# Euclidean Spherical Mechanics – Spacetime Metrics – (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.

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 – Spacetime Metric – Radius of Sphere – ($\vec{dr^2}$) – is a scalar (direction independent) and is described in Rectangular ($d\vec{x}\,,d\vec{y}\,,d\vec{z}\,,{dt}$) Cartesian Coordinates

$\vec{dr^2}\;=\;\vec{c^2_{Light}}\,{dt^2_{Light}}\;=\; [\vec{dx^2}\;+\;\vec{dy^2}\;+\;\vec{dz^2}]$ $\,\,\,---> {m^2}$

Differential – Spacetime Metric – Radius of Sphere – ($\vec{dr^2}$) – is a scalar (direction independent) and is described in Rectangular ($d\vec{x}\,,d\vec{y}\,,d\vec{z}\,,{dt}$) Cartesian Coordinates

$\vec{dr^2}\;=\; [\vec{ds^2}\;-\;\vec{ds^2_{Map}}_{\theta \, \phi}]\;=\; [\vec{dx^2}\;+\;\vec{dy^2}\;+\;\vec{dz^2}]$ $\,\,\,---> {m^2}$

$\vec{dr^2}\;=\;\vec{c^2_{Light}}\,{dt^2_{Light}}\;=\; [\vec{c^2_{Light}}\,{dt'^2_{Light(s)}}\;+\;\vec{c^2_{Light}}\,{dt^2_{Map}}]$  $\,\,---> {m^2}$

Differential – Spacetime Metric Radius of Sphere – ($\vec{dr^2}$) – measured in the “Proper Observer” center of mass frame of reference, and described as function of the Euclidean Metric Radius of Sphere ($\vec{ds^2}$), relative to the “External Observer” frame of reference; and likewise is described as a function of the Map/Patch/Manifold ($\vec{ds^2_{Map}}_{\theta \, \phi}$) “Geodesic” Metric, relative to the “Equal Observer (Co-variant)” frame of reference

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

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Differential – Spacetime Metric Radius of Sphere – ($\vec{dr^2}$) – “Special Relativity – Lorentz Transformation” from the “External Observer” frame of reference, into the “Proper Observer” center of mass frame of reference

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

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Differential – Spacetime Metric Radius of Sphere – ($\vec{dr^2}$) – “General Relativity – Euclidean Transformation” from the “External Observer” frame of reference, into the “Proper Observer” center of mass frame of reference

$\vec{dr^2}\;=\;\frac{3\,\vec{ds^2}\;\;-\;\;[\frac{(1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{c_{Light}}{|v|_{CM}})}]\,\vec{ds^2_{Map}}_{\theta \, \phi}}{(1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}})}\;=\;\frac{3\,\vec{ds^2}\;\;+\;\;[\frac{(1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{c_{Light}}{|v|_{CM}})}]\,\vec{c^2_{Light}}\,{dt^2_{Map}}}{(1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}})}$ $\,\,\,---> {m^2}$

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### “Proper Observer” Differential “Light Clock” Time-Like Metric

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

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

Differential – Proper Observer “Light Clock” Time-Like Metric – (${dt^2_{Light}}$) – measured in the “Proper Observer” center of mass frame of reference, and described as function of the External Observer “Light Clock” Time-Like Metric ($\vec{dt'^2_{Light(s)}}$), relative to the “External Observer” frame of reference; and likewise is described as a function of the Map/Patch/Manifold Time-Like Metric ($\vec{dt^2_{Map}}_{\theta \, \phi}$), relative to the “Equal Observer (Co-variant)” frame of reference

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

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Differential – Proper Observer “Light Clock” Time-Like Metric – (${dt^2_{Light}}$) – “Special Relativity – Lorentz Transformation” from the “External Observer” frame of reference, into the “Proper Observer” center of mass frame of reference

${dt^2_{Light}}\;=\;\frac{{dt'^2_{Light(s)}}{(1\;\;+\;\;2\,(\frac{|v|_{CM}}{c_{Light}}))}\;\;+\;\;(\frac{\vec{|v|^2_{CM}}}{c^4_{Light}})\,{ds^2}}{(1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}})}$ $\,\,\,---> {s^2}$

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Differential – Proper Observer “Light Clock” Time-Like Metric – (${dt^2_{Light}}$) – “General Relativity – Euclidean Transformation” from the “External Observer” frame of reference, into the “Proper Observer” center of mass frame of reference

${dt^2_{Light}}\;=\;\frac{3\,{dt'^2_{Light(s)}}\;\;-\;\;(\frac{1}{c^2_{Light}})[\frac{(1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{c_{Light}}{|v|_{CM}})}]\,\vec{ds^2_{Map}}_{\theta \, \phi}}{(1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}})}\;=\;\frac{3\,{dt'^2_{Light(s)}}\;\;+\;\;[\frac{(1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{c_{Light}}{|v|_{CM}})}]\,{dt^2_{Map}}}{(1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}})}$

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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 Spacetime Metric Radius of Sphere – ($\vec{ds^2}$) – is a scalar (direction independent) and is described in Rectangular ($d\vec{x}\,,d\vec{y}\,,d\vec{z}\,,{dt}$) Cartesian Coordinates

$\vec{ds^2}\;=\;\vec{c^2_{Light}}\,{dt'^2_{Light(s)}}\;=\; [\vec{dx'^2_s}\;+\;\vec{dy'^2_s}\;+\;\vec{dz'^2_s}]$ $\,\,---> {m^2}$

Differential – Euclidean Spacetime Metric Radius of Sphere – ($\vec{ds^2}$) – is a scalar (direction independent) and is described in Rectangular ($d\vec{x}\,,d\vec{y}\,,d\vec{z}\,,{dt}$) Cartesian Coordinates

$\vec{ds^2}\;=\; [\vec{dr^2}\;+\;\vec{ds^2_{Map}}_{\theta \, \phi}]\;=\; [\vec{dx'^2_s}\;+\;\vec{dy'^2_s}\;+\;\vec{dz'^2_s}]$ $\,\,\,---> {m^2}$

$\vec{ds^2}\;=\;\vec{c^2_{Light}}\,{dt'^2_{Light(s)}}\;=\; [\vec{c^2_{Light}}\,{dt^2_{Light}}\;-\;\vec{c^2_{Light}}\,{dt^2_{Map}}]$  $\,\,---> {m^2}$

Differential – Euclidean Spacetime Metric Radius of Sphere – ($\vec{ds^2}$) – is a scalar (direction independent) and is described in Spherical ($d\vec{r}\,,d\vec{\theta}\,,d\vec{\phi}\,,{dt}$) Coordinates

$\vec{ds^2}\; = \; [\vec{d{r^2}}\;\;+\;\;\vec{ds^2_{Map}}_{\theta \, \phi}]\;\;=\;\; [\vec{d{r^2}}\;\;+\;\;\vec{d{r^2}}_{\theta}\;\;+\;\;\vec{d{r^2}}_{\phi}]\; =\;$ $\,\,\,---> {m^2}$

$\vec{ds^2}\;=\;\vec{c^2_{Light}}\,{dt'^2_{Light(s)}}\;=\; [{dr^2}\;\;+\;\;{r^2}\;d\theta^2_{Lat}\;\;+\;\;{r^2}\,sin^2\theta_{Lat}\,d\phi^2_{Lon}]\;$ $\,\,\,---> {m^2}$

Differential – Euclidean Spacetime Radius of Sphere – ($\vec{ds^2}$) – is a scalar (direction independent) and is described in Spherical ($d\vec{r}\,,d\vec{\theta}\,,d\vec{\phi}\,,{dt}$) Coordinates mixed with Rectangular ($d\vec{x}\,,d\vec{y}\,,d\vec{z}\,,{dt}$) Coordinates

$\vec{ds^2}\;=\; [{dx^2}\;\;+\;\;{dy^2}\;\;+\;\;{dz^2}\;\;+\;\;{r^2}\;d\theta^2_{Lat}\;\;+\;\;{r^2}\,sin^2\theta_{Lat}\,d\phi^2_{Lon}]\;$ $\,\,\,---> {m^2}$

$\vec{ds^2}\;=\;\vec{c^2_{Light}}\,{dt'^2_{Light(s)}}\;=\;[{dx'^2_{s}}\;+\;{dy'^2_{s}}\;+\;{dz'^2_{s}}] \;$ $\,\,\,---> {m^2}$

Differential – Euclidean Spacetime Metric Radius of Sphere – ($\vec{ds^2}$) – measured in the “External Observer” frame of reference, is described by one (1) component of “metric radius” ($\vec{d{r^2}}$), and one (1) “geodesic arc-length” component, which is the “Map/Patch/Manifold geodesic metric” ($\vec{ds^2_{Map}}_{\theta \, \phi}$) surface of the sphere, and changes as a function of the “Latitude Angle Metric” ($\vec{d{\theta}^2}$), and the “Longitude Angle Metric” ($\vec{d{\phi}^2}$)

$\vec{ds^2}\; = \; [\vec{d{r^2}}\;\;+\;\;\vec{ds^2_{Map}}_{\theta \, \phi}]\;\;=\;\; [\vec{d{r^2}}\;\;+\;\;{r}\,{\vec{d\Omega_{Map}}_{(\theta \phi)}}]\; =\;$ $\,\,\,---> {m^2}$

$\vec{ds^2}\;=\;\vec{c^2_{Light}}\,{dt'^2_{Light(s)}}\;=\; [{dr^2}\;\;+\;\;{r^2}\;d\theta^2_{Lat}\;\;+\;\;{r^2}\,sin^2\theta_{Lat}\,d\phi^2_{Lon}]\;$ $\,\,\,---> {m^2}$

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

Differential – Euclidean Spacetime Metric Radius of Sphere – ($\vec{ds^2}$) – measured in the “External Observer” frame of reference, and described as function of the Metric Radius of Sphere ($\vec{dr^2}$), relative to the “Proper Observer” center of mass frame of reference; and likewise is described as a function of the Map/Patch/Manifold “Geodesic” ($\vec{ds^2_{Map}}_{\theta \, \phi}$) Metric, relative to the “Equal Observer (Co-variant)” frame of reference

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

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Differential – Euclidean Spacetime Metric Radius of Sphere – ($\vec{ds^2}$) – “Special Relativity – Lorentz Transformation” from the “Proper Observer” center of mass frame of reference, into the “External Observer” frame of reference

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

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Differential – Euclidean Spacetime Metric Radius of Sphere – ($\vec{ds^2}$) – “General Relativity – Euclidean Transformation” from the “Proper Observer” center of mass frame of reference into the “External Observer” frame of reference

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

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### “External Observer” Differential “Light Clock” Time-Like Metric

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

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

Differential – External Observer “Light Clock” Time-Like Metric – (${dt'^2_{Light(s)}}$) – measured in the “External Observer” frame of reference, and described as function of the Proper Observer “Light Clock” Time-Like Metric (${dt^2_{Light}}$), relative to the “Proper Observer” center of mass frame of reference; and likewise is described as a function of the Map/Patch/Manifold Time-Like Metric ($\vec{dt^2_{Map}}_{\theta \, \phi}$), relative to the “Equal Observer (Co-variant)” frame of reference

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

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Differential – External Observer “Light Clock” Time-Like Metric – (${dt'^2_{Light(s)}}$) – “Special Relativity – Lorentz Transformation” from the “Proper Observer” center of mass frame of reference into the “External Observer” frame of reference

${dt'^2_{Light(s)}}\;=\;\frac{{dt^2_{Light}}{(1\;\;-\;\;2\,(\frac{|v|_{CM}}{c_{Light}}))}\;\;+\;\;(\frac{\vec{|v|^2_{CM}}}{c^4_{Light}})\,{dr^2}}{(1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}})}$ $\,\,\,---> {s^2}$

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Differential – External Observer “Light Clock” Time-Like Metric – (${dt'^2_{Light(s)}}$) – “General Relativity – Euclidean Transformation” from the “Proper Observer” center of mass frame of reference into the “External Observer” frame of reference

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

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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 Metric

Differential – Map/Patch/Manifold – “Angle Metric” – ($\vec{d\Omega^2_{Map}}_{(\theta \phi)}$) – is a scalar (direction independent) and is described in Spherical ($d\vec{\theta}\,,d\vec{\phi}\,,{(\sqrt{-1})dt}$) Coordinates

$\vec{d\Omega^2_{Map}}_{(\theta \phi)}\; =\; [\frac{\vec{d{r^2}}_{\theta}}{r^2}\;\;+\;\;\frac{\vec{d{r^2}}_{\phi}}{r^2}]\;=\;[\vec{d\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,\vec{d\phi^2_{Lon}}]\;$ $\,\,\,---> {radians^2}$

$\vec{d\Omega^2_{Map}}_{(\theta \phi)}\;\;=\;\;\frac{\vec{ds^2_{Map}}_{\theta \, \phi}}{r^2}\;\;=\;\;(-)(\frac{\vec{c^2_{Light}}}{r^2})\,{dt^2_{Map}}\;$ $\,\,\,---> {radians^2}$

Differential – Map/Patch/Manifold – “Angle Metric” – ($\vec{d\Omega^2_{Map}}_{(\theta \phi)}$) – is a “geodesic arc-length” angle component, on the surface of the sphere, and changes as a function of the “radius metric” ($\vec{dr^2}$), and changes as a function of the “Euclidean Radius metric” ($\vec{ds^2}$) of a symmetric sphere

$\vec{d\Omega^2_{Map}}_{(\theta \phi)}\;=\;(-1)[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}](\frac{\vec{dr^2}}{r^2})\;=\;(-1)[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}](\frac{\vec{ds^2}}{r^2})$ $\,\,\,---> {radians^2}$

$\vec{d\Omega^2_{Map}}_{(\theta \phi)}\;=\;(-1)[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}](\frac{\vec{ds^2}}{s^2})$

Differential – Map/Patch/Manifold – “Angle Spacetime Metric” – ($\vec{d\Omega^2_{Map}}_{(\theta \phi)}$) – 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 square of the Map/Patch/Manifold Angular “Spin/Rotation” Velocity” (${\omega^2_{\Omega}}$) in the “Proper Observer” frame, the square of the  “Map/Patch/Manifold Angular “Spin/Rotation” Velocity” (${\omega'^2_{\Omega(s)}}$) in the “External Observer” frame of reference

$\vec{d\Omega^2_{Map}}_{(\theta \phi)}\;\;=\;\;{\omega^2_{\Omega}}\,{dt^2_{Light}}\;\;=\;{dt^2_{Light}}\,[{\omega^2_{\theta}\;+\;(sin^2\theta_{Lat})\,\omega^2_{\phi}}]$

$\vec{d\Omega^2_{Map}}_{(\theta \phi)}\;\;=\;\;{\omega'^2_{\Omega(s)}}\,{dt'^2_{Light(s)}}\;\;=\;\;{dt'^2_{Light(s)}}\,[{\omega'^2_{\theta(s)}\;+\;(sin^2\theta_{Lat})\,\omega'^2_{\phi(s)}}]$

$\vec{d\Omega^2_{Map}}_{(\theta \phi)}\;\;=\;\;(-)(\frac{\vec{c^2_{Light}}}{r^2})\,{dt^2_{Map}}\;$ $\,\,\,---> {radians^2}$

Differential – Map/Patch/Manifold – “Angle Metric” – ($\vec{d\Omega^2_{Map}}_{(\theta \phi)}$) – is a “geodesic arc-length” angle component, on the surface of a symmetric sphere and is related to the “Gravitational Field” in the following

${g_{Gravity}}\;\;=\;\;\frac{m_{Net}\,G}{r^2}\;$ $\,\,\,---> \frac{m}{s^2}$

$\vec{d\Omega^2_{Map}}_{(\theta \phi)}\;\;=\;\;(-)(\frac{\vec{c^2_{Light}}}{r^2})\,{dt^2_{Map}}\;\;=\;\;(-){g_{Gravity}}\,(\frac{\vec{c^2_{Light}}}{m_{Net}\,G})\,{dt^2_{Map}}\;$

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### “Differential” Latitude “Location” Angle Metric

Differential – “Latitude “Location” Angle Metric” – ($\vec{d\theta^2_{Lat}}$) – is a scalar (direction independent) and is described along the “Latitude” Spherical ($d\vec{\theta}$) Coordinate. And, likewise is described by the spin, rotation, or torsion on the surface of a sphere, given by the square of the “Latitude Angular “Spin/Rotation” Velocity” (${\omega^2_{\theta}}$) in the “Proper Observer” frame, and the square of the “Latitude Angular “Spin/Rotation” Velocity” (${\omega'_{\theta(s)}}$) in the “External Observer” frame of reference

$\vec{d\theta^2_{Lat}}\; = \;(\frac{\vec{dr^2_{\theta}}}{r^2})$ $---> radians^2$

$\vec{d\theta^2_{Lat}}\;\; = \;\;{\omega^2_{\theta}}\,{dt^2_{Light}}\;\;=\;\;{\omega'^2_{\theta}}\,{dt'^2_{Light(s)}}$ $---> radians^2$

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### “Differential” Longitude “Location” Angle Metric

Differential – “Longitude “Location” Angle Metric” – ($\vec{d\phi^2_{Lon}}$) – is a scalar (direction independent) and is described along the “Longitude” Spherical ($d\vec{\phi}$) Coordinate. And, likewise is described by the spin, rotation, or torsion on the surface of a sphere, given by the square of the “Longitude Angular “Spin/Rotation” Velocity” (${\omega^2_{\phi}}$) in the “Proper Observer” frame, and the square of the “Longitude Angular “Spin/Rotation” Velocity” (${\omega'^2_{\phi(s)}}$) in the “External Observer” frame of reference

$\vec{d\phi^2_{Lon}}\; =\;(\frac{\vec{dr^2_{\phi}}}{r^2\,(sin^2\theta^2_{Lat})})$ $---> radians^2$

$\vec{d\phi^2_{Lon}}\;\; = \;\;{\omega^2_{\phi}}\,{dt^2_{Light}}\;\;=\;\;{\omega'^2_{\phi(s)}}\,{dt'^2_{Light(s)}}$ $---> radians^2$

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

Differential – Map/Patch/Manifold – “Geodesic” Metric – ($\vec{ds^2_{Map}}_{\theta \, \phi}$) – is a scalar (direction independent) and is described in Rectangular ($d\vec{x}\,,d\vec{y}\,,d\vec{z}\,,(\sqrt{-1}){dt}$) Cartesian Coordinates

$\vec{ds^2_{Map}}_{\theta \, \phi}\;=\;(-)\vec{c^2_{Light}}\,{dt^2_{Map}}\;=\; [\vec{dx^2_{Map}}\;+\;\vec{dy^2_{Map}}\;+\;\vec{dz^2_{Map}}]$ $\,\,---> {m^2}$

Differential – Map/Patch/Manifold – “Geodesic” Metric – ($\vec{ds^2_{Map}}_{\theta \, \phi}$) – is a scalar (direction independent) and is described in Rectangular ($d\vec{x}\,,d\vec{y}\,,d\vec{z}\,,(\sqrt{-1}){dt}$) Cartesian Coordinates

$\vec{ds^2_{Map}}_{\theta \, \phi}\;=\; [\vec{ds^2}\;-\;\vec{dr^2}]\;=\; [\vec{dx^2_{Map}}\;+\;\vec{dy^2_{Map}}\;+\;\vec{dz^2_{Map}}]$ $\,\,\,---> {m^2}$

$\vec{ds^2_{Map}}_{\theta \, \phi}\;=\;(-)\vec{c^2_{Light}}\,{dt^2_{Map}}\;=\; [\vec{c^2_{Light}}\,{dt'^2_{Light(s)}}\;-\;\vec{c^2_{Light}}\,{dt^2_{Light}}]$  $\,\,---> {m^2}$

Differential – Map/Patch/Manifold – “Geodesic” Metric – ($\vec{ds^2_{Map}}_{\theta \, \phi}$) – is a scalar (direction independent) and is described in Spherical ($d\vec{\theta}\,,d\vec{\phi}\,,(\sqrt{-1}){dt}$) Coordinates

$\vec{ds_{Map}}_{\theta \, \phi}\;=\; [\vec{ds^2}\;-\;\vec{dr^2}]\;=\;\; [\vec{d{r^2}}_{\theta}\;\;+\;\;\vec{d{r^2}}_{\phi}]\; =\;$ $\,\,\,---> {m^2}$

$\vec{ds^2_{Map}}_{\theta \, \phi}\;=\;(-)\vec{c^2_{Light}}\,{dt^2_{Map}}\;=\; [{r^2}\;d\theta^2_{Lat}\;\;+\;\;{r^2}\,sin^2\theta_{Lat}\,d\phi^2_{Lon}]\;$ $\,\,\,---> {m^2}$

Differential – Map/Patch/Manifold – “Geodesic” Metric – ($\vec{ds^2_{Map}}_{\theta \, \phi}$) – is a “geodesic arc-length” spatial component, on the surface of the sphere, and changes as a function of the “Latitude Angle Metric” ($\vec{d{\theta}^2}$), and the “Longitude Angle Metric” ($\vec{d{\phi}^2}$) of a symmetric sphere

$\vec{ds^2_{Map}}_{\theta \, \phi}\;=\;{r^2}\,{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}\;=\;(-)\vec{c^2_{Light}}\,{dt^2_{Map}}\;$ $\,\,\,---> {m^2}$

$\vec{ds^2_{Map}}_{\theta \, \phi}\;=\;{r^2}\,{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}\;=\;{r^2}\,(d\theta^2_{Lat}\;+\;(sin^2\theta_{Lat})\,d\phi^2_{Lon})$ $\,\,\,---> {m^2}$

Differential – Map/Patch/Manifold – “Geodesic” Metric – ($\vec{ds^2_{Map}}_{\theta \, \phi}$) – is a “geodesic arc-length” spatial component, on the surface of the sphere, and changes as a function of the “Radius Metric” ($\vec{dr^2}$), and changes as a function of the “Euclidean Radius Metric” ($\vec{ds^2}$) of a symmetric sphere

$\vec{ds^2_{Map}}_{\theta \, \phi}\;=\;{r^2}\,{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}\;=\;(-)\vec{c^2_{Light}}\,{dt^2_{Map}}\;$ $\,\,\,---> {m^2}$

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

Differential – Map/Patch/Manifold – “Geodesic” Metric– ($\vec{ds^2_{Map}}_{\theta \, \phi}$) – is a “geodesic arc-length” spatial component, on the surface of a symmetric sphere and is related to the “Gravitational Field Acceleration” in the following

$\vec{ds^2_{Map}}_{\theta \, \phi}\;=\;{r^2}\,{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}\;=\;(\frac{m_{Net}\,G}{{g_{Gravity}}})\,{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}\;\;$ $\,\,\,---> {m^2}$

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

Differential – Map/Patch/Manifold Time-Like Metric – (${dt^2_{Map}}$) – measured in the “Equal Observer (C0-variant)” Frame of Reference

${dt^2_{Map}}\;=\;(-)[\frac{\vec{ds^2_{Map}}_{\theta \, \phi}}{\vec{c^2_{Light}}}]\;=\;(-)[\frac{(\vec{ds})^2 \;\;-\;\; (\vec{dr})^2}{\vec{c^2_{Light}}}]\;$ $\,\,\,---> {s^2}$

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

${dt^2_{Map}}\;=\;(-)[\frac{{r^2}\,{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}}{\vec{c^2_{Light}}}]\;$ $\,\,\,---> {s}$

Differential – Map/Patch/Manifold Time-Like Metric – (${dt^2_{Map}}$) – 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

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

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

Differential – Map/Patch/Manifold Time-Like Metric – (${dt^2_{Map}}$) – measured in the “Equal Observer (C0-variant)” frame of reference, and described as a function of the “Synchronization Time Metric” (${d\tau^2_{Sync}}$), and relative to the “Proper Observer” center of mass frame of reference, and the “External Observer” frame of reference

${dt^2_{Map}}\;=\;\;{d\tau^2_{Sync}}[{\frac{2(\frac{c_{Light}}{|v|_{CM}})}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}]\;\;=\;\;{d\tau'^2_{Sync}}[{\frac{2(\frac{c_{Light}}{|v|_{CM}})}{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}]\;$ $\,\,\,---> {s^2}$

${dt^2_{Map}}\;=\;\;((\frac{{|v|^2_{CM}}}{{c^4_{Light}} }){dr^2})[{\frac{2(\frac{c_{Light}}{|v|_{CM}})}{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}]\;\;=\;\;((\frac{{|v|^2_{CM}}}{{c^4_{Light}} }){ds^2})[{\frac{2(\frac{c_{Light}}{|v|_{CM}})}{1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}]\;$ $\,\,\,---> {s}$

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## 4) The Gradient Gravitational “Acceleration” described in the form of a “Elastic Wave Equation” – in consideration for Special Relativity & General Relativity

The Inertial Mass Gradient Gravitational Field Acceleration ($\vec{g_{Gravity}}$) varies as a function of “space” and “time” in each spherical volume potential of the gravity field, such that the larger the volume potential, the slower the acceleration towards the center of the gradient gravity field; and the smaller the volume potential, the faster the acceleration towards the center of the gradient gravity field; and can be described in the form of the second order partial differential “Elastic Wave Equation.”

In a consideration for General Relativity, we will need to obtain the equations for the Gradient Gravitational Field Acceleration ($\vec{g_{Gravity}}$) as a function of the “Space-Time Metrics”, which were derived in Section 4, of the work:

Euclidean Spherical Mechanics – Euclidean/Minkowski Spacetime Metrics

Only the “Proper Observer” center of mass frame of reference, “Elastic Wave”, Gravitational Field Acceleration(${g_{Gravity}}$) will be described below. Limiting, the discussion to the “Proper Observer” center of mass frame of reference is done for the main reason, that is the frame that the mechanics and mathematics, of the classical discusions of gravity, are most commonly discussed.

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Proper ObserverGradient Gravitational Field Acceleration(${g_{Gravity}}$)second order partial differential “Elastic Wave Equation” function of Radius of Sphere (${dr\,=\,{c_{Light}}\,dt_{Light}}$) Space & Time Metric and in the “Proper Observer” frame of reference.

${g_{Gravity}}\;\;=\;\;(-)({m_{Net}\,G})\,[\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}]\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{dr^2}$$\,\,----> \,\, \frac{m}{s^2}$

${g_{Gravity}}\;\;=\;\;(-)(\frac{m_{Net}\,G}{c^2_{Light}})\,[\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}]\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{dt^2_{Light}}$

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Proper Observer – Gradient Gravitational Field Acceleration(${g_{Gravity}}$)second order partial differential “Elastic Wave Equation” function of Map/Patch/Manifold – “Geodesic” ($\vec{ds_{Map}}_{\theta \, \phi}\,=\,(\sqrt{-1}){c_{Light}}\,{dt_{Map}}$) “Equal Observer (Co-Variant)” Space & Time Metric and in the “Proper Observer” frame of reference.

${g_{Gravity}}\;\;=\;\;\frac{m_{Net}\,G}{r^2}\;=\;({m_{Net}\,G})\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{\vec{ds^2_{Map}}_{\theta \, \phi}}$$\,\,----> \,\, \frac{m}{s^2}$

${g_{Gravity}}\;\;=\;\;(-)(\frac{m_{Net}\,G}{c^2_{Light}})\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{dt^2_{Map}}$

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Proper Observer – Gradient Gravitational Field Acceleration(${g_{Gravity}}$) – second order partial differential “Elastic Wave Equation” function of Euclidean Radius of Sphere (${ds}\,=\,{c_{Light}}\,dt'_{Light(s)}$) Space & Time Metric and in the “External Observer” frame of reference.

${g_{Gravity}}\;\;=\;\;(-)({m'_{Rel}\,G})\,[\frac{\sqrt{(1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}})^3\,(1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}})}}{2(\frac{|v|_{CM}}{c_{Light}})}]\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{ds^2}$$\,\,----> \,\, \frac{m}{s^2}$

${g_{Gravity}}\;\;=\;\;(-)(\frac{m'_{Rel}\,G}{c^2_{Light}})\,[\frac{\sqrt{(1 \;\;-\;\;\frac{|v|_{CM}}{c_{Light}})^3\,(1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}})}}{2(\frac{|v|_{CM}}{c_{Light}})}]\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{dt'^2_{Light(s)}}$

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Proper Observer – Gradient Gravitational Field Acceleration(${g_{Gravity}}$)second order partial differential “Elastic Wave Equation” function of Map/Patch/Manifold – “Geodesic” ($\vec{ds_{Map}}_{\theta \, \phi}\,=\,(\sqrt{-1}){c_{Light}}\,{dt_{Map}})$ Space & Time Metric and in the “External Observer” frame of reference.

${g_{Gravity}}\;=\;\frac{m_{Net}\,G}{r^2}\;=\;({m'_{Rel}\,G})\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{\vec{ds^2_{Map}}_{\theta \, \phi}}\sqrt{1\;-\;\frac{|v|^2_{CM}}{c^2_{Light}}}$$\,\,----> \,\, \frac{m}{s^2}$

${g_{Gravity}}\;=\;(-)(\frac{m'_{Rel}\,G}{c^2_{Light}})\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{dt^2_{Map}}\sqrt{1\;-\;\frac{|v|^2_{CM}}{c^2_{Light}}}$

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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.

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

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{s}{{r_{Schwarzschild}}}))^2\;\;+\;\;({\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  source of gravity, which is the Black Hole Event Horizon Schwarzschild Radius (${r_{Schwarzschild}}\,=\,\frac{2\,{m_{Net}}\,G}{c^2_{Light}}$).

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

${r^2}\;\;=\;\;(\frac{4\,{m^2_{Net}}\,G^2}{c^4_{Light}})\,{e^{(-)\,[{{\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,{\phi^2_{Lon}]}}[{{\frac{1}{2}}(1\;\;+\;\; \frac{c_{Light}}{|\vec{v}|_{CM}})}]}}$  $\,\,----> \,\, {m^2}$

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Euclidean Radius of Sphere

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

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

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Map/Patch/Manifold – “Geodesic” Metric

${s^2_{Map}}_{\theta \, \phi}\,\,=\,\,{r^2}\,{\Omega^2_{Map}}_{(\theta \phi)}\,\,=\,\,{r^2}\, [{{\theta^2_{Lat}}\;\;+\;\;\sin^2\theta_{Lat}\,{\phi^2_{Lon}}}]$  $\,\,----> \,\, {m^2}$

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

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Gravitatational Field Acceleration

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

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${g_{Gravity}}\,\,=\,\,(\frac{m_{Net}\,G}{s^2})[1\;\;+\;\;{{\theta^2_{Lat}}\;\;+\;\;\sin^2\theta_{Lat}\,{\phi^2_{Lon}}}]$

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### Citation

Robert Louis Kemp; The Super Principia Mathematica – The Rage to Master Conceptual & Mathematical Physics – The General Theory of Relativity – “Euclidean Spherical Mechanics – Spacetime Metrics – (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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### 6 Responses to Euclidean Spherical Mechanics – Euclidean/Minkowski Spacetime Metrics

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4. Krixhna Vinjamuri says:

The Schwarzschild radius of a spherical Primordial Quantum Black Hole (PQBH) or its time reversed PQWhite Hole (PQWH) is 2planck lengths. In the Schwarzschild event sphere, the number of PQBH and PQWH are 8. In our Universe the number of these odd balls are about 10^182. Asymmetry between these two twins produced our beautiful Universe. Dr. Vinn