There is time speed at \(v_t=c\) from the perspective of an individual not effected by any field (gravitational, charge nor temperature).
There is time speed at \(v_t=0\) in a black hole due to immense field (gravitational, charge or temperature).
And there is time as experienced \(\cfrac{\partial}{\partial t}\) due to a time field as the result of passing wave particles at light speed with an oscillating orthogonal time component (time particles) or the presence of a BIG particle and in its surrounding field (eg. Earth and its gravitational field).
What is the difference between \(\cfrac{\partial}{\partial t}\) and \(v_t\)?
If these two can be independent, then it is possible that we don't experience time (ie no \(\cfrac{\partial}{\partial t}\) operator) when \(v_t\ne 0\).
If these two are not equivalent, then it is possible that even when \(v_t=0\) that the operator \(\cfrac{\partial}{\partial t}\) does not blow up to infinity, as the time duration \(t\) when \(v_t=0\) is also zero.
\(v_t\) is time speed measured from an outside reference. It is necessarily not zero (\(v_t\ne 0\)) when there is a passage of time of finite duration \(t\). \(\cfrac{\partial}{\partial t}\) is the change with time as we experience it ("inside reference").
Does the difference and non-difference matter?
