Gravitational potential and escape velocity are of course highly related. In General Relativity, at least for the simple case with one large mass such as the Earth, it seems that the time dilation due to the gravitational potential equals the time dilation that would result from movement at escape velocity.

escape velocity =

[1] for a Lorentz factor of

or

From the Schwarzchild metric for a massive symmetric sphere, not rotating,

proper time is coordinate time times

[2]
Is that equality obvious, or an obvious result of the kinetic energy to escape equaling the gravitational potential energy, or a result of the Equivalence Principal, or is it something deeper?