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enter image description hereThere is a person at rest and a train in motion with velocity $v$. Inside the train, there is a dog running with velocity $u$ respect to the train. The problem states the person inside the train has noticed 2 events, the dog run from one point inside the train to another point and the time Interval is $\Delta t$. The question is the time Interval measured by the observer outside. I felt tempted to simply consider the formula of time dilation. However, it was deduced by asuming the events in the moving frame were at the same place, but in this case the events have different locations in the train. What is wrong with my understanding of time dilation?

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The time dilatation tells you how are times between two different rest frames connected. So while $\Delta t$ passed for observer inside the train, the dilated time will pass for an outside observer. However, the difference will rise due to the simultaneity relativity. If the dog started to run from the observer the events dog started to run and an observer started to measure time will be simultaneous for outside observer too. However, once the dog reaches the event 2, the simultaneity of observer stopping time measurement and dog reaching the event will no longer be simultaneous for outside observer.

you must use the whole coordinate formula: $$ \Delta t´ = \gamma (\Delta t -\frac{v}{c^2}\Delta x)$$

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  • $\begingroup$ Thanks for the answer. I reached the same equation considering both of them study the dog for . However, it feels messy since the dilation formula was more attractive. $\endgroup$ – Omar May 3 at 6:06
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    $\begingroup$ Actually the dilatation formula is messier. Dilatation formula doesnt take relativity of simultaneity into an account and it leads to most of the misconceptions about STR. It is much clearer to simply set up a coordinate system in spacetime and do the lorentz transform between them and only at the end interpret the coordinates as time and space, instead of trying to work with the distinction the whole time. $\endgroup$ – Umaxo May 3 at 7:37
  • $\begingroup$ I'm having trouble about simultaneity. How could that eliminate the space interval? $\endgroup$ – Omar May 3 at 7:54
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According to time dilation, it doesn't matter where the events are happening inside the moving frame. For the observer inside the train say, it took T minutes for the dog to reach it's position, for the observer outside, it would have taken a different Ti minutes for the dog to reach it's position. Time dilation holds in this scenario

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  • $\begingroup$ Thanks! What about the more general formula including space like interval? How can I justify it is not included. The dog has moved from one place to another. $\endgroup$ – Omar May 3 at 6:12
  • $\begingroup$ You can follow @Umaxo's answer. I was about to give the same equation $\endgroup$ – Dani Akash May 3 at 6:31

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