 \, .
 \end{eqnarray}
Since we are interested in the orbital motion we can take $\dot{r} = 0$ and we end up with the following constraint equation,
 \begin{eqnarray}
    \frac{E^2}{f^2 \, \phi^2 \, \hat{g}_{tt}}  +  \frac{\ell^2}{f^2 \, \phi^2 \, \hat{g}_{\varphi\varphi}}   = -1
    \quad \Longrightarrow \quad
   - \frac{E^2}{f^2 \, Q^{-2}(r) \kappa_4^{-2} \, Q^2(r) \left( 1 - \frac{2 G M Q(r) }{r} \right)}  +  \frac{\ell^2}{f^2 \, Q^{-2}(r) \kappa_4^{-2}  \, r^2}   = -1
    \, .
    \label{constraint}
 \end{eqnarray}
In order to extract a simple relation for the ratio between $\ell^2$ and $E^2$, we take the derivative of equation (\ref{constraint}) respect to $r$,
\begin{eqnarray}
E^2 \frac{\frac{d}{d r} \left(  1 - \frac{ 2 G M Q(r) }{r} \right)}{\left(  1 - \frac{ 2 G M Q(r) }{r} \right)^2} +
\ell^2 \frac{d}{d r} \left( \frac{Q^2(r)}{r^2} \right) = 0
\quad \Longrightarrow \quad \frac{\ell^2}{E^2} = \frac{2 G M r^3}{(2 + r \gamma^* ) (r-G M (2+r \gamma^* ))^2}
\, .
\label{LE}
\end{eqnarray}

The physical velocity on the equatorial plane and along the $\varphi$-direction reads:
\begin{equation}
v=\frac{\sqrt{\hat{g}_{\varphi\varphi}}}{\sqrt{-\hat{g}_{tt}}}
\,
\frac{d\varphi}{dt}=\frac{\sqrt{\hat{g}_{\varphi\varphi}}}{\sqrt{-\hat{g}_{tt}}}
\,
\frac{\dot{\varphi}}{\dot{t}} \, ,
\label{vr}
\end{equation}
where dot stays for the derivative respect to the proper time $\tau$. Replacing $\dot{t}$ and $\dot{\varphi}$ in (\ref{dotti}) into (\ref{vr}) we get:
\begin{equation}
	v^2=-\frac{\hat{g}_{tt}}{\hat{g}_{\varphi\varphi}}\frac{\ell^2}{E^2} \, ,
	\label{v2gene}
\end{equation}
where we finally replace (\ref{LE}),
\begin{equation}
	v^2= \frac{GM(2+\gamma^*r)}{2( r- GM (2+ \gamma^*r) )} = \frac{G M Q(r)}{r - 2 G M Q(r) }
		\, .
\end{equation}
In the limit of $r\gg2GM$, namely far from the Schwarzschild radius, the velocity turns into:
\begin{eqnarray}
v^2= \frac{G M Q(r)}{r(1 - G M \gamma^*) } \, ,
\label{v2r2GM}
\end{eqnarray}
and if we also assume $GM\gamma^*\ll1$,
\begin{equation}
	v^2= \frac{ G M Q(r)}{r} = \frac{GM}{r}+\frac{GM\gamma^*}{2}
	\label{v2r}
\end{equation}
which asymptotically approaches the constant value:
\begin{eqnarray}
v^2 \, \rightarrow \, v^2_{\infty} =
\frac{GM\gamma^*}{2}.
\label{vinfinity}
\end{eqnarray}
Let us now express the velocity in terms of the physical length $\ell_r$ in place of the radial coordinate $r$. What we need is the physical radial length, namely
\begin{eqnarray}
\ell_r &= & \int \sqrt{ \hat{g}_{rr} } \,  dr + {\rm const}
=  \int \frac{ dr }{Q(r) \sqrt{1 - \frac{2 G M Q(r)}{r}}} + {\rm const}
\approx  \int \frac{ dr }{ \sqrt{1 - G M \gamma^* } \, (1+ \frac{\gamma^*}{2} r)} + {\rm const}
\nonumber \\
& = & \frac{2 \log (2 + \gamma^* \, r)}{\gamma^*  \sqrt{1 -  G M \gamma^*}} + {\rm const}
\, .
\end{eqnarray}
where in the last by one step we have integrated for $r \gg 2 G M$.
Finally, we fix the integration constant imposing that $\ell_r(r=0) = 0$,
\begin{eqnarray}
\ell_r = \frac{2 \log\left(1 + \frac{\gamma^*}{2} \, r \right)}{\gamma^*  \sqrt{1 -  G M \gamma^*}} \, .
\label{ellrr}
\end{eqnarray}
Notice that in the intermedium Newtonian regime, namely $r \ll 2/\gamma^*$, and for $GM\gamma^*\ll1$, $\ell_r \approx r$.
The inverse relation $r(\ell_r)$ reads:
\begin{eqnarray}
r(\ell_r)= \frac{2 \left(e^{\frac{1}{2} \gamma^*  \ell_r \sqrt{1 - G M \gamma^*}}-1\right)}{\gamma^* }
\approx \frac{2 \left(e^{\frac{1}{2} \gamma^*  \ell_r}-1\right)}{\gamma^* }
\, ,
\label{rlr}
\end{eqnarray}
where the last approximation comes again from $GM\gamma^*\ll1$
(notice that also $r(\ell_r = 0) = 0$).

Replacing (\ref{rlr}) in (\ref{v2r2GM}), we get the physical velocity square, namely
\begin{eqnarray}
v^2(\ell_r) = \frac{G M \gamma^*}{4} \frac{\left[ 1 + \coth \left( \frac{1}{4} \ell_r \gamma^* \sqrt{1 - G M \gamma^*} \right)\right] }{1 -  G M \gamma^*} \,  ,
\label{v2lr}
\end{eqnarray}
which further simplifies for $GM\gamma^*\ll1$,
\begin{eqnarray}
\boxed{v^2(\ell_r) = \frac{G M \gamma^*}{4} \left[ 1 + \coth \left( \frac{ \ell_r \gamma^*}{4} \right)\right] } \,  .
\label{v2lr2}
\end{eqnarray}
The above astonishing simple analytic result correctly interpolates between the Newtonian's velocity and the asymptotic constant value (\ref{vinfinity}). It deserve to be notice that for small $\gamma^*$, namely $\ell_c \gg r_g = 2 G M$ ($r_g$ is the Schwarzschild radius), the exact result (\ref{v2lr}) and the velocity (\ref{v2r}) are extremely close each other. Therefore, the following replacement is a good approximation of (\ref{v2r}),
\begin{eqnarray}
v^2(\ell_r) = \frac{GM}{\ell_r}+\frac{GM\gamma^*}{2} \,  .
\label{v2rlr}
\end{eqnarray}

\subsubsection{The orbital velocity in the metric (\ref{Qmetric})}
In this section we compute again the velocity square, but now for the metric (\ref{Qmetric}). This computation not only will provide a further check of our result (\ref{v2lr2}), but also will make more explicit the crucial role of the asymptotic singularity in $x=2/\gamma^* = \ell_{\rm c}$.

According to the previous section ($\dot{t}$ and $\dot{\varphi}$)(\ref{dotti}), the velocity (\ref{vr}), and the velocity square (\ref{v2gene}) are general and independent on the metric. However, the ratio $\ell^2/E^2$ it does depend on the metric. Indeed, the proper time gauge for the metric (\ref{Qmetric}) reads:
\begin{eqnarray}
 \frac{d \hat{s}^2}{d \lambda^2} = - 1 \quad \Longrightarrow \quad \hat{g}_{tt} \dot{t}^2 + \hat{g}_{xx} \dot{x}^2 + \hat{g}_{\varphi \varphi} \dot{\varphi}^2 = -1 \,
 \label{orbitalpropx}
 \end{eqnarray}
which, for $x =$const. and replacing the metric (\ref{Qmetric}) within, turns into:
 \begin{eqnarray}
&& \hat{g}_{tt} \left( \frac{E}{f \, \phi \, \hat{g}_{tt}} \right)^2
 + \hat{g}_{\varphi \varphi} \left(  - \frac{\ell}{f \, \phi \, \hat{g}_{\varphi\varphi}}  \right)^2 = -1
 \quad \Longrightarrow \quad
    \frac{E^2}{f^2 \, \phi^2 \, \hat{g}_{tt}}
 +  \frac{\ell^2}{f^2 \, \phi^2 \, \hat{g}_{\varphi\varphi}}   = -1
 \nonumber \\
 &&  \Longrightarrow \quad
  \frac{E^2}{f^2 \, Q^{-2}(x) \, Q^2(x) \left( 1 - \frac{2 G M}{x} \right)  }
 +  \frac{\ell^2}{f^2 \,  Q^{-2}(x) \, Q^2(x)  x^2}   = -1
 \quad  \Longrightarrow \quad
  \frac{E^2}{f^2 \,  \left( 1 - \frac{2 G M}{x} \right)  }
 +  \frac{\ell^2}{f^2 \,   x^2}   = -1 \, ,
 \label{constraint0}
 \end{eqnarray}
which is independent on the rescaling $Q(x)$. Taking the derivative of (\ref{constraint0}) respect to radial coordinate $x$, we find:
\begin{eqnarray}
\frac{\ell^2}{E^2} = - \frac{g_{tt}^\prime}{g_{\varphi \varphi}^\prime } \, \frac{g_{\varphi \varphi}^2}{g_{tt}^2} \, .
\label{lex}
\end{eqnarray}
where we defined:
\begin{eqnarray}
g_{tt} = - \left( 1 - \frac{2 G M}{x} \right) \, , \quad g_{ \varphi \varphi} = x^2 \, .
\label{Sch}
\end{eqnarray}
The one above is not just a definition, but the Schwarzschild metric before to introduce the rescaling $Q(x)$.
Substitution of (\ref{lex}) in the velocity square (\ref{v2gene}) and making use of (\ref{Sch}) together imply:
\begin{eqnarray}
v^2(x) = \frac{g_{tt}^\prime}{g_{\varphi \varphi}^\prime }\frac{g_{\varphi \varphi}}{g_{tt}} = \frac{G M}{x - 2 GM} \approx \frac{GM}{x} \, ,
\label{v2x}
\end{eqnarray}
where in the last equality we assumed $x \gg 2 G M$.

The result just found for the velocity square may seem trivial and obvious, but it is actually rich in geometric meaning. Indeed, it is exactly the Newtonian's result in the radial coordinate $x$. However, we must remind that the larger value for $x$ is $2/\gamma^*$ and, therefore, the minimum asymptotic value for the velocity square is $GM\gamma^*/2$ in perfect agreement with (\ref{vinfinity}). This is clearly due solely to the singular structure of the conformal geometry in the unattainable asymptotic point $x = 2/\gamma^*$.

In order to complete the section we now express the velocity square in terms of the physical length $\ell_x$ that we set about calculating,
\begin{eqnarray}
\ell_x = \int \sqrt{g_{xx}} \, dx + {\rm const.} = \int dx \frac{  Q(x) }{ \sqrt{ 1 - \frac{2 G M}{x} } }+ {\rm const.} \approx
\int Q(x) \, dx + {\rm const.}
= - \frac{2}{\gamma^*} \log \left( 1 - \frac{\gamma^*  x}{2} \right)
\, ,
\label{lx}
\end{eqnarray}
where again we assumed $x \gg 2 G M$ and we fixed the integration constant imposing $\ell_x(x=0)  = 0$.
Notice that $\ell_x \rightarrow +\infty$ for $x\rightarrow 2/\gamma^*$.

It is straightforward to invert (\ref{lx}),
\begin{eqnarray}
x(\ell_x) =  \frac{2}{\gamma^*} \left(1-e^{- \frac{\ell_x \gamma^*}{2}}  \right) \, .
\end{eqnarray}
Replacing the above expression in the velocity square (\ref{v2x}) we find:
\begin{eqnarray}
v^2(\ell_x) = \frac{G M}{x(\ell_x)} = \frac{G M \gamma^*}{4}  \left[1 + \coth \left( \frac{\ell_x \gamma^*}{4} \right) \right]\, ,
\label{v2lx2}
\end{eqnarray}
which of course agrees with (\ref{v2lr2}), which is also expressed in terms of the physical distance. Notice that $\ell_x \equiv \ell_r$ because there is only one physical observable distance in nature.

\section{Newtonian effective theory and Gravitational Potential}
In order to derive the effective gravitational potential we start from the orbital velocity in terms of the physical distance. Indeed, in Newtonian physics we only deal with physical lengths and the Lagrangian simply reads:
\begin{eqnarray}
\mathcal{L}_{\rm N} = \frac{1}{2} m \left( \frac{ d \vec{r}}{d t} \right)^2 - m \Phi(|\vec{r}|)
= \frac{1}{2} m \left(\dot{\ell}_r^2 + \ell_r^2 \dot{\varphi}^2\right)  - m \Phi( \ell_r)
 \, ,
\end{eqnarray}
where $|\vec{r}| = \ell_r$, $m$ is the mass of a probe particle, and we assumed to be on the equatorial plane $\theta = \pi/2$.
From the Lagrangian above the EoM, assuming $\dot{\ell}_r = 0$, is:
\begin{eqnarray}
\ell_r  \dot{\varphi}^2 =
\frac{v^2(\ell_r)}{\ell_r} = \frac{ \partial \Phi(\ell_r)}{\partial \ell_r} = - E_{r}(\ell_r)
\quad \Longrightarrow \quad
v^2(\ell_r) =  - \ell_r \, E_{r}(\ell_r)
\, ,
\label{v2N}
\end{eqnarray}
where for future reference we also defined the gravitational field $\vec{E} = - \vec{\nabla} \Phi$.

Therefore, the effective potential can be obtained simply integrating (\ref{v2lr2}) or (\ref{v2lx2}),
\begin{eqnarray}
\Phi(\ell_r)= \int_{+ \infty}^{\ell_r } d \ell_r^\prime \frac{v^2(\ell_r^\prime)}{\ell_r^\prime} \, .
\label{intv23}
\end{eqnarray}
However, the velocity in (\ref{intv23}) can be very well approximated making use of (\ref{v2rlr}), and the integral
(\ref{intv23}) can be easily computed to give the following result,
\begin{eqnarray}
\Phi(\ell_r) \approx  - \frac{GM}{\ell_r} + \frac{GM\gamma^*}{2} \log ( \ell_r ) + {\rm const.} \, .
\end{eqnarray}

Now we have to consider the contribution of all the stars in a galaxy gravitationally acting on a probe star. This consists on integrating the potential in cylindrical coordinates after having introduced the following three vectors: $\vec{R}$, which points from the center of the galaxy to the probe star, $\vec{R}^\prime$ from the center of the galaxy to one of its stars, and $\vec{r}$ pointing from a star in the galaxy to the probe star. Therefore, we have: $\vec{r} = \vec{R} - \vec{R}^\prime$ and the contribution to the potential due to any star in the galaxy is:
\begin{eqnarray}
\Phi(| \vec{R} - \vec{R}^\prime |) = - \frac{GM}{| \vec{R} - \vec{R}^\prime |} + \frac{GM\gamma^*}{2} \log (| \vec{R} - \vec{R}^\prime | ) + {\rm const.}
 \equiv \Phi_0 + \Phi^{\rm log}
\, .
\label{singcon}
\end{eqnarray}
Notice that we replaced $\ell_r$ with $| \vec{R} - \vec{R}^\prime |$ because the Newtonian effective theory is defined in flat spacetime.

Let us now consider a thin disk galaxy model with exponential  distribution of matter that decays at large distance.  We assume that the mass of each star is $M_{\odot}$(solar mass) and the distribution of stars is described as follows (in cylindrical coordinates: $R, \varphi, z$):
\begin{eqnarray}\label{rho}
\rho(R^\prime, z^\prime) = \Sigma_0 \, e^{ - \frac{R^\prime}{R_0}} \, \delta(z^\prime)
\, , \quad [\rho] = L^{-3} \, ,
\end{eqnarray}
where $z$ is the coordinate orthogonal to the galaxy plane. Moreover, $R_0$ is the radius of the galaxy and $\Sigma_0$ is related to the number of stars of mass comparable to the solar mass in the galaxy, i.e.
\begin{eqnarray}
N^* & = & \int_0^{+ \infty} d R^\prime \, R^ \prime  \int_0^{2 \pi} d \varphi^\prime \int_{- \infty}^{+ \infty} d z^\prime
\, \rho(R^\prime, z^\prime) \nonumber \\
&=&
\int_0^{+ \infty} d R^\prime \, R^ \prime \int_0^{2 \pi} d \varphi^\prime \int_{- \infty}^{+ \infty} d z^\prime
\Sigma_0 \, e^{ - \frac{R^\prime}{R_0}} \, \delta(z^\prime) = 2 \pi \Sigma_0 R_0^2 \quad \Longrightarrow \quad
\Sigma_0 = \frac{ N^*}{2\pi R_0^2} \,.
\label{sigma0}
\end{eqnarray}
In order to get the total contribution to the gravitational potential we have to integrate over all the star in the galaxy each of them of solar mass $M_{\odot}$, namely:
\begin{eqnarray}
&& \hspace{-0.5cm}
\Phi_{\rm T}(R, z) = \int_0^{+ \infty} d R^\prime \, R^ \prime \int_0^{2 \pi} d \varphi^\prime \int_{- \infty}^{+ \infty} d z^\prime
\, \rho(R^\prime, z^\prime)
\, \Phi(R, R^\prime, z, z^\prime) \, ,\\
&& \hspace{-0.5cm}
\Phi(R, R^\prime, z, z^\prime) = - \frac{ G M_\odot}{\left[ R^2 + R^{\prime 2} - 2 R \, R^\prime \cos \varphi^\prime + (z - z^\prime)^2 \right]^{\frac{1}{2}} }
+ \frac{ G M_\odot \gamma^*}{2} \log \left\{ \! \frac{
\left[ R^2 + R^{\prime 2} - 2 R \, R^\prime \cos \varphi^\prime + (z - z^\prime)^2 \right]^{\frac{1}{2}}}{\ell}
\! \right\}, \nonumber
\end{eqnarray}
where $R$ is the distance of the probe star from the galactic center in cylindrical coordinates and $R_0$ is the characteristic scale of the galaxy. Since $\Phi(R,R',z,z')$ consists of two parts, we will integrate the two contributions of the potential separately obtaining the two corresponding contributions to the velocity square. Finally, $\ell$ is the scale coming from the integration constant that can not be zero since the potential grows with the distance. However, we do not have to worry about such scale because it will disappear in the orbital velocity that is related to the force and not the potential.

For the Newtonian potential contribution to (\ref{singcon}), namely $\Phi_0=-GM/|\vec{R}-\vec{R'}|$, and assuming the density profile (\ref{rho}), the rotation velocity square of a probe star was computed in \cite{Mannheim:2010xw} and the result is:
\begin{equation}
	v^2_0=\frac{GN^*M_{\odot}R^2}{2R_0^3} \left[ I_0 \left( \frac{R}{2R_0} \right)K_0 \left( \frac{R}{2R_0} \right) -I_1\left( \frac{R}{2R_0} \right) K_1 \left( \frac{R}{2R_0} \right) \right] \, ,
	\label{v2M}
\end{equation}
where $I_0$, $I_1$ are the modified Bessel functions of first kind and $K_0, K_1$ are the modified Bessel functions of second kind. In (\ref{v2M}) $M = N^*M_{\odot}$ is the mass of the all stars in the galaxy.

In order to compute the logarithmic contribution to the potential in (\ref{singcon}), namely
\begin{eqnarray}
\Phi^{\log} = \frac{GM\gamma^*}{2} \log \frac{|\vec{R}-\vec{R'}|}{\ell} \, ,
\label{Phi0}
\end{eqnarray}
we can use the Gaussian theorem
\begin{eqnarray}
{\rm div} \vec{E} = - 4 \pi G \rho \, ,
\label{Gauss}
\end{eqnarray}
to sum over all the stars in the galactic disc.
Notice that we can assume the sources of the logarithmic potential to be wires because of the cylindrical symmetry of the galaxy.

Due to the above logarithmic correction (\ref{Phi0}), the gravitational field in cylindrical coordinates (we here fix the origin in $\vec{R}' = 0$) is attractive and reads:
\begin{eqnarray}
E_R = - \frac{\partial \Phi^{\log} (R)}{\partial R} = - \frac{GM\gamma^*}{2 R} .
\label{Eg}
\end{eqnarray}
Integrating (\ref{Gauss}) on a three-dimensional volume $V$ with boundary $\partial V$ in Cylindrical coordinates, we can infer about the energy density $\rho_{\rm s}( \vec{x} ) = \rho_0 \delta(x) \delta(y)$ of a single wire-like source, namely
\begin{eqnarray}
&&\int_V {\rm div} \vec{E} \, dv = {\rm Flux}\Big|_{\partial V} (E) = - 4 \pi G \int_V dv \rho_{\rm s}( \vec{x} ) \, , \nonumber \\
&&
2\pi R\Delta z\,   E_R  = - 4 \pi  G \rho_0 \, \Delta z \, .
\label{IntGauss}
\end{eqnarray}
Replacing (\ref{Eg}) in (\ref{IntGauss}) we finally find $\rho_0$,
\begin{eqnarray}
 \rho_0 = - \frac{R \, E_R}{2} = \frac{M \gamma^*}{4},
\end{eqnarray}
and the potential can be recast in the following form in terms of the energy density,
\begin{eqnarray}
\Phi^{\log} = 2 G \rho_0 \, \log \frac{|\vec{R}-\vec{R'}|}{\ell} \, .
\label{Phi0rho}
\end{eqnarray}

If the gravitational sources and the probe star are all located in the same plane (we here assume the galactic disk to be in $z=0$ in cylindrical coordinates), then $\Phi^{\log}$ is analogous to the Newtonian potential of $N^*$ massive infinite wires each with uniform density $\rho_0$ and generating a logarithmic gravitational potential.

Assuming the principle of linearity of the gravitational forse and then of the gravitational potential, we can now apply again the Gauss' theorem to all the stars in the galaxy that are descried by the energy density profile in cylindrical coordinates:
\begin{eqnarray}
\rho_{\rm s}( \vec{x} ) = \rho_0 \delta(x) \delta(y) \qquad \longrightarrow \qquad
\rho_{N^*}(R) = \rho_0 \, \Sigma_0 \, e^{- \frac{R}{R_0}}  =
\frac{M_{\odot}\gamma^*}{4} \Sigma_0 \, e^{- \frac{R}{R_0}}  \, , \quad [\rho_{N^*}] = M L^{-3} \, ,
\label{rhow}
\end{eqnarray}
where we assumed any star to have mass $M_{\odot}$.
Notice that (\ref{rhow}) is an energy density while (\ref{rho}) is a density distribution.

Finally, the Gaussian theorem making use of the above energy density (\ref{rhow}) gives:
\begin{eqnarray}
- 2 \pi R\Delta z \, E^{\rm T}_R(R)  = 4\pi G\int_0^R\int_0^{2\pi}  R' dR' d\varphi  \, \rho_{N^*}(R')\int_0^{\Delta z}dz\quad \Longrightarrow \quad
E_R^{\rm T} (R) = - \frac{4\pi G}{R}\int_0^R \rho_{N^*}(R') R' dR' \, .
\label{TotV}
\end{eqnarray}
Using (\ref{v2N}) and upon integration of (\ref{TotV}), the contribution to the rotation velocity square (\ref{v2N}) due to the logarithmic term in the potential reads,
\begin{eqnarray}
v^{2}_{\rm log} & = & - E_R^{\rm T} (R) R = \pi GM_{\odot}\gamma^*\Sigma_0\int_0^Re^{- \frac{R^\prime}{R_0} } R^\prime dR^\prime =\frac{GM_{\odot}\gamma^*}{2} \, 2\pi\Sigma_0R_0^2 \, \left[ 1- \left( 1+\frac{R}{R_0} \right)e^{- \frac{R}{R_0} } \right]\nonumber \\
&=&\frac{GN^*M_{\odot}\gamma^*}{2} \left[ 1- \left( 1+\frac{R}{R_0} \right)e^{- \frac{R}{R_0} } \right].
\label{Dv2}
\end{eqnarray}
Finally, taking the sum of (\ref{v2M}) and (\ref{Dv2}) the total contribution to the velocity square reads:
\begin{eqnarray}
\hspace{-0.4cm}
\label{vtot}
\boxed{v^2(R) =
\frac{GN^*M_{\odot}R^2}{2R_0^3} \left[ I_0 \left( \frac{R}{2R_0} \right)K_0 \left( \frac{R}{2R_0} \right) -I_1\left( \frac{R}{2R_0} \right) K_1 \left( \frac{R}{2R_0} \right) \right]
+\frac{GN^*M_{\odot}\gamma^*}{2} \left[ 1- \left( 1+\frac{R}{R_0} \right)e^{- \frac{R}{R_0}} \right] }
\end{eqnarray}
which is constant for large $R$, namely
\begin{eqnarray}
v^2(R) \,\, \rightarrow \,\, \frac{G N^*M_{\odot} \gamma^*}{2}   \quad {\rm for} \quad R \rightarrow + \infty.
\end{eqnarray}

\section{The Tully-Fisher relation}

As we have said several times, in conformal gravity we are free to rescale the metric by an overall factor that will depend on at least one undetermined length scale. In our model the length scale is $\ell_c = 2/\gamma^*$, which turns out to be of the same order of magnitude of the galaxy (see next section).
However, if we focus our attention on a single star in the galaxy we can with equal naturalness fix $\ell_c$ to be comparable with either the Schwarzschild radius of the star or the galaxy extension.
Indeed, these two are the characteristic scales of the system. On the other hand if we were dealing with a single star in an empty Universe, it would be natural to select $\ell_c$ proportional to the Schwarzschild radius of the star.
Therefore, conceptually there is nothing wrong in selecting the free scale to be proportional to the galaxy extension, and actually it seems the natural choice whether we are interested to the global properties of the galaxies.
Furthermore, in conformal gravity we have an extra scalar field, the dilaton, that does not propagate (the perturbation can always be fixed to zero by the mean of conformal symmetry), but satisfies its on equation of motion whose solutions show up extra scales simply because of dimensional reasons and in accordance with the Mach's mechanical view of the Universe. In other words, the dilaton is responsible for the gravitational interaction from small to large distances through the presence of pole-like singularities, which are weighted by dimensional parameters, in the solution of its equation of motion.

The arguments above have an observational counterpart in the Tully-Fisher relation that relates the asymptotic velocity of a probe star to the Newton's constant, the mass of the galaxy, and the Milgrom's parameter $a_0$, namely
\begin{eqnarray}
v^4 =  a_0 G M \, , \quad [ a_0 ] = L \, T^{-2} \, , \quad [ G ] = L^3 \, M^{-1} \,T^{-2} \, ,
\label{v4}
\end{eqnarray}
where $M = N^*M_{\odot} + M_{\rm HI}$, $M_{\rm HI}$ is the mass of the Helium gas (see next section for more details).
Comparing the letter expression (\ref{v4}) with (\ref{vinfinity}) we finally get:
\begin{eqnarray}
\gamma^* = \sqrt{ \frac{4 a_0}{G M }} \, , \quad [ \gamma^*] = L^{-1} \, ,
\label{gammavalue}
\end{eqnarray}
which depends on the mass of the galaxy whether we assume $a_0$ to be a universal constant.

For the value of $a_0$ obtained by fitting the galactic rotation curves with the MOND theory \cite{Milgrom:1983zz}, namely $a_0 = 1.2 \times 10^{-10} {\rm m} \, {\rm s}^{-2}$, and for a galaxy made of $10^{12}$ solar mass stars we get:
\begin{eqnarray}
 \gamma^* \approx 10^{-21} {\rm m}^{-1} \, \quad \Longrightarrow \quad \ell_c \approx 10^{21} {\rm m} \, .
\end{eqnarray}

In conformal gravity $\gamma^*$ is one of the two free parameters to be obtained by fitting the observational data and assuming dependence on the mass of the entire galaxy like in (\ref{gammavalue}).

In the next section we will get a universal vale for $a_0$ from our model fitting $175$ galaxies.

\section{ Fitting of the galactic rotation curves and universality}

In order to completely specify the velocity square (\ref{vtot}),
we need: $N^*$ (the number of stars in the galaxy), $R_0$ (the effective scale of the galactic disk), and the free scale in our model, namely $\gamma^*$.
Moreover,
we have to consider the contribution to the velocity due to the gas Helium (HI). If we apply to the HI the disk model with exponential profile,
the contribution of HI to $v^2$ will be described by the same formula (\ref{vtot}). Therefore, the total $v^2$ reads:
\begin{eqnarray}
v^2_{\rm tot}=v^2(N^*, R_0, \gamma^*) + v^2(N_{\rm HI}, R_{\rm H0}, \gamma^*),
\end{eqnarray}
where $N_{\rm HI}= M_{\rm HI}/M_{\odot}$ represents the fraction of total mass of the HI gas respect to the solar mass and $R_{\rm H0}$ is the effective radius of the HI gas' cloud.

In our analysis we used the data from the SPARC database \cite{Lelli:2016zqa} that includes: the rotation-curves data, which the reader can find in the plots in Appendix \ref{AC}, the total luminosity ratio $L/L_{\odot}$, and the disk radius $R_0({\rm kpc})$ for 175 galaxies (see Appendix \ref{AB}). The database includes also $M_{\rm HI}$, while $R_{\rm H0}$ will be determined shortly.
Of course, the mass $M_{\odot}$ and the luminosity $L_{\odot}$ of the sun, and the luminosity of all the galaxies $L$ are known observed quantities. All these parameters are given in Appendix \ref{AB}.

The number of stars $N^*$ is related to the mass to luminosity ratio $M/L$, which is our second fitting parameter, the ration $M_{\odot}/L_{\odot}$, and the ration $L/L_{\odot}$, namely
\begin{eqnarray}
N^* = \frac{M}{M_\odot} = \frac{\frac{M}{L}}{\frac{M_\odot}{L_\odot}} \, \frac{L}{L_\odot} \, ,
\end{eqnarray}
in which $M_\odot/L_\odot$, and $L/L_\odot$ are known and given in the table in Appendix \ref{AB}.
Therefore, fitting $M/L$ is equivalent to the fitting of $N^*$.
Since we assume that there is no dark matter, the fitting results of $M/L$ should be close to $1$ rather than over $10$ like in Newtonian dynamics.

In the database \cite{Lelli:2016zqa} we can also find the mass $M_{\rm HI}$. However, in order to also include the amount of primordial Helium, we have to multiply HI times the factor $1.4$. Therefore, the total amount of Helium is:
\begin{eqnarray}
M^{\rm TOT}_{\rm HI} = 1.4 \, M_{\rm HI} \, .
\end{eqnarray}

In the SPARC database \cite{Lelli:2016zqa} one can find the the radius $R_{\rm H}$ defined to be one for which the density of HI is equal to the value $M_{\odot}/{\rm pc}^2$.
Therefore, we can infer about the effective radius $R_{\rm H0}$ of the Helium gas using the exponential density profile (\ref{rho}) and (\ref{sigma0}),
\begin{eqnarray}\label{rh0}
\Sigma_{\rm H0}e^{-R_{\rm H}/R_{\rm H0}}=\frac{N_{\rm HI}}{2\pi R^2_{\rm H0}}e^{-R_{\rm H}/R_{\rm H0}}=\frac{1}{{\rm pc}^2}
\quad \Longrightarrow \quad R_{\rm H0} \, ,
\end{eqnarray}
where the parameters $N_{\rm HI}$, which can be identified with the dimensionless quantity $M_{\rm HI}$, is available in the Appendix \ref{AB}.
However, equation (\ref{rh0}) is ambiguous because it usually has two solutions. Moreover, for some galaxies, equation (\ref{rh0}) has no solutions, which implies that for these galaxies the measurements of $N_{\rm HI}$ and $R_{\rm H}$ are not accurate enough or the distribution of HI does not fit the disk model properly. Therefore, we choose $R_{\rm H0}=4 R_0$ as an effective radius of the HI disk consistently with other papers in literature \cite{Mannheim:2010xw,Li:2019ksm}.

The results for the fitting parameters $M/L$ and $\gamma^*$ are given in the Appendix \ref{AB},
while the fitting of the rotation curves are displayed in the Appendix \ref{AC}.

The fitting results show that our model fits the rotation velocity data for most of the typical spiral galaxies (including S0, Sa, Sb, Sc, Sab, Sbc, and Scd type) and it fits very well some late spiral type galaxies (Sd, Sdm, and Sm). In particular for the velocity data at large scale ($R>2R_0$).

 As we expected, the fitting results for the mass to luminosity ratio (of luminous mass) are close to $1$. Moreover, in the plots in Appendix \ref{AC}, we can see that the Newtonian contribution dominates the rotation velocity at small scale ($R\lesssim 2R_0$), while the conformally modified geometry determines the the value of the velocity square asymptotically.  Our model (\ref{vtot}) interpolates between the two regimes.

However, there are some galaxies to which our model cannot fit very well.

This is the case of the galaxies NGC3949, NGC3953, and NGC4051. However, for such galaxies we have only few data and in particular we lack of data points at large radius. In this case the fitting results for $\gamma^*$ is actually 0.

For some spiral galaxies, e.g. NGC2955, NGC5005, NGC6195, UGC2916, UGC3546, UGC5253, and UGC11914,  the rotation velocity data tend to be flat at very small scale ($R\ll 2R_0$). Therefore, we think that the rotation curves cannot be consistent with the exponential profile for the matter density adopted.

For the irregular galaxies, Im (irregular Magellanic), BCD (irregular blue compact dwarf), and weak spiral types (Sm, Sd, and, Sdm), for instance: CamB, DDO161, F574-2, NGC2366, NGC3741, NGC4068, PGC51017, UGC2455, UGC4483, some fits are bad and usually the fitting results of the mass to luminosity ratio are anomalously small. However, this should be related to the irregular mass distribution of these galaxies that affects the irregular motion of matters.

Finally, having at our disposal the fittings values for $\gamma^*$ and $M/L$ ($L$ is an obseved quantity) we can now extract the {\em universal} parameter $a_0$ using the Tully-Fisher relation (\ref{gammavalue}).
The total mass in (\ref{gammavalue}) consists of the two contributions, stars and Helium, namely
\begin{eqnarray}
M=L\cdot \left(\frac{ M }{L} \right) +1.4 \, M_{\rm HI} \, .
\end{eqnarray}
Let us consider the following generalization of equation (\ref{gammavalue}), namely
\begin{eqnarray}
\gamma^* = \left( \frac{4 a_0}{G M } \right)^k \, ,
\label{gammavalue2}
\end{eqnarray}
where the constant $k$ has to be determined by means of the fitting. Hence, taking the ``log'' of both sides we get:
\begin{eqnarray}
\log \gamma^* = k \left( \log 4 a_0 -  \log GM \right) \, ,
\end{eqnarray}
in which the fitting parameters are $a_0$ and $k$.
The fitting results are shown in Fig.\ref{tf} (notice that we removed the seven points for which $\gamma^*=0$),
\begin{figure}[ht]
	\centering
	\includegraphics[width=0.5\linewidth]{a0.png}
	\caption{This plot shows the fitting of the relation between $\gamma^*$ and $M$. The fitting function is $y=k(b-x)$, where $y=\log((\gamma^*)\cdot {\rm kpc})$, $x=\log(GM/10^{-10}{\rm ms}^{-2}{\rm kpc}^2)$ and $b=\log(4 a_0/10^{-10}{\rm m s}^{-2})$. The results for the two fitting parameters are: $k=0.582$ and $b=0.573$.}
	\label{tf}
\end{figure}
where
\begin{eqnarray}
\gamma^*\propto M^{-0.582} \, ,  \quad  \boxed{a_0= 0.935 \times10^{-10} {\rm m/s}^2= 9.35 \times10^{-11} {\rm m/s}^2 } \, .
\end{eqnarray}
The $3\sigma$ confidence intervals of $k$ and $a_0$ are: $0.582 \pm 0.057$ and $(9.35 \pm 2.22)\times10^{-11}{\rm m/s}^2$ respectively. Notice that according to (\ref{gammavalue}) $k$ is compatible with $1/ 2$.

\section*{Conclusions}
We provided a geometrical mechanism capable to overcome the long standing issue of the galactic rotation curves without need of any kind of exotic dark matter. We are aware that dark matter is a proposal to rid out of multiple issues in cosmology and astrophysics while there is no need if it in the colliders' physics, but we found extremely interesting the outcome of this project from both the theoretical and observational sides. From the theoretical point of view the simple scalar-tensor Einstein's theory of gravity provides a kind of non-modified gravitational theory ghost-free and free of other instabilities. Indeed, the presence of the dilaton field on one side allows for other vacua without introducing other propagating degrees of freedom, on the other side introduces unattainable spacetime singularities that drastically modify the asymptotic spacetime structure from the micro to the macro.

Specifically, the effective Newtonian gravitational force, to which the stars of the galaxy are subject, is obtained starting from a ``unique'' (the metric depends only on one extra scale $\ell_{\rm c} = 2/\gamma^*$, see section (\ref{Unique})) spacetime geometry (\ref{Qmetric}) or (\ref{Qmetricr}) (in two different coordinate systems) for a single star and summing over all the stars in the galaxy.
The effective potential have the expected asymptotic logarithmic behaviour characteristic of the minimal confinement, and the velocity turns out to be constant (see formulas (\ref{v2r}) or (\ref{v2rlr}) and (\ref{vinfinity})) at large distance from the galactic center in agreement with the Tully-Fisher relation.

In force of the effective gravitational potential with logarithmic asymptotic behaviour we derived for a single source, we integrated on all the stars of the galaxy with exponential density profile to end up with the total potential. Hence, we obtained the orbital velocity of a probe star in the gravitational field of all the other stars in the galaxy (see (\ref{vtot})).
Afterwards, we tested the theory with $175$ galaxies making a fit of the parameters: $\gamma^*$ and the mass over the luminosity ratio. The outcome of the fits is given in the appendix (\ref{AC}).
One can notice that the fitting results for the ratio $M/L$ turned out to be close to $1$ consistently with the absence of dark matter.

Finally, using the observational Tully-Fisher relation we got the value for the universal parameter $a_0 = (9.35 \pm 2.22) \times 10^{-11} \, {\rm m/s^2}$.

\acknowledgments
This work was supported by the Basic Research Program of the Science, Technology, and Innovation Commission of Shenzhen Municipality (grant no. JCYJ20180302174206969).

redpajama_set_name: RedPajamaArXiv


Shop our "Fine N Dandy" Boot by Miss Macie! This bootie is perfect for the Western Fashionista in ALL of us! So cute & essential for Spring & Summer. Pairs wonderfully with denim shorts & graphic tees. True to size.


HelloWorld code is an initial android program, created with minimal effort, just use command from androidactivitycreator.py . This program just show a greeting, hello world. But you, the reader, can study from it, and can make it as a base from your next exciting program.
Layout code is just a simple modification to HelloWorld. In this code, we just add some UI Component, so in the feature, you, the reader will have no difficulties to create other layout.
Was this information helpfull ?


In the prior post, we asked a rather unpleasant question: where a government reasonably believes that hosting the Olympic Games would cause losses to the public fisc, but nonetheless submits a bid, has that government acted corruptly?
The Securities and Exchange Commission Wednesday charged global resources company BHP Billiton with violating the Foreign Corrupt Practices Act when it sponsored foreign government officials as guests at the 2008 Summer Olympics in Beijing.
BHP Billiton is in discussions to resolve an investigation of potential Foreign Corrupt Practices Act violations that relate to the company's sponsorship of the 2008 Beijing Olympics.


text:
Q: react-native array.map with asyncstorage.getItem I've been trying to retrive data from asyncstorage using react native.
Here is what i've done so far:
import React, { Component } from 'react';

import {
  Text,
} from 'react-native';

class GetData extends Component {
  constructor() {
    super()
    this.state = {
      dataLocalIds: [
        "data1Data",
        "data2Data",
        "data3Data",
        "data4Data",
        "data5Data",
        "data6Data"
      ],
    }

    this.state.dataLocalIds.map((value, index) => {
      this.data = this.getDatas(value, index);
    })
  }

  async getDatas(value, index) {
    try {
      const value = await AsyncStorage.getItem(value).then(val => {
        return JSON.parse(val)
      });
      return value
    } catch (err) {
      throw err
    }
  }

  renderScreen = () => {
      return (
        <Text> Hello World </Text>
      );
  }

  render() {
    return (
      this.renderScreen()
    );
  }
}

export default GetData;

Problem is: Apps crash with error, " the bind value at index 1 is null ".
Stacktrace does not point to lines inside my code. Instead, it points to sqlite and asyncstorage.
I really have no idea how to solve this issue. Help Would be appreciated.

A: Hi MacFlyer have you tried using multiget (https://facebook.github.io/react-native/docs/asyncstorage.html#multiget)?

A: Probably its causing due to your getData() method's argument name is value and also you are declaring a new variable const value inside the try block.
when value is passed to AsyncStorage it searches for reference to immediate scope (try block) if finds a variable is declared(JS parsing and tokenizing phase) but its value is undefined. so undefined is passed to the AsyncStorage.getItem method.
Try renaming one of those.

A: 1) this.myFunction = this.myFunction.bind(this) in the constructor could solve i think.
You have another problem.
Getdatas function is async and returns promise
2)
const promises = this.state.dataLocalIds.map((value, index) => this.getDatas(value, index));
const [data1, data2, ] = await Promise.all(promises)

3) the variable named value is already declared in geDatas scope
Try to fix and let us know

A: try something like:
constructor(){
    super();

    this.state={...}
    this.getDataFromAsyncStorage()
}
async getDataFromAsyncStorage() {
    let dataArray = [];
    for (let value of this.state.localIds){
        let data = await AsyncStorage.getItem(value);
        dataArray.push(data)
    }
    this.data = dataArray;
}

redpajama_set_name: RedPajamaStackExchange


text:
Master P Hosts Pop-Up Show At Savoy Nightclub In L.A. [PHOTOS]
Written By D.L. Chandler
HipHopWired Featured Video
Master P fans in California got quite the treat this weekend when the New Orleans Hip-Hop veteran thrilled them all with a pop-up performance in Los Angeles. The artist and mogul born Percy Miller rocked the stage at the Savoy nightclub, with the No Limit colonel running through a few of his biggest hits.
Unbeknownst to the partygoers at Savoy, the No Limit tank was revved up and ready to go behind the scenes with P jumping out the gate and surprising fans. Opening with the energetic "Make 'Em Say Ugh," the Savoy was officially all in from that point on.
P also rocked "Miss My Homies" and "Down For N*ggas" among other tracks, and he's also released a fresh single titled "Middle Finger" and sounds comfortable in the modern trap rap setting.
Check out a few clips from Master P's pop-up show at the Savoy in Los Angeles below. Hit the following pages to see images from the performance.
Photo: Kreative Kollective
los angeles pop up shop
1 2 3 4 5 6 7 8 9 10 11Next page »

redpajama_set_name: RedPajamaCommonCrawl


text:
Een  gebakkraam (ook oliebollenkraam of oudhollandse gebakkraam) is een kraam waar gefrituurde deeggerechten worden bereid en verkocht zoals olie- en krentenbollen, Berlinerbollen, appelbeignets en wafels.

De kramen zijn in Nederland te vinden op kermissen en staan gedurende de maanden rond de jaarwisseling (oktober t/m januari) in winkelstraten en op pleinen. In 2015 kocht ongeveer 50% van de huishoudens bij een winkel of kraam oliebollen, waar dat percentage in 2007 op 44 lag.

Oliebollentest
Bij de nationale AD-Oliebollentest werden tot 2017 in december oliebollen van gebakkramen en bakkerijen in Nederland getest op kwaliteit. Sinds 2018 wordt deze test gedaan door een groep van 12 vrijwilligers. De uitslag van de test is sindsdien te vinden op hun site.

Zie ook
 Poffertjeskraam

Horeca
Kermis
Oud en nieuw
Gebak

redpajama_set_name: RedPajamaWikipedia


text:
Can-Am unveils 2022 line-up of ATV and side-by-side vehicles
24th August 2021 22nd August 2021 webmaster
Can-Am is introducing two new Rotax engines to its lineup – the HD7/700 ACE and the HD9, replacing the HD5, HD8, and 800
As the off-road industry continues to grow, riders of all skill levels crave the best machine for their version of Off-Road Livin'. And Can-Am is once again delivering with its 2022 model year lineup of ATVs and side-by-side vehicles – from entry level to top performance. With more horsepower, better traction, enhanced comfort, and so much more.
For 2022, Can-Am will be the first side-by-side manufacturer to offer a 200hp engine coupled with an all-new pDrive roller clutch system, the entire dynamic changes through quicker acceleration and instant-on power at all time
And while the Can-Am Maverick delivers the best power-to-weight ratio in its class, handling remains critical to all-around performance. That is why BRP engineers worked tirelessly to ensure the lineup handles terrain with ease, thanks to Smart-Shox adaptive suspension, new 30 and 32-inch Maxxis Carnivore tires, a stronger frame, and higher ground clearance and stance.
"The 2022 Can-Am Maverick is the pinnacle of performance side-by-side vehicles," said Hailie Deegan, professional racer and Can-Am brand ambassador. "And the best part of working with Can-Am, is they keep making it better and better. There's no other choice if you're in the market for the best, and if you enjoy watching others try to keep up in the rearview."
The Rotax HD7 is set to become the new ruler of the lower-HP class. It will be available in the Can-Am Traxter, where it replaces the HD5 with 37% more power, and with a sportier calibration, it will power the Maverick Trail 700. The new engine is optimized for off-road usage, delivering an improved experience with less noise, less vibration, and a cooler environment for passengers resulting in a more comfortable ride.
And the Rotax HD9 option replaces the HD8 in the Can-Am Traxter lineup. It is a larger, more powerful engine that delivers 30%1 more horsepower than its predecessor. It promises the same dependability, with even more workability by way of class-leading power and low-end torque.
"With the addition of the HD7 and HD9 engine options to the already existing HD10, Can-Am is class leading in power and by far the best option to accomplish the task at hand, whether that's work or play," said Christian St-Onge Director Global Product Strategy Can-Am Off-Road Vehicles. "Regardless of your vehicle choice, this lineup is built to perform."
On top of the engine upgrades, Can-Am is fine-tuning its Traxter line-up with the expansion of the ABS on some HD9 and HD10 models. Those models benefit therefore from the innovative Brake and Traction Control System that offers maximum controllability in the toughest riding conditions. Those models fitted with ABS are no more limited to 60km/h.
On the ATV side, Can-Am is expanding its Visco-4Lok system. It works with the push of a button, where riders can engage a 4-wheel-drive lock mode, which instantly provides equal power to all four wheels for great traction. It will now be available on all 2022 Outlander and Renegade X mr packages, as well as the Outlander XU (650-1000), Outlander XXC, Outlander XT-P, and Outlander MAX Limited.
← HSE calls on farmers to make safety a priority after a series of serious incidents
New Holland dealer Russells celebrates centenary →
Make your soil pay with Cereals
14th April 2021 13th April 2021 webmaster
Cheffins to host timed online auction of machinery on behalf of Agrifac UK Ltd
Kubota announces further market expansion with Lister Wilder
1st November 2020 30th November 2020 webmaster


The Puerto Rico Treasury Department (the "Department") has issued Administrative Determination Number 17-07 ("AD 17-07") to provide guidance to merchants following recent amendments to Sections 4042.03 and 4042.04 of the Puerto Rico Internal Revenue Code of 2011 (the "Code").
Specifically, section 4042.03(a)(2) of the Code, as amended by Act 46 of 2017, establishes the requirement of paying SUT to the Department in two semimonthly payments during the month in which the taxable and payable event occurs. The Department has determined that the semimonthly deposit shall only be applicable to the SUT to be deposited with the Treasury. For now, the portion of SUT payable to the municipalities will remain payable with the SUT monthly return filing on or before the twentieth (20th) day of the following month in which the taxable transaction takes place.
"Large Taxpayers" as defined in Section 1010.01(a)(35) of the Code.
Merchants whose monthly volume of SUT deposited during the prior natural year exceed two thousand dollars ($2,000) a month (the "$2,000 Criteria").
The Department has determined that merchants that sell tangible personal property or taxable services subject to the 10.5% SUT in excess of $228,600 during the prior taxable year shall fall within the $2,000 Criteria and shall be subject to the semimonthly SUT remittal. For merchants whose operation began during the prior taxable year, the $2,000 Criteria shall be met if the total monthly sales for the months in operation average more than $19,050 (that is, $228,600 divided by 12 months).
The Department has further determined that for merchants dedicated exclusively to the sale of services between merchants ("B2B") or designated professional services subject to the 4% SUT, such merchants with sales in excess of $600,000 during the prior taxable year shall fall within the $2,000 Criteria and shall be subject to the semimonthly SUT remittal. For merchants whose operation began during the prior taxable year, the $2,000 Criteria shall be met if the total monthly sales for the months in operation average more than $50,000 (that is, $600,000 divided by 12 months).
The determination of whether a certain taxpayer is subject to semimonthly remittals of SUT shall be made once a year based on the information of total taxable sales realized during the prior taxable year. Once the determination has been made, the merchant shall be bound to carry on with the semimonthly payments for the entirety of the natural year. For purposes of determining the obligation to make deposits of SUT in semimonthly payments, the information reflected on the Monthly SUT Returns, Form SC 2915 filed via the Sistema Unificado de Rentas Internas ("SURI"), shall be used.
Finally, merchants subject to both the 10.5% and 4% SUT rates shall be obliged to make semimonthly payments if prior natural year sales exceeded $228,600.
1st Semimonthly Payment - 15th of each month.
2nd Semimonthly Payment - last day of the month.
Credit for taxes paid by a reseller in the purchase or import of inventory for resale.
Overpayment of prior period to be credited to the current tax liability (collectively, the "Applicable Credits").
Additionally, merchants subject to the semimonthly SUT payments must remit as 1st payment the lesser of: (1) eighty percent (80%) of the collected SUT over sales realized during days 1 through 14 of each month, net of Applicable Credits; or (2) half of seventy percent (70%) of SUT determined, net of Applicable Credits, according to the Monthly SUT Return corresponding to the same month of the prior year.
A ten percent (10%) penalty shall be imposed over the failed semimonthly tax deposit amount.
VII. Merchants currently under the Taxpayer Rehabilitation Program.
Notwithstanding the new provisions of Section 4042.03 of the Code regarding semimonthly SUT deposits, taxpayers under the Taxpayer Rehabilitation Program established under the Internal Revenue Circular Letter 17-05 shall continue to remit their SUT in weekly payments, as established in their payment agreement. However, such taxpayers must also comply with the 80/70 Rule and failure to comply will result in the penalty established in the foregoing section.
The Department has determined that in order for merchants to accommodate to the new amendments, the new semimonthly SUT payment obligation shall commence in: (1) August 2017 for "Large Taxpayers" and (2) September 2017 for those meeting the $2,000 Criteria.
The provisions of AD 17-07 were effective immediately.


Cyber security is a people problem more than a technology problem. A single compromised employee, whether malicious or negligent, can render the best systems worthless.
Although the media loves a shady computer hacker story — usually adorned with images of a man in a hoodie sitting in a dark room — the reality is that rogue employees can often do much more damage to an organisation's data integrity.
The insider threat can come from 'malicious' actors, whether that's an Edward Snowden, who leaks information for normative reasons, or someone who sells off personal information. But commonly it's just regular employees engaging in risky behaviour without realising.
"What is the biggest threat to the security of your business? Well in fact it is human apathy and ignorance," Latrobe University associate professor Sara Smyth told last week's ConnectExpo tech conference.
Clicking fake links in emails is another soft spot, as the Democrats found out in the United States election. Phishing is such a problem for Australia Post, where many employees don't seem to care about the risks, that its IT department recently took to testing whether they should scare reckless staff by making it appear their computer had been locked with ransomware.
While she does not advocate throwing the technology out the window — it's obviously necessary — it needs "to be combined with policies that promote security awareness alongside good behaviours that protect the overall security of the organisation".
There's even new software that can identify anomalous behaviour among employees — it looks weird if Jane from HR is accessing a certain file at 3am, for example — but what if a rogue employee is acting within the scope of their authorised activities?
Well-meaning staff who create cyber security risks through negligence are the most insidious, she says.
"That can result from a lack of understanding about internal security policies, or the absence of policies altogether, or a lack of communication of those policies to staff by management.
The good news is that negligence can at least be reduced through education and communication.
Prevention is the best way to mitigate against cyber threats — figure out what your risks are and put a plan in place.
Organisations need to question whether, in the event of malicious or inadvertent damage, from malware or hacking or some other disruption that takes their systems offline, their data is secure, she notes. Is the information only available to those who have the authority to access it? Who are the threat actors who could target your organisation? Are they insiders or outsiders, or both?
Has your agency been targeted in the past? Is there corporate memory of past incidents, and if not, can you learn from others who have been targeted?
Organisations need to work out what the risks are, but how those risks impact vital infrastructure.
Although it can be difficult to quantify how much money might be saved by investing in cyber security measures — and thus may not always look appealing to managers trying to reduce costs — a serious breach can put the entire organisation at risk. "It makes sense financially to pay money to secure that data," she argues.
"Who is protecting your organisation's data? Who is responsible for the day to day management of data security? Do you have someone internally? If not, do you need to hire someone outside the organisation?" questions Smyth.
She also recommends thinking about cyber insurance. The premium for cyber insurance, a relatively new phenomenon, will often be lowered if an organisation has already taken risk mitigation steps first.
"You need to have a process for reviewing and updating cyber security policies and communicating those to your staff. You need to measure and track whether those policies are in fact working, and you need to consider compliance measures, including those that are required by your insurance policy," she explains.
"You need, of course, to advise your employees and brief them about what to do in the event of an attack. You need to implement employee awareness and training seminars within your organisation," she notes.
It's important to ensure senior managers are setting the right tone for the organisation, communicating good cyber security policy and practicing what they preach, she adds.
Cyber security, she emphasises, is more than just an IT issue. It affects the entire organisation, and requires a response from the whole organisation.
"We know that cyber criminals spend a lot of time thinking about how to improve their success rates, reduce costs and take advantage of new innovations," Smyth points out.


text:
Extended Audio Sample
Share this sample:
Download the full audiobook now for just $7.95!
Add to Cart RESUME NOW!
( learn more )
Audiobooks > Juvenile Fiction > Fantasy & Magic > The Final Reckoning
Download The Final Reckoning Audiobook
Extended Sample Audio Sample Unavailable	Click for printable size audiobook cover
Author: Robin Jarvis Narrator: Roe Kendall Publisher: Blackstone Publishing Format: Unabridged Audiobook Delivery: Instant Download Audio Length: 10 hours Series: The Deptford Mice Trilogy Release Date: July 2007 ISBN: 9781481582469
4.00 (412 ratings) (rate this audio book)
Regular Price: $19.95 Add to Cart
FlexPass™ Price: $12.95$7.95$7.95 for new members!
Add to Cart RESUME NOW! ( learn more )
Add to Wish List Remove from Wishlist
The hideous spirit of Jupiter, the evil cat who was once lord of the sewer rats, has returned, more terrifying than ever. Bent on revenge for his death, he is determined to turn the world into an eternal winter of hatred. The Deptford Mice feel his chilling grasp everywhere they turn. Their allies are murdered, and the prophetic bats flee, sensing impending doom. A rat army thirsty for mouse blood is building underground. And most frightening of all is that the Starglass, their magical source of knowledge and power, has been stolen by a cruel and violent hand. The mice know they face the most desperate struggle of their lives. Will anyone survive?
Download and start listening now!
BK_BLAK_002310
Quotes & Awards
"Fans of the mice will find plenty of what they expect in this conclusion: lots of action, lots of gore, a large cast of characters, and a truly evil antihero."
"Jarvis brings the Deptford Mice trilogy to a stunning conclusion in a story that resonates with mysticism, fear, courage, and horror."
Booklist (starred review)
"A crescendo of savage horror, relieved only by a few grace notes of tender poignancy, make a gripping page-turner of this finale to the Deptford Mice trilogy...A superlative conclusion to a top-notch series."
Kirkus Reviews (starred review)
Listener Reviews
Comment (5000 characters max):
Tim | 10/25/2012
" The concluding story of the Deptford Mice and their dealings with Jupiter. "
Talkingtocactus | 3/9/2012
" ust got reminded of these by the fact they're doing a live version at wimbledon theatre. it was one of my favourites as a kid, i read all these deptford mice books multiple times. great stuff! "
Qt | 12/26/2011
" Dark and violent, this was a fitting end to the trilogy. A few things seemed like they needed more explaining, but I guess that's why there is a prequel trilogy! "
Virginia | 11/19/2011
" Conclusion was very well written - for the first time in this series had had several plot lines going at once which I thought added nicely to the adventure and plot anticipation. "
CarolinaQT | 5/17/2011
" I think the author was running out of ideas at this point. Final book in the trilogy was rushed and not that entertaining really. I really enjoyed the first two though. "
Donna | 5/8/2011
" A really fitting conclusion to the Deptford Mice trilogy. These books were really enjoyable. "
Donna | 12/9/2008
Virginia | 6/8/2008
Tim | 2/5/2008
> Hide All
Robin Jarvis was born in Liverpool, England, and studied graphic design in college. He worked in television and advertising before becoming a full-time author and illustrator. It was while working at a company that made characters for TV programs and advertising that he began making sketches of mice. From these drawings, the idea for the Deptford Mice was born. The Dark Portal, short-listed for the 1989 Smarties Book Prize in England, was followed by two more titles in the series, The Crystal Prison and The Final Reckoning. He lives in London.
About the Narrator
Roe Kendall (a.k.a. Vanessa Benjamin), a native of the British Isles, graduated from the Royal Academy of Dramatic Art, London. She has performed on stage, as a voice-over artist, and as an audiobook narrator.


First of all, who on the Republican side is asking for the Constitution to be changed? Did Vaughn's opponent, Rep. Tony Shipley (R-Kingsport) assert that the federal Constitution should somehow be changed to make immigration the purview of the States alone? No Republican in Tennessee-let alone Shipley-has made such an outlandish proposal. What many conservatives and Republicans do believe is that the federal government has failed in its responsibility to protect our borders and secure the several States from a tide of illegal immigration so great that it amounts to a foreign invasion on the scale of the Germanic invasions of Rome, and from which the federal government is obliged to defend them (see Article IV, Section 4 of the federal Constitution).
Representative Evans' letter was not, as the Nashville press seems to be painting it, a campaign letter. A campaign letter would not only be on campaign stationary or come in the form of a campaign flyer, and it would have said "vote to re-elect Josh Evans for State Representative." This letter did not do that or attempt to do that, it did state the committees that Evans serves on, did allude to some of the services his office might provide, and did state that he lives in Greenbrier, attends Ebenezer Baptist Church, and said that if the newcomer didn't have a church home, they were welcome at Ebenezer. Apparently, the one person in Robertson County who decided they didn't like that also decided that they were too much of a coward to tell their Representative they didn't care for it themselves. Instead, they had to go to WSMV to do it under the cloak of secrecy-you can just smell the ACLU in here somewhere.
Barack Obama has coattails alright-for Republicans-and the Tennessee and national GOP both know it, and so do the Democrats. That is why the Democrats have their "community organizing" lemmings in places like North Memphis, because they can't get their own disillusioned voters to the polls after these people discovered that the Mr. Yes We Can has turned out to be a tall glass full of Oh No You Don't. Those voters who did vote for Obama in 2008 and either voted for the first time or swung to the Democratic side from the Republicans are now just as likely to vote Republican as they are to vote for the President's party. That won't just have an impact on Congressional races, but could cause the Republican tide in many Tennessee General Assembly races to truly "go viral."
What national constituency maps really highlight-whether such models intend to or not-is the importance of elections for State Legislatures in Tennessee and around the country. Parties that control their respective Legislatures in any given State control the process whereby congressional and legislative district lines are redrawn after a federal census, which is one reason why Tennessee Republicans are so keen to let the world know that they are favored to pick up seats in the State House of Representatives this year. Whatever your political party, those who are in-th-know politically realize that there is much more at stake in the 2010 General Election than which political party controls Congress or the General Assembly, but literally which party could control those institutions for years, and perhaps decades to come.
Tennessee House Republicans, on the cusp of expanding their majority in the House to allow for the electing of officers of their choosing, do seem to have been trying to encourage lobbyists to give to Republicans. This was far less obvious and politically forceful, of course, that the Democrats used to tell lobbyists and their firms that it was in their interest to give to the majority party to remain in the good graces of their Leadership. Mike Turner, who is one of the most politically astute figures in the General Assembly, says Democrats contact lobbyists to "tell them to be fair."