**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

**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.

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 Spherical Space***“Proper Observer”**Differential – ** Spacetime Metric – Radius of Sphere** – () – is a scalar (direction independent) and is described in Rectangular () Cartesian Coordinates

Differential – ** Spacetime Metric – Radius of Sphere** – () – is a scalar (direction independent) and is described in Rectangular () Cartesian Coordinates

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

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

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

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

*“External Observer”***()**

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

*“Geodesic” Metric***frame of reference**

*“Equal Observer (Co-variant)”*

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

Differential – ** Spacetime Metric 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 – ** Spacetime Metric 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-Like Metric

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

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

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

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

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

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

*Map/Patch/Manifold Time-Like Metric***frame of reference**

*“Equal Observer (Co-variant)”*

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

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

**“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-Like Metric** – () –

**“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 Spacetime Metric Radius of Sphere** – () – is a scalar (direction independent) and is described in Rectangular () Cartesian Coordinates

Differential – ** Euclidean Spacetime Metric Radius of Sphere** – () – is a scalar (direction independent) and is described in Rectangular () Cartesian Coordinates

Differential – ** Euclidean Spacetime Metric Radius of Sphere** – () – is a scalar (direction independent) and is described in Spherical () Coordinates

Differential – ** Euclidean Spacetime Radius of Sphere** – () – is a scalar (direction independent) and is described in Spherical () Coordinates mixed with Rectangular () Coordinates

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

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

*“External Observer”*

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

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

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

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

*“Proper Observer”***()**

*Map/Patch/Manifold “Geodesic”***, relative to the**

*Metric***frame of reference**

*“Equal Observer (Co-variant)”*

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

Differential – ** Euclidean Spacetime Metric 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 Spacetime Metric 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-Like Metric

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

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

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

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

*“Light Clock” Time-Like Metric***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-Like Metric***frame of reference**

*“Equal Observer (Co-variant)”*

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

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

**“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-Like Metric** – () –

**“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 Metric**

Differential – ** Map/Patch/Manifold – “Angle Metric”** – () – is a scalar (direction independent) and is described in Spherical () Coordinates

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

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

**() in the**

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

*“Proper Observer”**square of the*

**() in the**

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

*“External Observer”*

Differential – **Map/Patch/Manifold – “Angle Metric”** – () – is a “geodesic arc-length” angle component, on the surface of a symmetric sphere and is related to the “Gravitational Field” in the following

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

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

Differential – ** “Latitude “Location” Angle Metric”** – () – is a scalar (direction independent) and is described along the

*“Latitude”*Spherical () Coordinate. And, likewise is described by the spin, rotation, or torsion on the surface of a sphere, given by the square of the

**() in the**

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

*“Proper Observer”***() in the**

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

*“External Observer”*

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

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

Differential – ** “Longitude “Location” Angle Metric” **– () – is a scalar (direction independent) and is described along the

*“Longitude”*Spherical () Coordinate. And, likewise is described by the spin, rotation, or torsion on the surface of a sphere, given by the square of the

**() in the**

*“Longitude Angular “Spin/Rotation” Velocity”***frame, and the square of 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” Metric** – () – is a scalar (direction independent) and is described in Rectangular () Cartesian Coordinates

Differential – ** Map/Patch/Manifold – “Geodesic” Metric** – () – is a scalar (direction independent) and is described in Rectangular () Cartesian Coordinates

Differential – ** Map/Patch/Manifold – “Geodesic” Metric** – () – is a scalar (direction independent) and is described in Spherical () Coordinates

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

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

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

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

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

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

**Frame of Reference**

*“Equal Observer (C0-variant)”*

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

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

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

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

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

*“External Observer”*

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

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

**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** () varies as a function of

**and**

*“space”***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**

*“time”*

*“Elastic Wave Equation.”*In a consideration for General Relativity, we will need to obtain the equations for the *Gradient Gravitational Field Acceleration*** **() 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 – **() will be described below. Limiting, the discussion to the

**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.**

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

** Proper Observer – Gradient Gravitational Field Acceleration – **()

**second order partial differential**

*–***function of**

*“Elastic Wave Equation”***()**

*Radius of Sphere***and in the**

*Space & Time Metric**frame of reference.*

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

** Proper Observer – Gradient Gravitational Field Acceleration – **()

**second order partial differential**

*–***function of**

*“Elastic Wave Equation”***()**

*Map/Patch/Manifold – “Geodesic”***and in the**

*“Equal Observer (Co-Variant)” Space & Time Metric**frame of reference.*

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

**Proper Observer – Gradient Gravitational Field Acceleration – **()

**– second order partial differential**

**function of**

*“Elastic Wave Equation”***()**

*Euclidean**Radius of Sphere***and in the**

*Space & Time Metric***frame of reference.**

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

**Proper Observer – Gradient Gravitational Field Acceleration – **()

**second order partial differential**

*–***function of**

*“Elastic Wave Equation”***(**

*Map/Patch/Manifold – “Geodesic”**and in the*

**Space & Time Metric****frame of reference.**

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

**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 source of gravity, which is the Black Hole Event Horizon Schwarzschild Radius ().

*Radius of Sphere*

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

*Euclidean Radius of Sphere*

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

*Map/Patch/Manifold – “Geodesic” Metric*

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

*Gravitatational 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 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

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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