A big confusion regarding to the 2 thought experiments of Relativity of Simultaneity leading to other confusions

Question

I will just present my arguments because the correct arguments would be known facts.

So in the experiment where a flash of light is given off at the center of a moving train just as the 2 observers pass each other ( one observer is standing at the midpoint of the train and the other observer on the platform), the platform observer sees the rear end of the train moving towards the flash of light and front end moving away from the speed of light. As the speed of light is constant and light has to move considerably a less distance to meet the rear end , the ground observer says that the flash of light reaches the rear end first and then reaches the front end. But the train observer is at rest with respect to train and the distance between the front end and him and rear end and him is the same. Also the speed of light is the same in his frame and hence he says the light flashes reach the rear and the front end at the exactly the same time ( this point is crucial).

Now taking the other experiment in which the flashes of light hit the ends of the train and the platform just as the 2 observers (1 at the middle of the train and the 2nd at the centre of the platform). The platform observer sees the light flashes at the same time because the flashes travel at the same speed the same distance (Important point in this ground frame discussion , as I think is that the ground observer is at rest w.r.t to the ground and hence sees the flashes at the same time). But since the train observer is moving towards the front end and away from the rear end , he sees the light flash from the front end first (this is the argument followed). But in the frame of of the train , the train observer is at rest (just as the ground observer is at rest w.r.t to the ground and the train observer was at rest w.r.t to the train in previous example). So since in the frame of the train the train observer is at rest and the distance between the front end and him and rear end and him is the same and the speed of light is the same in his frame , he should say that the flashes of light reach him at the same time (just as he had said in the previous example ).

He should say that the flashes reach him at the same time (case is exactly similar to the previous case of train observer. In both the cases he is at rest w.r.t ti the train ).

So, as far as I think:

1)The ground observer should say -"According to me I see the flashes at the same time but the train observer should see the flashes at different times." 2)And the train observer should say that " I see the flashes at the same time but the ground observer shouldn't. "

So the ground observer should say that the flashes are seen by the train observer at different times but the train observer says that he sees the flashes at the same time.

So why and where I am wrong ? I must be missing some ridiculously simple fact ? I am sorry if it's too ridiculous a question but I am missing some simple fact and unable to find it.

Also following from the 2nd experiment and lines below it in bold (my conclusions) , I say that that if the ground observer measures time on his clock to be Ta and predicts that time on train observer's clock to be $T_{ab}$, then he would say $T_{ab}>T_a$ (This should be time dilation. $T_{ab}$ is precisely calculated using Lorentz transformations). But the train observer would measure time in his frame to be $T_b$ and this $T_b$ should be equal to $T_a$.

So time dilation (similar arguments fir length contraction) should be that the ground observer should say " according to me a certain amount of time has passed in my frame but according to me the time passed in the train observer's frame should be more than that passed in my own frame (for the same 2 events that occurred in my and in his frame). But the train observer would maintain to say that i experienced exactly the same time in my frame as is experienced by the ground frame observer in his own frame , but it is the ground observer that according to me has measured a greater time.

In a nutshell I am saying that both the observer's experience the same time in their own frames (no time dilation in their own frame w.r.t to they themselves) but according to them the other one should experience a longer time interval ( time dilation ).

Is this reasoning wrong and if so where ?

Forgive me if my arguments are ridiculously silly because I am a beginner to Relativity !!

Answer 1

Lets start with the ground observer. Suppose that in his reference frame, the two flashes happen simultaneously. The light beams approach him from both sides with the same speed, so he sees the flashes happen at the same time.

But the train is moving, and so the ground observer sees that the first beam reaches the person in the middle of the wagon before the second.

Now consider the train observer. Because we have already presupposed that the two flashes happen simultaneously at the ground reference frame, by the relativity of simultaneity they happen one by one in the train frame. So despite the fact that the light still approaches the observer with the same speed from both sides, the train observer does see them consequently simply because they do happen consequently, and the ground observer is not mistaken.

The same thing happens if we suppose that both flashes happen simultaneously in the train reference system, but in this case you have to exchange the two observers.

Thus Special Relativity triumphs once again to survive another day until another young enthusiast questions its self-consistency :)

Answer 2

The key to understanding time dilation lies in the idea that acceleration/deceleration is not relative, while velocity/momentum is relative. I didn't read your whole post but i can almost guarantee this is the confusion you're having.