Same Old, Same Old

We have seen this before, with conservation of energy where $c^2$ is a constant,

${v}^{2}_{t}+{v}^{2}_{s} = {c}^{2}$

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.