bgq said:
Each of the observers in the train "observes" that the clock of the other is slower. Observer in train A observes that clock of B is slower than clock of A, but observer in train B observes that clock of A is slower than that of B. There is a clear disagreement about what each observes.
There is never any disagreement about what any observers actually see, measure or observe.
bgq said:
According to my understanding of physics and sciences, we do calculations to know what could be observed even if observations are not exist. If calculations do not reveal what could be observed (or at least what being true), then there is no point to do any calculations, and all formulas of physics has nothing to do with the real world!
This quote seems at odds with the first quote. First you said there is disagreement about observations and now you say that observations should reflect the truth about what is going on in the real world. What you need to understand is that there are different ways to calculate what observers will see, measure and observe but those calculations can have completely different numbers until you get to the final result. You also need to understand that time dilation is not one of those things that observers can see. Don't confuse the observed slowing down of a moving clock with time dilation.
Consider a new situation where two observers are approaching each other from a great distance apart. They will each observe the other ones clock as running faster than their own and then after they pass each other, they will each observe the other ones clock running slower than their own. These are just normal relativistic Doppler effects, correct?
Yet, if you apply the rules of Special Relativity and select any arbitrary inertial frame to analyze the situation, the time dilation of each clock remains the same while they approach, pass, and depart from each other. Do you understand that? The time dilation doesn't change while they each see the others clock first running faster than their own and then slower.
So, for example, if we analyze this new situation from a frame in which both observers are traveling at the same speed toward each other, we will calculate that they both have the same time dilation, even though they both first see each others clock running faster than their own and then running slower than their own after they pass.
Then if we use a second frame in which one of the observers is at rest, we will calculate that his clock has no time dilation while the other clock has an even greater time dilation than in the first frame and yet they still both see each others clock running faster then slower.
And if we use a third frame in which the other observer is at rest, we will exchange all the calculations between the two observers from the second frame.
And we can pick other frames in which both travelers are moving but at different speeds and then we will calculate they each have different time dilations.
So in each of these different frames, the time dilations of the two observers is a constant during the scenario for each observer and yet they still both see the other ones clock first running faster and then running slower than their own by exactly the same ratios.
The time dilation is a calculation based on the speed of the clock in the arbitrarily selected frame and is not something that anyone can ever see or measure or observe.
