Einstein explained the special relativity (roughly) like this: He first assumed the speed of light would not change regardless the speed of its source. Then he described a person on a moving train was tossing a ball up and down. Because the speed of light would not change, the time was slower than the time on the stationary ground. But speed is relevant to its reference. If the train was the reference, then the the ground was moving. Based on the relativity, the time on the ground should be getting slower than the time on the train. The paradox: how could the both times were getting slower than each other?
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First thing first... no one gets younger! Someone has shared the Einstein's paper, read through it to get the best understanding and you will see that it comes from Math and fundamental principles of Physics.
Now, I will try to clarify the theory a bit.
The dude on the train is moving with the train.In order to keep the speed of light the same, the time has to slow down. This does not mean that he is getting younger, it means he will get old slower. Massive difference.
The dude on the ground will look like he is moving but he is not moving (massive difference) and therefore time will not slow down for him and he will get old as normal.
The frame of reference does not change the reality. If both of them are moving really really fast with the same velocity. Then they will both observe each other to be stationary. But time will still slow down for both of them because they are both moving.
Reality and observation are two different things. If Earth was moving a bit faster around the sun, we would be aging a bit slower. The only way we can observe that we are moving is by looking at other stars and planets. Imagine a scenario, if instead of the Sun, we were revolving around a black hole (really really fast) and there were no stars or planet that we could see. We wouldn't be able to observe that we are moving but time would have still slowed down for us then the rest of the Universe.
Fun fact: Time dilation is a real thing. Our GPS uses that, otherwise we would be miles away from the observed location if we don't factor that in.
In first place: the game is not so simple. Einstein developed both Special da General Relativity in a mathematical way, not only based on conceptual arguments. If it interests you, his original paper in Special Relativity is On The Electrodynamics of Moving Bodies
Now, back to your question. The key to understanding it is accepting that time is a relative concept. When you are talking to someone face to face, you are at his/her front and he/she is at your front. Both of you are in fornt of one another, no problem with this.
Now, if you both take a step to the right, the other one will be seen at left. So you are at his/her left, he/she is at your left. Both interpretations are true and depend on the frame of reference you are considering.
Time works the same way. In one frame of reference, person one is younger than person two. In the other frame of reference, person two is younger than person one. There is no problem with this. When they both reach the same velocity and stand next to each other, general relativity will already have explained the age of each one and both will agree about their age (because they will be in the same frame of reference).
And an important remark: no one is getting younger. Time itself runs in a different speed for each one (like, everything in motion happens at a different speed relative to the observer's frame of reference), so that their biology also runs in a different speed. Therefore, the two people age at different rates. It would be more appropriate to say that one of them is getting old faster than the other (in a certain frame of reference).
There was an invalid argument in Einstein's 1905 paper:
Albert Einstein, On the Electrodynamics of Moving Bodies, 1905: "From this there ensues the following peculiar consequence. If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by tv^2/2c^2 (up to magnitudes of fourth and higher order), t being the time occupied in the journey from A to B." http://www.fourmilab.ch/etexts/einstein/specrel/www/
"the clock moved from A to B lags behind the other which has remained at B"
does not follow from Einstein's 1905 postulates - the argument is INVALID. This means that the conclusion is unacceptable, no matter whether the postulates are true or false.
The following two conclusions, in contrast, VALIDLY follow from the postulates (they will be true if the postulates are true):
Conclusion 1: The clock moved from A to B lags behind the other which has remained at B, as judged from the stationary system.
Conclusion 2: The clock which has remained at B lags behind the clock moved from A to B, as judged from the moving system.
Conclusions 1 and 2 (symmetrical time dilation) in their combination give no prediction for the readings of the two clocks as they meet at B. In contrast, the INVALIDLY deduced conclusion provides a straightforward prediction - the moving clock is SLOW, the stationary one is FAST (asymmetrical time dilation). The famous "travel into the future" is a direct implication - the slowness of the moving clock means that its (moving) owner can remain virtually unchanged while sixty million years are passing for the stationary system:
Thibault Damour: "The paradigm of the special relativistic upheaval of the usual concept of time is the twin paradox. Let us emphasize that this striking example of time dilation proves that time travel (towards the future) is possible. As a gedanken experiment (if we neglect practicalities such as the technology needed for reaching velocities comparable to the velocity of light, the cost of the fuel and the capacity of the traveller to sustain high accelerations), it shows that a sentient being can jump, "within a minute" (of his experienced time) arbitrarily far in the future, say sixty million years ahead, and see, and be part of, what (will) happen then on Earth. This is a clear way of realizing that the future "already exists" (as we can experience it "in a minute")." http://www.bourbaphy.fr/damourtemps.pdf