# 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

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

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## (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.

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### 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

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

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

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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

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

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

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

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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}$

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## (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}$

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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}$

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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!

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## (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}$

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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}$

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

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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}$

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

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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}$

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## (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}$

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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}$

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## (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}$

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## (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}$

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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}$

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## (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.

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

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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

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

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

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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}$

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

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

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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}$

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

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

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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}$

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

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

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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}$

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

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

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## (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.

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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}$

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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}$

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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}$

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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}$

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## (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}$

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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}$

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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}$

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## (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.

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

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

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

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

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

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

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

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

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

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

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

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

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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

${\mu_{L-Density}}_{BH}\,\,=\,\,6.73297478332358\times 10^{26} \frac{kg}{m}$

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### 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

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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

This entry was posted in The General Theory of Relativity and tagged , , , , , , , , , , . Bookmark the permalink.

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

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• Doctor Who Matt Smith,
Thanks for your comment, I think. So how can I get an “A” in your book. What do you mean by this statememt “I am not necessarily sure of just how you seem to connect the details which produce your conclusion”?

Best

Kemp