[hamdani yusuf]

Consider this scenario:
The left picture is a light clock inside a space lab which we consider stationary. The right picture shows the same clock when observed by a space ship moving to the left at velocity v.
https://en.wikipedia.org/wiki/Time_dilation#Simple_inference
If another space ship is also observing the light clock at the same speed, but the direction is up, will they agree on the amount of time dilation?

[Halc]

If another space ship is also observing the light clock at the same speed, but the direction is up, will they agree on the amount of time dilation?

Under SR conditions, dilation is a function of speed, so the direction makes no difference. It also makes no difference if it is a light clock, any other clock, or the rate of the paint peeling.

[hamdani yusuf]

let's say that the stationary observer measures that light takes 1 microsecond to travel from bottom mirror to top mirror. It also takes 1 microsecond to travel from top to bottom mirror. So the clock has 2 microseconds period.
How will the events appear to the observer moving horizontally to the light clock? Is it the same for the observer moving vertically?
If the space between the mirrors is filled with transparent medium which makes light travel at half of its speed in vacuum, what would be the difference?

[Halc]

Re-read my prior reply, which applies to all these questions.

[hamdani yusuf]

Re-read my prior reply, which applies to all these questions.

That's an assumption. Can you prove it using equations? Is it compatible with Galilean, Lorentzian, or Einsteinean relativity?

[Halc]

Re-read my prior reply, which applies to all these questions.

That's an assumption.

It's not an assumption, but rather something that follows directly from the two premises of SR (one of which is Galilean relativity). If you want to call something that has been repeatedly demonstrated 'assumptions', you can choose to do so.

Can you prove it using equations?

λ = 1/√(1-v²/c²)
There's nothing in that equation that changes with choice of direction of motion (λ is a scalar, not a vector, and v is a speed, not a velocity in any equation that squares it) or choice of process (paint peeling, light clock through material with high refractive index) to be dilated. Hence none of those factors effect the dilation due to motion of the process.

Is it compatible with Galilean, Lorentzian, or Einsteinean relativity?

Yes. Galileo never had reason to assume Einstein's second postulate. It didn't work with sound so he had no reason to suspect there was no speed-of-medium involved with light. Lorentz's and Einstein's relativity make the exact same empirical predictions, so yes to that part.

[hamdani yusuf]

λ = 1/√(1-v²/c²)
There's nothing in that equation that changes with choice of direction of motion

The commonly used symbol is gamma. One method to derive its value is using a light clock moving perpendicular to the light trajectory. The constraint to keep the light speed c leads to the conclusion that time has dilated by gamma factor.
The question is, do we get the same result if the clock is moving in the same direction as the light trajectory, while keeping the light speed at constant value c?

[Janus]

λ = 1/√(1-v²/c²)
There's nothing in that equation that changes with choice of direction of motion

The commonly used symbol is gamma. One method to derive its value is using a light clock moving perpendicular to the light trajectory. The constraint to keep the light speed c leads to the conclusion that time has dilated by gamma factor.
The question is, do we get the same result if the clock is moving in the same direction as the light trajectory, while keeping the light speed at constant value c?

For the round trip, yes.  The individual legs of the light trajectory will take different different amounts of time however.  Going in one direction the light has to chase after a mirror going in the same direction, and going in the other direction the mirror is traveling in the opposite direction of the the light and they close in on each other. ( the total summed time of the two legs will be the same as in the typical light clock setup.
Basically: If you set up a spherical light clock (one where the light bounces back and forth between the center and inner wall of a sphere), all light paths, no matter what direction they are emitted, return to the center at the same time as long as they were emitted from the center at the same time.
 
This means that if you are only interested in time dilation, you have to deal with round trip time, as when you deal with just one leg, you also need to take Relativity of Simultaneity into account.

[hamdani yusuf]

https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment#Light_path_analysis_and_consequences

In the equations above, transversal time is dilated by gamma, while longitudinal time is dilated by gamma squared.
To make transversal time equals longitudinal time, the longitudinal path length must be contracted by gamma.

[Janus]

https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment#Light_path_analysis_and_consequences

In the equations above, transversal time is dilated by gamma, while longitudinal time is dilated by gamma squared.
To make transversal time equals longitudinal time, the longitudinal path length must be contracted by gamma.

Which is is what Relativity predicts will happen.

[hamdani yusuf]

Which is is what Relativity predicts will happen.

Yes.  So far looks good.
Let's take further test. There are two light clocks with exactly same length put side to side. In the first clock, the space between mirrors is kept at vacuum, while the second clock is filled with transparent medium which makes light travel at half of c. 
The stationary observer sees that every 4 microseconds both clocks tick simultaneously by photons hitting their bottom mirrors. Does vertically moving observer sees the same phenomenon? What about the horizontally moving observer?

[Halc]

Let's take further test. There are two light clocks with exactly same length put side to side. In the first clock, the space between mirrors is kept at vacuum, while the second clock is filled with transparent medium which makes light travel at half of c. 
The stationary observer sees that every 4 microseconds both clocks tick simultaneously by photons hitting their bottom mirrors.

If one two otherwise identical light clocks has a high refractive index medium, then they're not both going to tick every 4 usec.  The one is going to tick at half the rate of the other.  Maybe the mirrors are closer if you want that?

If you're going to actually do the calculation of the time in each direction, you'll have to remember that the medium with the refractive index is moving in the alternate frames. That means that if the medium is moving fast enough, the light can actually make negative progress.

Does vertically moving observer sees the same phenomenon? What about the horizontally moving observer?

See reply 21, which says I don't need to compute the details to answer this question.

If the comoving clocks tick at the same rate for one observer, then they will for any observer.  They may not tick simultaneously in most frames since they're not in the same location and thus relativity of simultaneity may have the tick events simultaneous in some frames and not in others.

[Janus]

Which is is what Relativity predicts will happen.

Yes.  So far looks good.
Let's take further test. There are two light clocks with exactly same length put side to side. In the first clock, the space between mirrors is kept at vacuum, while the second clock is filled with transparent medium which makes light travel at half of c. 
The stationary observer sees that every 4 microseconds both clocks tick simultaneously by photons hitting their bottom mirrors. Does vertically moving observer sees the same phenomenon? What about the horizontally moving observer?

Like Halc said, you'd have to make the one set of mirrors closer to each other if you want the stationary observer to see the clocks tick in sync. 
The fact that the light traveling through the medium doesn't travel at c relative to the stationary observer ( as measured by said observer will not alter the fact that our moving observer will measure the "slower" clock as having the same time dilation as the vacuum clock, no matter which direction he is moving relative to it.  You need to apply relativistic velocity addition to the light passing through the medium.  So, for example in the case where the light path is parallel to the relative motion of v, then:  Our observer will measure the light traveling at (0.5c+v)/(1+0.5c(v)/c^2) relative to himself in one direction, and (0.5c-v)/(1-0.5c(v)/c^2). For an observer traveling perpendicular to this, you have to use the orthographic version of this equation.
While the math is a bit more involved,  If the "stationary" observer measures the clocks as ticking at the same rate, so will the moving observer, just time dilated.
Do not get sidetracked by the fact that c = the speed of light ( in a vacuum). Is the value c that is important and what determines the value of time dilation etc, not whether or not light itself in actually traveling at c. In those scenarios where the light doesn't travel at c, you still need to use c in the equations.

[hamdani yusuf]

If one two otherwise identical light clocks has a high refractive index medium, then they're not both going to tick every 4 usec.  The one is going to tick at half the rate of the other.  Maybe the mirrors are closer if you want that?

No. Every 4 microsecond, vacuum clock ticks twice, while filled clock only ticks once

[Janus]

If one two otherwise identical light clocks has a high refractive index medium, then they're not both going to tick every 4 usec.  The one is going to tick at half the rate of the other.  Maybe the mirrors are closer if you want that?

No. Every 4 microsecond, vacuum clock ticks twice, while filled clock only ticks once

This is a minor detail that has no significance tin terms of the larger picture of whether or not an observer with relative motion with respect to the clocks measure them as undergoing the same time dilation.

[hamdani yusuf]

We have seen that constant c implies that transversal clock experiences time dilation. Since longitudinal clock must have equal time dilation, it must also experience length contraction.
For speed less than c, vectorial addition formula can be derived by above constraints, which says that if the transversal clocks tick simultaneously at certain interval, the longitudinal clocks must also tick simultaneously at the same interval.

[hamdani yusuf]

 Clock A is considered stationary. clock B moves to the right at speed half c. According to A, clock B ticks slower. But according to B, clock A ticks slower.
Add clock C, which moves to the right at a quarter of c relative to A. According to C, clock A and B tick equally slower than clock C.

[yor_on]

Nice question Jeffrey. You're getting good at them.

[yor_on]

philosophically I don't have one answer. It depends on your frame of reference.

But logically, you have to expect the universe to have a mean of joining them. The reason why that is is that without a logic our whole idea of how it works falls apart.
=

Btw: Questions are so much better than answers :)

I'll tell you one thing. Had this dream, was on a shore at twilight, getting dark. It's was mysterious, and I heard and saw people acting out of their own necessities and needs. I stopped looking at them, instead looking up, because it was all unknown to me. And I saw the constellations, didn't recognize them but they were so beautiful. I lost myself in them.


Don't read this as something questioning any answer made here. I don't really know, I wish I did. It would make my life worthwhile.

[yor_on]

There is one other thing to define it from. What we call proper time, your wrist watch. It doesn't lie to you. If we accept that and then define the universe from it it becomes splintered. It's dependent on your frame of reference. so that is one definition, but then we have the one in where you talk to your friend over your smartphone, asking him if he sees the same as you. That's the universe, and you will agree assuming you're in a 'similar' frame of reference. Without it you will have to look for something more and that is where Lorentz transformations come in. Those are are more or less metaphysical, based on equation. Never the less they give you a logic.
=

The end result is a logic very close to magic. Isn't it?
===

But not really, it will give you a origin of a event. Then you will need to translate it into your own, or more correctly others. frame of reference. Don't remember the name we use for that but it works. Been a long time since I looked at this, and its all your fault Jeffrey :). It doesn't tell you what 'time' is globally, everything included, so you can't use it to define one frame of reference as being 'before' or 'after', at least I don't think you can do it that way, because the only definition I know of is 'proper time' and the Lorenz transformations of the same.

thinking some more of it. you will be able to define a logic and a origin, but that's all. and that is your question if I got it right?

Damn, there is something called the Andromeda paradox if I remember right.

https://medium.com/mathadam/the-andromeda-paradox-b4bb30a0e372

It doesn't state that the logic is wrong, that's what you need to see, but it does question what time is. Globally.

The point there is that you won't back up time, from any of those frames of reference. Assuming you and your friend meets up, 'Bob and Alice' you will be in a same frame of reference, with the 'invasion' or whatever event you looked at being in both of your past. Another name for time is 'entropy' and what may differ that from time is that it is specific to any frame of reference. Everything has a entropy. As far as I get it.