# 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.

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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.

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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.

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

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

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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!

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

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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.

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

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

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

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

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

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## (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}$

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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.

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

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

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

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## (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}$

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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.

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

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

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

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

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

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

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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

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

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

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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

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

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

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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

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

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

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

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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.

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

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

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

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

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

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## (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}$

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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.

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

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

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

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## (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}$

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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.

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

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

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

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

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

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

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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

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

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

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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’

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

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

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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’

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

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

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

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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

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

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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.

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### 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

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Best,

Author: Robert Louis Kemp

http://www.SuperPrincipia.com