What is the most amazing fact about relativity?
Time dilation is not reciprocal because special relativity also deals with invariants and absolutes and therefore, “it’s not all relative”
Consider the twin paradox: imagine two identical twins, one stays at home, on earth, and the other makes a journey into space. The traveling twin turns on his engines so as to accelerate. He moves with uniform velocity. He turns around (reverses acceleration) and heads his rocketship back to return home. During the turn-around period, he changes reference frames. He moves again with uniform velocity in a different reference frame. Thus the clocks of the two twins do not synchronize anymore. He turns on his engines so as to decelerate and lands on earth. He meets his earth-dwelling twin and finds he is much older than he is.
Ahha! If all motion is relative, then one could argue — and indeed respected people did argue this (e.g. most notable are Herbert Dingle’s objections) that from the point of view of the traveling twin, it is the stay-at-home twin who is making a round-trip because according to special relativity, time dilation is reciprocal. Hence, the clock of the stay-at-home twin should run slower than the clock of the traveling twin, and in the end, when the twins meet again, the traveling twin discovers that the earth-dwelling twin is younger.
But time dilation is not reciprocal. If the stay-at-home twin is older than the traveling twin, the traveling twin cannot be older than the stay-at-home twin. Gray hair is a fact of experience. We cannot switch between the twins: from all points of view, the reference frame of the stay-at-home twin is the one in which time is not dilated. In this regard, special relativity deals with invariant and so-called absolute quantities.
The above non-reciprocal nature of time dilation is manifested in the way we mathematically solve the twin paradox using special relativity formulas.
Using the formulas of special relativity, we solve the problem once in the reference frame of the stay-at-home twin, and then we solve it again in the reference frame of the traveling twin.
Although the values the two twins put in the time dilation formula are quite different because each twin measures distances and times in a different way in his reference frame, we nonetheless obtain the same final result in both cases: the stay-at-home twin is older than the traveling twin when they both meet again.
My answer to the questions in the comments: The newest important feature of time to emerge from the special theory of relativity is the clock (twin) paradox: a comparison of one clock in a moving system with many clocks in the rest system; there is no reciprocity, but one-many relationship.
According to Newtonian kinematics, it was not so important to distinguish between the local time, the time as measured by a clock transported along some path, and the global time, stipulated to be equal to the absolute time: they always agreed.
Unlike pre-relativistic kinematics, in the Minkowski formalism of special relativity, the proper time between two events, as measured for example by an ideal clock traveling between the two events, depends on the history of the clock, i.e. on its path through spacetime. This is the essence of the twin paradox (this is an explanation given by Prof. John Stachel).
Credit: Gali Weinstein