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.24.2 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.