Time Dilation Question on Special Relativity

This is a Time Dilation Question asked by Nick Percival to Robert Kemp

Q1) Time Dilation: What is the physics meaning of SR’s time dilation equation?

Does it describe physical, asymmetric effects?

If yes, does it mean that clocks traveling with a relative velocity will accumulate proper time at different rates?

If yes, does that mean that time itself is affected by relative velocity?

OR

Does it say that two observers in different inertial frames will JUST observe the other’s clock to be running slow with no attendant physical effects similar to when two twins separate and each observes the other to “shrink” whereas no physical effects occur?

“Time Dilation” in Special Relativity (SR) is different than “Time Dilation” in General Relativity (GR).

Time Dilation in Special Relativity (SR) operates under the guidelines that there is no “Gravitational Field” present, or that there are no Gravitational influences affecting the frame of reference, time, motion, and mass of an object at rest or in uniform motion.

General Relativity is just the opposite. It assumes that there is a “Gravitational Field” present, and that there are Gravitational influences that affect the frame of reference, time, motion, and mass of an object at rest or in uniform motion.

Q)   Does it describe physical, asymmetric effects?

I am not sure that I would use the word asymmetry, that word could be confusing to some. But “Yes” it is argued that moving clocks “physically” slow down. Whether you describe that as symmetry or asymmetry is another question.

Q)   If yes, does it mean that clocks traveling with a relative velocity will accumulate proper time at different rates?

“Yes.

Also let’s define the frame of reference for our clocks and declare that it is the “Proper Observer” Frame of Reference. The Proper observer assumes that his clocks are ticking at normal rates.

And there is another frame of reference known as the “External Observer” Frame of Reference. And let’s assume that the External Observer is in a frame of reference that is at rest watching the “Proper Observer” move either away from him, or move either toward him.

Now let’s look at the math, without deriving the result. Let’s assume the classical Special Relativity (SR) Time Dilation equation.

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

(Frame of Reference Independent) The isotropic speed of light constant    – $\vec{c_{Light}}$ $\;\;-->\,\frac{m}{s}$

(Frame of Reference Independent) The Center of Mass Velocity for the moving frame    – $\vec{|v|_{CM}}$ $\;\;-->\,\frac{m}{s}$

(Frame of Reference Independent) The Square of the Center of Mass Velocity for the moving frame expressed in three dimensions   –

$\vec{|v|^2_{CM}}\;\;=\;\;{|v|^2_{CM(x)}} \;\;+\;\; {|v|^2_{CM(y)}} \;\;+\;\;{|v|^2_{CM(z)}}$ $\;\;-->\,\frac{m^2}{s^2}$

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

Now, let’s consider the Proper Observer Frame of Reference

The Time for Proper Observer     –   $\Delta{t}$ $\;\;-->\,{s}$

The Time for the External Observer     – $\Delta{t'}$ $\;\;-->\,{s}$

Now, let’s consider the experiment where the “Proper Observer – (S)” Frame of Reference transforms into the “External Observer – (S’)” Frame of Reference ― (S  -> S’)

Special Relativity in Three Dimensions – Lorentz Transformation

From the measuring apparatus of the Proper Observer, he will conclude that time runs normal in his frame, and that the External Observer measures time moving slower or faster depending on the Center of Mass Velocity for the frame  – $\vec{|v|_{CM}}$ .

$\Delta{t'}\;\;=\;\;\frac{\Delta{t}}{\sqrt{1 \;\;-\;\; \frac{{|v|^2_{CM}}}{{c^2_{Light}}}}}$  $\;\;-->\,{s}$

The actual Time measurement in the Proper Observer Frame is this; if you include synchronization or “twin” synchronization.

$\Delta{t'}\;\;=\;\;\frac{\Delta{t}\;\;-\;\; \Delta{\tau_{Sync}}}{\sqrt{1 \;\;-\;\; \frac{{|v|^2_{CM}}}{{c^2_{Light}}}}}\;\;=\;\;\frac{\Delta{t}\;\;-\;\; [\frac{\vec{|v|_{CM}}\;\;\vec{u_{Frame}}}{c^2_{Light}}]\;\Delta{t}}{\sqrt{1 \;\;-\;\; \frac{{|v|^2_{CM}}}{{c^2_{Light}}}}}$  $\;\;-->\,{s}$

$\Delta{t'}\;\;=\;\;\frac{\Delta{t}\;\;-\;\; [\frac{\vec{|v|_{CM}}}{c^2_{Light}}]\;\;\vec{r}}{\sqrt{1 \;\;-\;\; \frac{{|v|^2_{CM}}}{{c^2_{Light}}}}}$  $\;\;-->\,{s}$

“Proper Observer – (S)” Frame of Reference transforms into the “External Observer – (S’)” Frame of Reference ― (S  ® S’)

Lorentz Transformation (“Fluid Medium” Distance Space) ― “Fluid Medium” ― “Rest Frame”®

$\vec{r'}\;\;=\;\;\frac{\vec{r}\;\;-\;\; {\vec{|v|_{CM}}\;\Delta{t}}}{\sqrt{1 \;\;-\;\; \frac{{|v|^2_{CM}}}{{c^2_{Light}}}}}$  $\;\;-->\,{m}$

Described in Rectangular (x, y, & z) Cartesian Coordinates

$\vec{x'}\;\;=\;\;\frac{\vec{x}\;\;-\;\; {\vec{|v|_{CM(x)}}\;\Delta{t}}}{\sqrt{1 \;\;-\;\; \frac{[ {|v|^2_{CM(x)}} \;\;+\;\; {|v|^2_{CM(y)}} \;\;+\;\;{|v|^2_{CM(z)}}]}{{c^2_{Light}}}}}$  $\;\;-->\,{m}$

$\vec{y'}\;\;=\;\;\frac{\vec{y}\;\;-\;\; {\vec{|v|_{CM(y)}}\;\Delta{t}}}{\sqrt{1 \;\;-\;\; \frac{[ {|v|^2_{CM(x)}} \;\;+\;\; {|v|^2_{CM(y)}} \;\;+\;\;{|v|^2_{CM(z)}}]}{{c^2_{Light}}}}}$  $\;\;-->\,{m}$

$\vec{z'}\;\;=\;\;\frac{\vec{z}\;\;-\;\; {\vec{|v|_{CM(z)}}\;\Delta{t}}}{\sqrt{1 \;\;-\;\; \frac{[ {|v|^2_{CM(x)}} \;\;+\;\; {|v|^2_{CM(y)}} \;\;+\;\;{|v|^2_{CM(z)}}]}{{c^2_{Light}}}}}$  $\;\;-->\,{m}$

Now, let’s consider the External Observer Frame of Reference

The Time for Proper Observer     –   $\Delta{t}$ $\;\;-->\,{s}$

The Time for the External Observer     – $\Delta{t'}$ $\;\;-->\,{s}$

Now, let’s consider the experiment where the “External Observer – (S’)” Frame of Reference transforms into the “Proper Observer – (S)” Frame of Reference ― (S’  -> S)

Special Relativity in Three Dimensions – Lorentz Transformations

From the measuring apparatus of the External Observer, he will conclude that time runs normal in his frame, and that the Proper Observer measures time moving slower or faster depending on the Center of Mass Velocity for the frame  – $\vec{|v|_{CM}}$ .

$\Delta{t}\;\;=\;\;\frac{\Delta{t'}}{\sqrt{1 \;\;-\;\; \frac{{|v|^2_{CM}}}{{c^2_{Light}}}}}$  $\;\;-->\,{s}$

The actual Time measurement in the External Observer Frame is this; if you include synchronization or “twin” synchronization.

$\Delta{t}\;\;=\;\;\frac{\Delta{t'}\;\;+\;\; \Delta{\tau'_{Sync}}}{\sqrt{1 \;\;-\;\; \frac{{|v|^2_{CM}}}{{c^2_{Light}}}}}\;\;=\;\;\frac{\Delta{t'}\;\;+\;\; [\frac{\vec{|v|_{CM}}\;\;\vec{u'_{Frame}}}{c^2_{Light}}]\;\Delta{t'}}{\sqrt{1 \;\;-\;\; \frac{{|v|^2_{CM}}}{{c^2_{Light}}}}}$  $\;\;-->\,{s}$

$\Delta{t}\;\;=\;\;\frac{\Delta{t'}\;\;+\;\; [\frac{\vec{|v|_{CM}}}{c^2_{Light}}]\;\;\vec{r'}}{\sqrt{1 \;\;-\;\; \frac{{|v|^2_{CM}}}{{c^2_{Light}}}}}$  $\;\;-->\,{s}$

“External Observer – (S’)” Frame of Reference transforms into the “Proper Observer – (S)” Frame of Reference ― (S’  ® S)

Lorentz Transformation (“Fluid Medium” Distance Space) ― “Fluid Medium” ― “Rest Frame”

$\vec{r}\;\;=\;\;\frac{\vec{r'}\;\;+\;\; {\vec{|v|_{CM}}\;\Delta{t'}}}{\sqrt{1 \;\;-\;\; \frac{{|v|^2_{CM}}}{{c^2_{Light}}}}}$  $\;\;-->\,{m}$

Described in Rectangular (x, y, & z) Cartesian Coordinates

$\vec{x}\;\;=\;\;\frac{\vec{x'}\;\;+\;\; {\vec{|v|_{CM(x)}}\;\Delta{t'}}}{\sqrt{1 \;\;-\;\; \frac{[ {|v|^2_{CM(x)}} \;\;+\;\; {|v|^2_{CM(y)}}\;\;+ \;\;{|v|^2_{CM(z)}}]}{{c^2_{Light}}}}}$  $\;\;-->\,{m}$

$\vec{y}\;\;=\;\;\frac{\vec{y'}\;\;+\;\; {\vec{|v|_{CM(y)}}\;\Delta{t'}}}{\sqrt{1 \;\;-\;\; \frac{[ {|v|^2_{CM(x)}} \;\;+\;\; {|v|^2_{CM(y)}}\;\;+ \;\;{|v|^2_{CM(z)}}]}{{c^2_{Light}}}}}$  $\;\;-->\,{m}$

$\vec{z}\;\;=\;\;\frac{\vec{z'}\;\;+\;\; {\vec{|v|_{CM(z)}}\;\Delta{t'}}}{\sqrt{1 \;\;-\;\; \frac{[ {|v|^2_{CM(x)}} \;\;+\;\; {|v|^2_{CM(y)}} \;\;+\;\;{|v|^2_{CM(z)}}]}{{c^2_{Light}}}}}$  $\;\;-->\,{m}$

Q)  Does it say that two observers in different inertial frames will JUST observe the other’s clock to be running slow with no attendant physical effects similar to when two twins separate and each observes the other to “shrink” whereas no physical effects occur?

The two observers in different inertial frames of reference will observe the other’s clock to be running slow, and also the physical effects also occur for the frame that is in motion.

Unified Gravitational Vortex Theory (GVT)

Description Abstract

This paper presents that an Inertial Mass Gravitation Vortex, is described by two Gravitational Attraction Forces, known as the Self Gravitational Field Force (Self Gravity) and the Newtonian Gravitational Field Force. These two attraction forces exist for every gravitational attraction and interaction between masses in an isolated and conserved inertial mass gravitational field vortex system.

The Newtonian Gravitational Field Force is responsible for the specific individual gravitational attraction between an orbiting mass and the conserved inertial mass vortex system. The Self Gravitational Field Force is responsible for the total curvature and total gravitational attraction of the conserved inertial mass vortex system.

This model predicts that the Newtonian Gravitational Field Force is a limiting force, when compared to the Self Gravitational Field Force (Self Gravity), of the total gravitational attraction of a conserved inertial mass vortex system.

Additionally this paper presents that the Kepler “Evolutionary Attraction Rate” of the gradient gravity field is not infinitely small but has a finite curvature described by the Schwarzschild radius of the gravity vortex.

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

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

Power Point Side Presentation with Animations –

Video of Newton & Self Gravitational Force (Self Gravity) Lecture

PDF of Presentation   –  PDF – Newton & Self Gravitational Force (Self Gravity) Lecture

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

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

Best,

Author: Robert Louis Kemp

http://www.SuperPrincipia.com

Three (3) Part Video Presentation to – Natural Philosophy Alliance (NPA)

Unified Gravitational Vortex Theory (GVT)

Description Abstract

The Gravitational Vortex Theory (GVT), was first discussed analytically, by Johannes Kepler (1609), then by Rene Descartes (1637), next by Isaac Newton (1726), then by Albert Einstein (1915), followed by Steven Rado (1994), further described by the Super Principia Mathematica (2010).

The Gravitational Vortex Theory (GVT) involves building a conceptual and mathematical model, which can be tested by experimentation, which describes how mass objects interact and behave in nature, when subject to the Vacuum of space and time, and the gravitational attraction; which in turn produces a myriad of linear and rotational physics, that results due solely to interaction of mass attracting mass, and mass interacting with the Aether; which fills the cosmos.

The Gravitational Vortex Theory (GVT) uses the Steven Rado Aethro-Kinematics & Dynamics model of an isotropic, omni-directional, random, and homogeneous, gaseous Aether, combined with the gravitational mechanics mathematics of Kepler, and Newton, to derive a new model and a novel way for describing gravitation theory.

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

Part #1 – March 24, 2012

Unified Gravitational Vortex Theory – Part 1

Power Point Side Presentation with Animations –

Unified Gravitational Vortex Theory – Part #1 (Slide Show)

PDF of Presentation   –  Unified Gravitational Vortex Theory – Part #1 (PDF)

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

Part #2 – May 5, 2012

Unified Gravitational Vortex Theory – Part 2

Power Point Side Presentation with Animations –

Unified Gravitational Vortex Theory – Part #2 (Slide Show)

PDF of Presentation   –  Unified Gravitational Vortex Theory – Part #2 (PDF)

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

Part #3 – May 19, 2012

Unified Gravitational Vortex Theory – Part 3

Power Point Side Presentation with Animations –

Unified Gravitational Vortex Theory – Part #3 (Slide Show)

PDF of Presentation   –   Unified Gravitational Vortex Theory – Part #3 (PDF)

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

Best,

Author: Robert Louis Kemp

http://www.SuperPrincipia.com

Total Mechanical Energy Conservation – Escape Velocity & Binding Energy – Einstein Field Equation

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”.

Under the condition where, only conservative forces do work, the “Total Mechanical Energy” of an isolated “Net Inertial Mass” system body remains “constant” and “conservative”. The “Total Mechanical Energy” of an isolated system is constant, means that any increase in “Kinetic Energy” is always accompanied by a decrease in the “Potential Energy” of the system.

This fundamental principle of the conservation of energy, of the “Total Mechanical Energy”, of any isolated and adiabatic system, is one of the most fundamental and important concepts of physics; which is why I am writing a second article on the subject.

The conservation of energy is also stated conceptually: Energy is never created or destroyed in the system; but energy changes from one form, into another form. For example, mechanical energy can be transformed into electrical energy, which can be further transformed into chemical energy.

In this work, a practical application of the Total Mechanical Energy is discussed; which includes a generalization of the Einstein Field Equation – Total Mechanical Energy.

In this work only the “Total Mechanical Energy” and that being the net sum of the “Net Kinetic Energy” (${T_{Kinetic-Energy}}$), and the “Gravitational Potential Energy” (${V_{Potential-Energy}}$) associated with the “Gravitational Force” ($\vec{F_{Gravity}}$) of the system mass body is considered.

Total Mechanical Energy

${E_{Total}}\,\,=\,\, {T_{Kinetic-Energy}}\;\;+\;\;{V_{Potential-Energy}}$

${E_{Total}}\,\,=\,\,{T_{KE - Initial}}\,\,+\,\,{V_{PE - Initial}}\,\,=\,\,{T_{KE - Final}}\,\,+\,\,{V_{PE - Final}}$

Now, consider that a “test mass” object is located in a gradient gravitational field, and is projected out from a “lower gravitational potential” of the gradient, into a “higher gravitational potential” in the field; with a certain “Kinetic Energy” (${T_{Kinetic-Energy}}$).

The “test mass” object has an initial “Kinetic Energy” (${T_{Kinetic-Energy}}$) and an initial “Speed”, as it moves up the gradient escaping the field, with decreasing speed, until the increase in the “Potential Energy” (${V_{Potential-Energy}}$), equals to the value of a decrease in the “Kinetic Energy” (${T_{Kinetic-Energy}}$), of the conserved system.

The “test mass” object, then stops and falls back towards the center of the gradient field, losing potential energy, and gaining in kinetic energy. At an infinite distance from the center of the gradient gravity potential, the “Potential Energy” (${V_{Potential-Energy}}=0$) is equal to zero (0).

When the “test mass” object is located at a specific potential in the gradient gravitational field with a negative “Potential Energy” (${V_{Potential-Energy}}=(-)\frac{m_{Mass}\,m_{Net}\,G}{r}$), this is considered the “strongly bound” to the potential,  energy condition; and is the greatest possible decrease in the energy of the conserved system.

The “Escape Energy” in this case, would be the greatest possible increase that the “test mass” can achieve, moving in an opposite direction through the gradient field, and is given by the positive value of the, “Potential Energy” (${V_{Potential-Energy}}=(+)\frac{m_{Mass}\,m_{Net}\,G}{r}$), which is equal and opposite, to the value of the negative potential energy, “strongly bound” condition.

The greatest possible increase in “Potential Energy” (${V_{Potential-Energy}}$), equals to the greatest possible decrease in the “Kinetic Energy” (${T_{Kinetic-Energy}}$). If the “test mass” has an initial “Kinetic Energy” (${T_{Kinetic-Energy}}\geq(-){V_{Potential-Energy}}$), that is greater than the negative of the “Potential Energy” ($(-){V_{Potential-Energy}}$), the “test mass” object will never stop and fall towards lower gravity gradient potentials, but “Escapes” into outer space.

Now, let’s consider “Ionization” of the “Hydrogen Atom”, and the, “Gravitational Vortex” model in the “Ground State”. The electron exists in a spherical volume, surrounding a smaller spherical volume of the proton; A simple “Hydrogen Atom” is comprised of one electron unit, and one proton unit. The electron existing in a spherical volume spins and takes (720) degrees to make a return to its initial starting point; which is considered a complete rotation; and is not the familiar (360) degrees. This is also called the “Spin 1/2” quantum state.

The familiar (360) degrees complete rotation of the the two spin state system, is folded in, and is described on spherical surface, and is output or result, in the form of the  Geodesic Arc-Length – Map/Patch/Manifold Angle (${\Omega_{Map}}_{(\theta \phi)}$).

This means that the electon must have two spin states. The electron in spherical volume about the proton is described by two independent rotations; one spin rotation in the “latitude” (${\theta_{Lat}}\,=\,{\omega_{\theta}}\Delta{t_{Light}}$) direction, and another spin rotation of the sphere, in the “longitude” (${\phi_{Lon}}\,=\,{\omega_{\phi}}\Delta{t_{Light}}$) direction; and is described by the Geodesic Arc-Length – Map/Patch/Manifold Angle (${\Omega^2_{Map}}_{(\theta \phi)}$) Metric on the surface of an electron spheroid, given by the following.

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

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

The Geodesic Arc-Length – Map/Patch/Manifold Angle (${\Omega_{Map}}_{(\theta \phi)}$), can be used to derive an equation for the total “spin states” on the surface of an electron spheroid; resulting in the following range of values for rotation.

$\;\;\;\;0\;\;\leq\;\;{\Omega_{Map}}_{(\theta \phi)}\;\;\leq\;\;2\pi$$\,\,----> \,\,{radians}$

In the lowest spherical potential of the “Hydrogen Atom”, the “Ground State”, the radius of the orbit, potential, or volume, equals the “Bohr Radius” (${r}=5.29\times10^{-11}\,m$) of the atom.

The various energy states (cases or conditions) that exists along a “Potential Energy” curve, for the “electron” and “proton” atomic system, has the same form, as that for a “satellite” orbiting the “earth”.

The “Electrostatic Potential Energy” (${V_{Electric-Potential}}$) of the “Hydrogen Atom”, two-body interacting system is described similarily to the “Gravitational Potential Energy” (${V_{Gravity-Potential}}$), being the energy of an infinite number of concentric spherical shells of gradient potentials of energy.

For the “Hydrogen Atom” system mass body, the “Ground State” potential, is the “energy state”, of the greatest quantity of attraction energy of system; and is considered to be a “Bound Strongly” to an “Electrostatic Potential” energy field condition; described by the following equation.

${V_{Electric-Potential}}\,=\,(-)\frac{q_{electron}\,\,q_{proton}}{4\pi\,\epsilon_{\circ}\,r}$$\,\,----> \,\, \frac{kg\,m^2}{s^2}$

${V_{Electric-Potential}}\;\;=\;\;(-)4.36\,\times\,10^{-18}\,\frac{kg\,m^2}{s^2}\,\,=\,\,(-)27.2\,eV$

Because the “electron” is considered to be orbiting the “proton” in an “Electrostatic Potential”; the magnitude of the “Total Mechanical Energy” of the “electron” in the “Orbiting Energy” condition of the “Hydrogen Atom” system, must therefore, equal to the “Kinetic Energy”; which must equal to the one half of the magnitude of the “Potential Energy”.

This means that electrons, protons, and hydrogen atoms, similar to planets, moons, and solar systems, due to their, mass and energy, “warp”, space and time, in their local vicinities, producing curvature in the physical form of, a gradient field of “Potential Energy Potentials”, and a gradient field of “Kinetic Energy Potentials”.

The minimum additional energy a “satellite” orbiting the “earth”, or an “electron” orbiting the “proton”, must be given to “Escape” from the “earth”, or to “Escape” the “proton”, is the “Escape Energy” condition. The “Kinetic Energy” of the orbiting electron or satellite in a fixed potential, must therefore be doubled during the “Escape Energy” condition; given by the following equations.

Gravitational System – Escape Energy – Earth & SatelliteCircular Orbit

${T_{Kinetic-Energy}}\;\;=\;\;(-)\frac{1}{2}{V_{Gravity-Potential}}$$\,\,----> \,\, \frac{kg\,m^2}{s^2}$

$\frac{1}{2}{m_{satellite}}{|\vec{v}|^2_{CM}}\;\;=\;\;\frac{1}{2}\frac{{m_{satellite}}\,(m_{earth} + m_{satellite})\,G}{r}$

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

Gravitational System – Escape Energy – Proton & ElectronCircular Orbit

${T_{Kinetic-Energy}}\;\;=\;\;(-)\frac{1}{2}{V_{Gravity-Potential}}$$\,\,----> \,\, \frac{kg\,m^2}{s^2}$

$\frac{1}{2}{m_{electron}}{|\vec{v}|^2_{CM}}\;\;=\;\;\frac{1}{2}\frac{{m_{electron}}\,(m_{electron} + m_{proton})\,G}{r}$

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

Electrostatic System – Escape Energy – Proton & ElectronCircular Orbit

${T_{Kinetic-Energy}}\;\;=\;\;(-)\frac{1}{2}{V_{Electric-Potential}}$$\,\,----> \,\, \frac{m^2}{s^2}$

$\frac{1}{2}{m_{electron}}{|\vec{v}|^2_{CM}}\,=\,(-)\frac{1}{2}\frac{q_{electron}\,\,q_{proton}}{4\pi\,\epsilon_{\circ}\,r}$

${T_{Kinetic-Energy}}=(-)\frac{1}{2}{V_{Electric-Potential}}=2.18\,\times\,10^{-18}\,\frac{kg\,m^2}{s^2}=13.6\,eV$

This result of the “Escape Energy” condition, is that the “Kinetic Energy” is one half (1/2) the magnitude of the “Potential Energy” holds for any “Orbiting” condition, where the “Central Force” of attraction, that being the Gravitational Force (${F_{Gravity}}\propto\frac{1}{r^2}$) or the Electrostatics Force (${F_{Electrostatic}}\propto\frac{1}{r^2}$); is inversely proportional to the square of the distance, relative to the center of any gradient field, of energy potentials.

The “Total Mechanical Energy” of the “electron” in “Orbit Condition” about the proton in the Hydrogen Atom, system is given by the following.

${E_{Total}}\,\,=\,\,{T_{Kinetic-Energy}}\,\,+\,\,{V_{Electric-Potential}}$

${E_{Total}}\,\,=\,\,(-){{T_{Kinetic-Energy}}\,\,=\,\,\frac{1}{2}{V_{Electric-Potential}}}$

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

${E_{Total}}\,\,=\,\,(-)\frac{1}{2}{m_{electron}}{|\vec{v}|^2_{CM}}\,=\,(-)\frac{1}{2}\frac{q_{electron}\,\,q_{proton}}{4\pi\,\epsilon_{\circ}\,r}$

The approximate value of the “Total Mechanical Energy” in “Ground State” of the “Hydrogen Atom” two-body system, is given by the following.

${E_{Total}}\;\;=\;\;(-)2.18\,\times\,10^{-18}\,\frac{kg\,m^2}{s^2}\;\;=\;\;(-)13.6\,eV$

In order to “ionize”, remove, or liberate the electron from the “Ground State” electrostatic energy potential, of the “Hydrogen” atomic system; the atom, must be given an “additional” amount of “supplied energy”, this surplus in energy is called the “Binding Energy” (${E_{Binding-Energy}}=13.6\,eV$), of the atom.

Thus, the energy required to remove or liberate the electron from the atom, is called “ionization”; and the “Binding Energy” is also called the “Ionization Energy”.

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

The Electrostatic FieldTotal Mechanical Energy – “Orbiting” in gradient field, of energy potential, condition:

$\;\;\;\;(-)\frac{q_{electron}\,\,q_{proton}}{4\pi\,\epsilon_{\circ}\,r}\;\;\leq\;\;{E_{Total}}\;\;\leq\;\;0$

The Electrostatic FieldTotal Mechanical Energy“Escaping” from gradient field, of energy potential, condition:

$\;\;\;\;(-)\frac{1}{2}\frac{q_{electron}\,\,q_{proton}}{4\pi\,\epsilon_{\circ}\,r}\;\;\leq\;\;{E_{Total}}\;\;\geq\;\;0$

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

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

(1) The Total Mechanical Energy Conservation – Euclidean Spherical Mechanics

Now that all of the physics terms and basic concepts have been discussed, let’s complete the conceptual and mathematical discussion of the Total Mechanical Energy Conservation (${E_{Total}}={T_{Kinetic-Energy}}+{V_{Potential-Energy}}$) in a consideration for various “Bound Potential Energy” and “Escape Potential Energy” conditional cases, and in consideration for “The General Theory of Relativity”.

The “math” and “physics” that will be discussed in the next few sections, are employed from work, that was derived in the articles below:

A Theory of Gravity for the 21st Century – The Gravitational Force and Potential Energy – in consideration with Special Relativity & General Relativity

Euclidean Spherical Mechanics – Total Mechanical Energy Conservation – in consideration with General Relativity

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

Total Mechanical Energy Conservation

${E_{Total}}\,\,=\,\, {T_{Kinetic-Energy}}\;\;+\;\;{V_{Gravity-Potential}}$

${E_{Total}}\,\,=\,\, {T_{Kinetic-Energy}}\;\;+\;\;(\frac{{m_{mass}}}{{m_{Net}}}){V_{Self-Potential}}$

Next substituting the appropriate “Kinetic Energy” and “Gravitational Potential Energy” terms yields:

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\,\,=\,\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\;\;-\;\;\frac{m_{Net}\,G}{r}]$ $\,\,----> \,\, \frac{m^2}{s^2}$

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\,\,=\,\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\,\,-\,\,{g_{Gravity}}\,{r}]$ $\,\,----> \,\, \frac{m^2}{s^2}$

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\,\,=\,\,\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\,\,-\,\,{v^2_{Gravity}}]$ $\,\,----> \,\, \frac{m^2}{s^2}$

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

The Total Mechanical Energy Conservation described in “Classical Newtonian Mechanics” “ordinary” mathematical form.

$\frac{E_{Total}}{m_{Mass}}\,\,=\,\,\frac{1}{m_{Mass}}[{T_{Kinetic-Energy}}\;\;+\;\;{V_{Gravity-Potential}}]$

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\,\,=\,\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\;\;-\;\;\frac{m_{Net}\,G}{r}]$

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

The Total Mechanical Energy Conservation described in “Spherical Euclidean Mechanics & General Relativity” “ordinary” mathematical form; and as a function of the Gradient Gravitational Field Inertial Potential (${v^2_{Gravity}}\,=\,\frac{m_{Net}\,G}{r}$), and the Speed of Light Inertia Potential (${c^2_{Light}}$).

$\frac{E_{Total}}{m_{Mass}}\,\,=\,\,\frac{1}{m_{Mass}}[{T_{Kinetic-Energy}}\;\;+\;\;{V_{Gravity-Potential}}]$

$\frac{E_{Total}}{m_{mass}}\,\,=\,\,[\frac{1}{2}\,{c^2_{Light}}\,[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}}{2\,{v^2_{Gravity}}}))^2\,\,+\,\,({\Omega_{Map}}_{(\theta \phi)})^2}]^2\,\,\,-\,\,\,{v^2_{Gravity}}]$

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

The Total Mechanical Energy Conservation described in “Spherical Euclidean Mechanics & General Relativity” “ordinary” mathematical form; and as a function of the Semi-Major Radius (${r}$) which is measured from the center of the field and relative to the Schwarzschild Radius (${r_{Schwarzschild}}\,=\,2\frac{m_{Net}\,G}{c^2_{Light}}$) Black Hole Event Horizon.

$\frac{E_{Total}}{m_{Mass}}\,\,=\,\,\frac{1}{m_{Mass}}[{T_{Kinetic-Energy}}\;\;+\;\;{V_{Gravity-Potential}}]$

$\frac{E_{Total}}{m_{mass}}\,\,=\,\,[\frac{m_{Net}\,G}{r_{Schwarzschild}}\,[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{r}{r_{Schwarzschild}}))^2\,\,+\,\,({\Omega_{Map}}_{(\theta \phi)})^2}]^2\,\,-\,\,\frac{m_{Net}\,G}{r}]$

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

The Total Mechanical Energy Conservation described in “Spherical Euclidean Mechanics & General Relativity” “ordinary” mathematical form; and as a function of the Linear Mass Density (${\mu_{L-Density}}=\frac{m_{Net}}{r}$) which is measured from the center of the field and relative to the Black Hole Linear Mass Density (${\mu_{L-Density}}_{BH}=\frac{m_{Net}}{r_{Schwarzschild}}$) constant.

$\frac{E_{Total}}{m_{Mass}}\,\,=\,\,\frac{1}{m_{Mass}}[{T_{Kinetic-Energy}}\;\;+\;\;{V_{Gravity-Potential}}]$

$\frac{E_{Total}}{m_{mass}}\,\,=\,\,[{\mu_{L-Density}}_{BH}\,G\,[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{\mu_{L-Density}}_{BH}}{{\mu_{L-Density}}}))^2\,\,+\,\,({\Omega_{Map}}_{(\theta \phi)})^2}]^2\,\,-\,\,{\mu_{L-Density}}\,G]$

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

Black Hole Linear Mass Density

${\mu_{L-Density}}_{BH}=\frac{m_{Net}}{r_{Schwarzschild}}=\frac{1}{2}\frac{c^2_{Light}}{G}=6.73297478332358\times 10^{26}\frac{kg}{m}$

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

Where the Geodesic Arc-Length – Map/Patch/Manifold Angle (${\Omega^2_{Map}}_{(\theta \phi)}$) Metric on the surface of a spheroid is given by the following.

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

$\;\;\;\;0\;\;\leq\;\;{\Omega_{Map}}_{(\theta \phi)}\;\;\leq\;\;2\pi$$\,\,----> \,\,{radians}$

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

(2) The Net Kinetic Energy in direct relation to the – Stress Energy of the Einstein Field Equation

The General Relativity – Einstein Field Equation determines the metric of a “gravity vortex” in the “Vacuum of Space-time” for a given arrangement of “Stress-Energy” (${T_{Energy}}_{(\theta \phi)}$) in any localized region of space and time, and where matter and energy is condensed and localized, space and time there is warped, bent, or curved.

The General Relativity – Einstein Field Equation declares that the attractive force of gravity, is communicated in time, to any unit of matter, located within a gradient gravitational field, at the “Speed of Light”. This is in contradiction with Isaac Newton’s concept of the attractive force of gravity, in which he theorized, that gravity was being communicated to any unit of matter, located within a gradient gravitational field, “Instantaneously”.

The General Relativity – Einstein Field Equation also declares that the attractive “Gravitational Force” that mass, exerts on other mass objects, located at some distance away from their respective centers, is due to the curvature of the “Vacuum of Space-time”. The attractive Gravitational Force is therefore, the result of the masses and their energies, warping and curving, space, and time in their respective vicinities.

All mass or matter, curves space and time in its local vicinity, in the form of a field of gradient energy potentials. The mass includes the mass of the electron, the mass of the proton, the mass of the hydrogen atom, the mass of a molecule, the mass of the earth, the mass of the moon, the mass of the sun, and the mass of the galaxy; all warp and curve, space, and time in their respective vicinities.

Thus, the masses immersed in the gradient gravitational field vortex, will follow a geodesic or curved path, created by the presence of mass and energy warping and curving the “Vacuum of Space-time” in the local vicinity of mass and energy.

The General Relativity – Einstein Field Equation also predicts that the “Vacuum of Spacetime”, also is known as the “Universal Vacuum Energy”, is an elastic continuum, from which a “gravity vortex” of gradient energy potentials evolves; and is measured by the localized evolution and condensing of the “Stress ‘Kinetic ‘Energy” of the universal vacuum energy, into matter, or mass units.

The “Stress ‘Kinetic ‘Energy” is a measure of an elastic energy of the universal vacuum, that changes in direct proportion to the distance, and time, or space-time, as measured from the center of the gradient gravity field; and likewise is directly proportional to the “Net Inertial Mass” (${m_{Net}}$) of the gradient gravitational field.

The “Stress Energy” of the “Einstein Field Equation” in “General Relativity” measures the elasticity of space and time surrounding a system mass body, and manifest in the form of an infinite series of concentric spherical shells, which, warp, deform, and curve, space and time, in the local vicinity of the “Net Inertial Mass” (${m_{Net}}$), as spherical gradient grid; and is described by a linear relationship between the “Stress Pressure” which is pressure acting normal to a spheroid surface and the “Strain Pressure” which is pressure acting tangential to a spheroid surface.

The “Stress Energy” (${T_{Energy}}_{(\theta \phi)}$) of the Einstein Field Equation is actually a measure of the “Kinetic Energy” and not the “Potential Energy” of an isolated system mass body. The “Stress Energy” (${T_{Energy}}_{(\theta \phi)}$) of the Einstein Field Equation is a measure of infinite number of concentric spherical gradient gravitational field of “Kinetic Energy Potentials”.

The “Kinetic Energy Potentials” of a gradient gravitational field, are not the same as the gradient gravitational field “Potential Energy Potentials”; although both field potentials exists at the same location in space, relative to the center of the gradient field.

The General Relativity – Einstein Field Equation predicts that since the “Vacuum of Spacetime”, is an elastic continuum, from which evolves a “Gravity Vortex” of matter, and is measured by the localized condensing of the “Stress Energy” (${T_{Energy}}_{(\theta \phi)}$) of the universal vacuum into units of “mass”.

The “Stress ‘Kinetic ‘Energy” (${dT_{Energy}}_{(\theta \phi)}$) of the universal vacuum changes in direct proportion to the distance and time (space-time), and produces a constant; such that there is a “Cosmic ‘Dark’ Vacuum Force” (${F_{Dark-Force}}$) that is a constant of Nature.

${F_{Dark-Force}}\,=\,2\pi(\frac{{dT_{Energy}}_{(\theta \phi)}}{{dG_{Space}}_{(\theta \phi)}})_{Source}\,=\,2\pi(\frac{{dT_{Ricci-Energy}}_{(\theta \phi)}}{{dR_{(\theta \phi)}}})_{Maximum}$

${F_{Dark-Force}}\;=\;Constant\;=\;3.0256479774082\times10^{+43}\;\; \frac{kgm}{s^2}$

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

Where the “Energy Source” and “Distance” of “Minimum Curvature” is described by the following Einstein, universal “vacuum force” constant ratio;

$(\frac{{dT_{Energy}}_{(\theta \phi)}}{{dG_{Space}}_{(\theta \phi)}})_{Source}$

And the “Energy Field” and “Distance” of “Maximum Curvature” is described by the following Ricci/Riemannian, universal “vacuum force” constant ratio;

$(\frac{{dT_{Ricci-Energy}}_{(\theta \phi)}}{{dR_{(\theta \phi)}}})_{Maximum}$

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

“Vacuum Energy” and “Vacuum Force” is an underlying background energy that permeates all of mass, space, and time in the universe. The “vacuum energy” on a fundamental level is the substratum substance of matter, space, time. Therfore, the “vacuum” in nature is, not empty at all, but is seething with energy, even when devoid of inertial matter. This means that the concept of “vacuum” must be described with the physics, such that it supports both Baryonic Matter, and Non Baryonic Matter. The Non Baryonic Matter is thus, the fundamental energy of the vacuum!

The Vacuum energy of the universe which shall be synonymous with the Aether, is a gas comprised of discrete units and according to ancient and medieval science, the Aether also spelled æther or ether, is the material that fills all regions of the Universe!

The “Vacuum Force” is the cause of the constant flow of a ‘Aether” fluid into a partial vacuum, or region of low pressure. The pressure gradient between this region and the ambient pressure will propel matter toward the low pressure area.

The Vacuum “Dark” Cosmic Force of Spacetime (${F_{Dark-Force}}$) is a “universal constant” force of interaction upon matter, and is pervasive throughout the universe, and acts on all mass bodies at all times; and is invariant to all frames of reference, and is independent of any changes in mass, space, or time.

The Vacuum “Dark” Cosmic Force of Spacetime (${F_{Dark-Force}}$) is the “constant” force of the universal continuum, the “Vacuum of Space-time”. The Vacuum “Dark” Cosmic Force of Spacetime (${F_{Dark-Force}}$) is the force of interaction upon matter, and represents where matter, energy, space, and time, are intertwined in the substratum of the vacuum; the elastic energy of the universe.

Definition:

The strength of the Cosmic “Dark” Vacuum Force (${F_{Dark-Force}}$) is a universal constant force that acts on all bodies immersed in the universe, at all times; and is equal to the square of the Aether Gravitation Light Force (${F^2_{Light-Force}}$) divided by four (4) times the Inertial Mass Gravitational Self Force (${4F_{Self-Gravity-Force}}$); and likewise is equal to one fourth, the fourth power of the Speed of Light ($\frac{c^4_{Light}}{4}$) divided by the Universal Gravitational Constant (G).

Cosmic “Dark” Vacuum Force

${F_{Dark-Force}}\;=\;\frac{1}{4}(\frac{c^4_{Light}}{G})\;\;=\;\;3.0256479774082\times10^{+43}\;\;\frac{kgm}{s^2}$

${F_{Dark-Force}}\;=\;\frac{1}{4}(\frac{c^4_{Light}}{G})\;=\;\frac{1}{4}(\frac{F^2_{Light-Force}}{F_{Self-Gravity-Force}})$

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

Cosmic “Dark” Vacuum Force – constant

$\vec{F_{Dark-Force}}\;\;=\;\;(-){\mu^2_{L-Density}}_{BH}\,G\;\hat{a}_{r}=\;\;(-)\frac{m_{Net}^2\,G}{r^2_{Schwarzschild}}\;\hat{a}_{r}$$---> \frac{kgm}{s^2}$

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

Where the Aether Gravitational Light Force ($\vec{F_{Light-Force}}$) is given by the following.

$\vec{F_{Light-Force}}\;\;=\;\;{\mu_{L-Density}}\,{c^2_{Light}}\;\hat{a}_{r}\;\;=\;\;\frac{m_{Net}\,c^2_{Light}}{r}\;\hat{a}_{r}$ $---> \frac{kgm}{s^2}$

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

Where the Inertial Mass Gravitational Self Force ($\vec{F_{Self-Force}}$) is given by the following.

$\vec{F_{Self-Force}}\;\;=\;\;(-){\mu^2_{L-Density}}\,G\;\hat{a}_{r}\;\;=\;\;(-)\frac{m^2_{Net}\,G}{r^2}\;\hat{a}_{r}$ $---> \frac{kgm}{s^2}$

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

The “Stress Energy” (${T_{Energy}}_{(\theta \phi)}$) of the Einstein Field Equation

The “Stress” Kinetic Energy (${T_{Energy}}_{(\theta \phi)}$) measures “Kinetic Energy Potentials of a gradient gravity field, and describes the “gravitational vortex” energy of curved space-time gravity potentials, in the presence matter and energy; which includes both energy and momentum densities, as well as stress pressure, and shear pressure.

Drawing further upon the analogy with Newtonian gravity, it is natural to assume that the field equation for gravity describes this “Stress” Kinetic Energy (${T_{Energy}}_{(\theta \phi)}$), on the surface of a spherical gravitational field of “Kinetic Energy Potentials”; which can be considered states in an Ideal Gas Equation of State of the “Vacuum of Space-time”.

The Vacuum of Spacetime is an elastic energy, from which a “gravity vortex” evolves, of concentric spherical shells of energy potentials; which can be considered “gas energy states”. These “gas energy states” are also described by an “Ideal Gas Equation of State”, where the gas “Pressure” (${P_{ressure}}_{(\theta \phi)}$), “Volume” (${g_{Vol}}_{(\theta \phi)}$), and gas “Temperature” (${T_{emp}}_{(\theta \phi)}$), are all conjugate of one another.

Ideal Gas Equation for Vacuum of Spacetime – Kinetic Stress Energy

${T_{Energy}}_{(\theta \phi)}\;= {P_{ressure}}_{(\theta \phi)}{g_{Vol}}_{(\theta \phi)} = N k_{B}\,{T_{emp}}_{(\theta \phi)}$ $---> \frac{kgm^2}{s^2}$

Where, the symbol (N) denotes the total number of constituents, of the “Gravitational Vortex” system body.

And, the Boltzmann Constant is given by

$k_{B} = 1.3806503 \times 10^{-23} \frac{kg m^2}{s^2 K}$

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

Vacuum of Spacetime – Einstein Field Equation – “Space & Time” Stress Energy

The “Stress Energy” (${dT_{Energy}}_{(\theta \phi)}$) changes linearly with increasing gradient field “distance” (${{dG_{Space}}_{(\theta \phi)}}\,=\,{r}\,{\Omega_{Map}}_{(\theta \phi)}$) from the center of the gradient gravity field, and behaves similar to the classical, Robert Hooke, “Elastic Force”.

In essence, Robert Hooke, predicts that the “Hooke Elastic Force” of a “Stretched Spring” (${F_{Hooke-Force}}\propto{r}$), changes linear, with the “distance” (${r}$) of a stretched spring; and  the “Hooke Elastic “Stress” Energy” of a “Spring” (${T_{Hooke-Energy}}\propto{r^2}$), changes with the “square of the distance” (${r^2}$), of the stretched spring.

However; the Albert Einstein, Field Equation, predicts that there is an “Elastic Vacuum Force” (${F_{Dark-Force}}={Constant}$) associated with the  “Stressed Space & Time” of a gradient gravitational field of energy potentials, that is constant. And the  “Elastic “Stress” Energy” of the “Vacuum of Space-time” (${T_{Einstein-Energy}}\propto{r}$) changes linearly with distance (${r}$), from the center of the gradient field, of energy potentials.

The above equation, explains is why “Dark Energy” behaves like a force!

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

Vacuum of Spacetime – Einstein Field Equation –  differential “Stress Energy” changes with differential “Geodesic Arc-Length” distance on surface of spheroid surface

$\int{dT_{Energy}}_{(\theta \phi)}\;= \; (\frac{F_{Dark-Force}}{2\pi})\int{{dG_{Space}}_{(\theta \phi)}}$ $---> \frac{kgm^2}{s^2}$

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

$\int{dT_{Energy}}_{(\theta \phi)}\;= \;\frac{1}{8\pi}(\frac{c^4_{Light}}{G})[\int{dR_{(\theta \phi)}}\;\;-\;\;(\frac{R_{Heat}}{2}\,\,-\,\,\Lambda_{Einstein})\,\int{{dg_{Vol}}_{(\theta \phi)}}]$

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

${T_{Energy}}_{(\theta \phi)}\;= \; {F_{Dark-Force}}[\frac{{G_{Space}}_{(\theta \phi)}}{2\pi}]$

${T_{Energy}}_{(\theta \phi)}\,\,=\,\, \frac{1}{4} ( \frac{c^4_{Light}}{G})[\frac{1}{2\pi}[{R_{(\theta \phi)}}\;-\;{{g_{Vol}}_{(\theta \phi)}}(\frac{R_{Heat}}{2}\,\,-\,\,\Lambda_{Einstein})]]$

${T_{Energy}}_{(\theta \phi)}\;=\; \frac{1}{4} ( \frac{c^4_{Light}}{G})[{r}(\frac{{\Omega_{Map}}_{(\theta \phi)}}{2\pi})]\,\,=\,\, \frac{1}{4} ( \frac{c^4_{Light}}{G})[\frac{1}{2\pi}[{R_{(\theta \phi)}}\;-\;\frac{{g_{Vol}}_{(\theta \phi)}}{S^2_{Expansion}}]]$

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

Vacuum of Spacetime – Einstein Field Equation – “Matter & Kinetic Energy” Stress Energy

Next, showing the relation between the Net Kinetic Energy (${T_{Kinetic-Energy}}$) gravity gradient kinetic potentials, and the Stress Energy (${T_{Energy}}_{(\theta \phi)}$) “Kinetic Energy Potentials” of the Vacuum of Spacetime.

${T_{Energy}}_{(\theta \phi)}\;=\; {T_{Kinetic-Energy}}[(\frac{{\Omega_{Map}}_{(\theta \phi)}}{2\pi})]$

${T_{Energy}}_{(\theta \phi)}\,\,=\,\,\frac{1}{2}\,{m_{Net}\,c^2_{Light}}\,[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}\,r}{2\,{m_{Net}}\,G}))^2\,\,+\,\,({\Omega_{Map}}_{(\theta \phi)})^2}]^2\,[(\frac{{\Omega_{Map}}_{(\theta \phi)}}{2\pi})]$

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

Vacuum of Spacetime – Kinetic Stress Energy – Black Hole Event Horizon

Next, letting the Semi-Major Radius (${r}$) equal to the Schwarzschild Semi-Major Radius (${r}\,=\,{r_{Schwarzschild}}\,=\,2\frac{m_{Net}\,G}{c^2_{Light}}$) Black Hole Event Horizon, distance of the gradient gravity field

${T_{Energy}}_{(\theta \phi)}\;=\; \frac{1}{4} ( \frac{c^4_{Light}}{G})[{r}(\frac{{\Omega_{Map}}_{(\theta \phi)}}{2\pi})]\,\,=\,\,\frac{1}{2}\,{m_{Net}\,c^2_{Light}}\,[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}\,r}{2\,{m_{Net}}\,G}))^2\,\,+\,\,({\Omega_{Map}}_{(\theta \phi)})^2}]^2[(\frac{{\Omega_{Map}}_{(\theta \phi)}}{2\pi})]$

${{T_{Energy}}_{(\theta \phi)}}_{BH}\;=\,\,\frac{1}{2}\,{m_{Net}\,c^2_{Light}}\,[(\frac{{\Omega_{Map}}_{(\theta \phi)}}{2\pi})]$

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

${{T_{Energy}}_{(\theta \phi)}}_{BH}\;\;=\;\;\frac{m_{Net}^2\,G}{r_{Schwarzschild}}\,[(\frac{{\Omega_{Map}}_{(\theta \phi)}}{2\pi})]$$---> \frac{kgm^2}{s^2}$

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

(3) The Total Mechanical Energy Conservation in direct relation to the – Einstein Field Equation of General Relativity

Total Mechanical Energy Conservation

$\frac{E_{Total}}{m_{Mass}}\,\,=\,\,\frac{1}{m_{Mass}}[{T_{Kinetic-Energy}}\;\;+\;\;{V_{Gravity-Potential}}]$

Next, substituting the “Stress Kinetic Energy” into the above equation

$\frac{E_{Total}}{m_{Mass}}\,\,=\,\,\frac{1}{m_{Mass}}[{T_{Energy}}_{(\theta \phi)}(\frac{2\pi}{{\Omega_{Map}}_{(\theta \phi)}})\;\;+\;\;{V_{Gravity-Potential}}]$

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

Total Mechanical Energy Conservation – Einstein Field Equation

$\frac{{E_{Total}}_{(\theta \phi)}}{m_{Mass}}\,\,=\,\,\frac{1}{m_{Mass}}[ {T_{Energy}}_{(\theta \phi)}\;\;+\;\;{V_{Gravity-Potential}}[(\frac{{\Omega_{Map}}_{(\theta \phi)}}{2\pi})]]$

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

Total Mechanical Energy Conservation – Einstein Field Equation

$\frac{{E_{Total}}_{(\theta \phi)}}{m_{Mass}}=[\frac{1}{4}(\frac{c^4_{Light}}{m_{Mass}G})[\frac{1}{2\pi}[{R_{(\theta \phi)}}-{{g_{Vol}}_{(\theta \phi)}}(\frac{R_{Heat}}{2}-\Lambda_{Einstein})]]-\frac{m_{Net}\,G}{r}\,[(\frac{{\Omega_{Map}}_{(\theta \phi)}}{2\pi})]]$

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

Total Mechanical Energy Conservation – Einstein Field Equation

$\frac{{E_{Total}}_{(\theta \phi)}}{m_{Mass}}\,\,=\,\,[\frac{1}{2}\,(\frac{m_{Net}}{m_{Mass}})\,{c^2_{Light}}\,[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}\,r}{2\,{m_{Net}}\,G}))^2\,\,+\,\,({\Omega_{Map}}_{(\theta \phi)})^2}]^2\,\,\,-\,\,\,\frac{m_{Net}\,G}{r}]\,[(\frac{{\Omega_{Map}}_{(\theta \phi)}}{2\pi})]$

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

(4) The Total Mechanical Energy Conservation – Energy States

The Total Mechanical Energy Conservation of a “Gradient Gravitational Field” of energy potentials, and the “Inertial Net Mass”, of the isolated system is described by a range of “Energy States”; given by the following conditional cases.

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

(Case #1) The Total Mechanical Energy Conservation – “Strongly Bound Condition” to a Gravitational Potential “Energy State”

In the case where “mass” and “energy” are in a “Bound/Strongly” condition to a “Gravitational Potential” of a gradient gravitational field, the Total Mechanical Energy Conservation (${E_{Self-Total}}={V_{Gravity-Potential}}$), is equal to the Gravitational Potential Energy located at a specific potential, of the gravity field gradient.

And the Net Kinetic Energy (${T_{Kinetic-Energy}}=0$), at that potential, is equal to zero (0).

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

Total Mechanical Energy Conservation

$\frac{E_{Total}}{m_{Mass}}\,\,=\,\,\frac{1}{m_{Mass}}[{T_{Kinetic-Energy}}\;\;+\;\;{V_{Gravity-Potential}}]$$\,\,----> \,\, \frac{m^2}{s^2}$

Next, substituting the, Net Kinetic Energy (${T_{Kinetic-Energy}}=0$), yields the Total Mechanical Energy Conservation given by;

Total Mechanical Energy Conservation“Bound/Strongly” to Gravitational Field Condition

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\,\,=\,\,\frac{V_{Gravity-Potential}}{m_{Mass}}$

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\,\,=\,\,(-)\frac{m_{Net}\,G}{r}$

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

(Case #2) The Total Mechanical Energy Conservation – “Orbiting Condition” in Gravitational Potential “Energy State”

In the case where “mass” and “energy” are in “Orbiting” condition in a “Gravitational Potential” of a gradient gravitational field, the Total Mechanical Energy Conservation (${E_{Total}}$) equals to one half the Gravitational ($\frac{1}{2}{V_{Gravity-Potential}}$) Potential Energy, and is equal to negative of the Kinetic Energy ($(-){T_{Kinetic-Energy}}$).

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

Total Mechanical Energy Conservation

$\frac{E_{Total}}{m_{Mass}}\,\,=\,\,\frac{{T_{Kinetic-Energy}}\;\;+\;\;{V_{Gravity-Potential}}}{m_{Mass}}$$\,\,----> \,\, \frac{m^2}{s^2}$

Next, setting the Total Mechanical Energy Conservation “Orbital Energy” equals to one half the Gravitational Potential Energy (${V_{Gravity-Potential}}$), and negative of the Kinetic Energy (${T_{Kinetic-Energy}}$), given by the following;

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\,\,=\,\,(-){{T_{Kinetic-Energy}}\,\,=\,\,\frac{1}{2}{V_{Gravity-Potential}}}$

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

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\,\,=\,\,(-)\frac{1}{2}{|\vec{v}|^2_{CM}}\,\,=\,\,(-)\frac{1}{2}{v^2_{Gravity}}\,\,=\,\,(-)\frac{1}{2}\frac{m_{Net}\,G}{r}$

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

Gravitational Field – Mechanical “Orbital Energy” – where the Kinetic Energy (${T_{Kinetic-Energy}}$) equals to negative one half the Gravitational Potential Energy (${V_{Gravity-Potential}}$). Potential Energy Potentials.

$\frac{T_{Kinetic-Energy}}{m_{Mass}}\;\;=\;\;(-)\frac{1}{2}\frac{V_{Gravity-Potential}}{m_{Mass}}$$\,\,----> \,\, \frac{m^2}{s^2}$

$\frac{1}{2}{|\vec{v}|^2_{CM}}\;\;=\;\;\frac{1}{2}{v^2_{Gravity}}\;\;=\;\;\frac{1}{2}\frac{m_{Net}\,G}{r}$

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

Gravitational Field – Mechanical “Orbital Energy” – where the Kinetic Energy (${T_{Kinetic-Energy}}$) equals to negative one half the Gravitational Potential Energy (${V_{Gravity-Potential}}$). Kinetic Energy “Spacetime” Potentials.

$\frac{T_{Kinetic-Energy}}{m_{Mass}}\;\;=\;\;(-)\frac{1}{2}\frac{V_{Gravity-Potential}}{m_{Mass}}$$\,\,----> \,\, \frac{m^2}{s^2}$

$\frac{1}{2}{|\vec{v}|^2_{CM}}\;\;=\;\;\frac{1}{2}{v^2_{Gravity}}_{Spacetime}\;\;=\;\;\frac{1}{2}\,{c^2_{Light}}\,[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}\,r}{2\,{m_{Net}}\,G}))^2\,\,+\,\,({\Omega_{Map}}_{(\theta \phi)})^2}]^2$

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

(Case #3) The Total Mechanical Energy Conservation – “Minimum Escape Condition” to “Unbound” from a Gravitational Potential “Energy State”

In the case where “mass” and “energy” are in “Minimum Escape” condition to “Unbound” from a “Gravitational Potential” of a gradient gravitational field, the Total Mechanical Energy Conservation is equal to zero (${E_{Total}}=0$) .

In the case where “mass” and “energy” are in “Minimum Escape” condition, this is also known as the “Escape Energy” condition, and is a measure of the “Escape Velocity” from a “Gradient Gravitational Field Potential”; and the Net Kinetic Energy equals to negative of the Gravitational Potential Energy (${T_{Kinetic-Energy}}=(-){V_{Gravity-Potential}}$), and located at a specific potential, of the gravity field gradient.

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

Total Mechanical Energy Conservation

$\frac{E_{Total}}{m_{Mass}}\,\,=\,\,\frac{{T_{Kinetic-Energy}}\;\;+\;\;{V_{Gravity-Potential}}}{m_{Mass}}$$\,\,----> \,\, \frac{m^2}{s^2}$

Next, setting the Total Mechanical Energy Conservation “Escape Energy” equal to zero (0) is given by the following;

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\,\,=\,\,0$

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

Gravitational Field – Mechanical “Minimum Escape Energy”equals to Gradient Gravitational Field “Potential Energy” Potentials

$\frac{T_{Kinetic-Energy}}{m_{Mass}}\;\;=\;\;(-)\frac{V_{Gravity-Potential}}{m_{Mass}}$$\,\,----> \,\, \frac{m^2}{s^2}$

$\frac{1}{2}{|\vec{v}|^2_{CM}}\;\;=\;\;{v^2_{Gravity}}\;\;=\;\;\frac{m_{Net}\,G}{r}$

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

The Escape Velocity from a local gravitational “Potential Energy” potential is then given by:

${|\vec{v}|_{CM}}\;\;=\;\;{v_{Gravity}}\,\sqrt{2}\;\;=\;\;\sqrt{2\,\frac{m_{Net}\,G}{r}}$ $\,\,----> \,\, \frac{m}{s}$

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

Gravitational Field – Mechanical “Minimum Escape Energy” – equals to Gradient Gravitational Field “Kinetic Energy” Vacuum of Space-time Potentials

$\frac{T_{Kinetic-Energy}}{m_{Mass}}\;\;=\;\;(-)\frac{V_{Gravity-Potential}}{m_{Mass}}$$\,\,----> \,\, \frac{m^2}{s^2}$

$\frac{1}{2}{|\vec{v}|^2_{CM}}\;\;=\;\;{v^2_{Gravity}}_{Spacetime}\;\;=\;\;\frac{1}{2}\,{c^2_{Light}}\,[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}\,r}{2\,{m_{Net}}\,G}))^2\,\,+\,\,({\Omega_{Map}}_{(\theta \phi)})^2}]^2$

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

The Escape Velocity from a local gravitational “Kinetic Energy” space-time potential is then given by:

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

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

(Case #4) The Total Mechanical Energy Conservation – “Totally Free Condition” from Gravitational “Potential Energy” – “Kinetic Energy” State Potentials

In the case where “mass” and “energy” are in “Totally Free Condition” from a “Gravitational Potential”, and from the “gravitational attraction” effects, of a gradient gravitational field. This energy state is complete kinetic energy, where the Total Mechanical Energy Conservation (${E_{Self-Total}}={T_{Kinetic-Energy}}$), is equal to the Net Kinetic Energy located at specific “Kinetic Energy Potentials”, of the gravity field gradient, relative to the “Schwarzschild Radius” (${r_{Schwarzschild}}\,=\,2\frac{m_{Net}\,G}{c^2_{Light}}$) “Black Hole Event Horizon”; which is located at the core, the most minimum distance and volume of the gradient field.

And the Gravitational Potential Energy (${V_{Gravity-Potential}}=0$), at that potential, is equal to zero (0).

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

Total Mechanical Energy Conservation

$\frac{E_{Total}}{m_{Mass}}\,\,=\,\,\frac{{T_{Kinetic-Energy}}\;\;+\;\; {V_{Gravity-Potential}}}{m_{Mass}}$$\,\,----> \,\, \frac{m^2}{s^2}$

Next, substituting the, Gravitational Potential Energy (${V_{Gravity-Potential}}=0$), yields the Total Mechanical Energy Conservation at specific “Kinetic Energy Potentials”, given by the following;

Total Mechanical Energy Conservation“Totally Free” from Gravitational Potential Energy Condition – Kinetic Energy Potentials

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\,\,=\,\,\frac{T_{Kinetic-Energy}}{m_{Mass}}$

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\,\,=\,\,\frac{1}{2}{|\vec{v}|^2_{CM}}\,\,=\,\,\frac{1}{2}\,{c^2_{Light}}\,[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}\,r}{2\,{m_{Net}}\,G}))^2\,\,+\,\,({\Omega_{Map}}_{(\theta \phi)})^2}]^2$

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

The Escape Velocity from a local gravitational “Kinetic Energy” potential is then given by:

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

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

The above “New” equation for describing the condition when the “The Total Mechanical Energy” is equal to the “Total Kinetic Energy” for any “mass” or “energy”, located at any potential of the gradient gravitational field; although the “mass” or “energy”, is totally free from the gravitating influences of the field. The energy potentials in this conditions are “Kinetic Energy Potentials”.

The above equation allows for prediction of the “Total Kinetic Energy” at each potential of the gradient gravitational field, located at some distace relative to the, “Black Hole Event Horizon” “Schwarzschild Radius” (${r_{Schwarzschild}}\,=\,2\frac{m_{Net}\,G}{c^2_{Light}}$).

This “energy state” condition, is free from “gravitational attraction” influences! This is therefore a gradient field of  “Kinetic Energy” Potentials, and not a gradient field of “Potential Energy” Potentials.

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

(Case #5) The Total Mechanical Energy Conservation – “Partially/Weakly Bound Condition” to a Gravitational Potential “Energy State”

In the case where “mass” and “energy” are in “Partially/Weakly Bound” condition to a “Gravitational Potential” of “Potential Energy” of a gradient gravitational field, the Total Mechanical Energy Conservation (${E_{Total}}={T_{Kinetic-Energy}}+{V_{Potential-Energy}}$) is equal to the net sum of the Net Kinetic Energy plus the Gravitational Potential Energy, located at each potential of the gradient gravity field.

The “Partially/Weakly Bound” condition in a gravitational potential, is where there is sufficient energy to escape the field, yet remains very loosely bound to the, “Gradient Gravitational Field Potential”.

In the “Partially/Weakly Bound” condition in a gravitational potential, the Net Kinetic Energy is greater than or equal to negative of the Gravitational Potential Energy (${T_{Kinetic-Energy}}\geq(-){V_{Gravity-Potential}}$) located at a specific potential, of the gravity field gradient.

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

Total Mechanical Energy Conservation

$\frac{E_{Total}}{m_{Mass}}\,\,=\,\,\frac{{T_{Kinetic-Energy}}\;\;+\;\;{V_{Gravity-Potential}}}{m_{Mass}}$$\,\,----> \,\, \frac{m^2}{s^2}$

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

Total Mechanical Energy Conservation “Partially/Weakly Bound” to Gravitational Potential Energy Condition

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\;\;=\;\;\frac{T_{Kinetic-Energy}}{m_{Mass}}\;\;+\;\;\frac{V_{Gravity-Potential}}{m_{Mass}}$

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

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\;\;=\;\;[\frac{1}{2}{|\vec{v}|^2_{CM}}\;\;-\;\;\frac{m_{Net}\,G}{r}]$

$\frac{E_{Self-Total}}{m_{Net}}\,\,=\,\,\frac{E_{Total}}{m_{mass}}\,\,=\,\,[\frac{1}{2}\,{c^2_{Light}}\,[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}\,r}{2\,{m_{Net}}\,G}))^2\,\,+\,\,({\Omega_{Map}}_{(\theta \phi)})^2}]^2\,\,\,-\,\,\,\frac{m_{Net}\,G}{r}]$

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

General Constants

Gravitational Universal Constant

${G}\;\;=\;\;6.67384 \times 10^{-11} \frac{m^3}{kg\,s^2}$

Speed of Light in vacuum constant

${c_{Light}}\;\;=\;\;2.99792459 \times 10^{8} \frac{m}{s}$

Cosmic “Dark” Vacuum Force universal constant

${F_{Dark-Force}}\;\;=\;\;\frac{1}{4}(\frac{c^4_{Light}}{G})\;\;=\;\;3.0256479774082\times10^{+43}\;\;\frac{kgm}{s^2}$

Black Hole Linear Mass Density

${\mu_{L-Density}}_{BH}\;\;=\;\;\frac{1}{2}\frac{c^2_{Light}}{G}=6.73297478332358\times 10^{26}\frac{kg}{m}$

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

Closing Remarks:

The above work discusses a new set of equations for describing the, “Total Mechanical Energy Conservation” of an isolated system body in regards to specific “Energy States” of various “Potential Energy Potentials” and “Kinetic Energy Potentials”. The above energy equations were also described in terms of the “Einstein Field Equation”, the “Universal Cosmic “Dark” Vacuum Force”, and the “Vacuum Energy”.

The “Einstein Field Equation” and the “Vacuum Energy” was shown to be an “Elastic “Stress” Kinetic Energy” equation that predicts that there are “Kinetic Energy Potentials”, and a warping, deforming, and curving of space and time in the vacuum, and in the local vicinity of any mass object.

The “Net Inertial Mass” was shown conceptualy and mathematically, to warp space and time, and create a gradient energy field of “Potential” and “Kinetic” energy potentials surrounding the mass object. And, at the core of the gradient field of potentials is the “Black Hole Event Horizon”, having a specific “Schwarzschild” “Radius” and “Volume”.

It is at the core of the gradient gravity field, the location of the “Black Hole Event Horizon”, which is the lowest potential of the gradient field, and the source of gravitation, where “mass” and “space” of an orgainzed unit or system body, are in universal continuum.

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

Citation

Robert Louis Kemp; The Super Principia Mathematica – The Rage to Master Conceptual & Mathematical Physics – The General Theory of Relativity – “Total Mechanical Energy Conservation – Escape Velocity & Binding Energy – Einstein Field Equation” – Online Volume – ISBN 978-0-9841518-2-0, Volume 3; July 2010

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

Best,

Author: Robert Louis Kemp

http://www.SuperPrincipia.com

Total Mechanical Energy Conservation in General Relativity

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”.

Under the condition where, only conservative forces do work, the “Total Mechanical Energy” of an isolated “Net Inertial Mass” system body remains “constant” and “conservative”. The “Total Mechanical Energy” of an isolated system is constant, means that any increase in “Kinetic Energy” is always accompanied by a decrease in the “Potential Energy” of the system.

This fundamental principle of the conservation of energy, of the “Total Mechanical Energy”, of any isolated and adiabatic system, is one of the most fundamental and important concepts of physics.

In nature there are always “non-conservative” energy and forces present; the most common type is the frictional force and heat energy. Another type of non-conservative energy and force is when a spring is stretched beyond its elastic limit; it can become permanently deformed, and the work done in stretching the spring is not recoverable, when the spring is released. The heat energy in both “non-conservative” cases is considered the “Internal Heat Energy” of the system.

An “Adiabatic” system, a term used to denote the total quantity of “Internal Heat Energy” in the system, such that no “additional” heat energy enters, nor is there a “deficiency” of heat energy, that leaves the system. The “Adiabatic – Internal” heat energy present in the system is constant and is directly proportional to the kinetic energy of the system.

Thus, for any isolated “Net Inertial Mass” system, the disappearance of any “mechanical energy”, is always accompanied by the appearance of “Internal Energy” of the system, and is measured by an increase in the “Temperature” of the system. This internal energy of the surroundings consists of the “Kinetic Energy” and “Potential Energy” of the molecular motion of the system, and is also a measure of the “Total Mechanical Energy” of the system mass body.

When the concept of “total energy” is generalized to include the above “Internal Energy”, the “Total Mechanical Energy” of an object, plus its surrounding is constant and conserved, and does not change even when friction is present.

It is always possible to account for any increases or decreases in the “mechanical energy” of the system, by the appearance or disappearance of “mechanical energy” somewhere else. For example, energy of a system is often decreased because of some form of radiation: i.e., there are water waves produced by a ship, or sound waves produced from the collision of two objects, or the electromagnetic waves which are produced by accelerated charges in a simple radio antenna.

The conservation of energy is also stated conceptually: Energy is never created or destroyed in the system; but energy changes from one form, into another form. For example, mechanical energy can be transformed into electrical energy, which can be further transformed into chemical energy.

Therefore, a generalized statement of the conservation of energy can be written in the following way. Let the “Total Mechanical Energy” (${E_{Total}}$) be the total energy of a given system, and let the “Power” (${P_{Power}}$) be the power input or output the system; and is defined as the rate at which energy is input or output the system. Thus, the value of the “Power Outflow” (${P_{Output}}=(-)\frac{dE_{Total}}{dt}$) is negative if energy is flowing out of the system; and the “Power Inflow” (${P_{Input}}=\frac{dE_{Total}}{dt}$) is positive if energy is flowing into of the system.

The significance of this statement is that the total amount of energy of the system is always exactly accounted for, by energy flow into or out of the system. Energy is never created or destroyed in the system, though it may change from one form into another form.

In this work only the “Mechanical Energy” of the “Net Kinetic Energy”(${T_{Kinetic-Energy}}$), and the “Gravitational Potential Energy” (${V_{Potential-Energy}}$) associated with the ‘Gravitational Force” ($\vec{F_{Gravity}}$) of the system mass body is considered.

Total Mechanical Energy

${E_{Total}}\,\,=\,\, {T_{Kinetic-Energy}}\;\;+\;\;{V_{Potential-Energy}}$

${E_{Total}}\,\,=\,\,{T_{KE - Initial}}\,\,+\,\,{V_{PE - Initial}}\,\,=\,\,{T_{KE - Final}}\,\,+\,\,{V_{PE - Final}}$

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

(1) The “Anisotropic” Net Kinetic Energy Relative to the Center of Mass of an Isolated Net Inertial Mass System Body – & consideration for “Special Relativity”

The Rectilinear “Anisotropic” Net Kinetic Energy (${T_{Kinetic-Energy}}$) is a “scalar quantity” measure of the mass-energy of directional (anisotropic) motion, relative to the center of mass of the system; defined as the product of the Net Inertial Mass (${m_{Net}}$), multiplied by one half the square of the Average Rectilinear Center of Mass Velocity (${|\vec{v}|^2_{CM}}$) of an isolated system body; given by the following equation.

The Rectilinear “Anisotropic” Net Kinetic Energy (${T_{Kinetic-Energy}}$) – in the Proper Observer “center of mass” frame of reference is given by the following.

${T_{Kinetic-Energy}} \,\, =\,\,\frac{1}{2}\,{m_{Net}}\,{|\vec{v}|^2_{CM}}\,\, =\,\,\frac{\vec{p^2_{Net-Momentum}}}{2\,m_{Net}}$$\,\,----> \,\, \frac{kg\,m^2}{s^2}$

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

Where the Net Inertial Mass – (${m_{Net}}$) – in the Proper Observer “center of mass” frame of reference is given by the following.

${m_{Net}}\, \,=\,\,\displaystyle\sum_{i=1}^N {m_{i}}\,\,=\,\,[ {m_{1}} + {m_{2}} + {m_{3}} + ...... + {m_{N}}]\,\,$ $---> {kg}$

And, the Relativistic Net Inertial Mass – (${m'_{Rel}}$) – in the External Observer frame of reference is given by the following.

${m'_{Rel}}\, \,=\,\,\frac{m_{Net}}{\sqrt{1\;\;-\;\;\frac{{|\vec{v}|^2_{CM}}}{c^2_{Light}}}}\,\,$ $---> {kg}$

The Relativistic Rectilinear “Anisotropic” Net Kinetic Energy (${T'_{Kinetic-Energy}}$) – in the External Observer frame of reference is given by the following.

${T'_{Kinetic-Energy}} \,\, =\,\,\frac{T_{Kinetic-Energy}}{\sqrt{1\;\;-\;\;\frac{{|\vec{v}|^2_{CM}}}{c^2_{Light}}}} \,\, =\,\,\frac{1}{2}\,{m'_{Rel}}\,{|\vec{v}|^2_{CM}}\,\, =\,\,\frac{\vec{p'^2_{Rel-Momentum}}}{2\,m'_{Rel}}$

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

The square of the Average Rectilinear Center of Mass Velocity(${|\vec{v}|^2_{CM}}$) – is an “invariant” scalar quantity of “inertia motion” that is equal to all observers and frames of reference.

${|\vec{v}|^2_{CM}} \,\, =\frac{\vec{p^2_{Net-Momentum}}}{m^2_{Net}}\,\, =\frac{\vec{p'^2_{Rel-Momentum}}}{m'^2_{Rel}}$

${|\vec{v}|^2_{CM}} \,\, =\frac{2\,T_{Kinetic-Energy}}{m_{Net}}\,\, =\frac{2\,T'_{Kinetic-Energy}}{m'_{Rel}}$

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

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

The Average Rectilinear Center of Mass Velocity(${|\vec{v}|_{CM}}$) – of the mass bodies of a net mass system is a vector quantity, that describes direction dependent motion, or “Anisotropic Motion”, and measures “equal distance changing in equal times” motion of an isolated system mass body; and is described in two ways:

The first (1) description of the Center of Mass Velocity(${|\vec{v}|_{CM}}$) vector that measures “equal distance changes in equal times”, and is a measure of mass in “Anisotropic” motion, relative to the “center of mass” or “Barycenter” of the system, which is considered at “Rest”; and is constant, invariant, and frame of reference independent.

The second (2) description of the Center of Mass Velocity(${|\vec{v}|_{CM}}$) vector that “measures equal distance changes in equal times”, and is the measure of the “Anisotropic” motion” of the “Center of Mass”, or “Barycenter” of the system; and is considered to be in “Uniform Rectilinear Motion”, relative to an “External Observer” frame of reference; and is constant, invariant, and frame of reference independent .

The Average Rectilinear Center of Mass Velocity(${|\vec{v}|_{CM}}$) – measures “equal distance changes in equal times” motion, and is defined as the Net Inertial Rectilinear Momentum (${\vec{p_{Net-Momentum}}}$) divided by the Net Inertial Mass (${m_{Net}}$) of an isolated system body; and is an “invariant” vector quantity of “rectilinear motion” that is equal to all observers and frames of reference.

${|\vec{v}|_{CM}} \,\, =\frac{\vec{p_{Net-Momentum}}}{m_{Net}}\,\, =\frac{\vec{p'_{Rel-Momentum}}}{m'_{Rel}}$

${|\vec{v}|_{CM}} \,\, =\sqrt\frac{2\,T_{Kinetic-Energy}}{m_{Net}}\,\,\hat{a}_{r}\,\, =\sqrt\frac{2\,T'_{Kinetic-Energy}}{m'_{Rel}}\,\,\hat{a}_{r}$

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

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

(2) The Total Mechanical Energy Conservation – Euclidean Spherical Mechanics – The General Theory of Relativity

Now that all of the physics terms required for a complete description have been determined, let’s complete the discussion of the Total Mechanical Energy Conservation (${E_{Total}}={T_{Kinetic-Energy}}+{V_{Potential-Energy}}$) in a consideration for General Relativity.

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

Total Mechanical Energy Conservation

${E_{Total}}\,\,=\,\, {T_{Kinetic-Energy}}\;\;+\;\;{V_{Gravity-Potential}}$

${E_{Total}}\,\,=\,\, {T_{Kinetic-Energy}}\;\;+\;\;(\frac{{m_{mass}}}{{m_{Net}}}){V_{Self-Potential}}$

Next substituting the appropriate “Kinetic Energy” and “Gravitational Potential Energy” terms yields:

${E_{Total}}\,\,=\,\,[\frac{1}{2}{m_{mass}}{|\vec{v}|^2_{CM}}\;\;-\;\;\frac{{m_{mass}\,m_{Net}\,G}}{r}]$ $\,\,----> \,\, \frac{kg\,m^2}{s^2}$

${E_{Total}}\,\,=\,\,{m_{mass}}[\frac{1}{2}{|\vec{v}|^2_{CM}}\,\,-\,\,{g_{Gravity}}\,{r}]$ $\,\,----> \,\, \frac{kg\,m^2}{s^2}$

${E_{Total}}\,\,=\,\,\,{m_{mass}} [\frac{1}{2}{|\vec{v}|^2_{CM}}\,\,-\,\,{v^2_{Gravity}}]$ $\,\,----> \,\, \frac{kg\,m^2}{s^2}$

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

From the mathematics derived below it will be shown that the Total Mechanical Energy Conservation is given by;

${E_{Total}}\,\,=\,\,[\frac{1}{2}{{m_{mass}}\,c^2_{Light}}\,[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}\,r}{2\,{m_{Net}}\,G}))^2\;\;+\;\;({\Omega_{Map}}_{(\theta \phi)})^2}]^2\;\;-\;\;\frac{{m_{mass}\,m_{Net}\,G}}{r}]$

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

(3) The Total “Self” Mechanical Energy Conservation – Euclidean Spherical Mechanics – The General Theory of Relativity

Now that all of the physics terms required for a complete description have been determined, let’s complete the discussion of the Total “Self” Mechanical Energy Conservation (${E_{Self-Total}}={T_{Kinetic-Energy}}+{V_{Potential-Energy}}$) in a consideration for General Relativity.

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

Total “Self” Mechanical Energy Conservation

${E_{Self-Total}}\,\,=\,\, {T_{Kinetic-Energy}}\;\;+\;\;{V_{Self-Potential}}$

${E_{Self-Total}}\,\,=\,\, {T_{Kinetic-Energy}}\;\;+\;\;(\frac{{m_{Net}}}{{m_{mass}}}){V_{Gravity-Potential}}$

Next substituting the appropriate “Kinetic Energy” and “Gravitational Potential Energy” terms yields:

${E_{Self-Total}}\,\,=\,\,[\frac{1}{2}{m_{Net}}{|\vec{v}|^2_{CM}}\;\;-\;\;\frac{m^2_{Net}\,G}{r}]$ $\,\,----> \,\, \frac{kg\,m^2}{s^2}$

${E_{Self-Total}}\,\,=\,\,{m_{Net}}\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\,\,-\,\,{g_{Gravity}}\,{r}]$ $\,\,----> \,\, \frac{kg\,m^2}{s^2}$

${E_{Self-Total}}\,\,=\,\,\,{m_{Net}}\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\,\,-\,\,{v^2_{Gravity}}]$ $\,\,----> \,\, \frac{kg\,m^2}{s^2}$

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

From the mathematics derived below it will be shown that the Total “Self” Mechanical Energy Conservation is given by;

${E_{Self-Total}}\,\,=\,\,[\frac{1}{2}{{m_{Net}}\,c^2_{Light}}\,[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}\,r}{2\,{m_{Net}}\,G}))^2\;\;+\;\;({\Omega_{Map}}_{(\theta \phi)})^2}]^2\;\;-\;\;\frac{{m^2_{Net}\,G}}{r}]$

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

(4) The “Anisotropic” Net Kinetic Energy Relative to the Center of Mass of an Isolated Net Inertial Mass System Body – & consideration for “General Relativity”

In order to complete the discussion of the Rectilinear “Anisotropic” Net Kinetic Energy (${T_{Kinetic-Energy}}$) in a consideration for General Relativity, we will need to obtain the “Spherical Mechanics” fundamentals of the “Differential” – Map/Patch/Manifold – “Angle Metric” ($\vec{d\Omega^2_{Map}}_{(\theta \phi)}$) which is a “geodesic arc-length” angle component, on the surface of the sphere, as discussed in Section 3, of the work:

Euclidean Spherical Mechanics – Euclidean/Minkowski Spacetime Metrics

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

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

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

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

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

Next, solving for the Average Rectilinear Center of Mass Velocity – (${|\vec{v}|_{CM}}$) in the above equation.

Average Rectilinear Center of Mass Velocity – (${|\vec{v}|_{CM}}$) – is invariant and is measured to have the same value to all observers, and frames of reference; as described in the following equations.

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

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}$

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

The Rectilinear “Anisotropic” Net Kinetic Energy (${T_{Kinetic-Energy}}$) is a “scalar quantity” measure of the mass-energy of directional (anisotropic) motion, relative to the center of mass of the system; and is described in consideration for General Relativity; given by the following equations.

The Rectilinear “Anisotropic” Net Kinetic Energy (${T_{Kinetic-Energy}}$) – is expressed in “differential” mathematical form; and is given by the following.

${T_{Kinetic-Energy}}\,\, =\,\,\frac{{m_{Net}}\,c^2_{Light}}{2}[\frac{(\vec{d\Omega_{Map}}_{(\theta \phi)})^2}{2\,(\frac{\vec{dr}}{r})^2\;\;+\;\;(\vec{d\Omega_{Map}}_{(\theta \phi)})^2}]^2 \,\, =\,\,\frac{{m_{Net}}\,c^2_{Light}}{2}[\frac{(\vec{d\Omega_{Map}}_{(\theta \phi)})^2}{2\,(\frac{\vec{ds}}{s})^2\;\;+\;\;(\vec{d\Omega_{Map}}_{(\theta \phi)})^2}]^2$

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

The Rectilinear “Anisotropic” Net Kinetic Energy (${T_{Kinetic-Energy}}$) – in the Proper Observer “center of mass” frame of reference is expressed in “ordinary” mathematical form; and is given by the following.

${T_{Kinetic-Energy}} \,\, =\,\,\frac{1}{2}\,{m_{Net}}\,{|\vec{v}|^2_{CM}}\,\, =\,\,\frac{{m_{Net}}\,c^2_{Light}}{2}[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}\,r}{2\,{m_{Net}}\,G}))^2\;\;+\;\;({\Omega_{Map}}_{(\theta \phi)})^2}]^2$

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

The Relativistic Rectilinear “Anisotropic” Net Kinetic Energy (${T'_{Kinetic-Energy}}$) – in the External Observer frame of reference is expressed in “ordinary” mathematical form; and is given by the following.

${T'_{Kinetic-Energy}} \,\, =\,\,\frac{1}{2}\,{m'_{Rel}}{|\vec{v}|^2_{CM}}\,\, =\,\,\frac{1}{2}(\frac{{m_{Net}}\,c^2_{Light}}{{\sqrt{1\;\;-\;\;\frac{{|\vec{v}|^2_{CM}}}{c^2_{Light}}}}})[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{{c^2_{Light}}\,r}{2\,{m_{Net}}\,G}))^2\;\;+\;\;({\Omega_{Map}}_{(\theta \phi)})^2}]^2$

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

(5) The Total Mechanical Energy Conservation – Euclidean Spherical Mechanics – The General Theory of Relativity

Next, in returning, and in order to complete the discussion of the Total Mechanical Energy Conservation (${E_{Total}}={T_{Kinetic-Energy}}+{V_{Potential-Energy}}$) 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 was discussed in Section 4, of the work:

Euclidean Spherical Mechanics – Euclidean/Minkowski Spacetime Metrics

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

Now that all of the physics terms required for a complete description have been determined, let’s complete the discussion of the Total Mechanical Energy Conservation (${E_{Total}}={T_{Kinetic-Energy}}+{V_{Potential-Energy}}$) in a consideration for General Relativity.

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

Total “Self” Mechanical Energy Conservation

${E_{Self-Total}}\,\,=\,\, {T_{Kinetic-Energy}}\;\;+\;\;{V_{Self-Potential}}$

${E_{Self-Total}}\,\,=\,\, {T_{Kinetic-Energy}}\;\;+\;\;(\frac{{m_{Net}}}{{m_{mass}}}){V_{Gravity-Potential}}$

Next substituting the appropriate “Kinetic Energy” and “Gravitational Potential Energy” terms yields:

${E_{Self-Total}}\,=\,{m_{Net}}\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\,-\,{g_{Gravity}}\,{r}]\,=\,{m_{Net}}\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\,-\,{v^2_{Gravity}}]$

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

(5.1) The Total Mechanical Energy Conservation – Proper Observer frame – Space & Time

Next substituting the appropriate “Kinetic Energy” and “Gradient Gravitational Field Acceleration” terms yields:

Gradient Gravitational Field Acceleration($\vec{g_{Gravity}}-->\frac{m}{s^2}$)function of differential Radius of Sphere (${dr\,=\,{c_{Light}}\,dt_{Light}}$) Space & Time Metric – Proper Observer (Center of Mass) Frame of Reference – Equation of Motion

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

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

Total “Self” Mechanical Energy Conservation – in “differential” mathematical form, and described as a function of Radius of Sphere (${dr\,=\,{c_{Light}}\,dt_{Light}}$) “Space” (distance).

${E_{Self-Total}}\,\,=\,\,[\frac{1}{2}({{m_{Net}}\,c^2_{Light}})[\frac{{d\Omega^2_{Map}}_{(\theta \phi)}}{2\,{({\frac{\vec{dr}}{r}})^2}\;\;+\;\;{d\Omega^2_{Map}}_{(\theta \phi)}}]^2\;\;+\;\;({m^2_{Net}\,G})\,[\frac{1 \;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}]\,(r)\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{dr^2}]$

Total “Self” Mechanical Energy Conservation – in “differential” mathematical form, and described as a function of Radius of Sphere (${dr\,=\,{c_{Light}}\,dt_{Light}}$), and “Time”.

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

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

(5.2) The Total Mechanical Energy Conservation – Proper Observer & Equal Observer (Co-Variant frame) – “Interior Surface” – Space & Time

Next substituting the appropriate “Kinetic Energy” and “Gradient Gravitational Field Acceleration” terms yields:

${E_{Self-Total}}\,=\,{m_{Net}}\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\,-\,{g_{Gravity}}\,{r}]\,=\,{m_{Net}}\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\,-\,{v^2_{Gravity}}]$

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

Gradient Gravitational Field Acceleration(${g_{Gravity}}-->\frac{m}{s^2}$)function of the differential Map/Patch/Manifold – “Geodesic” ($\vec{ds_{Map}}_{\theta \, \phi}\,=\,(\sqrt{-1}){c_{Light}}\,{dt_{Map}})$ Space & Time Metric – Equal Observer (Co-Variant) Frame of Reference – Equation of Motion

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

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

Total “Self” Mechanical Energy Conservation – in “differential” mathematical form, and described as a function of Map/Patch/Manifold – “Geodesic” Arc-Length ($\vec{ds_{Map}}_{\theta \, \phi}\,=\,(\sqrt{-1}){c_{Light}}\,{dt_{Map}}$), “Interior Surface”, “Space” (distance).

${E_{Self-Total}}\,\,=\,\,[\frac{1}{2}({{m_{Net}}\,c^2_{Light}})[\frac{{d\Omega^2_{Map}}_{(\theta \phi)}}{2\,{({\frac{\vec{dr}}{r}})^2}\;\;+\;\;{d\Omega^2_{Map}}_{(\theta \phi)}}]^2\;\;-\;\;({m^2_{Net}\,G})\,(\frac{{s_{Map}}_{\theta \, \phi}}{{\Omega_{Map}}_{(\theta \phi)}})\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{\vec{ds^2_{Map}}_{\theta \, \phi}}]$

Total “Self” Mechanical Energy Conservation – in “differential” mathematical form, and described as a function of Map/Patch/Manifold – “Geodesic” Arc-Length ($\vec{ds_{Map}}_{\theta \, \phi}\,=\,(\sqrt{-1}){c_{Light}}\,{dt_{Map}})$, and, “Interior Surface”, “Time”.

${E_{Self-Total}}\,\,=\,\,[\frac{1}{2}({{m_{Net}}\,c^2_{Light}})[\frac{{d\Omega^2_{Map}}_{(\theta \phi)}}{2\,{({\frac{\vec{dt_{Light}}}{t_{Light}}})^2}\;\;+\;\;{d\Omega^2_{Map}}_{(\theta \phi)}}]^2\;\;+\;\;(\sqrt{-1})(\frac{m^2_{Net}\,G}{c_{Light}})\,(\frac{t_{Map}}{{\Omega_{Map}}_{(\theta \phi)}})\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{dt^2_{Map}}]$

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

(5.3) The Total Mechanical Energy Conservation – External Observer frame – Space & Time

Next substituting the appropriate “Kinetic Energy” and “Gravitational Potential Energy” terms yields:

${E_{Self-Total}}\,=\,{m_{Net}}\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\,-\,{g_{Gravity}}\,{r}]\,=\,{m_{Net}}\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\,-\,{v^2_{Gravity}}]$

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

Gradient Gravitational Field Acceleration($\vec{g_{Gravity}}-->\frac{m}{s^2}$)function of the differential Euclidean Radius of Sphere (${ds}\,=\,{c_{Light}}\,dt'_{s}$) Space & Time Metric – External Observer (Spherical Surface) Frame of Reference – Equation of Motion

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

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

Next, employing the, Map/Patch/Manifold – “Geodesic” Arc Length – ($\vec{s_{Map}}_{\theta \, \phi}$)

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

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

Total “Self” Mechanical Energy Conservation – in “differential” mathematical form, as a function of Euclidean Radius of Sphere (${ds\,=\,{c_{Light}}\,dt'_{Light(s)}}$) “Space” (distance).

${E_{Self-Total}}\,\,=\,\,[\frac{1}{2}({{m_{Net}}\,c^2_{Light}})[\frac{{d\Omega^2_{Map}}_{(\theta \phi)}}{2\,{({\frac{\vec{ds}}{s}})^2}\;\;+\;\;{d\Omega^2_{Map}}_{(\theta \phi)}}]^2\;\;+\;\;({m^2_{Net}\,G})\,[\frac{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}{2(\frac{|v|_{CM}}{c_{Light}})}]\,(s)\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{ds^2}]$

Total “Self” Mechanical Energy Conservation – in “differential” mathematical form, as a function of Euclidean Radius of Sphere (${ds\,=\,{c_{Light}}\,dt'_{Light(s)}}$), and “Time”.

${E_{Self-Total}}\,\,=\,\,[\frac{1}{2}({{m_{Net}}\,c^2_{Light}})[\frac{{d\Omega^2_{Map}}_{(\theta \phi)}}{2\,{({\frac{\vec{dt'_{Light(s)}}}{t'_{Light(s)}}})^2}\;\;+\;\;{d\Omega^2_{Map}}_{(\theta \phi)}}]^2\;\;+\;\;(\frac{m^2_{Net}\,G}{c_{Light}})\,[\frac{\sqrt{1 \;\;-\;\;\frac{|v|^2_{CM}}{c^2_{Light}}}}{2(\frac{|v|_{CM}}{c_{Light}})}]\,({t_{Light}})\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{dt^2_{Light}}]$

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

(5.4) The Total Mechanical Energy Conservation – External Observer & Equal Observer (Co-Variant frame) – “Exterior Surface” – Space & Time

Next substituting the appropriate “Kinetic Energy” and “Gradient Gravitational Field Acceleration” terms yields:

${E_{Self-Total}}\,=\,{m_{Net}}\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\,-\,{g_{Gravity}}\,{r}]\,=\,{m_{Net}}\,[\frac{1}{2}{|\vec{v}|^2_{CM}}\,-\,{v^2_{Gravity}}]$

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

Gradient Gravitational Field Acceleration(${g_{Gravity}}-->\frac{m}{s^2}$)function of the differential Map/Patch/Manifold – “Geodesic” ($\vec{ds_{Map}}_{\theta \, \phi}\,=\,(\sqrt{-1}){c_{Light}}\,{dt_{Map}})$ Space & Time Metric – Equal Observer (Co-Variant), Frame of Reference – Equation of Motion

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

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

Total “Self” Mechanical Energy Conservation – in “differential” mathematical form, and described as a function of Map/Patch/Manifold – “Geodesic” Arc-Length ($\vec{ds_{Map}}_{\theta \, \phi}\,=\,(\sqrt{-1}){c_{Light}}\,{dt_{Map}}$) “Exterior Surface”, “Space” (distance).

${E_{Self-Total}}\,\,=\,\,[\frac{1}{2}({{m_{Net}}\,c^2_{Light}})[\frac{{d\Omega^2_{Map}}_{(\theta \phi)}}{2\,{({\frac{\vec{ds}}{s}})^2}\;\;+\;\;{d\Omega^2_{Map}}_{(\theta \phi)}}]^2\;\;-\;\;({m^2_{Net}\,G})\,(\frac{{s_{Map}}_{\theta \, \phi}}{{\Omega_{Map}}_{(\theta \phi)}})\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{\vec{ds^2_{Map}}_{\theta \, \phi}}]$

Total “Self” Mechanical Energy Conservation – in “differential” mathematical form, and described as a function of Map/Patch/Manifold – “Geodesic” Arc-Length ($\vec{ds_{Map}}_{\theta \, \phi}\,=\,(\sqrt{-1}){c_{Light}}\,{dt_{Map}})$, and, “Exterior Surface”, “Time”.

${E_{Self-Total}}\,\,=\,\,[\frac{1}{2}({{m_{Net}}\,c^2_{Light}})[\frac{{d\Omega^2_{Map}}_{(\theta \phi)}}{2\,{({\frac{\vec{dt'_{Light(s)}}}{t'_{Light(s)}}})^2}\;\;+\;\;{d\Omega^2_{Map}}_{(\theta \phi)}}]^2\;\;+\;\;(\sqrt{-1})(\frac{m^2_{Net}\,G}{c_{Light}})\,(\frac{t_{Map}}{{\Omega_{Map}}_{(\theta \phi)}})\,\frac{\vec{d\Omega^2_{Map}}_{(\theta \phi)}}{dt^2_{Map}}]$

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

(6) The Total “Specific” Mechanical Energy Conservation

The Total “Specific” Mechanical Energy Conservation is the net sum of the Gravitational Potential Energy (${V_{Potential-Energy}}$) for a general gradient gravitational field, where the source of gravity is the Net Inertial Mass (${m_{Net}}$), plus the Net Kinetic Energy (${T_{Kinetic-Energy}}$) of the inertial mass system body; that whole term divided by the “Net Inertial Mass” or the “Test Inertial Mass” of the system.

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

Gravitational Field Acceleration

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

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

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

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

Total Mechanical Energy – Specific

$\frac{E_{Total}}{m_{Mass}}\,\,=\,\, \frac{{T_{Kinetic-Energy}}\;\;+\;\;{V_{Potenatial-Energy}}}{m_{Mass}}$ $\,\,----> \,\, \frac{kg\,m^2}{s^2}$

Total Mechanical Energy – Specific

$\frac{E_{Total}}{m_{mass}}\,\,=\,\, (\frac{1}{m_{mass}})[{T_{Kinetic-Energy}}\;\;+\;\;{V_{Potenatial-Energy}}]$ $\,\,----> \,\, \frac{kg\,m^2}{s^2}$

$\frac{E_{Total}}{m_{mass}}\,\,=\,\,[\frac{{|\vec{v}|^2_{CM}}}{2}\;\;-\;\;{v^2_{Gravity}}]\,\,=\,\,[\frac{{|\vec{v}|^2_{CM}}}{2}\;\;-\;\;{g_{Gravity}}\,{r}]$ $\,\,----> \,\, \frac{kg\,m^2}{s^2}$

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

Total Mechanical Energy – Specific

$\frac{E_{Total}}{m_{mass}}\,\,=\,\,[\frac{{|\vec{v}|^2_{CM}}}{2}\;\;-\;\;{g_{Gravity}}\,{r}]$ $\,\,----> \,\, \frac{kg\,m^2}{s^2}$

$\frac{E_{Total}}{m_{mass}}\,\,=\,\,[(\frac{c^2_{Light}}{2})[\frac{({\Omega_{Map}}_{(\theta \phi)})^2}{2\,(ln(\frac{r}{{r_{Schwarzschild}}}))^2\;\;+\;\;({\Omega_{Map}}_{(\theta \phi)})^2}]^2\;\;-\;\;\frac{m_{Net}\,G}{r}]$ $\,\,----> \,\, \frac{kg\,m^2}{s^2}$

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

The above work discusses a new set of equations for describing the familiar, Total Mechanical Energy Conservation of an isolated system body. The above energy equations are described mainly in the Proper Observer frame of reference. However, it is clear to see that the Total Mechanical Energy equations can be transformed, and written in the External Observer’s frame of reference.

The next, Energy Conservation that needs to be discussed that was not discussed in this work, is the Total Light Energy Content of the dynamical mass system body; which will be discussed in another work.

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

Citation

Robert Louis Kemp; The Super Principia Mathematica – The Rage to Master Conceptual & Mathematical Physics – The General Theory of Relativity – “Total Mechanical Energy Conservation in General Relativity– Online Volume – ISBN 978-0-9841518-2-0, Volume 3; July 2010

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

Best,

Author: Robert Louis Kemp

http://www.SuperPrincipia.com

Relativistic Gravitational Force – The Non Inertial Frame of Reference in General Relativity

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, with the identical properties to a “Gravitational Vortex”, 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 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”.

Previously, in the work; A Theory of Gravity for the 21st Century”, it was demonstrated conceptually and mathematically that the “Potential Energy” is associated with the work done by a “central conservative force”, namely the “Gravitational Force.” Various other types of “central conservative forces” include: the Elastic Spring Force, the Electrostatics Force, and the Magnetostatics Force.

Any and every conserved and isolated “Net Inertial Mass” system body, can be modeled as a “vortex” system body, that is spheroid in nature, and is described by a gradient field, comprised of an infinite amount of “spherical shell potentials” relative to the center of the system. The gradient gravity field is described by concentric spherical volumetric potential shells of “Gravitational Potential Energy” and a conservative “Self Gravitational Force” at each potential.

It was also demonstrated that, for a general gradient gravitational field, the conservative “Self” Gravitational Potential Energy (${V_{Self-Potential}}$) of each concentric spherical shell potential of the gradient field, is associated with the Inertial Mass “Self “Gravitational Force ($\vec{F_{Self-Gravity}}$), where the source of gradient gravity field is the Net Inertial Mass (${m_{Net}}$).

And, lastly it was shown that the “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$) is a gravitational field parameter that varies, in direct proportion to the square of the Net Inertial Linear Mass Density (${\mu^2_{L-Density}}$); and is described mathematically in terms of “Relativistic” frames of reference, observers, and their respective motions, below.

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

In the work, it will be shown that the “Total Gravitational Force” ($\vec{F_{Self-Gravity-Total}}$) is conserved in nature, and is the net sum, in one frame and is the net difference in another frame, of the “Central Force” ($\vec{F_{Self-Gravity}}$), plus the “Surface “Tidal” Force” ($\vec{F_{Self-Gravity}}_{\theta \, \phi}$) of the isolated net inertial gravitational field, system mass body; i.e Gravitational Vortex.

The “Surface “Tidal” Force” ($\vec{F_{Self-Gravity}}_{\theta \, \phi}$) tidal forces arise on the interior and exterior surface of the sphere, of a fixed energy potential; determined by the fixed radius. The interior and exterior surface of a sphere, or gradient energy potential, can also be considered a Proper Observer and an External Observer frame of reference respectively.

In the work, a set of Euclidean Gravitational Force Transformation Equations, will be described, that provide Relativistic Force Transformation, into any frame of reference, from another frame of reference. The study of General Relativity concentrates its discussion on the parameters of force and acceleration, and the equations are described, in what is known classically as, the “Non-Inertial” or “Accelerating” frame of reference. This is very similar to the Lorentz Transformation equations of Special Relativity; where the Lorentz Transformation equations are described in the “Inertial Frame” or “Non-Accelerating” frame of reference.

In this model there is a Proper Observer located at the mean center of the sphere, and a Proper Observer located at the “Interior Surface” of the sphere. And likewise, there is an External Observer located at the surface of the sphere, and an External Observer located at the “Exterior Surface” of the sphere.

In classical discussions of “Gravitation” it is very common to think, model, and analyze the effects of “gravitation” on any “Net Inertial System Mass Body” with a “Gradient Gravitation Field of Potential of Energy” from the stand point of either, a fixed static or a fixed rotating frame of reference.

In this work it will be considered conceptually and mathematically, that the “Gravitational Force” is a “Relativistic Force”, and it will described, what happens when a “Gradient Gravity Field” is in motion with respect to different observers and different frames of reference.

It will be demonstrated that the Gravitational Force strength of Gradient Gravitational Field Energy Potential increases, when the velocity motion of the gradient field, or the net inertial mass as a whole unit, increases velocity.

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

Aphorism:

An External Observer, frame of reference, measures, the strength of the “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$) in a gradient gravitational field, in “Relativistic Motion” to “increase”. Any increase or decrease in the relative velocity of a Net Inertial Mass or Gradient Gravitational Field of Energy Potential, causes the Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$) at that potential to increase or decrease.

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

$\vec{F'_{Self-Gravity}}\,\,=\,\,(-)\frac{\vec{F_{Self-Gravity}}}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}\,\,=\,\,\frac{m^2_{Net}\,G}{r^2}[\frac{1}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}]\,\hat{a}_{r}$

$\frac{\vec{F'_{Self-Gravity}}}{\vec{F_{Self-Gravity}}}\,\,=\,\,(-)[\frac{1}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}]$

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

Aphorism: Principle of Equivalence

The Principle of Equivalence states that any experiments performed in a “non-inertial” or uniformly accelerating frame of reference, with linear acceleration defined as the rate of velocity change ($a = \frac{d|v|_{CM}}{dt}$), are indistinguishable from the same experiments performed in a “non-accelerating” inertial frame of reference which is located in a gravitational field of Potential Energy, where the uniform acceleration of the gravitational field ($g\,=\,\frac{m_{Net}\,G}{r^2}\,=\,(-)a = \frac{d|v|_{CM}}{dt}$), ) is equal to the linear acceleration.

There are very many “thought experiments” involving moving elevators located in gravitational fields, that have been discussed to analyze the above “Principle of Equilavence” over the years, which completely suffice the above mentioned aphorism.

Now, let’t try a different “thought experiment”; for describing non-inertial or accelerating frames of reference.

Imagine if you will, a Symmetrically Spherical Spaceship that contains astronauts.

Next, imagine that Spherical Spaceship is equipped with a device that allows the Spherical Spaceship to rotate, and create any uniform acceleration or “g-force” towards the center of the Spherical Spaceship; as well as boost in any x, y, or z direction with great ease.

Next, imagine that the Spherical Spaceship is able to rotate at the same uniform acceleration or “g-force” as the earth; where astronauts within the Spherical Spaceship experience our familiar (g=9.8 m/s^2) or (32 ft/s^2). The Spherical Spaceship generates a uniform acceleration or “g-force”, and a “Self Gravitational Force” such that astronauts aboard the Spherical Spaceship always feel weight, like they are on earth, no matter how far the spaceship is from the earth.

Next, imagine that the Spherical Spaceship is equipped with a “boost accelerator” such that any velocity of the Spaceship, less than the speed of light ($(\frac{v_{Ship}}{c_{Light}}) < 1$) can be selected.

Now, let’s do the experiment.

Now, allow the Spherical Spaceship to generate “g-force” = 9.8 m/s^2, and boost to a velocity of (0.7${c_{Light}}$). Now ask; Do the astronauts aboard the Spherical Spaceship still weigh the same, or is their weight different? Is the Self Gravitational Force at each potential of the gradient gravity field the same? Does the uniform velocity of the Spherical Spaceship affect the “g-force” that the astronauts experience? Among other questions?

The answer is; the astronauts should weigh more; and the faster the Spherical Spaceship travels relative to the speed of the light, the heavier the astronauts become.

It turns out, based on the mathematics, the “Gravitational Force” of the Gradient Energy Potential, increases, as the velocity of the spaceship, relative to the speed of light, increases.

It seems like we may never see the far reaches of the galaxy!

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

(1) The “Total” Conservative “Self Gravitational Force” of an “Inertial Mass” Gradient Gravitational Field – Proper Observer “CM” Frame of Reference

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

Where the “Total” Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity-Total}}$) – is conserved, and is equal to the net sum of the “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$), sum the Map/Patch/Manifold – Geodesic Arc-Length” Inertial Mass “Self” Gravitational “Surface-Tidal” Force ($\vec{F_{Self-Gravity}}_{\theta \, \phi}$), in the Proper Observer “center of mass” frame of reference, is given by the following.

$\vec{F_{Self-Gravity-Total}}\,\,=\,\,\vec{F_{Self-Gravity}}\,\,+\,\,\vec{F_{Self-Gravity}}_{\theta \, \phi}$

$\vec{F_{Self-Gravity-Total}}\,\,=\,\,(-)[{\mu^2_{L-Density}}\,\,+\,\,{\mu^2_{L-Density}}_{\theta \, \phi}]{G}\,\,\hat{a}_{r}$

$\vec{F_{Self-Gravity-Total}}\,\,=\,\,(-)[\frac{m^2_{Net}\,G}{r^2}\,\,+\,\,\frac{m^2_{Net}\,G}{{s^2_{Map}}_{\theta \, \phi}}]$

$\vec{F_{Self-Gravity-Total}}\,\,=\,\,(-)\frac{m^2_{Net}\,G}{r^2}[1\,\,+\,\,\frac{1}{[\vec{\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,\vec{\phi^2_{Lon}}]}]$

$\vec{F_{Self-Gravity-Total}}\,\,=\,\,(-)\frac{m^2_{Net}\,G}{r^2}[{\frac{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]$

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

(2) “Self Gravitational Force” of an “Inertial Mass” Gradient Gravitational Field – in the “Proper Observer” “CM” frame of reference

Where the Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$) – in the Proper Observer “center of mass” frame of reference, is given by the following.

$\vec{F_{Self-Gravity}}\,\,=\,\,(-){m_{Net}}\,{g_{Gravity}}\,\,\hat{a}_{r}\,\,=\,\,(\frac{m_{Net}}{m_{test-mass}})\,\vec{F_{Gravity}}$$\,\,----> \,\, \frac{kg\,m}{s^2}$

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

Where the Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$) – is given as a function of the Map/Patch/Manifold – Geodesic Arc-Length” Inertial Mass “Self” Gravitational “Surface-Tidal” Force ($\vec{F_{Self-Gravity}}_{\theta \, \phi}$), in in the Proper Observer “center of mass” frame of reference, is given by the following.

$\vec{F_{Self-Gravity}}\,\,=\,\,(-)\vec{F_{Self-Gravity}}_{\theta \, \phi}\,[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}]$

$\vec{F_{Self-Gravity}}\,\,=\,\,{\mu^2_{L-Density}}_{\theta \, \phi}\,[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}]\,\,\hat{a}_{r}\,\,=\,\,(\frac{m^2_{Net}\,G}{{s^2_{Map}}_{\theta \, \phi}})\,[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}]\,\,\hat{a}_{r}$

$\vec{F_{Self-Gravity}}\,\,=\,\,(-){\mu^2_{L-Density}}\,{G}\,\,\hat{a}_{r}\,\,=\,\,(-)\frac{m^2_{Net}\,G}{r^2}\,\hat{a}_{r}$

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

(3) Map/Patch/Manifold – Geodesic “Arc-Length” Self Gravitational “Surface-Tidal” Force – of an “Inertial Mass” Gradient Gravitational Field – in the “Proper Observer” frame of reference

Where the Map/Patch/Manifold – Geodesic Arc-Length” Inertial Mass “Self” Gravitational “Surface-Tidal” Force ($\vec{F_{Self-Gravity}}_{\theta \, \phi}$) – is the force that exists on the “internal” surface of a sphere, or fixed gravitational potential, in the Proper Observer “center of mass” frame of reference, is given by the following.

$\vec{F_{Self-Gravity}}_{\theta \, \phi}\,=\,(-){m_{Net}}\,{g_{Gravity}}_{\theta \, \phi}\,\,\hat{a}_{r}\,=\,(\frac{m_{Net}}{m_{test-mass}})\,\vec{F_{Gravity}}_{\theta \, \phi}$$---> \,\, \frac{kg\,m}{s^2}$

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

Aphorism:

The strength of the Map/Patch/Manifold – Geodesic Arc-Length” Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}_{\theta \, \phi}$) is a “tidal force” measure of the force of attraction and interaction of “mass towards mass”, on the surface of a sphere of fixed potential, and varies inversely with the net sum, of the square of the Latitude Location Angle ($\vec{\theta}^2_{Lat}$), sum the square Longitude Location Angle ($\sin^2{\theta}\vec{\phi}^2_{Lon}$), on the surface of a sphere, and relative to the center of the fixed potential.

$\vec{F_{Self-Gravity}}_{\theta \, \phi}\,\,\propto\,\,(-)\frac{1}{r^2}[\frac{1}{[\vec{\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,\vec{\phi^2_{Lon}}]}]$

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

$\vec{F_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(-)\vec{F_{Self-Gravity}}\,[{\frac{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]$

$\vec{F_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,{\mu^2_{L-Density}}\,{G}\,[{\frac{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\,\hat{a}_{r}\,\,=\,\,(\frac{m^2_{Net}\,G}{r^2})\,[{\frac{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\,\hat{a}_{r}$

$\vec{F_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(-){\mu^2_{L-Density}}_{\theta \, \phi}\,{G}\,\,\hat{a}_{r}\,\,=\,\,(-)\frac{m^2_{Net}\,G}{{s^2_{Map}}_{\theta \, \phi}}\,\,\hat{a}_{r}$

$\vec{F_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(-)\frac{m^2_{Net}\,G}{r^2}[\frac{1}{[\vec{\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,\vec{\phi^2_{Lon}}]}]$

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

(4) The Conservative “Self Gravitational Force” – Proper Observer – Frame of Reference Transformation Equations – S’ -> S

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

Below is a set of “Euclidean Gravitational Force Transformations” equations that, provides a measured value of force, in the form of a mathematical transformation from the External Observer frame of reference into the Proper Observer “center of mass” frame of reference. S’ -> S

Where the Inertial Mass “Self” Gravitational Force ($\vec{F'_{Self-Gravity}}$) – measured in the External Observer frame of reference, is transformed in the Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$), Proper Observer “center of mass” frame of reference, given by the following. S’ -> S

$\vec{F_{Self-Gravity}}\,\,=\,\,(-)\vec{F'_{Self-Gravity}}\,{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}$

$\vec{F_{Self-Gravity}}\,\,=\,\,{\mu'^2_{L-Density}}\,{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}\,\,\hat{a}_{r}\,\,=\,\,(\frac{m'^2_{Rel}\,G}{s^2}){({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}\,\,\hat{a}_{r}$

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

Where the Inertial Mass “Self” Gravitational Force ($\vec{F'_{Self-Gravity}}$) – measured in the External Observer frame of reference, is transformed into the Map/Patch/Manifold – Geodesic” “Self” Gravitational “Surface” Force ($\vec{F_{Self-Gravity}}_{\theta \, \phi}$), Proper Observer “center of mass” frame of reference, given by the following. S’ -> S

$\vec{F_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(-)\vec{F'_{Self-Gravity}}\,[{\frac{(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})^2\,(1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{|v|_{CM}}{c_{Light}})}}]$

$\vec{F_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,{\mu'^2_{L-Density}}\,{G}\,[{\frac{(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})^2\,(1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\,\hat{a}_{r}\,\,=\,\,(\frac{m'^2_{Rel}\,G}{s^2})\,[{\frac{(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})^2\,(1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\,\hat{a}_{r}$

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

Where the Inertial Mass “Self” Gravitational Force ($\vec{F'_{Self-Gravity}}$) – measured in the External Observer frame of reference, is transformed into the “Total” Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity-Total}}$) – which is conserved, and is equal to the net sum of the “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$), sum the Map/Patch/Manifold – Geodesic” “Self” Gravitational “Surface-Tidal” Force ($\vec{F_{Self-Gravity}}_{\theta \, \phi}$), in the Proper Observer “CM” frame of reference, given by the following. S’ -> S

$\vec{F_{Self-Gravity-Total}}\,\,=\,\,\vec{F_{Self-Gravity}}\,\,+\,\,\vec{F_{Self-Gravity}}_{\theta \, \phi}$

$\vec{F_{Self-Gravity-Total}}\,\,=\,\,(-)\vec{F'_{Self-Gravity}}[{\frac{{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}(1\;\;+\;\;3\frac{|v|_{CM}}{c_{Light}})}{2(\frac{|v|_{CM}}{c_{Light}})}}]$

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

(5) The “Total” Conservative “Self Gravitational Force” of an “Inertial Mass” Gradient Gravitational Field – External Observer Frame of Reference

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

Where the “Total” Inertial Mass “Self” Gravitational Force ($\vec{F'_{Self-Gravity-Total}}$) – is conserved, and is equal to the “net difference” of the inertial mass “Self” Gravitational Force ($\vec{F'_{Self-Gravity}}$), sum the Map/Patch/Manifold – Geodesic Arc-Length” Inertial Mass “Self” Gravitational “Surface” Force ($\vec{F'_{Self-Gravity}}_{\theta \, \phi}$), in the External Observer frame of reference, is given by the following.

$\vec{F'_{Self-Gravity-Total}}\,\,=\,\,\vec{F'_{Self-Gravity}}\,\,-\,\,\vec{F'_{Self-Gravity}}_{\theta \, \phi}$

$\vec{F'_{Self-Gravity-Total}}\,\,=\,\,(-)[{\mu'^2_{L-Density}}\,\,-\,\,{\mu'^2_{L-Density}}_{\theta \, \phi}]{G}\,\,\hat{a}_{r}$

$\vec{F'_{Self-Gravity-Total}}\,\,=\,\,(-)[\frac{m'^2_{Rel}\,G}{s^2}\,\,-\,\,\frac{m'^2_{Rel}\,G}{{s^2_{Map}}_{\theta \, \phi}}]$

$\vec{F'_{Self-Gravity-Total}}\,\,=\,\,(-)[\frac{m'^2_{Rel}\,G}{s^2}\,\,-\,\,\frac{m'^2_{Rel}\,G}{r^2}\frac{1}{[\vec{\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,\vec{\phi^2_{Lon}}]}]$

$\vec{F'_{Self-Gravity-Total}}\,\,=\,\,\frac{m'^2_{Rel}\,G}{s^2}[{\frac{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]$

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

(6) “Self Gravitational Force” of an “Inertial Mass” Gradient Gravitational Field – in the “External Observer” frame of reference

Where the Inertial Mass “Self” Gravitational Force ($\vec{F'_{Self-Gravity}}$) – in the External Observer frame of reference, is given by the following.

$\vec{F'_{Self-Gravity}}\,\,=\,\,(-){m'_{Rel}}\,{g'_{Gravity}}\,\,\hat{a}_{r}\,\,=\,\,(\frac{m'_{Rel}}{m'_{test-mass}})\,\vec{F'_{Gravity}}$$\,\,----> \,\, \frac{kg\,m}{s^2}$

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

Where the Inertial Mass “Self” Gravitational Force ($\vec{F'_{Self-Gravity}}$) – is given as a function of the Map/Patch/Manifold – Geodesic Arc-Length” Inertial Mass “Self” Gravitational “Surface” Force ($\vec{F'_{Self-Gravity}}_{\theta \, \phi}$), in in the External Observer “center of mass” frame of reference, is given by the following.

$\vec{F'_{Self-Gravity}}\,\,=\,\,(-)\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}]\,\,\hat{a}_{r}$

$\vec{F'_{Self-Gravity}}\,\,=\,\,{\mu'^2_{L-Density}}_{\theta \, \phi}\,{G}\,[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}]\,\,\hat{a}_{r}\,\,=\,\,(\frac{m'^2_{Rel}\,G}{{s^2_{Map}}_{\theta \, \phi}})\,[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\,-\,\frac{|v|_{CM}}{c_{Light}}}}]\,\hat{a}_{r}$

$\vec{F'_{Self-Gravity}}\,\,=\,\,(-){\mu'^2_{L-Density}}\,{G}\,\,\hat{a}_{r}\,\,=\,\,(-)\frac{m'^2_{Rel}\,G}{s^2}\,\,\hat{a}_{r}$

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

(7) Map/Patch/Manifold – Geodesic “Arc-Length” Self Gravitational “Surface-Tidal” Force – of an “Inertial Mass” Gradient Gravitational Field – in the “External Observer” frame of reference

Where the Map/Patch/Manifold – Geodesic Arc-Length” Inertial Mass “Self” Gravitational “Surface-Tidal” Force ($\vec{F_{Self-Gravity}}_{\theta \, \phi}$) – is the force that exists on the “external” surface of a sphere, or fixed gravitational potential, in the External Observer frame of reference, is given by the following.

$\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,=\,(-){m'_{Rel}}\,{g'_{Gravity}}_{\theta \, \phi}\,\,\hat{a}_{r}\,=\,(\frac{m'_{Rel}}{m'_{test-mass}})\,\vec{F'_{Gravity}}_{\theta \, \phi}$$---> \,\, \frac{kg\,m}{s^2}$

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

Aphorism:

The strength of the Map/Patch/Manifold – Geodesic Arc-Length” Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}_{\theta \, \phi}$) is a “tidal force” measure of the force of attraction and interaction of “mass towards mass”, on the surface of a sphere of fixed potential, and varies inversely with the net sum, of the square of the Latitude Location Angle ($\vec{\theta}^2_{Lat}$), sum the square Longitude Location Angle ($\sin^2{\theta}\vec{\phi}^2_{Lon}$), on the surface of a sphere, and relative to the center of the fixed potential.

$\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,\,\propto\,\,(-)\frac{1}{r^2}[\frac{1}{[\vec{\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,\vec{\phi^2_{Lon}}]\,({1\,\,-\,\,\frac{|v|^2_{CM}}{c^2_{Light}}})}]\,\,\hat{a}_{r}$

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

$\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(-)\vec{F'_{Self-Gravity}}\,[{\frac{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]$

$\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,{\mu'^2_{L-Density}}\,{G}\,[{\frac{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\,\hat{a}_{r}\,\,=\,\,(\frac{m'^2_{Rel}\,G}{s^2})\,[{\frac{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\hat{a}_{r}$

$\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(-){\mu'^2_{L-Density}}_{\theta \, \phi}\,{G}\,\,\hat{a}_{r}\,\,=\,\,(-)\frac{m'^2_{rel}\,G}{{s^2_{Map}}_{\theta \, \phi}}\,\,\hat{a}_{r}$

$\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(-)\frac{m^2_{Net}\,G}{r^2}[\frac{1}{[\vec{\theta^2_{Lat}}\;\;+\;\; \sin^2\theta_{Lat}\,\vec{\phi^2_{Lon}}]\,({1\,\,-\,\,\frac{|v|^2_{CM}}{c^2_{Light}}})}]\,\,\hat{a}_{r}$

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

(8) The Conservative “Self Gravitational Force” – External Observer – Frame of Reference Transformation Equations – S -> S’

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

Below is a set of “Euclidean Gravitational Force Transformations” equations that, provides a measured value of force, in the form of a mathematical transformation from the Proper Observer center of mass frame of reference, into the External Observer frame of reference. S -> S’

Where the Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$) – measured in the Proper Observer frame of reference, is transformed in the Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$), External Observer “center of mass” frame of reference, given by the following. S’ -> S

$\vec{F'_{Self-Gravity}}\,\,=\,\,(-)\frac{\vec{F_{Self-Gravity}}}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}$

$\vec{F'_{Self-Gravity}}\,\,=\,\,{{\mu^2_{L-Density}}\,G}\,[\frac{1}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}]\,\,\hat{a}_{r}\,\,=\,\,\frac{m^2_{Net}\,G}{r^2}[\frac{1}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}]\,\hat{a}_{r}$

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

Where the Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$) – measured in the Proper Observer frame of reference, is transformed into the Map/Patch/Manifold – Geodesic Arc-Length” Inertial Mass “Self” Gravitational “Surface-Tidal” Force ($\vec{F_{Self-Gravity}}_{\theta \, \phi}$), External Observer frame of reference, given by the following.  S -> S’

$\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(-)\,[{\frac{\vec{F_{Self-Gravity}}}{2(\frac{|v|_{CM}}{c_{Light}})\,(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}}]$

$\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,{\mu^2_{L-Density}}\,{G}\,[{\frac{1}{2(\frac{|v|_{CM}}{c_{Light}})\,(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}}]\,\,\hat{a}_{r}\,\,=\,\,(\frac{m^2_{Net}\,G}{r^2})\,[{\frac{1}{2(\frac{|v|_{CM}}{c_{Light}})\,(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}}]\,\hat{a}_{r}$

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

Where the Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$) – measured in the Proper Observer frame of reference, is transformed into the “Total” Inertial Mass “Self” Gravitational Force ($\vec{F'_{Self-Gravity-Total}}$) – which is conserved, and is equal to the net difference of the “Self” Gravitational Force ($\vec{F'_{Self-Gravity}}$), sum the Map/Patch/Manifold – Geodesic Arc-Length” “Self” Gravitational “Surface-Tidal” Force ($\vec{F'_{Self-Gravity}}_{\theta \, \phi}$), in the External Observer “center of mass” frame of reference, given by the following. S -> S’

$\vec{F'_{Self-Gravity-Total}}\,\,=\,\,\vec{F'_{Self-Gravity}}\,\,-\,\,\vec{F'_{Self-Gravity}}_{\theta \, \phi}$

$\vec{F'_{Self-Gravity-Total}}\,\,=\,\,(-)\frac{\vec{F_{Self-Gravity}}}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}\,\,-\,\,(-)\,[{\frac{\vec{F_{Self-Gravity}}}{2(\frac{|v|_{CM}}{c_{Light}})\,(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}}]$

$\vec{F'_{Self-Gravity-Total}}\,\,=\,\,(-){\vec{F_{Self-Gravity}}}\,[{\frac{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}}){({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}}}]$

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

General Constants

Gravitational Constant

${G}\;=\;6.67384 \times 10^{-11} \frac{m^3}{kg\,s^2}$

Speed of Light in vacuum constant

${c_{Light}}\;=\;2.99792459 \times 10^{8} \frac{m}{s}$

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

The above work completes a new theory of Relativistic Gravitation; and produces a complete conceptual and mathematical model of matter, space, and time in non-inertial or accelerated frames of reference. The above work opens the door to discuss new concepts and mathematics of gravity, in consideration for Special Relativity and General Relativity; the Super Special Theory of Relativity.

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

Citation

Robert Louis Kemp; The Super Principia Mathematica – The Rage to Master Conceptual & Mathematical Physics – The General Theory of Relativity – “Relativistic Gravitational Force – The Non Inertial Frame of Reference in General Relativity– Online Volume – ISBN 978-0-9841518-2-0, Volume 3; July 2010

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

Best,

Author: Robert Louis Kemp

http://www.SuperPrincipia.com

A Theory of Gravity for the 21st Century

The “Central Conservative” Gravitational Force and Potential Energy – in consideration with Special Relativity and General Relativity

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, with the identical properties to a “Gravitational Vortex”, 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 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”.

In this Gravitational Vortex Model, it is necessary to model gravity, because it is gravity that binds us to the earth, that binds the earth, and other planets to the solar system. In general, it is the “Gravitational Force” that is responsible for why all things fall down on planet earth, it is responsible for the formation of galaxies, the evolution of stars, and what keeps the bones in our bodies firm and rigid.

In this work, it will be demonstrated conceptually and mathematically that the “Potential Energy” is associated with the work done by a “central conservative force”, namely the “Gravitational Force.” Various other types of “central conservative forces” include: the Elastic Spring Force, the Electrostatics Force, and the Magnetostatics Force.

All “central conservative forces” can be generalized to model a system, such that there exists an associated “Potential Energy” function, where the work done by the “Central Force”, equals a decrease in the “Potential Energy” of the system. In the case of the “Gradient Gravitational Field Force”, work is done by the force of gravity, decreasing the “Potential Energy” of any mass object, located anywhere within the gradient field; and where the strength of “Gradient Gravitational Field Force”, and the “Potential Energy” decreases, with increasing distance from the center of the field.

If the only “central conservative force” acting on an isolated “Net Inertial Mass” system body is the “Gravitational Force”, then it is the force of gravity that is doing work; and according to the conservation of energy, that work is also equal to the increase in the “Kinetic Energy” of the Net Inertial Mass system body.

The required condition for a force to be “central conservative force” is, if the total work, the force does, on any object located in its surroundings, is moved around any closed path, at any speed, and returns to its initial position, and the work done at the end of the process is equal zero (0), then the force acting on the object is considered a “central conservative force.”

The work done by a “central conservative force” with proper mathematical formalism is denoted with a negative value; and because of conservation when the potential energy increases, the kinetic energy decreases by an equal amount; and vice versa. Therefore, the “Total Mechanical Energy” is conserved, being the sum of the “Kinetic Energy” and the “Potential Energy” which when summed together mathematically remains a constant value.

The “Total Mechanical Energy” of an isolated system is conserved, because any decrease in the potential energy is balanced by an increase in the Kinetic Energy; and vice versa; as described by the equation below.

Total Mechanical Energy

${E_{Total}}\,\,=\,\, {T_{Kinetic-Energy}}\;\;+\;\;{V_{Potential-Energy}}$

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

(1) The Conservative “Central Force” of an “Inertial Mass” Gradient Gravitational Field

Any and every conserved and isolated “Net Inertial Mass” system body, can be modeled as a “vortex” body that is spheroid in nature, and is described by a gradient field, comprised of an infinite amount of “spherical shell potentials” relative to the center of the system. The gradient gravity field is described by concentric spherical volumetric potential shells of “Gravitational Potential Energy” and a conservative “Central Gravitational Force” at each potential.

For a general gradient gravitational field, the conservative Gravitational Potential Energy (${V_{Gravity-Potential}}$) of each concentric spherical shell potential, is associated with the Inertial Mass Gravitational Force ($\vec{F_{Gravity}}$), where the source of gradient gravity field is the Net Inertial Mass (${m_{Net}}$).

In this “Gravitational Vortex” model, at the origin of every “Net Inertial Mass Gradient Gravity Field’ there is a “Schwarzschild Radius Black Hole Event Horizon.” The most minimum spatial distance of the gradient gravitational field is the Schwarzschild Radius (${r_{Schwarzschild}}\,=\,\frac{2\,{m_{Net}}\,G}{c^2_{Light}}$) Black Hole Event Horizon.

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

The Conservative “Central Force” of an “Inertial Mass” Gradient Gravitational Field

The Inertial Mass Gravitational Force ($\vec{F_{Gravity}}$) is a measure of the force of attraction and interaction, of “mass towards mass”, and is a conservative central force, exerted by the Net Inertial Mass (${m_{Net}}$) on any other “test” mass (${m_{mass}}$) body, of the “collective” net mass system body; which includes its “Self” attraction and interaction force.

The strength of the Inertial Mass Gravitational Force ($\vec{F_{Gravity}}$) varies inversely with the square of the Semi-Major radius ($\frac{1}{r^2}$) distance, relative to the center of the Gradient Gravitational Field.

Also, the Inertial Mass Gravitational Force ($\vec{F_{Gravity}}$) is a conservative central force that comes in two forms, the “Newtonian” Gravitational Force ($\vec{F_{Gravity}}$), and the “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$).

Video of Newton & Self Gravitational Force (Self Gravity) Lecture************************************************************

The Inertial Mass Gravitational Force ($\vec{F_{Gravity}}$) which is described mathematically below, is an “inflow” radial vector, given by the following equations.

Aphorism:

The strength of the “Newtonian” Gravitational Force ($\vec{F_{Gravity}}$) is a measure of the force of attraction and interaction of “mass towards mass”, and varies in direct proportion to the product of the Net Inertial Mass (${m_{Net}}$) multiplied by the orbiting “test” mass (${m_{mass}}$), and varies inversely with the square of the Semi-Major radius ($\frac{1}{r^2}$) distance, relative to the center of the Gradient Gravitational Field.

${F_{Gravity}}\,\,\propto\,\,\frac{m_{mass}\,m_{Net}}{r^2}$

Inertial Mass “Newtonian” Gravitational Force

$\vec{F_{Gravity}}\,\,=\,\,(-)\frac{m_{mass}\,m_{Net}\,G}{r^2}\,{\hat{a}_r}$ $\,\,----> \,\, \frac{kg\,m}{s^2}$

$\vec{F_{Gravity}}\,\,=\,\,(-){m_{mass}}\,{g_{Gravity}}\,{\hat{a}_r}\,\,=\,\,(-)\frac{{m_{mass}}\,{v^2_{Gravity}}}{r}\,{\hat{a}_r}$ $\,\,----> \,\, \frac{kg\,m}{s^2}$

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

Aphorism:

The strength of the “Self” Gravitational Force ($\vec{F_{Gravity}}$) is a measure of the force of attraction and interaction of “mass towards mass”, and varies directly with the square of the Linear Mass Density ($\mu^2_{L-Density}$); and likewise varies in direct proportion to the square of the Net Inertial Mass (${m^2_{Net}}$), and varies inversely with the square of the Semi-Major radius ($\frac{1}{r^2}$) distance, relative to the center of the Gradient Gravitational Field.

${F_{Self-Gravity}}\,\,\propto\,\,\frac{m^2_{Net}}{r^2}\,\,=\,\,\mu^2_{L-Density}$

Inertial Mass “Self” Gravitational Force

$\vec{F_{Self-Gravity}}\,\,=\,\,(-)\frac{m^2_{Net}\,G}{r^2}\,{\hat{a}_r}$ $\,\,----> \,\, \frac{kg\,m}{s^2}$

$\vec{F_{Self-Gravity}}\,\,=\,\,(-){m_{Net}}\,{g_{Gravity}}\,{\hat{a}_r}\,\,=\,\,(-)\frac{{m_{Net}}\,{v^2_{Gravity}}}{r}\,{\hat{a}_r}$ $\,\,----> \,\, \frac{kg\,m}{s^2}$

$\vec{F_{Self-Gravity}}\,\,=\,\,(-){\mu^2_{L-Density}\,G}\,{\hat{a}_r}\,\,=\,\,(-){\mu_{L-Density}\,{v^2_{Gravity}}}\,{\hat{a}_r}$

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

Black Hole Event HorizonInertial Mass “Self” Gravitational Force

$\vec{F_{Self-Gravity}}\,\,=\,\,(-)\frac{m^2_{Net}\,G}{r^2_{Schwarzschild}}\,{\hat{a}_r}\,\,=\,\,(-)\frac{1}{4}\frac{c^4_{Light}}{G}\,{\hat{a}_r}$$\,\,----> \,\, \frac{kg\,m}{s^2}$

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

(2)The “Gravitational Acceleration” of an “Inertial Mass” Gradient Gravitational Field

The Inertial Mass Gradient Gravitational Field Acceleration ($\vec{g_{Gravity}}$) describes the acceleration of mass and energy, towards the center of the gradient gravity field, and towards ever decreasing and smaller volumes of spherical gradient shell potentials.

The Inertial Mass Gradient Gravitational Field Acceleration ($\vec{g_{Gravity}}$) varies 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.

The Inertial Mass Gradient Gravitational Field Acceleration ($\vec{g_{Gravity}}$) is defined as the ratio of the Inertial Mass Gravitational Force ($\vec{F_{Gravity}}$) divided by the Mass (${m_{mass}}$) of the system; and likewise the Gravitational Field Acceleration ($\vec{g_{Gravity}}$) diminishes as the square of the Semi-Major radius (${r^2}$) distance from the center of the Gradient Gravitational Field increases.

Aphorism:

The motion of the Gradient Gravitational Field ($\vec{g_{Gravity}}$) Acceleration is a measure of the acceleration of the attraction and interaction of “mass towards mass”, and varies directly proportional to the Net Inertial (${m_{Net}}$) Mass, and varies inversely with the square of the Semi-Major radius ($\frac{1}{r^2}$) distance, relative to the center of the Gradient Gravitational Field.

${g_{Gravity}}\,\,\propto\,\,\frac{m_{Net}}{r^2}$

Inertial Mass – Gradient Gravitational Field Acceleration

$\vec{g_{Gravity}}\,\,=\,\,(-)\frac{\vec{F_{Gravity}}}{m_{mass}}\,\,=\,\,(-)\frac{\vec{F_{Self-Gravity}}}{m_{Net}}$$\,\,----> \,\, \frac{m}{s^2}$

$\vec{g_{Gravity}}\,\,=\,\,\frac{m_{Net}\,G}{r^2}\,{\hat{a}_r}\,\,=\,\,\frac{{v^2_{Gravity}}}{r}\,{\hat{a}_r}$$\,\,----> \,\, \frac{m}{s^2}$

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

Black Hole Event Horizon – Gradient Gravitational Field Acceleration

The  Black Hole Event Horizon – Gradient Gravitational Field ($\vec{g_{Gravity}}_{BH}$) Acceleration can be thought of as the measure of the amount of curvature, and the measure of the acceleration that any observer standing at the surface of the “Black Hole Event Horizon”, would feel being pulled towards the center of the black hole.

The  Black Hole Event Horizon – Gradient Gravitational Field ($\vec{g_{Gravity}}_{BH}$) Acceleration is the “Quantized” and maximum amount of acceleration of a gradient gravitational field system, where the measure of the “Black Hole Acceleration” is inversly proportional to the Net Inertial Mass  (${m_{Net}}$) of the system.

Such that the larger the Net Inertial Mass  (${m_{Net}}$) of the gradient gravitational field the slower the Black Hole Acceleration ($\vec{g_{Gravity}}_{BH}$), rate relative to the “Black Hole Event Horizon – Potential” and towards the center of the gradient field.

Likewise, such that the smaller the Net Inertial Mass  (${m_{Net}}$) of the gradient gravitational field the faster the Black Hole Acceleration ($\vec{g_{Gravity}}_{BH}$), rate relative to the “Black Hole Event Horizon – Potential” and towards the center of the gradient field.

The  Black Hole Event Horizon – Gradient Gravitational Field ($\vec{g_{Gravity}}_{BH}$) Acceleration can also be thought of as the measure of the “Quantized” amount warping, or curvature, in the form of the acceleration of a “Spherical” disturbance, in the “Vacuum of Space-time” and in the local vicinity, of the Net Inertial Mass; the source of the gradient gravitational field system body.

Aphorism:

The motion of the Black Hole Event Horizon – Gradient Gravitational Field ($\vec{g_{Gravity}}_{BH}$) Acceleration is equal to the measure of the acceleration of the attraction and interaction of “mass towards mass”, at the “Black Hole Potential”, and is idependent of the “distance” or “radius” of the “Black Hole” and is dependent, and varies inversely proportional to the Net Inertial ($\frac{1}{m_{Net}}$) Mass, of the Gradient Gravitational Field, system body.

$\vec{g_{Gravity}}_{BH}\,\,\propto\,\,\frac{1}{m_{Net}}$

$\vec{g_{Gravity}}_{BH}\,\,=\,\,\frac{m_{Net}\,G}{r^2_{Schwarzschild}}\,{\hat{a}_r}\,\,=\,\,\frac{1}{4}\frac{c^4_{Light}}{m_{Net}\,G}\,{\hat{a}_r}$$\,\,----> \,\, \frac{m}{s^2}$

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

For example: Electrons have faster or larger Black Hole Acceleration ($\vec{g_{Gravity}}_{BH}$) towards their Black Hole centers than do Protons, Plantets and Suns.

Protons have faster or larger Black Hole Acceleration ($\vec{g_{Gravity}}_{BH}$) towards their Black Hole centers than do Neutrons.

And the Earth, has a faster or larger Black Hole Acceleration ($\vec{g_{Gravity}}_{BH}$), towards its Black Hole center, than the Sun does.

Furthermore the Sun, has a faster or larger Black Hole Acceleration ($\vec{g_{Gravity}}_{BH}$), towards its Black Hole center, than the Galaxy does.

This means that, an object with the right amount of force and acceleration, that being larger than the Black Hole event Horizon acceleration, should be able to escape the death grips of the Black Hole!

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

(3) The “Potential Energy” of an “Inertial Mass” Gradient Gravitational Field

For a general gradient gravitational field, the conservative Gravitational Potential Energy (${V_{Gravity-Potential}}$) of each concentric spherical shell potential, is associated with the Inertial Mass Gravitational Force ($\vec{F_{Gravity}}$), where the source of gradient gravity field is the Net Inertial Mass (${m_{Net}}$), and manifests, when a “Gravity Force” acts upon a “mass” object within the gradient field, that tends to move it to a lower energy location within the field.

The Gravitational Potential Energy (${V_{Gravity-Potential}}$) is a measure of the “work energy”, of the relative spatial separation, of the attraction and interaction of “mass towards mass”. The Gravitational Potential Energy (${V_{Gravity-Potential}}$) is the work done in the gravity field, by the Net Inertial Mass (${m_{Net}}$) of the system body, moving “mass towards mass” from infinite places in the universe!

The change in the Gravitational Potential Energy (${dV_{Gravity-Potential}}$) is the measure of the work done by the Inertial Mass Gravitational Force ($\vec{F_{Gravity}}$) of a general gradient gravitational field; and the work done by the force, is integrated over changes in the Semi-Major radius distance ($\vec{dr}$), relative to the center of the gradient gravity field.

The Gravitational Potential Energy (${V_{Gravity-Potential}}$) can be described physically as the energy associated with each “surface potential” of the infinite concentric “thin” spherical shells, that make up the gradient gravity field, and is the potential energy difference between the energy of a “mass” object in a given gradient field position, and its energy at some other reference position within the field; given by the following equations.

${dV_{Gravity-Potential}}\,\,=\,\,(-)\vec{F_{Gravity}}\,\vec{dr}\,\,=\,\,(+)[\frac{m_{mass}\,m_{Net}\,G}{r^2}]\,{dr}$

${V_{Gravity-Potential}}\,\,=\,\,\int{dV_{Potential-Energy}}\,\,=\,\,(-)\int{\vec{F_{Gravity}}\,\vec{dr}}$$\,\,----> \,\, \frac{kg\,m^2}{s^2}$

${V_{Gravity-Potential}}\,\,=\,\,(+)({m_{mass}\,m_{Net}\,G})\int{\frac{dr}{r^2}}\,\,=\,\,(+)({m_{mass}\,m_{Net}\,G})\int_\infty^r {\frac{dr}{r^2}}$

${V_{Gravity-Potential}}\,\,=\,\,(-)\frac{{m_{mass}\,m_{Net}\,G}}{r}$$\,\,----> \,\, \frac{kg\,m^2}{s^2}$

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

Aphorism:

The energy content of the Gradient Gravitational Field “Newtonian” Potential Energy (${V_{Gravity-Potential}}$) is a measure of the energy potential, of the relative spatial separation, of the attraction and interaction of “mass towards mass”, and varies in direct proportion to the product of the Net Inertial Mass (${m_{Net}}$) multiplied by the test mass (${m_{mass}}$), and varies inversely with the Semi-Major radius ($\frac{1}{r}$) distance, relative to the center of the Gradient Gravitational Field.

${V_{Gravity-Potential}}\,\,\propto\,\,\frac{m_{mass}\,m_{Net}}{r}$

Gradient Gravitational Field “Newtonian” Potential Energy

${V_{Gravity-Potential}}\,\,=\,\,(-)\frac{{m_{mass}\,m_{Net}\,G}}{r}$$\,\,----> \,\, \frac{kg\,m^2}{s^2}$

${V_{Gravity-Potential}}\,\,=\,\,(-)({m_{mass}\,v^2_{Gravity}})\,\,=\,\,(-)({m_{mass}\,g_{Gravity}\,{r}})$

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

Aphorism:

The energy content of the Gradient Gravitational Field “Self” Potential Energy (${V_{Self-Potential}}$) is a measure of the energy potential, of the relative spatial separation, of the attraction and interaction of “mass towards mass”, and varies directly proportional to the square of the Net Inertial Mass (${m^2_{Net}}$), and varies inversely with the linear, Semi-Major radius ($\frac{1}{r}$) distance, relative to the center of the Gradient Gravitational Field.

${V_{Self-Potential}}\,\,\propto\,\,\frac{m^2_{Net}}{r}$

Gradient Gravitational Field “Self” Potential Energy

${V_{Self-Potential}}\,\,=\,\,(-)\frac{m^2_{Net}\,G}{r}$$\,\,----> \,\, \frac{kg\,m^2}{s^2}$

${V_{Self-Potential}}\,\,=\,\,(-)({m_{Net}\,v^2_{Gravity}})\,\,=\,\,(-)({m_{Net}\,g_{Gravity}\,{r}})$

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

Black Hole Event Horizon – Gradient Gravitational Field “Self” Potential Energy

${V_{Self-Potential}}_{BH}\,\,=\,\,(-)\frac{m^2_{Net}\,G}{r_{Schwarzschild}}\,\,=\,\,(-)\frac{1}{2}{m_{Net}\,c^2_{Light}}$$\,\,----> \,\, \frac{kg\,m^2}{s^2}$

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

(4) The “Inertia Potential” of an “Inertial Mass” Gradient Gravitational Field

The Inertial Mass Gradient Gravitational Field “Inertia” Potential (${v^2_{Gravity}}$) is a measure of the square of the orbiting/spin/rotation squared tangential velocity of each potential of the gradient gravity field; and describes the squared velocity “inertia” potential, of the relative spatial separation, of the attraction and interaction of “mass towards mass”; and towards the center of the gradient gravity field, and towards ever decreasing and smaller volumes, of spherical gradient shell potentials.

The Inertial Mass Gradient Gravitational Field “Inertia” Potential (${v^2_{Gravity}}$) varies in each spherical volume potential of the gravity field, such that the larger the volume potential, the slower the squared velocity “inertia” towards the center of the gradient gravity field; and the smaller the volume potential, the faster the squared velocity “inertia” towards the center of the gradient gravity field.

The Inertial Mass Gradient Gravitational Field “Inertia” Potential (${v^2_{Gravity}}$) is a measure of the inertia of motion in gravity field, and is defined as the ratio of the Gravitational Potential Energy (${V_{Gravity-Potential}}$) divided by the Mass (${m_{mass}}$) of the system; and likewise the Gravitational Field Potential (${v^2_{Gravity}}$) diminishes as the linear Semi-Major radius (${r}$) distance from the center of the Gradient Gravitational Field increases.

Aphorism:

The inertia motion of the Gradient Gravitational Field “Inertia” Potential (${v^2_{Gravity}}$) is a measure of the orbiting/spin/rotation squared velocity potential, of the relative spatial separation, of the attraction and interaction of “mass towards mass”, and varies in direct proportion to the Linear Mass Density ($\mu_{L-Density}$); and likewise varies in direct proportion to the Net Inertial Mass (${m_{Net}}$), and varies inversely with the linear Semi-Major radius ($\frac{1}{r}$) distance, relative to the center of the Gradient Gravitational Field.

${v^2_{Gravity}}\,\,\propto\,\,\frac{m_{Net}}{r}\,\,=\,\,\mu_{L-Density}$

Inertial Mass – Gradient Gravitational Field “Inertia” Potential

${v^2_{Gravity}}\,\,=\,\,(-)\frac{V_{Gravity-Potential}}{m_{mass}}\,\,=\,\,(-)\frac{V_{Self-Potential}}{m_{Net}}$$\,\,----> \,\, \frac{m^2}{s^2}$

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

${v^2_{Gravity}}\,\,=\,\,{\mu_{L-Density}\,G}$$\,\,----> \,\, \frac{m^2}{s^2}$

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

Black Hole Event Horizon – Gradient Gravitational Field “Inertia” Potential

${v^2_{Gravity}}\,\,=\,\,\frac{m_{Net}\,G}{r_{Schwarzschild}}\,\,=\,\,\frac{1}{2}{c^2_{Light}}$$\,\,----> \,\, \frac{m^2}{s^2}$

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

(5) The “Inertial” Linear Mass Density Gradient Gravitational Field Potentials

For a general gradient gravitational field, the conservative the “Self” Gravitational Potential Energy (${V_{Gravity-Potential}}$), and the Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$) of each concentric spherical shell potential of the gradient gravity field, varies in direct proportion to the Net Inertial Linear Mass Density (${\mu_{L-Density}}$) of the the gradient field, dependent only on the inverse distance from the center of the field.

This model predicts that, while considering the Net Inertial Mass (${m_{Net}}$) of the gradient gravity field constant, the Net Inertial Linear Mass Density (${\mu_{L-Density}}$), of the field varies from place to place or location to location within the gravity field.

The Net Inertial Linear Mass Density (${\mu_{L-Density}}$), measures a “greater” or “condensed” linear density, the closer the Semi-Major Radius (${r}\geq{r_{Schwarzschild}}$) distance is to the “mean center” and Schwarzschild Radius (${r_{Schwarzschild}}$) Black Hole Event Horizon of the system.

The Net Inertial Linear Mass Density (${\mu_{L-Density}}$), measures a “smaller” or “rarer” linear density, the further away the Semi-Major Radius (${r}\gg{r_{Schwarzschild}}$) distance is from the “mean center” and the Schwarzschild Radius (${r_{Schwarzschild}}$) Black Hole Event Horizon of the gradient gravity field system.

The Net Inertial Linear Mass Density (${\mu_{L-Density}}$), is actually the result of the “Linear Mass Density” “potential” difference between the maximum “Black Hole” Net Inertial Linear Mass Density (${\mu_{L-Density}}_{BH}$) constant, at the core of the gradient field, and its Net Inertial Linear Mass Density (${\mu_{L-Density}}$) at some other reference position within the field.

The Net Inertial Linear Mass Density (${\mu_{L-Density}}$), measures a “maximum” linear density, when the Semi-Major Radius (${r}={r_{Schwarzschild}}$) distance is equal to the Schwarzschild Radius (${r_{Schwarzschild}}$) of the gradient gravity field system; and measures zero (0) when the Semi-Major Radius (${r}=\infty$) distance is equal to infinity, or an infinite distance away from the “mean center” of the gradient gravity field system body.

Aphorism:

The dense intensity of Net Inertial Linear Mass Density (${\mu_{L-Density}}$) is a measure of the linear density of the gradient potentials of the gravity field, and varies in direct proportion to the ratio of the Net Inertial Mass (${m_{Net}}$), and inversely with increases or decreases in the linear Semi-Major radius ($\frac{1}{r}$) distance, relative to the center of the Gradient Gravitational Field. And likewise varies in direct proportion to the the Gravitational Field Potential (${v^2_{Gravity}}$); and further varies in direct proportion to the square root of the “Self” Gravitational Force ($\sqrt{F_{Self-Gravity}}$) of each concentric spherical shell potential of the gradient gravity field.

$\mu_{L-Density}\,\,\propto\,\,{v^2_{Gravity}}\,\,\propto\,\,\sqrt{F_{Self-Gravity}}$

The Net Inertial Linear Mass Density (${\mu_{L-Density}}$)

${\mu_{L-Density}}\,\,=\,\,\frac{m_{Net}}{r}\,\,=\,\,\frac{v^2_{Gravity}}{G}\,\,=\,\,\sqrt\frac{F_{Self-Gravity}}{G}$$\,\,----> \,\, \frac{kg}{m}$

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

(6) The “Maximum – Black Hole – Event Horizon” Linear Mass Density Gradient Gravitational Field Potential Constant

The “Maximum – Black Hole – Event Horizon” Net Inertial Linear Mass Density(${\mu_{L-Density}}_{BH}$)

The “Maximum” Net Inertial Linear Mass Density (${\mu_{L-Density}}_{BH}$), measures a “maximum” linear density, when the Semi-Major Radius (${r}={r_{Schwarzschild}}$) distance is equal to the Schwarzschild Radius (${r_{Schwarzschild}}$) Black Hole Event Horizon, and represents the “lowest potential” or the “smallest volume” of the gradient gravity field system mass body.

In this Gravitational Vortex Model, the “Maximum” Net Inertial Linear Mass Density (${\mu_{L-Density}}_{BH}$), which is located in the “lowest potential” of the gradient field, is also known as the “Black Hole” Net Inertial Linear Mass Density; an it measures a “constant” value.

The “Maximum” Net Inertial Linear Mass Density (${\mu_{L-Density}}_{BH}$), exists at the core center, of every Net Inertial Mass (${m_{Net}}$), and is the core of every gradient gravitational field; and represents the: smallest volume, greatest Gravitational Force, largest Inertia Potential, greatest Potential Energy, largest Gravitational Acceleration, fastest Orbiting Velocity, and the shortest Orbital Period, of the gradient gravity field.

In this “Gradient Vortex Gravitational Field” model, the “Black Hole” Net Inertial Linear Mass Density (${\mu_{L-Density}}_{BH}$)” is a constant value, that is spatially located at the Black Hole Event Horizon” origin source, of the gravitational gradient field; and is the “vacuum energy” binding proportionality between “Matter/Mass” and the “Space” of the “Vacuum of Space-time”; and can be modeled as a “fabric continuum” or “vacuum energy” that permeates throughout the entire universe.

There is no place in the cosmos of the universe that is void of vacuum energy.

The “Black Hole” Net Inertial Linear Mass Density (${\mu_{L-Density}}_{BH}$)” is a direct measure of the vacuum of space-time continuum, where the Net Inertial Mass – (${m_{Net}}$) or “matter” of the gravitational field system body, is directly proportional to the “space” distance of the “source of the” gravity field; and where the minimum distance, and the lowest energy potential, is given by the Schwarzschild Radius (${r_{Schwarzschild}}$) Black Hole Event Horizon, of the gradient gravity field, described by the following relation, and equation.

${m_{Net}}\,\,=\,\,({Constant})\,{r_{Schwarzschild}}\,\,=\,\,({\mu_{L-Density}}_{BH})\,{r_{Schwarzschild}}$

The “Black Hole” Net Inertial Linear Mass Density (${\mu_{L-Density}}_{BH}$) is a gravitational field parameter where the ratio of the Net Inertial Mass(${m_{Net}}$) divided by the Schwarzschild Radius (${r_{Schwarzschild}}$) Black Hole Event Horizon, is a constant of nature.

“Black Hole” Net Inertial Linear Mass Density (${\mu_{L-Density}}_{BH}$)

${\mu_{L-Density}}_{BH}\,\,=\,\,\frac{m_{Net}}{r_{Schwarzschild}}\,\,=\,\,{Constant}$$\,\,----> \,\, \frac{kg}{m}$

${\mu_{L-Density}}_{BH}\,\,=\,\,\frac{m_{Net}}{r_{Schwarzschild}}\,\,=\,\,\frac{1}{2}\frac{c^2_{Light}}{G}\,\,=\,\,\sqrt\frac{F_{Self-Gravity}}{G}$$\,\,----> \,\, \frac{kg}{m}$

${\mu_{L-Density}}_{BH}\,\,=\,\,{\mu_{L-Density}}\,(\frac{r}{r_{Schwarzschild}})$

${\mu_{L-Density}}_{BH}\,\,=\,\,\frac{1}{2}\frac{c^2_{Light}}{G}\,\,=\,\,6.73297478332358\times 10^{26}$$\,\,----> \,\, \frac{kg}{m}$

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

In this “Gravitational Vortex” model, the “Black Hole” Net Inertial Linear Mass Density (${\mu_{L-Density}}_{BH}$)” is a gravitational field parameter, such that at the location of the “Black Hole Event Horizon”, all the “central conservative forces” of the system become equal.

${\mu_{L-Density}}_{BH}\,\,=\,\,\frac{m_{Net}}{r_{Schwarzschild}}\,\,=\,\,\sqrt\frac{F_{Self-Gravity}}{G}\,\,=\,\,\sqrt\frac{F_{Dark-Force}}{G}$$\,\,----> \,\, \frac{kg}{m}$

${\mu_{L-Density}}_{BH}\,\,=\,\,\frac{m_{Net}}{r_{Schwarzschild}}\,\,=\,\,\sqrt\frac{F_{Light-Force}}{2\,G}\,\,=\,\,\sqrt\frac{F_{Heat-Radiation}}{G}$

${\mu_{L-Density}}_{BH}\,\,=\,\,{\mu_{L-Density}}\,(\frac{r}{r_{Schwarzschild}})$

${\mu_{L-Density}}_{BH}\,\,=\,\,\frac{1}{2}\frac{c^2_{Light}}{G}\,\,=\,\,6.73297478332358\times 10^{26}$$\,\,----> \,\, \frac{kg}{m}$

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

(7) The “General Relativistic” Linear Mass Density Gradient Gravitational Field Potentials

Where the Net Inertial Mass – (${m_{Net}}$) – in the Proper Observer “center of mass” frame of reference is given by the following.

${m_{Net}}\, \,=\,\,\displaystyle\sum_{i=1}^N {m_{i}}\,\,=\,\,[ {m_{1}} + {m_{2}} + {m_{3}} + ...... + {m_{N}}]\,\,$ $---> {kg}$

And, the Relativistic Net Inertial Mass – (${m'_{Rel}}$) – in the External Observer frame of reference is given by the following.

${m'_{Rel}}\, \,=\,\,\frac{m_{Net}}{\sqrt{1\;\;-\;\;\frac{{|\vec{v}|^2_{CM}}}{c^2_{Light}}}}\,\,$ $---> {kg}$

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

Next, employing the – Map/Patch/Manifold – “Geodesic” Arc Length – ($\vec{s_{Map}}_{\theta \, \phi}$) – which is a “geodesic arc-length” spatial component, on the surface of the sphere, and changes as a function of the “Radius” ($\vec{r}$), and changes as a function of the “Euclidean Radius” ($\vec{s}$) of a symmetric sphere, as derived in Section 3, of the work:

Euclidean Spherical Mechanics – Euclidean/Minkowski Space-time Metrics

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

The Map/Patch/Manifold – “Geodesic” Arc Length – ($\vec{s_{Map}}_{\theta \, \phi}$)

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

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

Where the Net Inertial Linear Mass Density (${\mu_{L-Density}}$) – in the Proper Observer “center of mass” frame of reference, is given by the following.

${\mu_{L-Density}}\,\,=\,\,\frac{m_{Net}}{r}$$\,\,----> \,\, \frac{kg}{m}$

${\mu_{L-Density}}\,\,=\,\,{\mu'_{L-Density}}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})}\,\,=\,\,(\sqrt{-1})\,{\mu_{L-Density}}_{\theta \, \phi}\,\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}$

${\mu_{L-Density}}\,\,=\,\,\frac{m_{Net}}{r}\,\,=\,\,(\frac{m'_{Rel}}{s}){({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})}\,\,=\,\,(\sqrt{-1})\,(\frac{m_{Net}}{{s_{Map}}_{\theta \, \phi}})\,\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}$

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

Where the Map/Patch/Manifold – Geodesic Arc-Length” Net Inertial Linear Mass Density (${\mu_{L-Density}}_{\theta \, \phi}$) – in the Proper Observer “center of mass” frame of reference, is given by the following.

${\mu_{L-Density}}_{\theta \, \phi}\,\,=\,\,\frac{m_{Net}}{{s_{Map}}_{\theta \, \phi}}$$\,\,----> \,\, \frac{kg}{m}$

${\mu_{L-Density}}_{\theta \, \phi}\,\,=\,\,(\sqrt{-1})\,{\mu_{L-Density}}\,\sqrt{\frac{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}\,\,=\,\,(\sqrt{-1})\,{\mu'_{L-Density}}\,\sqrt{\frac{(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})^2\,(1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{|v|_{CM}}{c_{Light}})}}$

${\mu_{L-Density}}_{\theta \, \phi}\,\,=\,\,\frac{m_{Net}}{{s_{Map}}_{\theta \, \phi}}\,\,=\,\,(\sqrt{-1})\,(\frac{m_{Net}}{r})\,\sqrt{\frac{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}\,\,=\,\,(\sqrt{-1})\,(\frac{m'_{Rel}}{s})\,\sqrt{\frac{(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})^2\,(1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{|v|_{CM}}{c_{Light}})}}$

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

Where the Net Inertial Linear Mass Density (${\mu'_{L-Density}}$) – in the External Observer frame of reference, is given by the following.

${\mu'_{L-Density}}\,\,=\,\,\frac{m'_{Rel}}{s}$$\,\,----> \,\, \frac{kg}{m}$

${\mu'_{L-Density}}\,\,=\,\,\frac{{\mu_{L-Density}}}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})}\,\,=\,\,(\sqrt{-1})\,{\mu'_{L-Density}}_{\theta \, \phi}\,\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}$

${\mu'_{L-Density}}\,\,=\,\,\frac{m'_{Rel}}{s}\,\,=\,\,\frac{m_{Net}}{r}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^{-1}}\,\,=\,\,(\sqrt{-1})\,(\frac{m'_{Rel}}{\vec{s_{Map}}_{\theta \, \phi}})\,\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}$

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

Where the Map/Patch/Manifold – Geodesic Arc-Length” Net Inertial Linear Mass Density (${\mu'_{L-Density}}_{\theta \, \phi}$) – in the External Observer frame of reference, is given by the following.

${\mu'_{L-Density}}_{\theta \, \phi}\,\,=\,\,\frac{m'_{rel}}{{s_{Map}}_{\theta \, \phi}}$$\,\,----> \,\, \frac{kg}{m}$

${\mu'_{L-Density}}_{\theta \, \phi}\,\,=\,\,(\sqrt{-1})\,{\mu'_{L-Density}}\,\sqrt{\frac{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}\,\,=\,\,(\sqrt{-1})\,{\mu_{L-Density}}\,\sqrt{\frac{1}{2(\frac{|v|_{CM}}{c_{Light}})\,(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}}$

${\mu'_{L-Density}}_{\theta \, \phi}\,\,=\,\,\frac{m'_{rel}}{{s_{Map}}_{\theta \, \phi}}\,\,=\,\,(\sqrt{-1})\,(\frac{m'_{Rel}}{s})\,\sqrt{\frac{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}\,\,=\,\,(\sqrt{-1})\,(\frac{m_{Net}}{r})\,\sqrt{\frac{1}{2(\frac{|v|_{CM}}{c_{Light}})\,(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}}$

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

(8) The “Inertia Potential” of an “Inertial Mass” Gradient Gravitational Field – in consideration for Special Relativity & General Relativity

The Inertial Mass Gradient Gravitational Field “Inertia” Potential (${v^2_{Gravity}}$) is a measure of the square of the orbiting/spin/rotation squared tangential velocity of each potential of the gradient gravity field; and describes the squared velocity “inertia” potential, of the relative spatial separation, of the attraction and interaction of “mass towards mass”; and towards the center of the gradient gravity field, and towards ever decreasing and smaller volumes, of spherical gradient shell potentials.

The Inertial Mass Gradient Gravitational Field “Inertia” Potential (${v^2_{Gravity}}$) is a gravitational field parameter that varies, in direct proportion to the Net Inertial Linear Mass Density (${\mu_{L-Density}}$); and is described mathematically in terms of “Relativistic” frames of reference, observers, and their respective motions, below.

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

Where the Gradient Gravitational Field “Inertia” Potential (${v^2_{Gravity}}$) – in the Proper Observer “center of mass” frame of reference, is given by the following.

${v^2_{Gravity}}\,\,=\,\,{\mu_{L-Density}}\,{G}\,\,=\,\,\frac{m_{Net}\,G}{r}$$\,\,----> \,\, \frac{m^2}{s^2}$

${v^2_{Gravity}}\,\,=\,\,{\mu'_{L-Density}\,G}\,{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})}\,\,=\,\,(\sqrt{-1})\,{\mu_{L-Density}}_{\theta \, \phi}\,\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}$

${v^2_{Gravity}}\,\,=\,\,\frac{m_{Net}\,G}{r}\,\,=\,\,(\frac{m'_{Rel}\,G}{s}){({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})}\,\,=\,\,(\sqrt{-1})\,(\frac{m_{Net}\,G}{{s_{Map}}_{\theta \, \phi}})\,\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}$

${v^2_{Gravity}}\,\,=\,\,(-)\frac{V_{Gravity-Potential}}{m_{test-mass}}\,\,=\,\,(-)\frac{V_{Self-Potential}}{m_{Net}}$$\,\,----> \,\, \frac{m^2}{s^2}$

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

Where the Geodesic Arc-Length” Gradient Gravitational Field “Inertia” Potential (${v^2_{Gravity}}_{\theta \, \phi}$) – in the Proper Observer “center of mass” frame of reference, is given by the following.

${v^2_{Gravity}}_{\theta \, \phi}\,\,=\,\,{\mu_{L-Density}}_{\theta \, \phi}\,G\,\,=\,\,\frac{m_{Net}\,G}{{s_{Map}}_{\theta \, \phi}}$$\,\,----> \,\, \frac{m^2}{s^2}$

${v^2_{Gravity}}_{\theta \, \phi}\,\,=\,\,{\mu_{L-Density}}_{\theta \, \phi}\,{G}\,\,=\,\,(\sqrt{-1})\,{\mu_{L-Density}}\,G\,\sqrt{\frac{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}\,\,=\,\,\dots\dots\,\,=\,\,(\sqrt{-1}){\mu'_{L-Density}}\,{G}\,\sqrt{\frac{(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})^2\,(1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{|v|_{CM}}{c_{Light}})}}$

${v^2_{Gravity}}_{\theta \, \phi}\,\,=\,\,\frac{m_{Net}\,G}{{s_{Map}}_{\theta \, \phi}}\,\,=\,\,(\sqrt{-1})\,(\frac{m_{Net}\,G}{r})\,\sqrt{\frac{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}\,\,=\,\,(\sqrt{-1})\,(\frac{m'_{Rel}\,G}{s})\,\sqrt{\frac{(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})^2\,(1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{|v|_{CM}}{c_{Light}})}}$

${v^2_{Gravity}}_{\theta \, \phi}\,\,=\,\,(-)\frac{{V_{Gravity-Potential}}_{\theta \, \phi}}{m_{test-mass}}\,\,=\,\,(-)\frac{{V_{Self-Potential}}_{\theta \, \phi}}{m_{Net}}$$\,\,----> \,\, \frac{m^2}{s^2}$

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

Where the Gradient Gravitational Field “Inertia” Potential (${v'^2_{Gravity}}$) – in the External Observer frame of reference, is given by the following.

${v'^2_{Gravity}}\,\,=\,\,{\mu'_{L-Density}}\,{G}\,\,=\,\,\frac{m'_{Rel}\,G}{s}$$\,\,----> \,\, \frac{m^2}{s^2}$

${v'^2_{Gravity}}\,\,=\,\,\frac{{\mu_{L-Density}}\,G}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})}\,\,=\,\,(\sqrt{-1})\,{\mu'_{L-Density}}_{\theta \, \phi}\,{G}\,\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}$

${v'^2_{Gravity}}\,\,=\,\,\frac{m'_{Rel}\,G}{s}\,\,=\,\,\frac{m_{Net}\,G}{r}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^{-1}}\,\,=\,\,(\sqrt{-1})\,(\frac{m'_{Rel}\,G}{{s_{Map}}_{\theta \, \phi}})\,\sqrt{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}$

${v'^2_{Gravity}}\,\,=\,\,(-)\frac{V'_{Gravity-Potential}}{m'_{test-mass}}\,\,=\,\,(-)\frac{V'_{Self-Potential}}{m'_{Rel}}$$\,\,----> \,\, \frac{m^2}{s^2}$

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

Where the Map/Patch/Manifold – Geodesic Arc-Length” Gradient Gravitational Field “Inertia” Potential (${v'^2_{Gravity}}_{\theta \, \phi}$) – in the External Observer frame of reference, is given by the following.

${v'^2_{Gravity}}_{\theta \, \phi}\,\,=\,\,{\mu'_{L-Density}}_{\theta \, \phi}\,{G}\,\,=\,\,\frac{m'_{rel}\,G}{{s_{Map}}_{\theta \, \phi}}$$\,\,----> \,\, \frac{m^2}{s^2}$

${v'^2_{Gravity}}_{\theta \, \phi}\,\,=\,\,(\sqrt{-1})\,{\mu'_{L-Density}}\,{G}\,\sqrt{\frac{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}\,\,=\,\,(\sqrt{-1})\,{\mu_{L-Density}}\,{G}\,\sqrt{\frac{1}{2(\frac{|v|_{CM}}{c_{Light}})\,(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}}$

${v'^2_{Gravity}}_{\theta \, \phi}\,\,=\,\,\frac{m'_{rel}\,G}{{s_{Map}}_{\theta \, \phi}}\,\,=\,\,(\sqrt{-1})\,(\frac{m'_{Rel}\,G}{s})\,\sqrt{\frac{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}\,\,=\,\,(\sqrt{-1})\,(\frac{m_{Net}\,G}{r})\,\sqrt{\frac{1}{2(\frac{|v|_{CM}}{c_{Light}})\,(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}}$

${v'^2_{Gravity}}_{\theta \, \phi}\,\,=\,\,(-)\frac{{V'_{Gravity-Potential}}_{\theta \, \phi}}{m'_{test-mass}}\,\,=\,\,(-)\frac{{V'_{Self-Potential}}_{\theta \, \phi}}{m'_{Rel}}$$\,\,----> \,\, \frac{m^2}{s^2}$

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

(9) The Conservative “Self Gravitational Force” of an “Inertial Mass” Gradient Gravitational Field – in consideration for Special Relativity & General Relativity

Any and every conserved and isolated “Net Inertial Mass” system body, can be modeled as a “vortex” system body, that is spheroid in nature, and is described by a gradient field, comprised of an infinite amount of “spherical shell potentials” relative to the center of the system. The gradient gravity field is described by concentric spherical volumetric potential shells of “Gravitational Potential Energy” and a conservative “Self Gravitational Force” at each potential.

For a general gradient gravitational field, the conservative “Self” Gravitational Potential Energy (${V_{Self-Potential}}$) of each concentric spherical shell potential, is associated with the Inertial Mass “Self “Gravitational Force ($\vec{F_{Self-Gravity}}$), where the source of gradient gravity field is the Net Inertial Mass (${m_{Net}}$).

The Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$) is a gravitational field parameter that varies, in direct proportion to the square of the Net Inertial Linear Mass Density (${\mu^2_{L-Density}}$); and is described mathematically in terms of “Relativistic” frames of reference, observers, and their respective motions, below.

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

Where the Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}$) – in the Proper Observer “center of mass” frame of reference, is given by the following.

$\vec{F_{Self-Gravity}}\,\,=\,\,(-){\mu^2_{L-Density}}\,{G}\,\,\hat{a}_{r}\,\,=\,\,(-)\frac{m^2_{Net}\,G}{r^2}\,\hat{a}_{r}$$\,\,----> \,\, \frac{kg\,m}{s^2}$

$\vec{F_{Self-Gravity}}\,\,=\,\,(-){\mu'^2_{L-Density}\,G}\,{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}\,\hat{a}_{r}\,\,=\,\,{\mu^2_{L-Density}}_{\theta \, \phi}\,[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}]\,\,\hat{a}_{r}$

$\vec{F_{Self-Gravity}}\,=\,(-)\frac{m^2_{Net}\,G}{r^2}\,\,\hat{a}_{r}\,=\,(\frac{m'^2_{Rel}\,G}{s^2}){({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}\,\,\hat{a}_{r}\,=\,(\frac{m^2_{Net}\,G}{{s^2_{Map}}_{\theta \, \phi}})\,[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}}]\,\,\hat{a}_{r}$

$\vec{F_{Self-Gravity}}\,\,=\,\,(-){m_{Net}}\,{g_{Gravity}}\,\,\hat{a}_{r}\,\,=\,\,(\frac{m_{Net}}{m_{test-mass}})\,\vec{F_{Gravity}}$$\,\,----> \,\, \frac{kg\,m}{s^2}$

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

Where the Map/Patch/Manifold – Geodesic Arc-Length” Inertial Mass “Self” Gravitational Force ($\vec{F_{Self-Gravity}}_{\theta \, \phi}$) – in the Proper Observer “center of mass” frame of reference, is given by the following.

$\vec{F_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(-){\mu^2_{L-Density}}_{\theta \, \phi}\,{G}\,\,\hat{a}_{r}\,\,=\,\,(-)\frac{m^2_{Net}\,G}{{s^2_{Map}}_{\theta \, \phi}}\,\,\hat{a}_{r}$$\,\,----> \,\, \frac{kg\,m}{s^2}$

$\vec{F_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(-){\mu^2_{L-Density}}_{\theta \, \phi}\,{G}\,\,\hat{a}_{r}\,\,=\,\,{\mu^2_{L-Density}}\,{G}\,[{\frac{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\,\hat{a}_{r}\,\,=\,\,\dots\dots\,\,=\,\,{\mu'^2_{L-Density}}\,{G}\,[{\frac{(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})^2\,(1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\,\hat{a}_{r}$

$\vec{F_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(\frac{m^2_{Net}\,G}{r^2})\,[{\frac{1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\,\hat{a}_{r}\,\,=\,\,(\frac{m'^2_{Rel}\,G}{s^2})\,[{\frac{(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})^2\,(1\;\;+\;\;\frac{|v|_{CM}}{c_{Light}})}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\,\hat{a}_{r}$

$\vec{F_{Self-Gravity}}_{\theta \, \phi}\,=\,(-){m_{Net}}\,{g_{Gravity}}_{\theta \, \phi}\,\,\hat{a}_{r}\,=\,(\frac{m_{Net}}{m_{test-mass}})\,\vec{F_{Gravity}}_{\theta \, \phi}$$---> \,\, \frac{kg\,m}{s^2}$

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

Where the Inertial Mass “Self” Gravitational Force ($\vec{F'_{Self-Gravity}}$) – in the External Observer frame of reference, is given by the following.

$\vec{F'_{Self-Gravity}}\,\,=\,\,(-){\mu'^2_{L-Density}}\,{G}\,\,\hat{a}_{r}\,\,=\,\,(-)\frac{m'^2_{Rel}\,G}{s^2}\,\,\hat{a}_{r}$$\,\,----> \,\, \frac{kg\,m}{s^2}$

$\vec{F'_{Self-Gravity}}\,\,=\,\,(-)\frac{{\mu^2_{L-Density}}\,G}{({1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}})^2}\,\,\hat{a}_{r}\,\,=\,\,{\mu'^2_{L-Density}}_{\theta \, \phi}\,{G}\,[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}}]\,\,\hat{a}_{r}$

$\vec{F'_{Self-Gravity}}\,=\,(-)\frac{m'^2_{Rel}\,G}{s^2}\,\hat{a}_{r}\,=\,(-)\frac{m^2_{Net}\,G}{r^2}{({1\,-\,\frac{|v|_{CM}}{c_{Light}}})^{-2}}\,\hat{a}_{r}\,=\,(\frac{m'^2_{Rel}\,G}{{s^2_{Map}}_{\theta \, \phi}})\,[{\frac{2(\frac{|v|_{CM}}{c_{Light}})}{1\,-\,\frac{|v|_{CM}}{c_{Light}}}}]\,\hat{a}_{r}$

$\vec{F'_{Self-Gravity}}\,\,=\,\,(-){m'_{Rel}}\,{g'_{Gravity}}\,\,\hat{a}_{r}\,\,=\,\,(\frac{m'_{Rel}}{m'_{test-mass}})\,\vec{F'_{Gravity}}$$\,\,----> \,\, \frac{kg\,m}{s^2}$

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

Where the Map/Patch/Manifold – Geodesic Arc-Length” Inertial Mass “Self” Gravitational Force ($\vec{F'_{Self-Gravity}}_{\theta \, \phi}$) – in the External Observer frame of reference, is given by the following.

$\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(-){\mu'^2_{L-Density}}_{\theta \, \phi}\,{G}\,\,\hat{a}_{r}\,\,=\,\,(-)\frac{m'^2_{rel}\,G}{{s^2_{Map}}_{\theta \, \phi}}\,\,\hat{a}_{r}$$\,\,----> \,\, \frac{kg\,m}{s^2}$

$\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,{\mu'^2_{L-Density}}\,{G}\,[{\frac{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\,\hat{a}_{r}\,\,=\,\,{\mu^2_{L-Density}}\,{G}\,[{\frac{1}{2(\frac{|v|_{CM}}{c_{Light}})\,(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}}]\,\,\hat{a}_{r}$

$\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,\,=\,\,(\frac{m'^2_{Rel}\,G}{s^2})\,[{\frac{1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}}}{2(\frac{|v|_{CM}}{c_{Light}})}}]\,\hat{a}_{r}\,\,=\,\,(\frac{m^2_{Net}\,G}{r^2})\,[{\frac{1}{2(\frac{|v|_{CM}}{c_{Light}})\,(1\;\;-\;\;\frac{|v|_{CM}}{c_{Light}})}}]\,\hat{a}_{r}$

$\vec{F'_{Self-Gravity}}_{\theta \, \phi}\,=\,(-){m'_{Rel}}\,{g'_{Gravity}}_{\theta \, \phi}\,\,\hat{a}_{r}\,=\,(\frac{m'_{Rel}}{m'_{test-mass}})\,\vec{F'_{Gravity}}_{\theta \, \phi}$$---> \,\, \frac{kg\,m}{s^2}$

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

(10) 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 Space-time 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 discussions of gravity, are most commonly discussed.

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

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}$

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

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

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}$

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

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

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}$

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

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

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}$

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

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

General Constants

Gravitational Constant

${G}\;=\;6.67384 \times 10^{-11} \frac{m^3}{kg\,s^2}$

Speed of Light in vacuum constant

${c_{Light}}\;=\;2.99792459 \times 10^{8} \frac{m}{s}$

“Black Hole” Net Inertial Linear Mass Density Constant

${\mu_{L-Density}}_{BH}\,\,=\,\,6.73297478332358\times 10^{26} \frac{kg}{m}$

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

Citation

Robert Louis Kemp; The Super Principia Mathematica – The Rage to Master Conceptual & Mathematical Physics – The General Theory of Relativity – “The “Central Conservative” Gravitational Force and Potential Energy – in consideration with Special Relativity and General Relativity– Online Volume – ISBN 978-0-9841518-2-0, Volume 3; July 2010

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

The above work completes a new theory of Gravity for the 21st Century; and produces a complete conceptual and mathematical model of matter, space, and time. The above work opens the door to discuss new concepts and mathematics of gravity, in consideration for Special Relativity and General Relativity; the Super Special Theory of Relativity.

Best,

Author: Robert Louis Kemp

http://www.SuperPrincipia.com

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.

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

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

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

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

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

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

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

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

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

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

“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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

“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

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

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

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

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

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

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

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

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

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

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

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

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

“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

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

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

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

“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

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

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

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

“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

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

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

“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

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

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

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

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.

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

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}$

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

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

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}$

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

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

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}$

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

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

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}$

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

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

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}$

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

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}$

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

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}$

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

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 (${r_{Schwarzschild}}\,=\,\frac{2\,{m_{Net}}\,G}{c^2_{Light}}$).

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

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

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

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

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

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

Map/Patch/Manifold – “Geodesic” Metric

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

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

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

Gravitatational Field Acceleration

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

“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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

“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

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

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

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

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

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

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

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

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

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

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

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 “Latitude Angle” ($\vec{d{\theta}}$), and the “Longitude Angle” ($\vec{d{\phi}}$) of a symmetric sphere

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

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

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

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

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

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

“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

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

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

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

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

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

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

“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

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

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

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

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

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

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

“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

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

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

“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

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

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

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

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

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

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

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

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

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

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

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

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

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}$

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

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}$

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

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{r}{{r_{Schwarzschild}}}))^2\;\;+\;\;({\Omega_{Map}}_{(\theta \phi)})^2}]$$\,\,----> \,\, \frac{m}{s}$

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

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 (${r_{Schwarzschild}}\,=\,\frac{2\,{m_{Net}}\,G}{c^2_{Light}}$).

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

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

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

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

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

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

Map/Patch/Manifold – “Geodesic” Arc Length

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

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

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

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}$

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

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

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

Citation

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

http://www.SuperPrincipia.com

Vacuum Energy in the 21st Century

The “Vacuum” which is modeled as a “Aether Gas” whose constituents are the “Aetherons” must be re-interpreted; based on 21st Century Physics.

One of the main misunderstandings is that we now live in the 21st Century, and we understand a lot more; so based on the classical interpretation of the vacuum, we now have to discuss two (2) different types of vacuums.

The first (1st) type of “Vacuum” which is modeled as a “Aether Gas” whose constituents, should be named a “Baryonic Vacuum” and the second (2nd) type of vacuum should be named a “Non-Baryonic Vacuum.”

The first (1st) type of vacuum, the “Baryonic Vacuum” is the one that you are the most familiar with. And you are correct here in stating that a real or true “Baryonic Vacuum” will never exist, and is always a “partial vacuum.”

A “Baryonic Vacuum” is a volume of space that is filled with atomic matter such as atoms, electrons, protons, and neutrons.

Now let’s do a “thought experiment or Gedankenexperiment.”

Image we create an experiment where we build a “rectangular box” that can be sealed, and set this box in a room where we can turn on and off the lights in this room. Now returning to our “rectangular box” inside the box we insert a “flashlight” that we can turn on and off, and a “bell” that we can ring on and off.

The next condition is that we are external observers to the “rectangular box” and its contents, such that while standing in the room and viewing the box as an external observer, if the “flashlight” is turned on we can see the light; and if the “bell” is rung or turned on, we can hear the bell when it rings, while standing outside of the box and in the room.

The next condition is that we design the electronics of the “rectangular box” so that when the box is sealed, we can install a vacuum pump that will allow us to vacuum pump the “Baryonic Matter: air molecules, electrons, protons, and neutrons” out of the box. The second set of electronics provides us with a set of switches, that allow us to turn on and off the “flashlight”, and to turn on and off the “bell” without opening the box.
Ok, now that our experiment is set up, we are ready to do the experiment.

The first (1st) – “sealed box” experiment –

(1) Turn off the light in the room. (2) Turn “off” the vacuum pump. (3) let the “Baryonic Matter: air molecules, electrons, protons, and neutrons” remain in the box. (4) Turn “on” the flashlight; we see the light. (5) Turn “on” the bell, we hear the bell ringing.

The Second (2nd) – “sealed box” experiment –

(1) Turn off the light in the room. (2) Turn “on” the vacuum pump. (3) remove the “Baryonic Matter: air molecules, electrons, protons, and neutrons” from the box; this is a vacuum state. (4) Turn “on” the flashlight; we see the light. (5) Turn “on” the bell, we “do not” hear the bell ringing.

In this experiment we have results. Inside the box we have a vacuum sate. Although we suck or remove all of the “Baryonic Matter: air molecules, electrons, protons, and neutrons” from the box; in the real world we can never pump all of the “Baryonic Matter” from the box or vacuum. This is what is known as a “Partial Vacuum. Thus, all “Baryonic Vacuums” are not true or real vacuums but are only “Partial Vacuums”

Since the air molecules have been pumped out of the box we “do not” hear the bell because sound waves need “air molecules” to vibrate to produce sound. Since the air molecules have been “removed” or reduced to a very, very rare minimum number, the sound waves can’t propagate because there aren’t any molecules to vibrate in sufficient number. Remember the bell is still operating properly we just can’t hear it because there is nothing to wave!

However, we can still see the “flash light” shining brightly, but can’t hear the bell ringing; Why??

Here is where our second (2nd) type of vacuum the “Non-Baryonic Vacuum” must be discussed.

The second (2nd) type of vacuum which is modeled as a “Aether Gas” whose constituents, the “Non-Baryonic Vacuum” is the one that we are the most unfamiliar with. The “Non-Baryonic Vacuum” is a vacuum that is seething with a form of energy known as the Aether.

A “Non-Baryonic Vacuum” is a volume of space that is filled with Aether or “Non-Baryonic” matter such as photons, Bosons, Aetherons, Dark-Matter, and Dark Energy. This “Non-Baryonic Vacuum” is host to spontaneous emission of photon-photon pair production and matter-antimatter pair production.

This modern view of the vacuum supports spontaneous emission of photon-photon pair production in the “Non-Baryonic Vacuum.”

This “Non-Baryonic Vacuum” is also host to electromagnetic fields and is why we can see the shining light of the flashlight, when the “Baryonic Matter” is removed from the sealed rectangular box, in our thought or Gedankenexperiment.”

Finally, this “Non-Baryonic Vacuum” is always a real or true vacuum and is never a partial vacuum. Wherever you have a volume of space you have this “Non-Baryonic Vacuum” energy that cannot be removed from any container. An infinite amount of Matter and energy can be created from this “Non-Baryonic Vacuum”

I hope this brings enlightenment to your current understanding of the Vacuum Energy in the 21st century!

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