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This is my first year in Physics and we have been shown about S.R very superficially, just having a bunch of equations to apply. However, I feel rather confused about results such as time dilation, where proper time measured by the "moving" observer is lower that the one measured by the "stationary" one, yet Lorentz equations seem to seesay otherwise (t is multiplied by gamma, not the other way round).

I´ve seen a video that seems to provide some clues but I would need to fully know what x,t and x', t' mean. Correct me if I'm wrong. Everything happens in spacetime, and everyone has their own reference frame (R.F), which can be seen as a normal 3D space full of clocks everywhere. However, I'm not sure if both observers would see the 'same' clocks. Of course the time they measured is different, but I mean the 'physical' clock. For example, imagine the clock centered at the "stationary" frame of reference (the one using x,t) is yellow and the rest are black in his frame. Would the "moving" person see that difference in color too?

This might seem weird, but it's related with my question about what is t' exactly. Namely, does it mean the time that the "moving" observer would see in the same clock located at point x? Returning to the color difference, if the frame S measured time t at the yellow clock at x, is t' the time that appears on the yellow clock in S' frame? Also, I have the same problem regarding the meaning of x' and l'.

There seems to be some ambiguity regarding what an observer is: sometimes it looks like if it were a person and sometimes is the R.F. I think there would be a difference here because the person would have some time delays to record events (just as it happens in astronomy) whereas a R.F is something theoretical that could make any measurement anywhere in the universe. Which of these two is the one discussed in these transformations?

Besides the answers I may get, I definitely need a good textbook that explains and tries to give some intuition regarding S.R, especially considering the core concepts are usually overlooked. I got the one recommended by my University, but still has the same problem. In fact, many of the answers in this page are better explanations than my textbooks. Any suggestions would be much appreciated.

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$S'$ has different clocks than $S$. $S'$ has a lattice of stationary, synchronized clocks for doing thought experiments. Meanwhile, $S$'s clocks are all moving and out-of-synch with each other, according to $S$.

Well, he does have some use for $S$'s clocks. If he looks at one (black, or yellow), he sees it at $(t, x)$, then if he computes the Lorentz transform:

$$ t' = \gamma(t -\frac{vx}{c^2}) $$

it will agree with the clock reading.

When doing thought experiments, we are not concerned with light propagation and delay and what a single observer at the origin sees; rather, a frame has a lattice of rulers and clocks everywhere and every when, with which we can reconstruct all events' space and time coordinates, as needed.

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The moving reference frame (MRF) would perceive a color difference in the stationary clock depending on whether the stationary clock is travelling towards or away from the MRF due to light frequencies being either blueshifted or redshifted respectively (see the Doppler effect for more information).

Regarding any confusion about time dilation and the Lorentz factor, remember that the Lorentz factor γ is equivalent to the fraction:

$$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

where the value of the radicand is inversely proportional to your given velocity v and, hence, v is proportional to the total value of the fraction.

Given the time dilation equation for proper time: $$c\tau=\frac{ct}{\gamma}=ct\sqrt{1-\frac{v^2}{c^2}}$$ the value of γ must always be at least 1 so that dividing stationary time ct by γ will yield a lower number. Visually the MRF would see the stationary clock tick faster and the stationary observer would see MRF's clock tick slower.

One good textbook on both special and general relativity is A First Course In General Relativity - Bernard Schutz

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Well, I managed to figure it out. The key was to consider the conditions about 'same time', 'same position', depending on who was measuring the proper length or time.

This video was enlightening: https://www.youtube.com/watch?v=svwWKi9sSAA&t=491s

A very recommendable channel as well just to get a small introduction. (It's meant for laypersons but it's great anyways)

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