Given that time and space dimensions are orthogonal and that conservation of energy applies. We have
\(\nwarrow \quad +\quad \nearrow \quad =\quad \upuparrows \)
\( { v }_{ t }^{ 2 }+{ v }_{ s }^{ 2 }={ c }^{ 2 }\)
where \({ v }_{ t }\) is time velocity and \({ v }_{s }\) is velocity in the space dimension and c is the normal time velocity.
Differentiating with respect to time
\(2{ v }_{ t }.\cfrac { d{ v }_{ t } }{ d{ t } } +2{ v }_{ s }.\cfrac { d{ v }_{ s } }{ dt } =0\)
\(g=\cfrac { d{ v }_{ s } }{ dt } =-\cfrac { { v }_{ t } }{ { v }_{ s } } \cfrac { d{ v }_{ t } }{ dt } \)
since \({ v }_{ s }=\cfrac { dx }{ dt }\),
\(g=-{ v }_{ t }.\cfrac { d{ v }_{ t } }{ dx }\)
\(g=-\cfrac { 1 }{ 2 } .\cfrac { d{ v }_{ t }^{ 2 } }{ dx } but E=m{ v }_{ t }^{ 2 }\)
As such we have
\(g=-\cfrac { 1 }{ 2m } .\cfrac { dE }{ dx }\)
We see here that gravity g, is the result of half a change in a conservative field E over a distance x per unit mass. The negative sign implies g is in the direction of decreasing E.
This derivation is independent of any other mass. A change in field E can be the result of a chane in space density (as such a corresponding change in c, the speed of time) as when one pushes more space towards one side or where a massive body that compresses space around it. Space around a body relaxes to normal space as one move way from the massive body. As space is more compressed, c is smaller, and the direction of decreasing E is towards the surface of the massive body. This is consistent with our understanding that black holes are very dense and yet attracts every mass in its vicinity; as oppose to a void of no space. It is also consistent with our understanding of gravity around planets. What is new here, is space has mass, and that its density changes. Space can be compressed, expanded and made to flow.
