\({v}^{2}_{t}+{v}^{2}_{s} = {c}^{2}\)
differentiating both sides with respect to time,
\(2{v}_{t}\cfrac{d{v}_{t}}{dt}+2{v}_{s}\cfrac{d{v}_{s}}{dt} = 0\)
\({v}_{t}\cfrac{d{v}_{t}}{dx}\cfrac{dx}{dt}+{v}_{s}\cfrac{d{v}_{s}}{dt} = 0\)
\({v}_{t}\cfrac{d{v}_{t}}{dx}{v}_{s}+{v}_{s}\cfrac{d{v}_{s}}{dt} = 0\)
\({v}_{t}\cfrac{d{v}_{t}}{dx} = - \cfrac{d{v}_{s}}{dt}=-g\) --- (*)
And
\(E_t=mv_t^2\)
(*) becomes,
\(\cfrac{1}{2}\cfrac{d{v}^2_{t}}{dx}=\cfrac{1}{2m}\cfrac{d\,E_t}{dx} = -g\)
\(-\cfrac{d\,E_t}{dx}=2mg\)
where the minus sign indicates a decreasing time potential field \(E_t\) with \(x\) in the direction of \(g\).
In a gravity well, around a planet or black hole, we also has a time potential well. This time potential is at its minimum when gravity is at its maximum.
Good day.
