Dealing with Time Speed and a Fixed Reference

The problem in dealing with time is that all time measurement must be based on a standard second.  At different time speed however, the measured time duration is different over 1 second.

At time speed v which is less that normal time speed c,

$v.1 = { t }_{ v }$

At normal time speed, c

$c.1 = { t }_{ c }$

Both v and c are velocities defined as 

${ v, c } = \cfrac { d{ x } }{ d{ t }_{ c } }$

which are then velocities defined at time speed c.  And so we have this relationship

$\cfrac { c }{ v } =\cfrac { { t }_{ c } }{ { t }_{ v } } \quad ,\quad \cfrac { c }{ v } .\cfrac { 1 }{ d{ t }_{ c } } =\cfrac { 1 }{ d{ t }_{ v } }$


Consider, the work done equation as applied to a mass m, along the time dimension.

$E=\int { { F }_{ v }dx }$  and that force is the rate of change of momentum

${ F }_{ v }=\cfrac { d{ P }_{ v } }{ d{ t }_{ v } } =\cfrac { mdv }{ d{ t }_{ v } }$

Notice the expression is $d{ t }_{ v }$, this is to say a person will still observe the same fact that, force is the rate of of change momentum no matter what time speed he is travelling at.  In order to derive a consistent expression however, we will need to change $d{ t }_{ v }$ to $d{ t }_{ c }$, as we observe the universe travelling at time speed c.

Till the next post.