\(g_T=D.\cfrac{d(d_s(x))}{dx}\)
And from the post "Possibilities, Possibilities, And God Created More Possibilities",
\({d}_{s}(T)=\cfrac{({d}_{T_{max}}-{d}_{n})}{T_{max}}.T+d_n\)
In general,
\({d}_{s}=\cfrac{\partial (d_s)}{\partial T}.T+d_n\)
which is basically the equation for a straight line when \(\cfrac{\partial (d_s)}{\partial T}\) is a constant.
\(\cfrac{\partial ({d}_{s})}{\partial x}=\cfrac{\partial (d_s)}{\partial T}.\cfrac{d T(x)}{d x}+\cfrac{\partial }{\partial x}\left\{\cfrac{\partial (d_s)}{\partial T}\right\}.T\)
\(\cfrac{\partial ({d}_{s})}{\partial x}=\cfrac{\partial (d_s)}{\partial T}.\cfrac{d T(x)}{d x}+\cfrac{\partial }{\partial T}\left\{\cfrac{\partial (d_s)}{\partial x}\right\}.T\)
If we assume that \(d_s\) itself does not varies in space, but its distribution is wholly due to the variations in temperature in space, ie. when
\(\cfrac{d T(x)}{d x}=0\)
\(\cfrac{\partial ({d}_{s})}{\partial x}=0\)
ie. \(d_s\) is uniform but redistribute solely due to \(T\), so
\(\cfrac{\partial }{\partial T}\left\{\cfrac{\partial (d_s)}{\partial x}\right\}.T=0\)
(Note: There can be other treatment than setting this term to zero.)
That is to say,
\(d_s(T)=d_s(T(x))=d_s(x)=d_s\)
\(\cfrac{\partial ({d}_{s}(T))}{\partial x}=\cfrac{\partial ({d}_{s}(x))}{\partial x}=\cfrac{\partial ({d}_{s})}{\partial x}=\cfrac{\partial (d_s)}{\partial T}.\cfrac{d T(x)}{d x}\)
As such,
\(g_T=D.\cfrac{\partial (d_s)}{\partial T}.\cfrac{\partial T(x)}{\partial x}\)
If we assume a linear temperature profile along a elements/metal rod of original length \(L_o\), then
\(\Delta L=L_o.\alpha\Delta T\)
where \(\alpha\) is the linear thermal expansion coefficient.
\(\cfrac{\Delta T}{\Delta L}=\cfrac{1}{L_o.\alpha}\) ie
\(\cfrac{d T(x)}{d x}=\begin{matrix} lim \\ \Delta L\rightarrow 0 \end{matrix}\cfrac{\Delta T}{\Delta L}=\cfrac{1}{L_o.\alpha}\)
\(g_T=D.\cfrac{\partial (d_s)}{\partial T}.\cfrac{1}{L_o.\alpha}\)
A force per unit volume develops,
\(\rho\sqrt{3}g_T=F_T=\sqrt{3}\rho D.\cfrac{\partial (d_s)}{\partial T}.\cfrac{1}{L_o.\alpha}\)
and \(\rho\) the density of the element. Note that \(\alpha\) is the linear coefficient not the volumetric coefficient. We are also assuming that the element expands uniformly in 3D, isotropic.
Thermal gravity then provides an explanation for the thermal expansion of elements, where for a sphere of radius 1m,
This force pulls on the element causing it to expand. Like a spring, a counter force, \(F_L\) acts against \(F_T\) due the increasing atomic distance in the lattice of the structure. The result is a sphere of greater radius, when both forces balances. \(F_T\) might be called thermal stress. The resulting expansion is of course thermal expansion.
\(\rho\) is specific to a particular element. \(D\) is the constant of proportionality in the gravity-space density formulation. \(\cfrac{\partial (d_s)}{\partial T}\) might be specific to a particular lattice structure and atom population. And \(\alpha\) is the constant of proportionality in this formula; at this point it may not be element specific. The point is to move all specificity to \(\cfrac{\partial (d_s)}{\partial T}\) or \(\rho\) instead, if possible. Note: \(\alpha\) is element specific, more accurately specific to a particular lattice structure. \(\cfrac{\partial (d_s)}{\partial T}\) is approximately that of free space as the atoms themselves are mainly free space.
