“Vortex Gravitation” Model – Einstein Field Equation – Differential Mathematical Form

What makes the Inertial Mass “Gravitation Vortex” model very powerful and useful is that it predicts Isaac Newton’s Graviation Laws and Einstein’s General Theory of Relativity.

The “Gravitation Vortex” model is described by the equations of General Relativity:

The vortex model must obey the Ideal Gas Law which satisfies the Einstein Field Equation of General Relativity

Equations of General Relativity in “Differential Form” Mathematical Mechanics

The General Relativity Gravitational Field Equation describes a “Gravitational Vortex” where the gradient Gravitational Field Potential is modeled as a Euclidean Sphere where the differential “Map/Patch/Manifold Geodesic Arc Length” ( ${dG_{Space}}_{(\theta \phi)}$ ) is the source of curvature of each gravitational gradient field potential and is a differential “Map/Patch/Manifold Geodesic Arc Length” in three (3) dimensional space ( ${dx}, {dy}, {dz}$ ), one dimension of time ( ${dt_{time}}$ ), and two (2) dimensional angles ( $d\theta_{Latitude}$ , $d\phi_{Longitude}$ ), and is directly proportional to the differential “Stress Energy” ( ${dT_{Energy}}_{(\theta \phi)}$ ) of a gravity vortex system body.

Source of Curvature of Spherical Gravitational Field Potential ― Map/Patch/Manifold “Field Equation” – Differential – Geodesic Arc Length Distance

${dG_{Space}}_{(\theta \phi)} \;=\; {r}{d\Omega_{Map}}_{(\theta \phi)}\;$ $---> m$

${dG_{Space}}_{(\theta \phi)} \;=\;[{dR_{(\theta \phi)}}\;-\;{dg_{Vol}}_{(\theta \phi)}(\frac{R_{Heat}}{2}\;-\;{\Lambda_{Einstein}})]$ $---> m$

${dG_{Space}}_{(\theta \phi)}\;=\;2\pi [\frac{{dT_{Energy}}_{(\theta \phi)}}{\frac{1}{4} ( \frac{c^4_{Light}}{G})}]\;=\;{[{dR_{(\theta \phi)}}\;-\;\frac{{dg_{Vol}}_{(\theta \phi)}}{S^2_{Expansion}}]}$ $---> m$

${dG_{Space}}_{(\theta \phi)}\;=\;2\pi [\frac{{dT_{Energy}}_{(\theta \phi)}}{F_{Dark-Force}}]$ $---> m$

“Differential” Map/Patch/Manifold Angle

${d\Omega_{Map}}_{(\theta \phi)}\;=\;\sqrt{d\theta^2_{Lat}\;+\;d\phi^2_{Lon} (sin^2\theta_{Lat})}$ $---> radians$

“Differential” Latitude Angle

${d\theta_{Lat}}\;=\;\frac{dr_{\theta}}{r}$ $---> radians$

“Differential” Longitude Angle

${d\phi_{Lon}}\;=\;\frac{dr_{\phi}}{r(sin\theta_{Lat})}$ $---> radians$

Thus, there is a difference between the Euclidean volume ( ${V_{ol}}\;=\;(\frac{4\pi r^3}{3})$ ) enclosed by this surface of a sphere, and the differential Map/Patch/Manifold Surface Volume ( ${dg_{Vol}}_{(\theta \phi)}$ ) segments of a sphere. The differential Map/Patch/Manifold Surface Volume ( ${dg_{Vol}}_{(\theta \phi)}$ ) is two (2) dimensional expressed as a function of geodesic arc length, angle, and area, in three (3) dimensions given by the following equations.

Euclidean Spherical Volume

${V_{ol}}\;=\;(\frac{4\pi r^3}{3})$ $---> {m^3}$

“Differential” Volume Element of Sphere

${dg_{Vol}}_{(\theta \phi)}\;=\;2(\frac{V_{ol}}{4\pi^2}){\Omega_{Map}}_{(\theta \phi)}{d\Omega_{Map}}_{(\theta \phi)}$ $---> {m^3}$

${dg_{Vol}}_{(\theta \phi)}\;=\;2(\frac{r^3}{3\pi}){\Omega_{Map}}_{(\theta \phi)}{d\Omega_{Map}}_{(\theta \phi)}$ $---> {m^3}$

${dg_{Vol}}_{(\theta \phi)}\;=\;(\frac{2r}{3\pi}){G_{Space}}_{(\theta \phi)}{dG_{Space}}_{(\theta \phi)}$ $---> {m^3}$

The General Relativity Einstein field equation determines the metric tensor of spacetime for a given arrangement of stress-energy in the spacetime. The Einstein field equations describe the fundamental force of gravitation as a curved spacetime caused by matter and energy.

The differential “Stress” Energy ( ${dT_{Energy}}_{(\theta \phi)}$ ) describes curved spacetime which includes matter and energy 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 differential “Stress” Energy ( ${dT_{Energy}}_{(\theta \phi)}$ ) as the source of gravity on the surface of a spherical gravitational field potential.

Ideal Gas Equation for Spacetime – Differential – Stress Energy

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

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

${dT_{Energy}}_{(\theta \phi)}\;=\; \frac{1}{4} ( \frac{c^4_{Light}}{G})[{r}(\frac{{d\Omega_{Map}}_{(\theta \phi)}}{2\pi})]$ $---> \frac{kgm^2}{s^2}$

${dT_{Energy}}_{(\theta \phi)}\;= {dP_{ressure}}_{(\theta \phi)}{dg_{Vol}}_{(\theta \phi)} = N k_{B} {dT_{emp}}_{(\theta \phi)}$ $---> \frac{kgm^2}{s^2}$

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

Where the Boltzman Constant is given by

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

Therefore, the Aether Gravitational Vortex model can be used to describe what is currently being predicted by recent experiments in cosmology and general relativity.

Cosmic “Dark” Vacuum Force

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

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

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

The classical Einstein Cosmological Constant is denoted by the symbol constant ( ${\Lambda_{Einstein}}$ ). Let’s start by saying that the Cosmological Constant is in no way “Constant” as demonstrated below. It is better to say that the Einstein Cosmological term is directly proportional to the “Vacuum Density” of Spacetime ( $\rho_{Vacuum-Density}$ ), and varies in inverse proportion to the “Net Inertial Mass” of the system. And likewise in inverse proportion to the distance ( $\frac{1}{r}$ ) as measured from the center of the vortex system.

Einstein Vacuum Density – Inverse Square Distance Relation

${\Lambda_{Einstein}}\;=\;\frac{1}{4}(\frac{c^2_{Light}}{{m_{Net}}G})(\frac{1}{r})$ $---> \frac{1}{m^2}$

${\Lambda_{Einstein}}\;=\; \frac{8\pi G}{c^2_{Light}}(\rho_{Vacuum-Density})$ $---> \frac{1}{m^2}$

The Einstein Field Equation of General Relativity also list a parametrer that is a “Heat Radiation Gravitation” term of the Spacetime. The Heat Radiation Scalar term varies in direct proportion to the forth power of the temperature gradient ( ${T^4_{Temp}}$ ) as measured from the center of the vortex system.

Heat Radiation Scalar – Inverse Square Distance Relation

$\frac{R_{Heat}}{2}\;=\; 8\pi [\frac{(\frac{\sigma_{Stefan}}{c_{Light}}){T^4_{Temp}}}{(\frac{c^4_{Light}}{4G})}]$ $---> \frac{1}{m^2}$

Where the Stefan Boltzman Heat Radiation Constant is given by

$\sigma_{Stefan}\;=\;\frac{2\pi^2{k^4_{B}}}{15 {h^3_{Planck}}{c^2_{Light}}}$ $---> \frac{kg}{s^3 K^4}$

$\sigma_{Stefan}\;=\;5.670373 \times 10^{-8} \frac{kg}{s^3 K^4}$

The expansion of three (3) dimensional spaces in concentric spherical volumetric shells of gravitational field potential away from the surface of the Schwarzschild Black Hole Event Horizon Radius ( ${r_{Schwarzschild}}\;=\;2(\frac{m_{Net}G}{c^2_{Light}})\;$ ), given by the following Spacetime Expansion ( ${S^2_{Expansion}}$ ) Metric equation below.

Schwarzschild Black Hole Event Horizon Radius

${r_{Schwarzschild}}\;=\;2(\frac{m_{Net}G}{c^2_{Light}})\;$ $---> {m}$

Square of the Gravitation Tangential Orbiting Velocity – (Scalar)

${v^2_{Gravity}} \;=\;\frac{K_{Gravity}}{r}\;=\;\frac{m_{Net}G}{r}\;=\;4\pi^2(\frac{r^{2}}{T^{2}_{Period}})$ $---> \frac{m^2}{s^2}$

Spacetime Expansion Metric

${S^2_{Expansion}}\;=\;\frac{{dg_{Vol}}_{(\theta \phi)}}{({dR_{(\theta \phi)}} \;-\;{dG_{Space}}_{(\theta \phi)})}\;$ $---> {m^2}$

${S^2_{Expansion}}\;=\;\frac{1}{(\frac{R_{Heat}}{2} \;-\;{\Lambda_{Einstein}})}\;$ $---> {m^2}$

${S^2_{Expansion}}\;=\;\frac{1}{(\frac{1}{r^2}\;-\;\frac{r_{Schwarzschild}}{r^3})}\;$ $---> {m^2}$

Ricci/Riemann Maximum Curvature of Spherical Gravitational Field Potential ― Map/Patch/Manifold “Field Equation” – Differential – Geodesic Arc Length Distance

${dR_{(\theta \phi)}} \;=\;{[{dG_{Space}}_{(\theta \phi)}\;+\;\frac{{dg_{Vol}}_{(\theta \phi)}}{S^2_{Expansion}}]}$ $---> m$

${dR_{(\theta \phi)}}\;=\;[{dG_{Space}}_{(\theta \phi)}\;+\; {dg_{Vol}}_{(\theta \phi)}(\frac{R_{Heat}}{2}\;-\;{\Lambda_{Einstein}})]$ $---> m$

${dR_{(\theta \phi)}} \;=\;{[{r}{d\Omega_{Map}}_{(\theta \phi)}\;+\;\frac{1}{3\pi}(\frac{r^3}{S^2_{Expansion}}){\Omega_{Map}}_{(\theta \phi)}{d\Omega_{Map}}_{(\theta \phi)} ]}$ $---> m$

${dR_{(\theta \phi)}} \;=\;{[2\pi [\frac{{dT_{Energy}}_{(\theta \phi)}}{\frac{1}{4} ( \frac{c^4_{Light}}{G})}]\;+\;\frac{{dg_{Vol}}_{(\theta \phi)}}{S^2_{Expansion}}]}$ $---> m$

General Constants

Gravitational Constant

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

Planck’s Constant

${h_{Planck}}\;=\;6.62606957 \times 10^{-34} \frac{kgm^2}{s}$

Speed of Light in vacuum constant

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

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Citation

Cite this article as:

Robert Louis Kemp; The Super Principia Mathematica – The Rage to Master Conceptual & Mathematical Physics – The General Theory of Relativity – ““Vortex Gravitation” Model – Einstein Field Equation – Differential Mathematical Form” – Online Volume – ISBN 978-0-9841518-2-0, Volume 3; July 2010

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Author: Robert Louis Kemp

http://www.SuperPrincipia.com

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