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

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}}

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(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}

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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}}

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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}

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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}

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(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.

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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}

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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}]

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(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.

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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}

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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}]

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(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

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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})

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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}

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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

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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

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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

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(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

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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.

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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}}]

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(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}}

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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}}]

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(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}}]

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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}}}

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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}}]

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(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}}]

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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)}}

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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}

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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}}]

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(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}}]

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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}}}

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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}}]

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(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.

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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}

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{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}}}]

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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}

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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}

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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.

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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 – “Total Mechanical Energy Conservation in General Relativity– Online Volume – ISBN 978-0-9841518-2-0, Volume 3; July 2010

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Best,

All Comments are welcome.

Author: Robert Louis Kemp

http://www.SuperPrincipia.com

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

  1. Pingback: Total Mechanical Energy Conservation – Escape Velocity & Binding Energy – Einstein Field Equation | Superprincipia's Blog

  2. Summary: When only conservative forces interact, the mechanical energy of an isolated system can not change.

  3. I regard something truly interesting about your site so I saved to fav.

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