The preceding argument was not very rigourous, but it served to show
the essential necessity for regarding the effective mass
of an object as a relative quantity.
Let's see what happens as we try to accelerate a mass to the
velocity of light: at first it picks up speed just as we have
been trained to expect by Galileo.
But as 
becomes appreciable, we begin to see an interesting
phenomenon: it gets harder to accelerate! (This is, after all,
what we mean by ``effective mass.'') As 
,
the multiplicative ``mass correction factor'' 
and eventually we can't get any more speed out of it, we just keep
pumping energy into the effective mass. This immediately suggests
a new way of looking at mass and energy, to be developed in the
following Section.
But first let's note an interesting side effect: the rate at which
a constant accelerating force produces velocity changes,
as measured from a nonmoving reference frame,
slows down by a factor 
; but the same factor
governs the time dilation of the ``speed'' of the clock
in the moving frame. So (as observed from a stationary frame)
the change in velocity per tick of the clock in the
moving frame is constant. This has no practical consequences
that I know of, but it is sort of cute.