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Let S be a manifold of dimension 4, \((c\:dt,dx_1,dx_2,dx_3)\). Every point x in S has an inner product \(g_{ij}\) with signature \((+,+,+,+)\) defined on a tangent space \(T_xS\) of S at x. The metric tensor \(g_{ij}\) is defined as

\begin{equation}\label{e00} \tag{1}
g_{ij}=A \delta_{ij}
\end{equation}

where \(\delta_{ij}\) is the Kronecker delta and A is a scalar defined as

\begin{equation}\label{e0} \tag{2}
A=1+\frac{T_g-V_g}{E_0}=1+\frac{\mathcal{L}}{E_0}
\end{equation}

where \(E_0\) is the rest mass energy of mass \(M\) generating the field, \(T_g\) is the kinetic energy of a virtual point mass [/latex]m_v\( with respect to [latex]M\), \(m_v\) is equal to the magnitude of \(M\), and \(V_g\) is the Newtonian gravitational potential of \(m_v\). The value \(\mathcal{L}\) is the Lagrangian for the point mass \(m_v\). Mass \(M\) is a homogeneous, non-spinning, non-charged spherical mass with radius \(R>0\).
The kinetic energy \(T_g\) is defined

\begin{equation} \tag{3}
T_g = \frac{m_v v^2}{2}
\end{equation}

The gravitational potential energy is defined
\(
\begin{equation} \tag{4}
V_g=-\frac{G M m_v}{r}
\end{equation}
\)
Where \(G\) is the Newtonian Gravitational constant and \(r\) is the distance from the center of mass of \(M\) to \(M_v\). Equation \ref{e0} in long form is
\begin{equation} \label{e1}\tag{5}
A=1+\frac{\frac{m_v v^2}{2} + \frac{G M m_v}{r}}{M c^2}
\end{equation}
Equation \ref{e1} reduced (Where \(m_v=M\)) is
\begin{equation} \label{e2}\tag{6}
A=1+\frac{v^2}{2 c^2}+\frac{G M}{r c^2}
\end{equation}
Equation \ref{e00} long form
\begin{equation}\label{e3}\tag{7}
g_{ij} =A \:\delta_{ij} =\left( 1+\frac{v^2}{2 c^2}+\frac{G M}{r c^2} \right)\delta_{ij}
\end{equation}
In a \(T_xS\) of \(S\) where \(M=0 \rightarrow v_2=0\) since there is no \(M\) for relative kinetic energy. Thus where \(M=0\), the metric tensor in equation \ref{e3} reduces to
\begin{equation}\label{e4}\tag{8}
g_{ij} = \delta_{ij}
\end{equation}
By definition, equation \ref{e4} is the metric for Euclidean space. For convenience any manifold where all \(T_xS\) have a metric tensor in the form of equation \ref{e4} is termed Exemplar \(\mathcal{E}\) and any manifold where all \(T_xS\) have a metric tensor in the form of equation \ref{e3} is termed Privo \(\mathcal{P}\). Of note, a spherical surface with center at the center of mass \(M\) has a constant A value and a constant metric \(g_{ij}=A_{constant} \delta_{ij}\) in \(S\) over the spherical surface. The constant metric on a spherical surface gives rise to an unexpected outcome when mapping an arc path of \(2 \pi\) radians in \(\mathcal{P}\) to an equivalent path in \(\mathcal{E}\).

Mapping \(\mathcal{P}\) to \(\mathcal{E}\)
The mapping \(\phi\) from a point x in \(\mathcal{P}\) to a point x in \(\mathcal{E}\) is
\begin{equation}\tag{9}
\phi:\mathcal{P} \rightarrow \mathcal{E}
\end{equation}
Defined by
\begin{equation}\tag{10}
\phi(x)=A \: \mathcal{P}_x
\end{equation}
Mapping from \(T_x\mathcal{P}\) at \(x\) to \(T_x\mathcal{E}\) at \(\phi(x)\) is defined by
\begin{equation}\tag{11}
d\phi_x:T_x\mathcal{P}\rightarrow T_{\phi(x)}\mathcal{E}
\end{equation}
Where
\begin{equation}\label{e5}\tag{12}
d\phi(x)=A \: d\mathcal{P}_x
\end{equation}
The inverse of \(\phi\) is
\begin{equation}\tag{13}
\phi^{-1}:\mathcal{E} \rightarrow \mathcal{P}
\end{equation}
Defined
\begin{equation}\tag{14}
\phi^{-1}(x)=A^{-1} \: \mathcal{E}_x
\end{equation}
The inverse of \(d\phi\) is
\begin{equation}\tag{15}
d\phi^{-1}_x:T_{\phi^{-1}(x)}\mathcal{E}\rightarrow T_x\mathcal{P}
\end{equation}
Defined
\begin{equation}\tag{16}
d\phi^{-1}(x)=A^{-1} \: d\mathcal{E}_x
\end{equation}

Arc Path on Concentric Spheres
When mapping an arc path of \(2 \pi\) radians in \(\mathcal{P}\) to \(\mathcal{E}\) the path in \(\mathcal{E}\) takes the form
\begin{equation}\tag{17}
s=\int_{0}^{1}\int_{0}^{2\cdot \mathrm{Pi}}A^2r\: d \theta\: dt
\end{equation}
Where \(s\) is the arc length, A is the scalar value from equation \ref{e3}, \(\theta\) is the angle traversed and \(t\) is the time elapsed to make a circle arc of \(2\pi\) radians. The limits of integration are \(0\rightarrow 2\pi\) for the arc angle \(\theta\) and \(0\rightarrow 1\) for a single period of time \(t\) to traverse the arc. For rigor, the value \(r\) is mapped from \(\mathcal{P}\) to \(\mathcal{E}\) using equation \ref{e5} where \(r\) in \(\mathcal{P}\) is
\begin{equation}\tag{18}
r=\sqrt{(dx)^2 + (dy)^2+(dz)^2}
\end{equation}
And time \(t\) in \(\mathcal{P}\) using equation \ref{e5} is \(A\:dt\) in \(\mathcal{E}\). The result is in \(\mathcal{P}\) the arc path returns to it starting position after \(2\pi\) radians, but when mapped to \(\mathcal{E}\) the arc path advances beyond its starting position. The advance is
\begin{equation}\tag{19}
advance = 2 \pi A^2 r – 2 \pi r = 2 \pi r (A^2-1)
\end{equation}
For a gravitational circular orbit the advance in arc length in \(\mathcal{E}\) is
\begin{equation} \label{e6}\tag{20}
advance = \frac{6 \pi G M}{c^{2}}+\frac{9 \pi G^{2} M^{2}}{2 r \,c^{4}}
\end{equation}
This is in strong agreement with General Relativity, Gravitation [pg 1110].

Equation \ref{e6} shows one contribution to the advance is an invariant length of value
\begin{equation}\label{e6a}\tag{21}
constant \: \: advance =\frac{6 \pi G M}{c^2}
\end{equation}
for any gravitational orbit about a fixed mass \(M\) independent of radial distance. This is \(6\pi\) times the characteristic length for relativistic effects in a spherical field. Gravitation [pg 1111]

Discrete Metric
Each sphere is a smooth manifold with a diffeomorphic metric. When transforming from one concentric spherical surface to another (or from one smooth manifold to another) the connection is not continuous, it is discrete. The uniform discreteness arises from the characteristic length. For an orbital about a massive object the characteristic length generates an invariant length added to all circumferential paths, regardless of radial distance of the orbit. This length for a gravitational orbit is equation \ref{e6a}. A corollary of the additional circumferential length is there are discrete lengths the radial distance can take on. The radial distance must be a multiple of equation \ref{e6a} divided by \(2/pi\) or
\begin{equation}\label{e6b}\tag{22}
Discrete \:\: Radial =\frac{3 G M}{c^2}
\end{equation}
Thus, the concentric spheres are not connected, but rather discrete with discrete radial distances. Each spherical surface has a tangent space at every location, but the tangent spaces are not continuous betweeen each sphere. Do not confuse this discreteness with non-orbital measurements. This only applies to orbits. This also implies equation \ref{e6b} is the minimum circular gravitational orbit allowed. Particles can come closer (e.g. an elliptical orbit) but the total circumference of the orbit will have an added length of equation \ref{e6a}.

Each massive particle has a set of smooth manifolds representing orbitals of discrete radial distances from the center of mass of the massive particle.