No Time, No Light, No Escape

Consider the time speed equation,

${ v }_{ t }= \sqrt {{ c }^{ 2 }-\cfrac { 2{ G }_{ o } }{ (x+r_{ e }) }  }$

Can space be so dense that at $x=0$ that time speed is zero.  That time itself stopped.  Theoretically, that can happen when

${ c }^{ 2 }=\cfrac { 2{ G }_{ o } }{ (r_{ e }) }$

that is when ${ r }_{ eo }$ is given by the expression

${ r }_{ eo }=\cfrac { 2{ G }_{ o } }{ { c }^{ 2 } }$        (*)

But first we have to calculate ${ G }_{ o }$.  Using earth as an example, where $g$ = 9.80665 ms^-2 on the surface of earth, and ${ r }_{ e }$ = 6371 km, from previously

$g=-{ G }_{ o }.\cfrac { 1 }{ { (x+{ r }_{ e }) }^{ 2 } }$, $x=0$

We have,

$|g|=\cfrac { { G }_{ o } }{ { r }_{ e }^{ 2 } }$

${ G }_{ o }=|g|.{ { r }_{ e }^{ 2 } }$

${ G }_{ o }$ = 9.80665*(6.371e6 )^2 = 3.980484e14

With $c$ = 299792458 ms^-2, from equation * when

${ r }_{ eo }$ = 0.00886 m time speed ${ v }_{ t }$ is zero.

This means that if earth is to be squeezed to a size of 8.86 mm, time would stand still on its surface.