But let's look at the boundary at \(x=L\). \(T\) is being accelerated towards this boundary, at the boundary itself, \(T=0\) but its velocity is not zero. If this suggests that materials are lost from the hot zone, they are not taking heat from it, because \(T=0\) and \(T\ge0\). So, at the boundary,
\(\cfrac{dT_L}{dt}=0\)
But when happens to all the power from the Sun (\(\cfrac{dT_S}{dt}\) at the Sun is finite)?
\(g_T\) can apply in two possible ways.
Firstly, \(g_T\) applies on \(T\) only, this is to suggest that \(T\) has an existence apart from \(m\), and that \(m\), the inertia along the gravity time dimension, is affected by \(g_T\) only after acquiring \(T\). In this scenario, a body dropped into a hot zone will not be accelerated outwards immediately. The body will reach thermal equilibrium and then be accelerated outwards. During the time the body is heating up it will drop further into the hot spot under \(g\). As it heats up \(g_T\) increases and it falls with less acceleration \(g-g_T\). There will be a point where \(g-g_T\) is zero. At thermal equilibrium with its surroundings, it will experience a net gravity outwards. So the body initially drops further towards the hot spot, stop and is then repelled away from the hot spot. It initially heats up but on its path away from the hot spot it is cooling down as it gains velocity; the temperature profile decreases with increasing distance from \(x\). Thermal energy is converted to kinetic energy in this thermal gravitational field.
Even after the temperature profile has been established \(g_T\) does not disappear. Otherwise, planetary rings and spheres are just transients. The temperature profile stop growing beyond \(x=L\) because
\(\cfrac{dT}{dt}\) is finite. Power is finite.
Secondly, it is possible that \(g_T\) affects \(m\) directly as long as it is in the hot zone where \(g_T\) has already been established. \(m\) will experience an net gravity due to thermal \(g_T\) and normal gravity \(g\) irrespective of its temperature. A body drop into a hot zone around a hot spot will be accelerated out immediately at the same time acquiring heat to establish thermal equilibrium with its changing ambient. Eventually, the body will start losing temperature as it gains speed under the effect of \(g_T- g\), at distances further and further from the hot spot where the ambient temperature is lower.
The difference is, in the first case each body has a personal space around it effected only by its \(T\), in the second case a third medium exist between bodies shared by all bodies, changes in this medium by any body is felt by all sharing the medium. It is more likely that space exist as a third medium effected by all and shared by all.
In both scenario, material are driven away from the hot zone. The sun is losing mass. \(g_T\) is likely to be the origin of solar wind.
In both scenario, the hot body under \(g_T-g\) gains speed but loses temperature (given time for the body to establish thermal equilibrium with its ambient) inside the thermal gravitational field.
\(\cfrac{1}{2}mv^2\rightleftharpoons T\)
Which is consistent with \(T\) being defined as an energy term,
\(T_c=\cfrac{1}{2}qv^2_{rc}\) and \(T_g=\cfrac{1}{2}mv^2_{rg}\)
in the post "Hot Topic, Mumble and Jumble..."
More importantly,
\(\cfrac{dT_S}{dt}\) at the Sun = rate of loss of kinetic energy at the boundary \(x=L\).
That's why the temperature profile around the Sun does not grow further, and its power (\(\frac{dT}{dt}\)) is finite. Power is loss from the hot zone in the form of kinetic energy. The power from the Sun equals the loss of mass with K.E pass \(x=L\). Something is wrong here.
Have a nice day.
