First off: There is a standard measurement of Time that does not rely on the Earth.
It is determined as 9,192,631,770 oscillations of specified transition in Cs-133 atom.
This is one second. Now the term second is a throw back to earth time. But the standard
itself is not. There is also the Natural unit of time which is the smallest time interval
thought to exist which is the Planck Time Tp = SQRT(hG/c^5). (Check out
Wikipedia).
It is in extreme gravity situations that Time itself is warped (remember Time is a part of
Space-Time and gravity affects Space-Time). The higher the gravity field the more time
is warped. Time relatively slows down when in higher gravity fields. (Lets not forget that
gravity itself is really a factor of how mass curves space-time.)
This is the same factor that accounts for extreme velocities causing time dilation.
Since the lowest measurement of time accepted at this day and age is the Planck time
there is no real definition of instantaneous. There is an ambiguity detailed in that.
That could be defined as the meaning of instantaneous but it is relative.
One's mass does not increase exponentially as one reaches the speed of light it is
through the
Lorentz Transformation that you can calculate relativistic mass. (Although Energy is probably the better usage here, along with momentum.)
When an object is levitated through magnetic levitation the mass of the item is still
the same as before the magnetic force was applied. Gravity being the weakest of the
4 fundamental forces and can easily be overcome by any of the others at small scales.
(Small scales here being less than galactic or when not near huge mass sources).
Basically I think the original idea introduced by the OP is one that is not practical.
I beleive most of
Gatt's argument is correct for why it would not work.
If you could get the ship to safely go relativistic speeds then you would have something.
The only problem is it is one way "time travel". (Unless you want to start talking about
time variant wormholes.)