Hak said:
Thank you. So it is related to the principle of relativity, which states that all laws of physics take the same form in every inertial frame of reference, and that consequently there is no such thing as absolute motion, no one moves and no one stands still intrinsically; so this fact implies that there can be no discrepancy between time measured with a light clock and time measured with an atomic clock, otherwise you could tell whether the system is intrinsically in motion, which goes against the principle of relativity? Now, the consequence of that would be that if one observer sees another in motion relative to him, that observer's time appears slowed down, and the thing is reciprocal, right? This is an anti-intuitive concept, as is the principle of constancy of the speed of light though, or is it?
It is helpful here to disentangle the notion of time into proper time, and coordinate time. Proper time is what a clock measures. Coordinate time is a convention that assigns time labels to events.
Proper time is always an interval, and is what a clock actually measures. A clock travels some path through space, and it keeps time, and clocks can be compared to other clocks when they are at the same location in space at the same time.
In the language usually used by textbooks, we talk about "world lines" or "time-like worldlines" that represent the motion of an object. Time-like worldlines represent physical objects that always move slower than the speed of light and can have a clock attached or associated with it. This clock essentially measures the "length" of the worldline. You might see a reference to "space-like worldlines". They're not needed for this discussion, except to say that they aren't relevant to clocks, rather they are relevant to rulers. Since we won't need them, I won't discuss them further.
With this view, the twin paradox is no more inconsistent than to say that if you drive in space a straight line from one city to another, with an odometer attached to y our car, your trip as measured by the odometer which measures the distance your car moves is shorter when you drive in a straight line than when you take a detour. Notably, though, there is an important sign difference in this simple analogy. In space, the path of straight-line motion minimizes the distance. In space-time, the path of "straight-line" motion, which is the natural inertial motion of a body with no forces acting on it, give the longest proper time, not the shortest. But aside from this key difference, the ideas are the same. There's no prior reason to expect that clocks that follow different paths and meet up again will read the same time when they re-unite wihtout additional assumptions, just as there is no reason to think that the odometer readings for two cars taking different paths through space will remain the same.
Coordinate time is typically invoked by creating an unlimited number of clocks, all arranged in some particular fashion in space. And, importantly, in order to define a coordinate system, these clocks must be synchronized. And this is where the fun starts.
In special relativity, different observers have different notions of synchronization, depending on their state of motion. Fully exploring the consequences of this turns out to be trickier than one imagines, and typically one does not imagine it's easy. In fact, I think much of resistance to this concept comes from the nagging feeling that one hasn't figured out all the logical consequences it entails. The only remark I can say is that while this is true, someone just introduced to relativity is unlikely to imagine all the logical consequence, and furthermore that one is unlikely to find all of the consequences written down in one place to be studied, the relativity of simultaneity is nonethelss a keystone result of special relativity.
Some of the more obvious issues arise with our notion of cause-and-effect. I won't go into them at this point, I'll just point out that it's one of many consequences, and it takes a considerable amount of work to understand how causality works in special relativity. I'll mention that this is typically done via "light cones", but I won't go into any more details than that. This is in contrast to the classical idea that "the past" (a singular agreed upon notion by everyone) can cause "the future". In special relativity the idea of "the past" as a singular agreed upon entity gets replaced with the "past light cone", which is unique to each observer.
So - to recap. The twin paradox isn't that hard to explain, (in my honest opinon) once one understand that simultaneity is relative. The hard part is understanding (again in my honest opinion) is understanding the relativity of simulttaneity and it's consequences.