From the post "Gravity Exponential Form",
\(g=\cfrac{{G}_{o}}{{r}_{e}({d}_{e}-{d}_{n})}.\cfrac{d({d}_{s}(x))}{dx}\),
this expression considered space density and conservation of energy, there is a discrepancy where
\({v}^{2}_{t}+\frac{1}{2}{v}^{2}_{s}={c}^{2}\) instead of \({v}^{2}_{t}+ {v}^{2}_{s}={c}^{2}\)
The extra factor of \(\frac{1}{2}\) in front of \(v^2_s\) is eventually absorbed into \(G_o\).
We will here consider
\({v}^{2}_{t}+\frac{1}{2}{v}^{2}_{rg}={c}^{2}\)
where \(v^2_{rg}\) is deemed to be thermal and is responsible for manifesting thermal gravity as time speed \(v_t\) slows in denser space. (\(v^2_{rc}\) was used in charges that annihilate along the charge time dimension. The charges collided with opposite velocities along the charge time dimension (matter/antimatter). The loss of K.E was the gain in rotational K.E, \(v^2_{rc}\), which manifest itself as \(T\) temperature.)
We have a similar expression for thermal gravity, \(g_T\)
\(g_T=\cfrac{{G}_{T}}{{x}_{o}({d}_{n}-{d}_{T_{max}})}.\cfrac{d({d}_{s}(x))}{dx}\)
where \(G_{T}\) is a constant, \(x_o\) is the radius of the hot spot.
\(T=T_{max}\), where \(x=x_o\) on the boundary of the hot spot.
\(d_{T_{max}}\) is the density of space at temperature \(T_{max}\) and \(d_n\) is the normal space density. The term,
\({d}_{n}-{d}_{T_{max}}>0\)
because temperature thins out space and the resulting thermal gravity is positive in the direction of \(x\). If we assume that \(d_s(x)\) is linear in \(T\).
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| Space Density, ds vs Temperature, T |
\({d}_{s}(T)=\cfrac{({d}_{T_{max}}-{d}_{n})}{T_{max}}.T+d_n\)
Space density varies with \(T\), temperature, given the temperature profile (1).
\({d}_{s}(x)=\cfrac{({d}_{T_{max}}-{d}_{n})}{T_{max}}.\cfrac{A}{\sqrt{x}}+d_n\)
\(\cfrac{d({d}_{s}(x))}{dx}=-A\cfrac{({d}_{T_{max}}-{d}_{n})}{2T_{max}}.\cfrac{1}{({x})^{{3}/{2}}}\)
And we have,
\(g_T=\cfrac{{G}_{T}}{{x}_{o}({d}_{n}-{d}_{T_{max}})}(-A\cfrac{({d}_{T_{max}}-{d}_{n})}{2T_{max}}.\cfrac{1}{({x})^{3/2}})\)
\(g_T={{G}_{To}}\cfrac{1}{{x}^{3/2}}\)
where \(G_{To}\) replaces \(\cfrac{A{G}_{T}}{2{x}_{o}{T_{max}}}\)
In fact, for \(n=1,2,3...\)
\({d}_{s}(T)=\cfrac{{d}_{T_{max}}-{d}_{n}}{T^n_{max}}.T^n+d_n\)
\(g_T=\cfrac{nA^n{G}_{T}}{2{x}_{o}{T^n_{max}}}\cfrac{1}{\sqrt[n+2]{x}}\)
The following is a set of curves showing the dependence of Space Density, \(d_s\) on Temperature \(T^n\).
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| Space Density vs Temperature |
It also show \(g_T\) for the case of linear dependence of \(d_s\) on \(T\). The graph below shows \(g_T\) in the first few \(n\) and \(g=e^{-x}\).
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| gT(x) field equation |
\({ d }_{ s }(T)=\cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ ln(\cfrac { 1 }{ T_{ max }^{ 2 } } ) } ln(\cfrac { 1 }{ T^{ 2 }(x) } )+{ d }_{ n }\)
then substitute (1) in,
\({ d }_{ s }(T)=\cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ -ln({ T_{ max }^{ 2 } } ) }( ln(x )-ln(A^2))+{ d }_{ n }\)
\(\cfrac{d({d}_{s}(x))}{dx}=-\cfrac{({d}_{T_{max}}-{d}_{n})}{ ln( { T_{ max }^{ 2 } } ) }.\cfrac{1}{x}\)
And we have,
\(g_{ T }=\cfrac { { G }_{ T } }{ { x }_{ o }({ d }_{ n }-{ d }_{ T_{ max } }) } (-\cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ ln({ T_{ max }^{ 2 } }) } ).\cfrac { 1 }{ x } \)
\(g_T={{G}_{To}}\cfrac{1}{{x}}\)
where \(G_{To}\) replaces \(\cfrac{{G}_{T}}{{x}_{o}{ ln({ T_{ max }^{ 2 } }) }}\)
then substitute (1) in,
\({ d }_{ s }(T)=\cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ -ln({ T_{ max }^{ 2 } } ) }( ln(x )-ln(A^2))+{ d }_{ n }\)
\(\cfrac{d({d}_{s}(x))}{dx}=-\cfrac{({d}_{T_{max}}-{d}_{n})}{ ln( { T_{ max }^{ 2 } } ) }.\cfrac{1}{x}\)
And we have,
\(g_{ T }=\cfrac { { G }_{ T } }{ { x }_{ o }({ d }_{ n }-{ d }_{ T_{ max } }) } (-\cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ ln({ T_{ max }^{ 2 } }) } ).\cfrac { 1 }{ x } \)
\(g_T={{G}_{To}}\cfrac{1}{{x}}\)
where \(G_{To}\) replaces \(\cfrac{{G}_{T}}{{x}_{o}{ ln({ T_{ max }^{ 2 } }) }}\)
So we have, \(g_T=\cfrac{G_{To}}{\sqrt[n+2]{x}}\), for \(n=1,2,3..\) or \(g_T={{G}_{To}}\cfrac{1}{{x}}\)
(Note: The \(G_{To}\)s are to be determined respectively.)
(Note: The \(G_{To}\)s are to be determined respectively.)
Each of these suggests a unique variation of space density, \(d_s(x)\), with temperature, \(T\). Which one is the right one? The only way to determine which is correct is to measure \(g_T\) or \(g_T-g\) and obtain its variation with \(T\) experimentally.
