Time dialtion when light travels a shorter distance compared to the observer

Question

I am just starting to learn these subjects on the internet and I don't get it.

Usually, they will tell you that time just slows down when you are approaching the speed of light. They would usually rationalize it using all sorts of thought experiments, but to me, they appear logically flawed.

I usually see 2 main thought experiments. One is about 2 mirrors moving close to the speed of light so that the distance which light travels is longer. Now, because speed of light is constant, I, as an observer, would conclude that time slows down because we have a bigger $x$ but same $c$, so we must have a bigger $t$ to compensate.

Here is where my confusion starts. Why did you choose an experiment where light travels a longer distance? I can pose the same experiment differently, where the 2 mirrors are heading towards each other, so that the light on the moving mirrors would travel a shorter distance than my own resting mirrors. Wouldn't this prove just the opposite? Here, we have the same speed $c$ but shorter $x$ for the moving objects. Wouldn't I, as an observer, conclude that $t$ must be smaller?

Another experiment which I often see is about someone chasing light. What if I considered a different thought experiment where I weren't chasing light but instead running away from it?

The problem I am seeing in all of these thought experiments is that they arbitrarily choose one direction to prove that time slows down when approaching the speed of light. But how does that solve the 2 phenomena when you are heading towards it vs running away? Wouldn't "time slow down" solve it only in one scenario but not in the other?

Just as I show with the experiment of mirrors I could create a scenario which would prove that time actually goes faster. Not more slowly.

All you have to do is play with the directions. Tell me what I am missing?

Answer 1

It is impossible to be sure what you have in mind, since you have not described your thought experiments with sufficient clarity. However, this might help you...

Imagine that two people, Como and Zaquette, stand together and flash a light which heads off to the right and left. After a second, the light will be 300,000,000m away from them in each direction.

Now repeat the experiment, but this time Zaquette walks after the light to the right at about 1m/s. After 1 second from Como's perspective, the light is 300,000,000m away in each direction. From Zaquette's perspective, however, the right hand beam has travelled about 299,999,999m, while the left hand beam is about 300,000,001m away. Since the speed of light is constant for Zaquette, she has to conclude that the right hand beam has travelled for slightly less than a second, while the left hand beam has travelled slightly more.

So, indeed, Zaquette's view of time in Como's frame depends on direction- in her forward direction of travel, time in Como's frame seems to have run ahead of time in hers, whereas in the reverse direction time in Como's frame seems to be running behind hers.

The effect I have described is known as the relativity of simultaneity. From it, all the other effects, such as time dilation, length contraction, the resolution to the twin paradox etc, can be explained. Unfortunately, it is an aspect of relativity that is often overlooked or misunderstood by people new to the subject, which leads to all sorts of confusion.

Note that had Zaquette walked to the left instead of the right, the effect would have been reversed in direction- time in Como's frame to the left would appear to her to be running ahead of time in Como's frame to the right. You should be able to see now why a moving clock always runs slow, because regardless of which direction Zaquette walks in, time in the stationary frame always seems to be moving ahead of her time in the direction she is headed.

The effect is symmetrical, since after a second in Zaquette's frame the light will be 300,000,000m from her in each direction, whereas Como will see that light to be 299,999,999m from him in one direction and 300,000,001m away from him in the other, so he will think Zaquette's clocks are out of synch just as she considers his to be.

The choice you mentioned of light clocks moving in a direction parallel to their length is one that is taken by many text books because it is an easy example to explain the time dilation effect, but you can take any scenario you like, and provided you think it through properly you will always arrive at the same results- if you do find yourself arriving at a result that appears at odds with SR then you will have made a mistake somewhere in your reasoning.

Answer 2

Einstein purposefully desingned the Einstein light clock in a certain way. A sensible way.

You purposefully designed the your light clock in a different way. Not very sensible way.

Both light clock are slower when they move.

Answer 3

Thought experiments can only get you so far. Time dilation is proved by real experiments, that are performed every day e.g. in the global positioning system, and at the LHC (large hadron collider). The world really does work this way.

The thought experiments are designed to be illustrative or to give you a sense of why the world works some way. If you're confused by some particular thought experiment, then move on and look at other ones. There are many good online resources about relativity.

For your specific question: if you're looking at a light clock that's aligned with the direction of motion (instead of perpendicular to it) then you have to take length contraction into account as well as time dilation. It makes things much more complicated, which is why the perpendicular clock is the one usually presented in explanations.

Edited to add: as @MarcoOcram points out below, the relativity of simultaneity also must be taken into account. Not everyone will agree that two events in different places are "at the same time".