Euclidean Spherical Mechanics – Spacetime Frames of Reference – (Differential Mathematical Form):

The study of Euclidean Spherical Mechanics, is a set of conceptual and mathematical tools, used to describe the physics of a spherically symmetric system mass body that creates its own gravitational field, while; at rest/static, in relativistic motion, spinning/rotating at rest, or spinning/rotating while in motion.

The Euclidean Spherical Mechanics takes into account the relativity of different measuring observers, and different frames of reference; a “proper observer” located at the center of the sphere, and an “external observer” located at the surface of the sphere.

The Euclidean Spherical Mechanics unifies and generalizes, the theories, concepts, and mathematics of “Special Theory of Relativity” and “General Theory of Relativity” into a single framework known as the “Super Special Theory of Relativity”.

Many attempts have been made to develop a space and time or “Metric Gravitation Theory” in competition to Einstein’s theory of “General Relativity.” See, Wiki article on: Alternatives to General Relativity.

In Euclidean Spherical Mechanics, an Euclidean Spacetime Metric describes a symmetrically spherical, space-time system body, where the “Speed of Light” is invariant or constant, and isotropic; and is described by a set of three (3) different inertial frames of reference, three (3) dimensions of space, two (2) dimensions of angle, and one (1) dimension of time.

The Euclidean Spacetime Metric of a sphere is the “net sum” of the square of the “Radial” “spatial component”, plus the square of the “Geodesic Arc-Length” (Map/Patch/Manifold) surface “spatial component”. But, before a complete mathematical description of the Euclidean Spacetime Metric is discussed, a more general description of each of the three (3) different “Spacetime” inertial frames of reference, which are the component parts of the Euclidean Spacetime Metric, is described in the work below.

A “Proper Observer” center of mass frame of reference, where the origin is located at the “Mean Center” or “Center of Mass” of the sphere; and is known as the internal part of the system.

An “External Observer” frame of reference, where the origin is located on the external surface of the sphere; and is known as the external part of the system.

And lastly, there is the “Equal Observer” (Co-variant)” inertial frame of reference that exist on the internal surface, and the external surface of the sphere; and measures equal values of “space” and “time” by both the “Proper Observer” and the “External Observer” frames of reference.

The Euclidean Spacetime Metric of a symmetrically spherical, space-time system body, measures the “Speed of Light” invariant or constant, and isotropic; and is described by the net sum of the square of the “Radial” spatial component sum the square of the “Geodesic Arc-Length” (Map/Patch/Manifold) surface spatial component, and is described by three (3) “Space-Time” frames of reference:

1) a four (4) dimensional spherical “Radial Space-Time” center of mass frame of reference

-Proper Observer “Center of Mass” Frame of Reference

$\vec{dr}\;=\;\vec{c_{Light}}\,{dt_{Light}}\;=\; \vec{dr}(d\vec{x}\,,d\vec{y}\,,d\vec{z}\,,{dt})$

The measurements made by the “Proper Observer” “center of mass” frame of reference will conclude that the “External Observer” frame of reference located on the surface of the sphere, has access to more spherical information; and measures “longer distances”, and “longer or slower” light clock times.

2) a four (4) dimensional spherical “Euclidean Radial Space-Time” frame of reference

-External Observer Frame of Reference

$\vec{ds}\;=\;\vec{c_{Light}}\,{dt'_{Light(s)}}\;=\; \vec{ds}(d\vec{x_{s}}\,,d\vec{y_{s}}\,,d\vec{z_{s}}\,,{dt'_{s}})\,=\,\vec{ds}(\vec{dr}\,,\vec{d\theta}\,,\vec{d\phi}\,,{dt'_{s}})$

The measurements made by the “External Observer” frame of reference will conclude that the “Proper Observer” center of mass frame of reference, has access to less surface information; and measures “shorter distances”, and “shorter or faster” light clock times.

3) a four (4) dimensional spherical surface “Map/Patch/Manifold – Geodesic Space-Time” frame of reference

-Equal Observer Frame of Reference (Covariant Frame)

$\vec{ds_{Map}}_{\theta \, \phi}\;=\;\vec{c_{Light}}\,{dt_{Map}}\;=\;\vec{c_{Light}}\,\sqrt{({dt'_{Light(s)}})^2\;-\;({dt_{Light}})^2}$

$\vec{ds_{Map}}_{\theta \, \phi}\;=\;\vec{ds_{Map}}_{\theta \, \phi}(d\vec{x_{Map}}\,,d\vec{y_{Map}}\,,d\vec{z_{Map}}\,,(\sqrt{-1}){dt_{Map}})\,=\,\vec{ds_{Map}}_{\theta \, \phi}(\vec{d\theta}\,,\vec{d\phi}\,,(\sqrt{-1}){dt_{Map}})$

And the “Equal Observer” “Co-variant” frame is measured equally by both the “Proper Observer” “Center of Mass” frame of reference, and the “External Observer” frames of reference; and measures “equal distances”, and “equal times” light clock times.

In theoretical and mathematical physics, the term “Co-variance” describes the “Invariance” of the form of physical parameters, terms, or laws, during differentiable coordinate transformations, and all frames of reference. Thus, a physical parameters, terms, or law expressed in generally “covariant” equations takes the same mathematical form in all coordinate systems, and all frames of reference.

In physics and mathematics of this work, a “Map/Patch/Manifold” geodesic is “curved line” or arc length mapped on a curved spherical surface, similar to the concept of a “straight line” mapped onto a flat surface. In relativistic physics, geodesics describe the motion of point particles under the influence of gravity alone. Geodesics are commonly seen in the study of Riemannian geometry or metric geometry. In general, the curvilinear path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all geodesics in curved spacetime.

In the mathematics and physics of topology, a surface is a two (2) dimensional topological manifold in three (3) dimensions of space. Common examples of surface topologies are those that surface as the boundaries of solid objects in ordinary three (3) dimensional Euclidean spaces.

To say that a surface is “two (2) dimensional” in three (3) dimensional space means that, about each point, on the surface, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, the surface area of a spherical ball is two (2) dimensional. The surface of the Earth can also be mapped onto a two (2) dimensional sphere, where the latitude and longitude provide the two (2) dimensional coordinates of the surface; although the spherical ball and the earth are described by three (3) dimensional spaces.

Now, let’s discuss each frame of reference of an Euclidean Symmetric Spherical system body individually.

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1) Proper Observer “Center of Mass” Frame of Reference

A “Proper Observer” frame of reference, origin is located at the “Mean Center” or the “Center of Mass” of the sphere; and is known as the internal part of the system.

“Proper Observer” Differential Spherical Space

Differential – Radius of Sphere – ( $\vec{dr}$ ) – is a vector (direction dependent) described in Rectangular ( $d\vec{x}\,,d\vec{y}\,,d\vec{z}\,,{dt}$ ) Cartesian Coordinates, and along Rectangular ( $\hat{a}_{x}\,,\,\hat{a}_{y}\,,\,\hat{a}_{z}$ ) Cartesian Axes

$\vec{dr}\;=\;\vec{c_{Light}}\,{dt_{Light}}\;=\; \begin{bmatrix} \vec{dx} \\[0.3em] \vec{dy} \\[0.3em] \vec{dz} \end{bmatrix}\;=\; \begin{bmatrix} {dx}\,\hat{a}_{x} \\[0.3em] {dy}\,\hat{a}_{y} \\[0.3em] {dz}\,\hat{a}_{z} \end{bmatrix}$ $\,\,\,---> {m}$

Differential – Radius of Sphere – ( $\vec{dr}$ ) – is a vector (direction dependent) described in Spherical ( $d\vec{r}\,,d\vec{\theta}\,,d\vec{\phi}\,,{dt}$ ) Coordinates, and along Rectangular ( $\hat{a}_{x}\,,\,\hat{a}_{y}\,,\,\hat{a}_{z}$ ) Cartesian Axes

$\vec{dr}\;=\; {dr}\,\hat{a}_{r}\;=\; \begin{bmatrix} [(sin\theta_{Lat}\,cos\phi_{Lon})\,{dr} \; -r(\,sin\theta_{Lat}\,sin\phi_{Lon})\,d\phi_{Lon}\; +r(cos\theta_{Lat}\,cos\phi_{Lon})\,d\theta_{Lat}]\,\hat{a}_{x} \\[0.3em] [(sin\theta_{Lat}\,sin\phi_{Lon})\,{dr} \; +r(\,sin\theta_{Lat}\,cos\phi_{Lon})\,d\phi_{Lon}\; +r(cos\theta_{Lat}\,sin\phi_{Lon})\,d\theta_{Lat}]\,\hat{a}_{y} \\[0.3em] [(cos\theta_{Lat})\,{dr} \;\;\;\;\; +(0)\,d\phi_{Lon}\;\;\;\;\; -r(sin\theta_{Lat})\,d\theta_{Lat}]\,\hat{a}_{z} \end{bmatrix}$

Differential – Radius of Sphere – ( $\vec{dr}$ ) – measured in the “Proper Observer” center of mass frame of reference, and described as function of the Euclidean Radius of Sphere ( $\vec{ds}$ ), relative to the “External Observer” frame of reference; and likewise is described as a function of the Map/Patch/Manifold ( $\vec{ds_{Map}}_{\theta \, \phi}$ ) “Geodesic” Arc Length, relative to the “Equal Observer (Co-variant)” frame of reference

$\vec{dr}\;=\;\vec{ds}\sqrt{\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}\;\;=\;\; {\frac{\vec{ds_{Map}}_{\theta \, \phi}}{(\sqrt{-1})}}\sqrt{\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}\;$ $\,\,\,---> {m}$

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Differential – Radius of Sphere – ( $\vec{dr}$ ) – “Special Relativity – Lorentz Transformation” from the “External Observer” frame of reference, into the “Proper Observer” center of mass frame of reference

$\vec{dr}\;=\;\frac{\vec{ds}\;\;+\;\;\vec{c_{Light}}\,{d\tau'_{Sync}}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}\;=\;\frac{\vec{ds}\;\;+\;\;\vec{|v|_{CM}}\,{dt'_{Light(s)}}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}\;=\;\frac{\vec{ds}\;\;+\;\;(\frac{|v|_{CM}}{c_{Light}})\,\vec{ds}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}$ $\,\,\,---> {m}$

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Differential – Radius of Sphere – ( $\vec{dr}$ ) – “General Relativity – Euclidean Transformation” from the “External Observer” frame of reference, into the “Proper Observer” center of mass frame of reference

$\vec{dr}\;=\;\frac{\vec{ds}\;\;+\;\;(\frac{1}{\sqrt{-1}})[\sqrt{\frac{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{c_{Light}}{|v|_{CM}})}}]\,\vec{ds_{Map}}_{\theta \, \phi}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}\;=\;\frac{\vec{ds}\;\;+\;\;\vec{c_{Light}}\,[\sqrt{\frac{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{c_{Light}}{|v|_{CM}})}}]\,{dt_{Map}}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}$ $\,\,\,---> {m}$

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“Proper Observer” Differential “Light Clock” Time

${dt_{Light}}\;=\;\frac{\vec{dr}}{\vec{c_{Light}}}\;=\;\frac{\sqrt{(\vec{ds})^2\;\;-\;\;(\vec{ds_{Map}}_{\theta \, \phi})^2}}{\vec{c_{Light}}}\,$ $\,\,\,---> {s}$

${dt_{Light}}\;=\;\sqrt{({dt'_{Light(s)}})^2\;\;+\;\;({dt_{Map}})^2}\;=\;\sqrt{({dt'_{Light(s)}})^2\;\;-\;\;\frac{(\vec{ds_{Map}}_{\theta \, \phi})^2}{c^2_{Light}}}$

Differential – Proper Observer “Light Clock” Time – ( ${dt_{Light}}$ ) – measured in the “Proper Observer” center of mass frame of reference, and described as function of the External Observer “Light Clock” Time ( $\vec{dt'_{Light(s)}}$ ), relative to the “External Observer” frame of reference; and likewise is described as a function of the Map/Patch/Manifold Time ( $\vec{dt_{Map}}_{\theta \, \phi}$ ), relative to the “Equal Observer (Co-variant)” frame of reference

${dt_{Light}}\;=\;{dt'_{Light(s)}}\sqrt{\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}\;\;=\;\;{dt_{Map}}\sqrt{\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}\;$ $\,\,\,---> {s}$

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Differential – Proper Observer “Light Clock” Time – ( ${dt_{Light}}$ ) – “Special Relativity – Lorentz Transformation” from the “External Observer” frame of reference, into the “Proper Observer” center of mass frame of reference

${dt_{Light}}\;=\;\frac{{dt'_{Light(s)}}\;\;+\;\;{d\tau'_{Sync}}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}\;=\;\frac{{dt'_{Light(s)}}\;\;+\;\;(\frac{|v|_{CM}}{c_{Light}})\,{dt'_{Light(s)}}}{\sqrt{1\;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}\;=\;\frac{{dt'_{Light(s)}}\;\;+\;\;(\frac{|v|_{CM}}{c^2_{Light}})\,{ds}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}$

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Differential – Proper Observer “Light Clock” Time – ( ${dt_{Light}}$ ) – “General Relativity – Euclidean Transformation” from the “External Observer” frame of reference, into the “Proper Observer” center of mass frame of reference

${dt_{Light}}\;=\;\;\frac{{dt'_{Light(s)}}\;\;+\;\;[\sqrt{\frac{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{c_{Light}}{|v|_{CM}})}}]\,{dt_{Map}}}{\sqrt{1\;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}\;=\;\frac{{dt'_{Light(s)}}\;\;+\;\;(\frac{1}{\sqrt{-1}})(\frac{1}{{c_{Light}}})[\sqrt{\frac{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{c_{Light}}{|v|_{CM}})}}]\,\vec{ds_{Map}}_{\theta \, \phi}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}$

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2) External Observer Frame of Reference

An “External Observer” frame of reference, origin is located at the “Map/Patch/Manifold surface” of the sphere; and is known as the external part of the system.

An “External Observer” frame of reference, is described by two (2) space-time components: (1) a “radial” component, and (2) a “geodesic arc length” component, which is the “Map/Patch/Manifold” surface of the sphere.

“External Observer” Differential Euclidean Spherical Space

Differential – Euclidean Radius of Sphere – ( $\vec{ds}$ ) – is a vector (direction dependent) described in Spherical ( $d\vec{r}\,,d\vec{\theta}\,,d\vec{\phi}\,,{dt}$ ) Coordinates, and along Spherical ( $\hat{a}_{r}\,,\,\hat{a}_{\theta}\,,\,\hat{a}_{\phi}$ ) Coordinate Axes

$\vec{ds}\; = \; \begin{bmatrix} \vec{d{r}} \\[0.3em] \vec{ds_{Map}}_{\theta \, \phi} \end{bmatrix}\;= \; \begin{bmatrix} \vec{d{r}} \\[0.3em] \vec{d{r}}_{\theta} \\[0.3em] \vec{d{r}}_{\phi} \end{bmatrix} =\; \begin{bmatrix} {d{r}}\,\hat{a}_{r} \\[0.3em] {d{r}}_{\theta}\,\hat{a}_{\theta} \\[0.3em] {d{r}}_{\phi}\,\hat{a}_{\phi} \end{bmatrix} = \; \begin{bmatrix} dr\,\hat{a}_{r} \\[0.3em] {r}\;d\theta_{Lat}\,\hat{a}_{\theta} \\[0.3em] {r}\,sin\theta_{Lat}\,d\phi_{Lon}\,\hat{a}_{\phi} \end{bmatrix}\;$ $\,\,\,---> {m}$

Differential – Euclidean Radius of Sphere – ( $\vec{ds}$ ) – is a vector (direction dependent) described in Rectangular ( $\hat{a}_{x}\,,\,\hat{a}_{y}\,,\,\hat{a}_{z}$ ) Cartesian Axes components, and along each Spherical ( $d\vec{r}\,,d\vec{\theta}\,,d\vec{\phi}\,,{dt}$ ) Coordinates

$\vec{ds}\;=\;{ds}\,\hat{a}_{r}\;=\;\begin{bmatrix} [(sin\theta_{Lat}\,cos\phi_{Lon})\,\hat{a}_{x} \; +\;(\,sin\theta_{Lat}\,sin\phi_{Lon})\,\hat{a}_{y}\; +\;(cos\theta_{Lat})\,\hat{a}_{z}]{dr} \\[0.3em] [(cos\theta_{Lat}\,cos\phi_{Lon})\,\hat{a}_{x} \;+\;(\,cos\theta_{Lat}\,sin\phi_{Lon})\,\hat{a}_{y}\; -\;(sin\theta_{Lat})\,\hat{a}_{z}]\,{d{r}}_{\theta}\\[0.3em] [(sin\phi_{Lon})\,\hat{a}_{x} \;\;\; +\;\;\;(cos\phi_{Lon})\,\hat{a}_{y}\;\;\; +\;\;\;(0)\,\hat{a}_{z})]\,{d{r}}_{\phi} \end{bmatrix}\;$

Differential – Euclidean Radius of Sphere – ( $\vec{ds}$ ) – is a vector (direction dependent) described in Spherical ( $d\vec{r}\,,d\vec{\theta}\,,d\vec{\phi}\,,{dt}$ ) Coordinates, and mixed with Rectangular Cartesian ( $d\vec{x}\,,d\vec{y}\,,d\vec{z}\,,{dt}$ ) Coordinates, and along Rectangular ( $\hat{a}_{x}\,,\,\hat{a}_{y}\,,\,\hat{a}_{z}$ ) Cartesian Axes

$\vec{ds}\;=\;\vec{c_{Light}}\,{dt'_{Light(s)}}\;=\;\begin{bmatrix} \vec{dx_{s}} \\[0.3em] \vec{dy_{s}} \\[0.3em] \vec{dz_{s}} \end{bmatrix}\; = \begin{bmatrix} ds(sin\theta_{Lat}\,cos\phi_{Lon})\,\hat{a}_{x} \\[0.3em] ds(sin\theta_{Lat}\,sin\phi_{Lon})\,\hat{a}_{y} \\[0.3em] ds(cos\theta_{Lat})\,\hat{a}_{z} \end{bmatrix}\;$ $\,\,\,---> {m}$

Differential – Euclidean Radius of Sphere – ( $\vec{ds}$ ) – measured in the “External Observer” frame of reference, is described by one (1) component of “radius” ( $\vec{d{r}}$ ), and one (1) “geodesic arc-length” component, which is the “Map/Patch/Manifold” ( $\vec{ds_{Map}}_{\theta \, \phi}$ ) surface of the sphere, and changes as a function of the “Latitude Angle” ( $\vec{d{\theta}}$ ), and the Longitude Angle ( $\vec{d{\phi}}$ )

$\vec{ds}\; = \; \begin{bmatrix} \vec{d{r}} \\[0.3em] \vec{ds_{Map}}_{\theta \, \phi} \end{bmatrix}\;= \; \begin{bmatrix} \vec{d{r}} \\[0.3em] {r}\,{\vec{d\Omega_{Map}}_{(\theta \phi)}} \end{bmatrix}\; =\; \begin{bmatrix} {d{r}}\,\hat{a}_{r} \\[0.3em] r(\sqrt{(d\theta_{Lat}\,\hat{a}_{\theta})^2 + (sin\theta_{Lat}\,d\phi_{Lon}\,\hat{a}_{\phi})^2})\,\hat{a}_{r} \end{bmatrix}\;$

Differential – Euclidean Radius of Sphere – ( $\vec{ds}$ ) – measured in the “External Observer” frame of reference, is described by one (1) component of radius ( $\vec{dr}$ ), and one (1) “geodesic arc-length” component, which is the “Map/Patch/Manifold” ( $\vec{ds_{Map}}_{\theta \, \phi}$ ) surface of the sphere, and changes as a function of the “radius” ( $\vec{dr}$ ) of the sphere

$\vec{ds}\; = \; \begin{bmatrix} \vec{d{r}} \\[0.3em] \vec{ds_{Map}}_{\theta \, \phi} \end{bmatrix}\;= \; \begin{bmatrix} \vec{d{r}} \\[0.3em] {r}\,{\vec{d\Omega_{Map}}_{(\theta \phi)}} \end{bmatrix}\; =\; \begin{bmatrix} {d{r}}\,\hat{a}_{r} \\[0.3em] (\sqrt{-1})[\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}]({dr})\;\hat{a}_{r}\end{bmatrix}\;$

Differential – Euclidean Radius of Sphere – ( $\vec{ds}$ ) – measured in the “External Observer” frame of reference, and described as function of the Radius of Sphere ( $\vec{dr}$ ), relative to the “Proper Observer” center of mass frame of reference; and likewise is described as a function of the Map/Patch/Manifold ( $\vec{ds_{Map}}_{\theta \, \phi}$ ) “Geodesic” Arc Length, relative to the “Equal Observer (Co-variant)” frame of reference

$\vec{ds}\;=\;\vec{dr}\sqrt{\frac{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}\;\;=\;\;{{\frac{\vec{ds_{Map}}_{\theta \, \phi}}{(\sqrt{-1})}}}\sqrt{\frac{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}\;$ $\,\,\,---> {m}$

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Differential – Euclidean Radius of Sphere – ( $\vec{ds}$ ) – “Special Relativity – Lorentz Transformation” from the “Proper Observer” center of mass frame of reference, into the “External Observer” frame of reference

$\vec{ds}\;=\;\frac{\vec{dr}\;\;-\;\;\vec{c_{Light}}\,{d\tau_{Sync}}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}\;=\;\frac{\vec{dr}\;\;-\;\;\vec{|v|_{CM}}\,{dt_{Light}}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}\;=\;\frac{\vec{dr}\;\;-\;\;(\frac{|v|_{CM}}{c_{Light}})\,\vec{dr}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}$ $\,\,\,---> {m}$

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Differential – Euclidean Radius of Sphere – ( $\vec{ds}$ ) – “General Relativity – Euclidean Transformation” from the “Proper Observer” center of mass frame of reference into the “External Observer” frame of reference

$\vec{ds}\;=\;\frac{\vec{dr}\;\;-\;\;(\frac{1}{\sqrt{-1}})[\sqrt{\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{c_{Light}}{|v|_{CM}})}}]\,\vec{ds_{Map}}_{\theta \, \phi}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}\;=\;\frac{\vec{dr}\;\;-\;\;\vec{c_{Light}}\,[\sqrt{\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{c_{Light}}{|v|_{CM}})}}]\,{dt_{Map}}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}$ $\,\,\,---> {m}$

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“External Observer” Differential “Light Clock” Time

${dt'_{Light(s)}}\;=\;\frac{\vec{ds}}{\vec{c_{Light}}}\;=\;\frac{\sqrt{(\vec{dr})^2 \;\;+\;\;(\vec{ds_{Map}}_{\theta \, \phi})^2}}{\vec{c_{Light}}}\,$ $\,\,\,---> {s}$

${dt'_{Light(s)}}\;=\;\sqrt{({dt_{Light}})^2\;\;-\;\;({dt_{Map}})^2}\;=\;\sqrt{({dt_{Light}})^2\;\;+\;\;\frac{(\vec{ds_{Map}}_{\theta \, \phi})^2}{c^2_{Light}}}$

Differential – External Observer “Light Clock” Time – ( ${dt'_{Light(s)}}$ ) – measured in the “External Observer” frame of reference, and described as function of the Proper Observer “Light Clock” Time ( ${dt_{Light}}$ ), relative to the “Proper Observer” center of mass frame of reference; and likewise is described as a function of the Map/Patch/Manifold Time ( $\vec{dt_{Map}}_{\theta \, \phi}$ ), relative to the “Equal Observer (Co-variant)” frame of reference

${dt'_{Light(s)}}\;=\;{dt_{Light}}\sqrt{\frac{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}\;\;=\;\;{dt_{Map}}\sqrt{\frac{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}\;$ $\,\,\,---> {s}$

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Differential – External Observer “Light Clock” Time – ( ${dt'_{Light(s)}}$ ) – “Special Relativity – Lorentz Transformation” from the “Proper Observer” center of mass frame of reference into the “External Observer” frame of reference

${dt'_{Light(s)}}\;=\;\frac{{dt_{Light}}\;\;-\;\;{d\tau_{Sync}}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}\;=\;\frac{{dt_{Light}}\;\;-\;\;(\frac{|v|_{CM}}{c_{Light}})\,{dt_{Light}}}{\sqrt{1\;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}\;=\;\frac{{dt_{Light}}\;\;-\;\;(\frac{|v|_{CM}}{c^2_{Light}})\,{dr}}{\sqrt{1\;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}$

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Differential – External Observer “Light Clock” Time – ( ${dt'_{Light(s)}}$ ) – “General Relativity – Euclidean Transformation” from the “Proper Observer” center of mass frame of reference into the “External Observer” frame of reference

${dt'_{Light(s)}}\;=\;\frac{\vec{dt_{Light}}\;\;-\;\;[\sqrt{\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{c_{Light}}{|v|_{CM}})}}]\,{dt_{Map}}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}\;=\;\frac{\vec{dt_{Light}}\;\;-\;\;(\frac{1}{\sqrt{-1}})(\frac{1}{{c_{Light}}})[\sqrt{\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{c_{Light}}{|v|_{CM}})}}]\,\vec{ds_{Map}}_{\theta \, \phi}}{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}$

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3) Equal Observer (Covariant Frame) Frame of Reference

An “Equal Observer (Co-variant)” frame of reference, origin is located at the interior and exterior “Map/Patch/Manifold surface” of the sphere; and measures equal distances by both the “Proper Observer” and the “External Observer” frames of reference.

An “Equal Observer (Co-variant)” frame of reference, describes a “geodesic arc length” component, which is the “Map/Patch/Manifold” surface of the sphere, and is comprised of two (2) space-time components: (1) a “latitude angle” component, and (2) a “longitude angle” component of a symmetric sphere.

-Equal Observer Frame of Reference (Covariant Frame)

“Equal Observer (Co-variant)” Differential Spherical Angle Space

“Differential” Map/Patch/Manifold Angle

Differential – Map/Patch/Manifold – “Angle” – ( $\vec{d\Omega_{Map}}_{(\theta \phi)}$ ) – is a vector (direction dependent) described in Spherical ( $d\vec{\theta}\,,d\vec{\phi}\,,(\sqrt{-1}){dt}$ ) Coordinates, and directed along Spherical ( $\hat{a}_{\theta}\,,\,\hat{a}_{\phi}$ ) Coordinate Axes

$\vec{d\Omega_{Map}}_{(\theta \phi)}\; = {d\Omega_{Map}}_{(\theta \phi)}\,\hat{a}_{r}\; =\; \begin{bmatrix} \vec{d\theta_{Lat}} \\[0.3em] \vec{d\phi_{Lon}} \end{bmatrix}\;= \; \begin{bmatrix} d\theta_{Lat}\,\hat{a}_{\theta} \\[0.3em] sin\theta_{Lat}\,d\phi_{Lon}\,\hat{a}_{\phi} \end{bmatrix}\;$ $\,\,\,---> {radians}$

Differential – Map/Patch/Manifold – “Angle” – ( $\vec{d\Omega_{Map}}_{(\theta \phi)}$ ) – is a vector (direction dependent) described in Rectangular ( $\hat{a}_{x}\,,\,\hat{a}_{y}\,,\,\hat{a}_{z}$ ) Cartesian Axes components, along each Spherical ( $d\vec{\theta}\,,d\vec{\phi}\,,(\sqrt{-1}){dt}$ ) Coordinates

$\vec{d\Omega_{Map}}_{(\theta \phi)}\; = {d\Omega_{Map}}_{(\theta \phi)}\,\hat{a}_{r}\; = \; \begin{bmatrix}[(cos\theta_{Lat}\,cos\phi_{Lon})\,\hat{a}_{x} \;+\;(\,cos\theta_{Lat}\,sin\phi_{Lon})\,\hat{a}_{y}\; -\;(sin\theta_{Lat})\,\hat{a}_{z}]\,d{\theta}\\[0.3em] [(sin\phi_{Lon})\,\hat{a}_{x} \;\;\; +\;\;\;(cos\phi_{Lon})\,\hat{a}_{y}\;\;\; +\;\;\;(0)\,\hat{a}_{z})]\,d{\phi} \end{bmatrix}\;$

$\vec{d\Omega_{Map}}_{(\theta \phi)}\;=\; \sqrt{\frac{\vec{d{r^2}}_{\theta}}{r^2}\;\;+\;\;\frac{\vec{d{r^2}}_{\phi}}{r^2}}\;=\;\sqrt{d\theta^2_{Lat}\;+\;(sin^2\theta_{Lat})\,d\phi^2_{Lon}}\;\;\hat{a}_{r}$

$\vec{d\Omega_{Map}}_{(\theta \phi)}\;\;=\;\;\frac{\vec{ds_{Map}}_{\theta \, \phi}}{r}\;\;=\;\;(\sqrt{-1})(\frac{\vec{c_{Light}}}{r})\,{dt_{Map}}\;$ $\,\,\,---> {radians}$

Differential – Map/Patch/Manifold – “Angle” – ( $\vec{d\Omega_{Map}}_{(\theta \phi)}$ ) – is a “geodesic arc-length” angle component, on the surface of the sphere, and changes as a function of the “radius” ( $\vec{dr}$ ), and changes as a function of the “Euclidean Radius” ( $\vec{ds}$ ) of a symmetric sphere

$\vec{d\Omega_{Map}}_{(\theta \phi)}\;=\;(\sqrt{-1})[\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}](\frac{dr}{r})\;\hat{a}_{r}\;=\;(\sqrt{-1})[\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}](\frac{ds}{r})\;\hat{a}_{r}$

Differential – Map/Patch/Manifold – “Angle” – ( $\vec{d\Omega_{Map}}_{(\theta \phi)}$ ) – is a “geodesic arc-length” angle component, on the surface of the sphere, and described by the spin, rotation, or torsion on the surface of a sphere, given by the “Map/Patch/Manifold Angular “Spin/Rotation” Velocity” ( ${\omega_{\Omega}}$ ) in the “Proper Observer” frame, the “Map/Patch/Manifold Angular “Spin/Rotation” Velocity” ( ${\omega'_{\Omega(s)}}$ ) in the “External Observer” frame of reference

$\vec{d\Omega_{Map}}_{(\theta \phi)}\;\;=\;\;{\omega_{\Omega}}\,{dt_{Light}}\;\;\hat{a}_{r}\;\;=\;{dt_{Light}}\,\sqrt{\omega^2_{\theta}\;+\;(sin^2\theta_{Lat})\,\omega^2_{\phi}}\;\;\hat{a}_{r}$

$\vec{d\Omega_{Map}}_{(\theta \phi)}\;\;=\;\;{\omega'_{\Omega(s)}}\,{dt'_{Light(s)}}\;\;\hat{a}_{r}\;\;=\;{dt'_{Light(s)}}\,\sqrt{\omega'^2_{\theta(s)}\;+\;(sin^2\theta_{Lat})\,\omega'^2_{\phi(s)}}\;\;\hat{a}_{r}$

$\vec{d\Omega_{Map}}_{(\theta \phi)}\;\;=\;\;(\sqrt{-1})(\frac{\vec{c_{Light}}}{r})\,{dt_{Map}}\;$ $\,\,\,---> {radians}$

Differential – Map/Patch/Manifold – “Angle” – ( $\vec{d\Omega_{Map}}_{(\theta \phi)}$ ) – is a “geodesic arc-length” angle component, on the surface of a symmetric sphere and is related to the isotropic light speed “Aether Gravitational Field” in the following

${g_{Aether}}\;\;=\;\; \frac{c^2_{Light}}{r}$ $\,\,\,---> \frac{m^2}{s^2}$

$\vec{d\Omega_{Map}}_{(\theta \phi)}\;\;=\;\;(\sqrt{-1})(\frac{\vec{c_{Light}}}{r})\,{dt_{Map}}\;\;=\;\;(\sqrt{-1})(\frac{\vec{g_{Aether}}}{c_{Light}})\,{dt_{Map}}\;$

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“Differential” Latitude “Location” Angle

Differential – “Latitude “Location” Angle” – ( $\vec{d\theta_{Lat}}$ ) – is a vector (direction dependent) described in Rectangular Cartesian ( $\hat{a}_{x}\,,\,\hat{a}_{y}\,,\,\hat{a}_{z}$ ) Coordinate Axes components, along the “Latitude” Spherical ( $d\vec{\theta}$ ) Coordinate

$\vec{d\theta_{Lat}}\; = {d\theta_{Lat}}\,\hat{a}_{\theta}\; = \; \begin{bmatrix}[(cos\theta_{Lat}\,cos\phi_{Lon})\,\hat{a}_{x} \;+\;(\,cos\theta_{Lat}\,sin\phi_{Lon})\,\hat{a}_{y}\; -\;(sin\theta_{Lat})\,\hat{a}_{z}]\, d{\theta} \end{bmatrix}\;$

$\vec{d\theta_{Lat}}\; = {d\theta_{Lat}}\,\hat{a}_{\theta}\;=\;(\frac{dr_{\theta}}{r})\,\hat{a}_{\theta}$ $---> radians$

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Differential – “Latitude “Location” Angle” – ( $\vec{d\theta_{Lat}}$ ) – described by the spin, rotation, or torsion on the surface of a sphere, given by the “Latitude Angular “Spin/Rotation” Velocity” ( ${\omega_{\theta}}$ ) in the “Proper Observer” frame, and the “Latitude Angular “Spin/Rotation” Velocity” ( ${\omega'_{\theta(s)}}$ ) in the “External Observer” frame of reference

$\vec{d\theta_{Lat}}\;\; = \;\;{\omega_{\theta}}\,{dt_{Light}}\,\,\hat{a}_{\theta}\;\;=\;\;{\omega'_{\theta}}\,{dt'_{Light(s)}}\,\,\hat{a}_{\theta}$ $---> radians$

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“Differential” Longitude “Location” Angle

Differential – “Longitude “Location” Angle” – ( $\vec{d\phi_{Lon}}$ ) – is a vector (direction dependent) described in Rectangular Cartesian ( $\hat{a}_{x}\,,\,\hat{a}_{y}\,,\,\hat{a}_{z}$ ) Coordinate Axes components, along the “Longitude” Spherical ( $d\vec{\phi}$ ) Coordinate

$\vec{d\phi_{Lon}}\; = \;{ d\phi_{Lon}}\,\hat{a}_{\phi}\; = \; \begin{bmatrix} [(sin\phi_{Lon})\,\hat{a}_{x} \;\;\; +\;\;\;(cos\phi_{Lon})\,\hat{a}_{y}\;\;\; +\;\;\;(0)\,\hat{a}_{z})]\,d{\phi} \end{bmatrix}\;$

$\vec{d\phi_{Lon}}\; = \;{d\phi_{Lon}}\,\hat{a}_{\phi}\;=\;(\frac{dr_{\phi}}{r(sin\theta_{Lat})})\,\hat{a}_{\phi}$ $---> radians$

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Differential – “Longitude “Location”Angle” – ( $\vec{d\phi_{Lon}}$ ) – described by the spin, rotation, or torsion on the surface of a sphere, given by the “Longitude Angular “Spin/Rotation” Velocity” ( ${\omega_{\phi}}$ ) in the “Proper Observer” frame, and the “Longitude Angular “Spin/Rotation” Velocity” ( ${\omega'_{\phi(s)}}$ ) in the “External Observer” frame of reference

$\vec{d\phi_{Lon}}\;\; = \;\;{\omega_{\phi}}\,{dt_{Light}}\,\,\hat{a}_{\phi}\;\;=\;\;{\omega'_{\phi(s)}}\,{dt'_{Light(s)}}\,\,\hat{a}_{\phi}$ $---> radians$

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“Equal Observer (Co-variant)” Differential Spherical Space

Differential – Map/Patch/Manifold – “Geodesic” Arc Length – ( $\vec{ds_{Map}}_{\theta \, \phi}$ ) – is a vector (direction dependent) described in Spherical ( $d\vec{\theta}\,,d\vec{\phi}\,,(\sqrt{-1}){dt}$ ) Coordinates, and along Spherical ( $\hat{a}_{\theta}\,,\,\hat{a}_{\phi}$ ) Coordinate Axes

$\vec{ds_{Map}}_{\theta \, \phi}\;=\;(\sqrt{-1})\,\vec{c_{Light}}\,{dt_{Map}}\;= \; \begin{bmatrix} \vec{d{r}}_{\theta} \\[0.3em] \vec{d{r}}_{\phi} \end{bmatrix}\;= \; {r}\begin{bmatrix} d\theta_{Lat}\,\hat{a}_{\theta} \\[0.3em] sin\theta_{Lat}\,d\phi_{Lon}\,\hat{a}_{\phi} \end{bmatrix}\;$ $\,\,\,---> {m}$

Differential – Map/Patch/Manifold – “Geodesic” Arc Length – ( $\vec{ds_{Map}}_{\theta \, \phi}$ ) – is a vector (direction dependent) described in Rectangular ( $\hat{a}_{x}\,,\,\hat{a}_{y}\,,\,\hat{a}_{z}$ ) Cartesian Axes components, and along each Spherical ( $d\vec{\theta}\,,d\vec{\phi}\,,(\sqrt{-1}){dt}$ ) Coordinates

$\vec{ds_{Map}}_{\theta \, \phi}\;=\;{ds_{Map}}_{\theta \, \phi}\,\hat{a}_{r}\;=\;\begin{bmatrix}[(cos\theta_{Lat}\,cos\phi_{Lon})\,\hat{a}_{x} \;+\;(\,cos\theta_{Lat}\,sin\phi_{Lon})\,\hat{a}_{y}\; -\;(sin\theta_{Lat})\,\hat{a}_{z}]\,{d{r}}_{\theta}\\[0.3em] [(sin\phi_{Lon})\,\hat{a}_{x} \;\;\; +\;\;\;(cos\phi_{Lon})\,\hat{a}_{y}\;\;\; +\;\;\;(0)\,\hat{a}_{z})]\,{d{r}}_{\phi} \end{bmatrix}\;$

Differential – Map/Patch/Manifold – “Geodesic” Arc Length – ( $\vec{ds_{Map}}_{\theta \, \phi}$ ) – is a vector (direction dependent) described in Spherical ( $d\vec{r}\,,d\vec{\theta}\,,d\vec{\phi}\,, (\sqrt{-1}){dt}$ ) Coordinates mixed with Rectangular Cartesian ( $d\vec{x}\,,d\vec{y}\,,d\vec{z}$ ) Coordinates, and directed along the Rectangular ( $\hat{a}_{x}\,,\,\hat{a}_{y}\,,\,\hat{a}_{z}$ ) Axes

$\vec{ds_{Map}}_{\theta \, \phi}\;=\;(\sqrt{-1})\,\vec{c_{Light}}\,{dt_{Map}}\;=\;\begin{bmatrix} \vec{dx_{Map}} \\[0.3em] \vec{dy_{Map}} \\[0.3em] \vec{dz_{Map}} \end{bmatrix}\; = \begin{bmatrix} {ds_{Map}}_{\theta \, \phi}(sin\theta_{Lat}\,cos\phi_{Lon})\,\hat{a}_{x} \\[0.3em] {ds_{Map}}_{\theta \, \phi}(sin\theta_{Lat}\,sin\phi_{Lon})\,\hat{a}_{y} \\[0.3em] {ds_{Map}}_{\theta \, \phi}(cos\theta_{Lat})\,\hat{a}_{z} \end{bmatrix}\;$

Differential – Map/Patch/Manifold – “Geodesic” Arc Length – ( $\vec{ds_{Map}}_{\theta \, \phi}$ ) – is a “geodesic arc-length” spatial component, on the surface of the sphere, and changes as a function of the “Latitude Angle” ( $\vec{d{\theta}}$ ), and the “Longitude Angle” ( $\vec{d{\phi}}$ ) of a symmetric sphere

$\vec{ds_{Map}}_{\theta \, \phi}\;=\;{r}\,{\vec{d\Omega_{Map}}_{(\theta \phi)}}\;=\;(\sqrt{-1})\,\vec{c_{Light}}\,{dt_{Map}}\;$ $\,\,\,---> {m}$

$\vec{ds_{Map}}_{\theta \, \phi}\;=\;{r}\,{\vec{d\Omega_{Map}}_{(\theta \phi)}}\;=\;({r}\,\sqrt{d\theta^2_{Lat}\;+\;(sin^2\theta_{Lat})\,d\phi^2_{Lon}})\;\hat{a}_{r}$ $\,\,\,---> {m}$

Differential – Map/Patch/Manifold – “Geodesic” Arc Length – ( $\vec{ds_{Map}}_{\theta \, \phi}$ ) – is a “geodesic arc-length” spatial component, on the surface of the sphere, and changes as a function of the “Radius” ( $\vec{dr}$ ), and changes as a function of the “Euclidean Radius” ( $\vec{ds}$ ) of a symmetric sphere

$\vec{ds_{Map}}_{\theta \, \phi}\;=\;{r}\,{\vec{d\Omega_{Map}}_{(\theta \phi)}}\;=\;(\sqrt{-1})\,\vec{c_{Light}}\,{dt_{Map}}\;$ $\,\,\,---> {m}$

$\vec{ds_{Map}}_{\theta \, \phi}\;=\;(\sqrt{-1})[\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}]({dr})\;\hat{a}_{r}\;=\;(\sqrt{-1})[\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}]({ds})\;\hat{a}_{r}$

Differential – Map/Patch/Manifold – “Geodesic” Arc Length– ( $\vec{ds_{Map}}_{\theta \, \phi}$ ) – is a “geodesic arc-length” spatial component, on the surface of a symmetric sphere and is related to the isotropic light speed “Aether Gravitational Field Acceleration” in the following

$\vec{ds_{Map}}_{\theta \, \phi}\;=\;{r}\,{\vec{d\Omega_{Map}}_{(\theta \phi)}}\;=\;(\frac{c^2_{Light}}{g_{Aether}})\,{\vec{d\Omega_{Map}}_{(\theta \phi)}}\;$ $\,\,\,---> {m}$

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“Equal Observer (Co-variant)” Differential Map/Patch/Manifold Time

Differential – Map/Patch/Manifold Time – ( ${dt_{Map}}$ ) – measured in the “Equal Observer (C0-variant)” Frame of Reference

${dt_{Map}}\;=\;\frac{\vec{ds_{Map}}_{\theta \, \phi}}{(\sqrt{-1})\,\vec{c_{Light}}}\;=\;\frac{\sqrt{(\vec{ds})^2 \;\;-\;\; (\vec{dr})^2}}{(\sqrt{-1})\,\vec{c_{Light}}}\;$ $\,\,\,---> {s}$

${dt_{Map}}\;=\;(\frac{1}{\sqrt{-1}})\sqrt{({dt'_{Light(s)}})^2\;-\;({dt_{Light}})^2}\;$ $\,\,\,---> {s}$

${dt_{Map}}\;=\;\frac{{r}\,{\vec{d\Omega_{Map}}_{(\theta \phi)}}}{(\sqrt{-1})\,\vec{c_{Light}}}\;$ $\,\,\,---> {s}$

Differential – Map/Patch/Manifold Time – ( ${dt_{Map}}$ ) – measured in the “Equal Observer (C0-variant)” Frame of Reference, and relative to the “Proper Observer” center of mass frame of reference, and the “External Observer” frame of reference

${dt_{Map}}\;=\;\;{dt_{Light}}\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}\;\;=\;\;{dt'_{Light(s)}}\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}\;$ $\,\,\,---> {s}$

${dt_{Map}}\;=\;\;{dr}\sqrt{\frac{2(\frac{|v|_{CM}}{c^3_{Light}})}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}\;\;=\;\;{ds}\sqrt{\frac{2(\frac{|v|_{CM}}{c^3_{Light}})}{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}\;$ $\,\,\,---> {s}$

Differential – Map/Patch/Manifold Time – ( ${dt_{Map}}$ ) – measured in the “Equal Observer (C0-variant)” frame of reference, and described as a function of the “Synchronization Time” ( ${d\tau_{Sync}}$ ), and relative to the “Proper Observer” center of mass frame of reference, and the “External Observer” frame of reference

${dt_{Map}}\;=\;\;{d\tau_{Sync}}\sqrt{\frac{2(\frac{c_{Light}}{|v|_{CM}})}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}\;\;=\;\;{d\tau'_{Sync}}\sqrt{\frac{2(\frac{c_{Light}}{|v|_{CM}})}{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}\;$ $\,\,\,---> {s}$

${dt_{Map}}\;=\;\;((\frac{{|v|_{CM}}}{{c^2_{Light}} }){dr})\sqrt{\frac{2(\frac{c_{Light}}{|v|_{CM}})}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}\;\;=\;\;((\frac{{|v|_{CM}}}{{c^2_{Light}} }){ds})\sqrt{\frac{2(\frac{c_{Light}}{|v|_{CM}})}{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}\;$ $\,\,\,---> {s}$

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4) Aether Gradient Gravitational Field Acceleration

The Aether Gravitational Field Acceleration – ( ${g_{Aether}}$ ) for a specific potential in the gradient field is, isotropic, homogeneous, and invariant to all observers, and frames of reference, as demonstrated mathematically below.

Aether Gravitational Field Acceleration – ( ${g_{Aether}}-->\frac{m}{s^2}$ ) – function of Map/Patch/Manifold – “Geodesic” ( $\vec{ds^2_{Map}}_{\theta \, \phi}\,=\,\sqrt(-1){c_{Light}}\,{dt_{Map}})$ Space & Time

${g_{Aether}}\;\;=\;\; \frac{c^2_{Light}}{r}\;\;=\;\;({c^2_{Light}})\,\frac{\vec{d\Omega_{Map}}_{(\theta \phi)}}{\vec{ds_{Map}}_{\theta \, \phi}}\;\;=\;\;(\sqrt{-1})(\vec{c_{Light}})\,\frac{\vec{d\Omega_{Map}}_{(\theta \phi)}}{{dt_{Map}}}$ $\,\,\,---> \frac{m}{s^2}$

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Aether Gravitational Field Acceleration – ( ${g_{Aether}}-->\frac{m}{s^2}$ ) – function of Radius of Sphere ( ${dr\,=\,{c_{Light}}\,dt_{Light}}$ ) Space & Time

${g_{Aether}}\;\;=\;\;(\sqrt{-1})({c^2_{Light}})[\sqrt{\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\frac{\vec{d\Omega_{Map}}_{(\theta \phi)}}{\vec{dr}}\;\;=\;\;(\sqrt{-1})(\vec{c_{Light}})[\sqrt{\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\frac{\vec{d\Omega_{Map}}_{(\theta \phi)}}{{dt_{Light}}}$

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Aether Gravitational Field Acceleration – ( ${g_{Aether}}-->\frac{m}{s^2}$ ) – function of Euclidean Radius of Sphere ( ${ds}\,=\,{c_{Light}}\,dt'_{Light(s)}$ ) Space & Time

${g_{Aether}}\;\;=\;\;(\sqrt{-1})({c^2_{Light}})[\sqrt{\frac{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\frac{\vec{d\Omega_{Map}}_{(\theta \phi)}}{\vec{ds}}\;\;=\;\;(\sqrt{-1})(\vec{c_{Light}})[\sqrt{\frac{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\frac{\vec{d\Omega_{Map}}_{(\theta \phi)}}{{dt'_{Light(s)}}}$

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5) “Equal Frames (Invariant)” Center of Mass Velocity

Average Rectilinear Center of Mass Velocity – ( ${|\vec{v}|_{CM}}$ ) – , of Fluid particles of a Euclidean Sphere

Average Rectilinear Center of Mass Velocity – ( ${|\vec{v}|_{CM}}$ ) – is measured to have the same value, and is “Equal” in all “Frames of Reference” and is (Invariant) to all observers, and frames of reference

${|\vec{v}|_{CM}} \,\, = \,\, \frac{\displaystyle\sum_{i=1}^N {m_{i}}{v_{i}}}{m_{Net}}\;\;=\;\; \frac{{m_{1}}{v_{1}} + {m_{2}}{v_{2}} + {m_{3}}{v_{3}} + ...... + {m_{N}}{v_{N}}}{{m_{1}} + {m_{2}} + {m_{3}} + ...... + {m_{N}}}$ $\,\,----> \,\, \frac{m}{s}$

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Average Rectilinear Center of Mass Velocity – ( ${|\vec{v}|_{CM}}$ ) – is invariant and is measured to have the same value, and is described in the “Proper Observer” center of mass frame of reference

${|\vec{v}|_{CM}} \,\, =\,\,(-){c_{Light}}[\frac{(\vec{d\Omega_{Map}}_{(\theta \phi)})^2}{2\,(\frac{\vec{dr}}{r})^2\;\;+\;\;(\vec{d\Omega_{Map}}_{(\theta \phi)})^2}]$ $\,\,----> \,\, \frac{m}{s}$

Integrating the differential terms in the numerator and the denominator of the above equation yields the following.

${|\vec{v}|_{CM}} \,\, =\,\,(-){c_{Light}}[\frac{(\vec{\int{d\Omega_{Map}}_{(\theta \phi)}})^2}{2\,(\int_C^r{\frac{\vec{dr}}{r}})^2\;\;+\;\;(\int{\vec{d\Omega_{Map}}_{(\theta \phi)}})^2}]$ $\,\,----> \,\, \frac{m}{s}$

${|\vec{v}|_{CM}} \,\, =\,\,(-){c_{Light}}[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{r}{{r_{Schwarzschild}}}))^2\;\;+\;\;({\Omega_{Map}}_{(\theta \phi)})^2}]$ $\,\,----> \,\, \frac{m}{s}$

${|\vec{v}|_{CM}} \,\, =\,\,(-){c_{Light}}[\frac{[{\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,{\phi^2_{Lon}}]}{2\,(ln(\frac{r}{{r_{Schwarzschild}}}))^2\;\;+\;\;[{\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,{\phi^2_{Lon}}]}]$ $\,\,----> \,\, \frac{m}{s}$

For any Net Inertial Mass ( ${m_{Net}}$ ) the radius of the Euclidean spherical source of gravity, is the Black Hole Event Horizon Schwarzschild Radius ( ${r_{Schwarzschild}}\,=\,\frac{2\,{m_{Net}}\,G}{c^2_{Light}}$ ).

${|\vec{v}|_{CM}} \,\, =\,\,(-){c_{Light}}[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}\,r}{2\,{m_{Net}}\,G}))^2\;\;+\;\;({\Omega_{Map}}_{(\theta \phi)})^2}]$ $\,\,----> \,\, \frac{m}{s}$

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Average Rectilinear Center of Mass Velocity – ( ${|\vec{v}|_{CM}}$ ) – is invariant and measured to have the same value, and is described in the “External Observer” frame of reference

${|\vec{v}|_{CM}}\,\,=\,\,(-){c_{Light}}[\frac{(\vec{d\Omega_{Map}}_{(\theta \phi)})^2}{2\,(\frac{\vec{ds}}{s})^2\;\;+\;\;(\vec{d\Omega_{Map}}_{(\theta \phi)})^2}]$ $\,\,----> \,\, \frac{m}{s}$

Integrating the differential terms in the numerator and the denominator of the above equation.

${|\vec{v}|_{CM}}\;\;=\;\;(-){c_{Light}}[\frac{(\vec{\int{d\Omega_{Map}}_{(\theta \phi)}})^2}{2\,(\int_C^s{\frac{\vec{ds}}{s}})^2\;\;+\;\;(\int{\vec{d\Omega_{Map}}_{(\theta \phi)}})^2}]$ $\,\,----> \,\, \frac{m}{s}$

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From the above mathematics, it can be derived, the radius of the Euclidean Sphere in a gravitational field, relative to the isotropic light speed “Aether Gravitational Field” and the source of gravity, which is the Black Hole Event Horizon Schwarzschild Radius ( ${r_{Schwarzschild}}\,=\,\frac{2\,{m_{Net}}\,G}{c^2_{Light}}$ ).

${r}\,\,=\,\,{r_{Schwarzschild}}\,{e^{(\sqrt{-1})\,{\Omega_{Map}}_{(\theta \phi)}\sqrt{{\frac{1}{2}}(1\;\;+\;\; \frac{c_{Light}}{|\vec{v}|_{CM}})}}}$ $\,\,----> \,\, {m}$

Radius of Sphere

${r}\;\;=\;\;(\frac{2\,{m_{Net}}\,G}{c^2_{Light}})\,{e^{(\sqrt{-1})\,[\sqrt{{\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,{\phi^2_{Lon}}}]\sqrt{{\frac{1}{2}}(1\;\;+\;\; \frac{c_{Light}}{|\vec{v}|_{CM}})}}}$ $\,\,----> \,\, {m}$

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Euclidean Radius of Sphere

${s}\,\,=\,\,{r}\,\sqrt{1\;\;+\;\;{\Omega^2_{Map}}_{(\theta \phi)}}$ $\,\,----> \,\, {m}$

${s}\,\,=\,\,{r_{Schwarzschild}}\,(\sqrt{1\;+\;{\theta^2_{Lat}}\;+\; \sin^2\theta_{Lat}\,{\phi^2_{Lon}}})\,{e^{(\sqrt{-1})\,[\sqrt{{\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,{\phi^2_{Lon}}}]\sqrt{{\frac{1}{2}}(1\;\;+\;\; \frac{c_{Light}}{|\vec{v}|_{CM}})}}}$

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Map/Patch/Manifold – “Geodesic” Arc Length

${s_{Map}}_{\theta \, \phi}\,\,=\,\,{r}\,{\Omega_{Map}}_{(\theta \phi)}\,\,=\,\,{r}\, \sqrt{{\theta^2_{Lat}}\;\;+\;\;\sin^2\theta_{Lat}\,{\phi^2_{Lon}}}$ $\,\,----> \,\, {m}$

${s_{Map}}_{\theta \, \phi}\,\,=\,\,(\sqrt{-1})\,{r}\,[\frac{ln(\frac{r}{{r_{Schwarzschild}}})}{\sqrt{{\frac{1}{2}}(1\;\;+\;\; \frac{c_{Light}}{|\vec{v}|_{CM}})}}]\,\,=\,\,(\sqrt{-1})\,(\frac{{m_{Net}}\,G}{v^2_{Gravity}})\,[\frac{ln(\frac{r}{{r_{Schwarzschild}}})}{\sqrt{{\frac{1}{2}}(1\;\;+\;\; \frac{c_{Light}}{|\vec{v}|_{CM}})}}]$

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Aether Gravitational Field Acceleration

${g_{Aether}}\,\,=\,\,\frac{c^2_{Light}}{r}\,\,=\,\,(\sqrt{-1})\,(\frac{c^2_{Light}}{{s_{Map}}_{\theta \, \phi}})\,[\frac{ln(\frac{r}{{r_{Schwarzschild}}})}{\sqrt{{\frac{1}{2}}(1\;\;+\;\; \frac{c_{Light}}{|\vec{v}|_{CM}})}}]$ $\,\,----> \,\, \frac{m}{s^2}$

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${g_{Aether}}\,\,=\,\,(\frac{c^2_{Light}}{s})\sqrt{1\;\;+\;\;{\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,{\phi^2_{Lon}}}$

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Citation

Cite this article as:

Robert Louis Kemp; The Super Principia Mathematica – The Rage to Master Conceptual & Mathematical Physics – The General Theory of Relativity – “Euclidean Spherical Mechanics – Spacetime Frames of Reference – (Differential Mathematical Form)” – Online Volume – ISBN 978-0-9841518-2-0, Volume 3; July 2010

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The above work completes the desire of Albert Einstein, which was to describe Special Relativity and General Relativity into a complete conceptual and mathematical model of matter, space, and time.

Best,

Author: Robert Louis Kemp

http://www.SuperPrincipia.com

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