Another way to look at the time dilation equation
\({ v }_{ t }^{ 2 }={c}^{2}-{\cfrac{2{G}_{o}}{{r}_{e}}}{e}^{-\cfrac{{g}_{o}{r}_{e}}{{G}_{o}}(x)}\)
is to see that \(\cfrac{1}{2}{ v }_{ t}^{ 2 }\) is kinetic energy per unit mass. On entering a gravitational field, part of its total energy across space and time dimensions is converted to the term,
\({\cfrac{2{G}_{o}}{{r}_{e}}}{e}^{-\cfrac{{g}_{o}{r}_{e}}{{G}_{o}}(x)}\)
If we compared time dilation equation to
\(\cfrac{1}{2}{ v }_{ t }^{ 2 }+\cfrac{1}{2}{ v }_{ s}^{ 2 }=\cfrac{1}{2}{c}^{2}\)
Then,
\(\cfrac{1}{2}{ v }_{ s}^{ 2 }={\cfrac{{G}_{o}}{{r}_{e}}}{e}^{-\cfrac{{g}_{o}{r}_{e}}{{G}_{o}}(x)}\)
\(|P.E|={\cfrac{{G}_{o}}{{r}_{e}}}{e}^{-\cfrac{{g}_{o}{r}_{e}}{{G}_{o}}(x)}={{g}_{o}{r}_{e}}{e}^{-\cfrac{1}{{r}_{e}}(x)}\) ----(*)
Traditionally we called expression (*) the Gravitational Potential Energy (per unit mass). It is actually energy converted from the total energy of a body in free space to Kinetic Energy in the space dimension, as it enters a gravitational field. GPE is a 'loss' from the total energy, so has a negative value, the body however gain an equivalent amount of Kinetic Energy. The velocity so developed is in the direction of gravity, towards denser space. Loss as GPE is zero where x is far away at infinity where KE is zero also. The odd thing was, why such a conversion from GPE to KE even occurs. The answer is denser space. A gravitational field is just a region of compressed space.
