Notice that time dilation \(\gamma \) is independent of \({ v }_{ s }\), this means a body experiences time dilation because of a change in gravity only, irrespective of its space velocity. However, as a body increased speed, the relative space density in the direction of its travel increases. The situation is analogous to a body at rest counting a number of smaller bodies around it per unit volume. If at speed v the body counts A then at speed 2v it will count 2A. The relative space density increases proportionally with its space velocity.
Consider the expression
\({ v }_{ s }tA{ d }_{ s }\), for time t, unit area A, where \({ v }_{ s }\) is space velocity and \({ d }_{ s }\) space density.
at speed \(B{ v }_{ s }\) we have \(B{ v }_{ s }tA{ d }_{ s }\) which is \({ v }_{ s }tA(B{ d }_{ s })\) which suggest a relative increase in space density
This leads us to,
\(\cfrac { { v }_{ s1 } }{ { v }_{ s2 } } =\cfrac { { d }_{ s1 } }{ { d }_{ s2 } }\)
but
\({ v }_{ t }^{ 2 }\propto \cfrac { 1 }{ { d }_{ s } }\)
as such
\(\cfrac { { v }_{ t1 }^{ 2 } }{ { v }_{ t2 }^{ 2 } } =\cfrac { { d }_{ s2 } }{ { d }_{ s1 } } =\cfrac { { v }_{ s2 } }{ { v }_{ s1 } }\)
which gives us
\({ v }_{ t1 }={ v }_{ t2 }\sqrt { \cfrac { { v }_{ s2 } }{ { v }_{ s1 } } }\)
This result is without any experimental proof! And is actually running away. In reality space does not pile up indefinitely in front of the moving body. Space like a light fluid, flows around the body as the body moves through it. However, an initial increase in space velocity causes time velocity to decrease resulting in a further increase in space velocity (via, \( { v }_{ t }^{ 2 }+{ v }_{ s }^{ 2 }={ c }^{ 2 }\) ) . An initial kick in the rear, nothing more.
