I've been reading up on the arrow of time and there is one thing that is being omitted from every explanation that I've found: time flow.
I am rewording this question to make it clearer. From the entropy definition of arrow of time $Δt=f(ΔS)$, where $S$ is entropy. But thermodynamically, $$ΔS=Q*(1/T_2-1/T_1),$$ where $T_1$ and $T_2$ are temperatures. Temperatures need to be integrated over volumes, but in equilibrium cases, which is what interests us, we can speak of the temperature at a point, $T_1(x,y,z)$ and $T_2(x,y,z)$. So we get
Obviously, function ΔS(x,y,z)!=const. Let's just take (x1,y1,z1) in vacuum and (x2,y2,z2) in the center of the Sun.
On the other hand, we observe that Δt is NOT dependent on (x,y,z) within an inertial reference frame.
That gives our function f very peculiar properties - it is actually an orthogonolizing transformation of spacetime that somehow cancels out spatial irregularities of entropy flow and creates uniform time flow out of it.
I have not found any satisfactory explanations or theories about f and its physical nature. Can someone enlighten me?