Possibilities, Possibilities, And God Created More Possibilities

$T(x)=\cfrac{A}{\sqrt{x}}$ --- (1)

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  $T$.

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  $T^n$.

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  $g=e^{-x}$.

gT(x) field equation

To obtain the answer calculated previously where  $g_T=\cfrac{A}{x}$.  We have to consider,

${ d }_{ s }(T)=\cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ ln(\cfrac { 1 }{ T_{ max }^{ 2 } } ) } ln(\cfrac { 1 }{ T^{ 2 }(x) } )+{ d }_{ n }$

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 $\cfrac{{G}_{T}}{{x}_{o}{ ln({ T_{ max }^{ 2 } }) }}$

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.)

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.