**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

**and**

*“Special Theory of Relativity”***into a single framework known as the**

*“General Theory of Relativity”*

*“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

**describes a symmetrically spherical, space-time system body, where the**

*Euclidean Spacetime Metric**“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

*“spatial component”, plus the square of the*

**“Radial”****“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

*and the*

**“Proper Observer”****frames of reference.**

*“External Observer”*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

*spatial component sum the square of the*

**“Radial”****“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*

The measurements made by the * “Proper Observer”* “center of mass” frame of reference will conclude that the

*frame of reference located on the surface of the sphere, has access to more spherical information; and measures*

**“External Observer”***“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**

The measurements made by the * “External Observer”* frame of reference will conclude that the

*center of mass frame of reference, has access to less surface information; and measures*

**“Proper Observer”***“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)*

And the * “Equal Observer” “Co-variant”* frame is measured equally by both the

*“Center of Mass” frame of reference, and the*

**“Proper Observer”***frames of reference; and measures*

**“External Observer”***“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.

********************************************************************

**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 Spherical Space***“Proper Observer”**Differential – ** Radius of Sphere** – () – is a vector (direction dependent) described in Rectangular () Cartesian Coordinates, and along Rectangular () Cartesian Axes

Differential – ** Radius of Sphere** – () – is a vector (direction dependent) described in Spherical () Coordinates, and along Rectangular () Cartesian Axes

Differential – ** Radius of Sphere** – () – measured in the

**center of mass frame of reference, and described as function of the**

*“Proper Observer”***(), relative to the**

*Euclidean Radius of Sphere***frame of reference; and likewise is described as a function of the**

*“External Observer”***()**

*Map/Patch/Manifold***, relative to the**

*“Geodesic” Arc Length***frame of reference**

*“Equal Observer (Co-variant)”*

************************************************************

Differential – ** Radius of Sphere** – () –

**“Special Relativity – Lorentz Transformation”**from the

**frame of reference, into the**

*“External Observer”***center of mass frame of reference**

*“Proper Observer”*

************************************************************

Differential – ** Radius of Sphere** – () –

**“General Relativity – Euclidean Transformation”**from the

**frame of reference, into the**

*“External Observer”***center of mass frame of reference**

*“Proper Observer”*

************************************************************

**“Proper Observer”** Differential “Light Clock” Time

*Differential “Light Clock” Time***“Proper Observer”**

Differential – *Proper Observer “Light Clock” Time* – () – measured in the

**center of mass frame of reference, and described as function of the**

*“Proper Observer”**(), relative to the*

*“Light Clock” Time***External Observer****frame of reference; and likewise is described as a function of the**

*“External Observer”***(), relative to the**

*Map/Patch/Manifold Time***frame of reference**

*“Equal Observer (Co-variant)”*

************************************************************

Differential – ** Proper Observer “Light Clock” Time** – () –

**“Special Relativity – Lorentz Transformation”**from the

**frame of reference, into the**

*“External Observer”***center of mass frame of reference**

*“Proper Observer”*************************************************************

Differential – ** Proper Observer “Light Clock” Time** – () –

**“General Relativity – Euclidean Transformation”**from the

**frame of reference, into the**

*“External Observer”***center of mass frame of reference**

*“Proper Observer”*********************************************************************

**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 Spherical Space***“External Observer”**Differential – ** Euclidean Radius of Sphere** – () – is a vector (direction dependent) described in Spherical () Coordinates, and along Spherical () Coordinate Axes

Differential – ** Euclidean Radius of Sphere** – () – is a vector (direction dependent) described in Rectangular () Cartesian Axes components, and along each Spherical () Coordinates

Differential – ** Euclidean Radius of Sphere** – () – is a vector (direction dependent) described in Spherical () Coordinates, and mixed with Rectangular Cartesian () Coordinates, and along Rectangular () Cartesian Axes

Differential – ** Euclidean Radius of Sphere** – () – measured in the

**frame of reference, is described by one (1) component of “radius” (), and one (1) “geodesic arc-length” component, which is the “Map/Patch/Manifold” () surface of the sphere, and changes as a function of the “Latitude Angle” (), and the Longitude Angle ()**

*“External Observer”*Differential – ** Euclidean Radius of Sphere** – () – measured in the

**frame of reference, is described by one (1) component of radius (), and one (1) “geodesic arc-length” component, which is the “Map/Patch/Manifold” () surface of the sphere, and changes as a function of the “radius” () of the sphere**

*“External Observer”*Differential – ** Euclidean Radius of Sphere** – () – measured in the

**frame of reference, and described as function of the**

*“External Observer”***(), relative to the**

*Radius of Sphere***center of mass frame of reference; and likewise is described as a function of the**

*“Proper Observer”***()**

*Map/Patch/Manifold***, relative to the**

*“Geodesic” Arc Length***frame of reference**

*“Equal Observer (Co-variant)”*

************************************************************

Differential – ** Euclidean Radius of Sphere** – () –

**“Special Relativity – Lorentz Transformation”**from the

**center of mass frame of reference, into the**

*“Proper Observer”***frame of reference**

*“External Observer”*

************************************************************

Differential – ** Euclidean Radius of Sphere** – () –

**“General Relativity – Euclidean Transformation”**from the

**center of mass frame of reference into the**

*“Proper Observer”***frame of reference**

*“External Observer”*

************************************************************

**“External Observer”** Differential “Light Clock” Time

*Differential “Light Clock” Time***“External Observer”**

Differential – **External Observer “Light Clock” Time** – () – measured in the

**frame of reference, and described as function of the**

*“External Observer”***(), relative to the**

*“Light Clock” Time***Proper Observer****center of mass frame of reference; and likewise is described as a function of the**

*“Proper Observer”***(), relative to the**

*Map/Patch/Manifold Time***frame of reference**

*“Equal Observer (Co-variant)”*

************************************************************

Differential – ** External Observer “Light Clock” Time** – () –

**“Special Relativity – Lorentz Transformation”**from the

**center of mass frame of reference into the**

*“Proper Observer”***frame of reference**

*“External Observer”*************************************************************

Differential – ** External Observer “Light Clock” Time** – () –

**“General Relativity – Euclidean Transformation”**from the

**center of mass frame of reference into the**

*“Proper Observer”***frame of reference**

*“External Observer”*********************************************************************

**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 Spherical Angle Space***“Equal Observer (Co-variant)”****“Differential” Map/Patch/Manifold Angle**

Differential – ** Map/Patch/Manifold – “Angle”** – () – is a vector (direction dependent) described in Spherical () Coordinates, and directed along Spherical () Coordinate Axes

Differential – ** Map/Patch/Manifold – “Angle”** – () – is a vector (direction dependent) described in Rectangular () Cartesian Axes components, along each Spherical () Coordinates

Differential – ** Map/Patch/Manifold – “Angle”** – () – is a “geodesic arc-length” angle component, on the surface of the sphere, and changes as a function of the “Latitude Angle” (), and the “Longitude Angle” () of a symmetric sphere

Differential – **Map/Patch/Manifold – “Angle”** – () – is a “geodesic arc-length” angle component, on the surface of the sphere, and changes as a function of the “radius” (), and changes as a function of the “Euclidean Radius” () of a symmetric sphere

Differential – ** Map/Patch/Manifold – “Angle”** – () – 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

**() in the**

*“Map/Patch/Manifold Angular “Spin/Rotation” Velocity”***frame, the**

*“Proper Observer”***() in the**

*“Map/Patch/Manifold Angular “Spin/Rotation” Velocity”***frame of reference**

*“External Observer”*

Differential – **Map/Patch/Manifold – “Angle”** – () – 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

************************************************************

**“Differential” Latitude “Location” Angle**

Differential – ** “Latitude “Location” Angle”** – () – is a vector (direction dependent) described in Rectangular Cartesian () Coordinate Axes components, along the “Latitude” Spherical () Coordinate

************************************************************

Differential – ** “Latitude “Location” Angle”** – () – described by the spin, rotation, or torsion on the surface of a sphere, given by the

**() in the**

*“Latitude Angular “Spin/Rotation” Velocity”***frame, and the**

*“Proper Observer”***() in the**

*“Latitude Angular “Spin/Rotation” Velocity”***frame of reference**

*“External Observer”*

************************************************************

**“Differential” Longitude “Location” Angle**

Differential – ** “Longitude “Location” Angle” **– () – is a vector (direction dependent) described in Rectangular Cartesian () Coordinate Axes components, along the “Longitude” Spherical () Coordinate

************************************************************

Differential – ** “Longitude “Location”Angle”** – () – described by the spin, rotation, or torsion on the surface of a sphere, given by the

**() in the**

*“Longitude Angular “Spin/Rotation” Velocity”***frame, and the**

*“Proper Observer”***() in the**

*“Longitude Angular “Spin/Rotation” Velocity”***frame of reference**

*“External Observer”*

************************************************************

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

*Differential Spherical Space***“Equal Observer (Co-variant)”**Differential – ** Map/Patch/Manifold – “Geodesic” Arc Length** – () – is a vector (direction dependent) described in Spherical () Coordinates, and along Spherical () Coordinate Axes

Differential – ** Map/Patch/Manifold – “Geodesic” Arc Length** – () – is a vector (direction dependent) described in Rectangular () Cartesian Axes components, and along each Spherical () Coordinates

Differential – ** Map/Patch/Manifold – “Geodesic” Arc Length** – () – is a vector (direction dependent) described in Spherical () Coordinates mixed with Rectangular Cartesian () Coordinates, and directed along the Rectangular () Axes

Differential – ** Map/Patch/Manifold – “Geodesic” Arc Length** – () – is a “geodesic arc-length” spatial component, on the surface of the sphere, and changes as a function of the “Latitude Angle” (), and the “Longitude Angle” () of a symmetric sphere

Differential – ** Map/Patch/Manifold – “Geodesic” Arc Length** – () – is a “geodesic arc-length” spatial component, on the surface of the sphere, and changes as a function of the “Radius” (), and changes as a function of the “Euclidean Radius” () of a symmetric sphere

Differential – ** Map/Patch/Manifold – “Geodesic” Arc Length**– () – 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

************************************************************

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

*Differential Map/Patch/Manifold Time***“Equal Observer (Co-variant)”**Differential – ** Map/Patch/Manifold Time** – () – measured in the

**Frame of Reference**

*“Equal Observer (C0-variant)”*

Differential – ** Map/Patch/Manifold Time** – () – measured in the

**Frame of Reference, and relative to the**

*“Equal Observer (C0-variant)”***center of mass frame of reference, and the**

*“Proper Observer”***frame of reference**

*“External Observer”*

Differential – ** Map/Patch/Manifold Time** – () – measured in the

**frame of reference, and described as a function of the**

*“Equal Observer (C0-variant)”***(), and relative to the**

*“Synchronization Time”***center of mass frame of reference, and the**

*“Proper Observer”***frame of reference**

*“External Observer”*

************************************************************

**4)**** Aether Gradient Gravitational Field Acceleration**

**4)**

**Aether Gradient Gravitational Field Acceleration**The *Aether Gravitational Field Acceleration** – *() 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** – *()

*(*

**– function of Map/Patch/Manifold – “Geodesic”**

**Space & Time**

************************************************************

* Aether Gravitational Field Acceleration – *()

*()*

**– function of Radius of Sphere**

**Space & Time**************************************************************

* Aether Gravitational Field Acceleration – *()

*()*

**– function of Euclidean Radius of Sphere**

**Space & Time**********************************************************************

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

**5)**

*Center of Mass Velocity***“Equal Frames (Invariant)”****Average Rectilinear ****Center of Mass Velocity **– () – **, of Fluid particles of a Euclidean Sphere**

**Average Rectilinear ****Center of Mass Velocity **– () – 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

************************************************************

**Average Rectilinear ****Center of Mass Velocity **– () – is *invariant* and is measured to have the same value, and is described in the ** “Proper Observer”** center of mass frame of reference

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

For any Net Inertial Mass () the radius of the Euclidean spherical source of gravity, is the Black Hole Event Horizon Schwarzschild Radius ().

************************************************************

**Average Rectilinear ****Center of Mass Velocity **– () – is *invariant* and measured to have the same value, and is described in the ** “External Observer”** frame of reference

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

For any Net Inertial Mass () the radius of the Euclidean spherical source of gravity, is the Black Hole Event Horizon Schwarzschild Radius ().

************************************************************

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 ().

**Radius of Sphere**

************************************************************

**Euclidean Radius of Sphere**

************************************************************

*Map/Patch/Manifold – “Geodesic” Arc Length*

************************************************************

*Aether Gravitational Field Acceleration*

************************************************************

************************************************************

**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**

************************************************************

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

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