Understanding the Train Experiment: Explaining Observers and Time Perception

[Doc Al]

You see Doc:

If I see your clocks going slow and I measured it as going the same speed as mine. Then it was an illusion.

That's a good definition of illusion. Note that when talking about observations made by observers in different frames we are almost always talking about measurements, not just what they literally "see". Generally we view a frame as having as many clocks and observers as we need so that for any event we want to measure, we have an observer right there--so there's no optical illusions or light travel time to worry about.

If I measured your clocks going slow and you agreed they were slow. Then it's a fact.

If you measure my clock going slow compared to yours, and I measure my clock going slow compared to yours, then the relative clock rates would be a frame-independent fact. (But clocks don't work that way!) Note that there are frame-independent quantities.

If I measure your clocks to be going slow and it's a fact but you don't agree it's a completely different ball game to comparing two perspectives at rest with one another. The third situation is obviously SR works. That is what I mean.

Now let me see if I have this right: in a vacuum if I saw two syncronized clocks at rest with respect to me, depending on my position and their distance apart, they might appear unsyncronized. I still see them operating at the same speed, so I could correct for this discrepancy if I know the co-ordinates of the clocks.

Yes. But when reporting observerations made within a frame we always assume that such corrections have already been made. (Except when literally talking about what people "see".) But you are correct. Example: If I see two explosions at the same time (meaning: the light from both explosions hits my eyes at the same time), it would be pretty foolish of me to claim that the explosions were simultaneous unless I know how far away they occurred.

But if I am moving I also need to be aware of length contraction. The clocks are operating at the same speed, varying distance makes them appear out of sync and operating at a different speed, but because of length contraction I can't simply correct for varying distance between myself and the clocks. Distance is also out of whack.

Careful here: Clocks that move with respect to you "really are" unsynchronized as far as you are concerned. It's not an optical illusion that goes away once you correct for distance and light speed. (We assume you've done all the corrections already before reporting your observations.)

So let's revisit the first experiment in this thread. I'm going to set it up a little differently though so I know exactly what is going on with length contraction.

I'm going to set my contractors to work taking off the top of the train and the seats so all we see from the train station platform is a moving stage. They will also set up two triggers 20ft apart on the port side of the train. While I'm doing that we'll set up two piles of gun powder 20 ft apart on the edge of the platform. The first trigger needs to be set up so it doesn't set off the first pile of gunpowder it passes. Otherwise my experiment is stupid. I want them both to be extruding from the train so they merely touch the piles as they pass, so the first trigger will need to be extended as it passes the first pile. I could possibly be more technical than this but that will do for me.

If relativity wasn't a true phenomena then they should be set off at the exact same time according all observers. This experiment is perfectly symetrical so we should see something strange happening with the triggers. Will they be set off at the same time and observers will not see the events occurring as they are, or would the triggers actually be hit at different times? I have no idea.

So we will have an observer on the train (F1) and one on the platform (F2) recording the time the first trigger sets off the first pile and and the same for the second pile (S1 on the train S2 on the platform).

We will set up a ref in the middle of the train (R1) and in the middle of the two piles of gunpowder (R2) to see exactly when their respective two observers see the trigger hit.

According to special relativity the train is shorter for the platform observers and the platform is shorter for the train observers. In Einstein's experiment, for the platform observers it is the back of the train that is shorter, so the hind trigger should hit first. From the train observer's perspective it is the opposite so the front trigger should hit first.

From F1 and F2's perspectives the first trigger should hit at the same time. From S1 and S2's perspective the second trigger should hit at the same time. I seem to be having difficulty deciding what all the other perspectives in this situation would be. Could you help me out?

I'm going to take a crack at rewording this setup so that it's clearer (at least to me). Let's assume that the train travels north. Let's arrange for two locations on the platform to be some distance (D) apart. Call them PN (platform north) and PS (platform south). Also define a location right in the middle: PM. Put clocks and observers at all three points. Also put some explosives at PN and PS.

On the train, let's do something similar. Let's have locations TN and TS some distance D' apart. And a spot right in the middle: TM. Again, clocks and observers everywhere. TN and TS have special triggers that set off the explosives at PN and PS respectively, when they pass.

I think what you want is that when the train passes the platform, TN & PN, TM & PM, and TS & PS all pass each other at the same time according to the platform clocks. So the two explosions happen at the same time according to the platform clocks. Right?

(If relativity didn't apply, then all clocks will always be synchronized and read the same time. Also, D' and D would be equal. And everyone will observe--measure--the two explosions happening at the same time.)

Given this setup, we can talk about train observations versus platform observations.

 

[Joanna Dark]

After consideration, in my first explanation of length contraction, I seem to have forgotten about time dilation. It's difficult to put them all together at the same time.

As for my experiment, it's quite unusual because, in my understanding, all three observers on the train should see the front trigger hit first and the platform observers see the second trigger hit first. There should be no way, according to SR, that R1 will see F1 and F3 react to the trigger at the same time.

The way I figure this is that from the train the gun powder on the platform is not 20ft apart. Vice versa from the platform. If F1 and F3 reacted to the triggers at the same time, they would be behaving as if the gun powder piles were 20ft apart.

The only possibilities, ignoring the platform observers for a minute, are that 1) all three on the train see the triggers hit at the same time, 2) that they all see the front trigger hit first or 3) that all three observers see the triggers hit at different times.

The first possibility violates SR and nothing I have seen in SR suggests that the third result is likely. The second possibility is that from the train the platform is "actually" shorter as evidenced by the fact the triggers did not hit simultaneously. That would mean that the train is "actually" shorter for the platform observers. But Doc you seem to think this violates SR.

I'm guessing that another possibility is available but I can't see it?

 

[Joanna Dark]

I'm ok with your letters

F1 = TN
F2 = PN
F3 = TS
F4 = PS

R1 = TM
R2 = PM

D = distance between the triggers
D1 = distance between gun powder

D and D1 equal 20ft at rest.

What actually happens I am not entirely sure of.

 

[Doc Al]

After consideration, in my first explanation of length contraction, I seem to have forgotten about time dilation. It's difficult to put them all together at the same time.

As for my experiment, it's quite unusual because, in my understanding, all three observers on the train should see the front trigger hit first and the platform observers see the second trigger hit first. There should be no way, according to SR, that R1 will see F1 and F3 react to the trigger at the same time.

It's not clear to me what your experiment is. Please restate it using the notation that I introduced in my last post, which I believe is unambigous.

The way I figure this is that from the train the gun powder on the platform is not 20ft apart. Vice versa from the platform. If F1 and F3 reacted to the triggers at the same time, they would be behaving as if the gun powder piles were 20ft apart.

If the distance between the gunpowder on the platform was 20ft, that distance would be less than 20ft as measured by the train observers.

The only possibilities, ignoring the platform observers for a minute, are that 1) all three on the train see the triggers hit at the same time, 2) that they all see the front trigger hit first or 3) that all three observers see the triggers hit at different times.

The first possibility violates SR and nothing I have seen in SR suggests that the third result is likely. The second possibility is that from the train the platform is "actually" shorter as evidenced by the fact the triggers did not hit simultaneously. That would mean that the train is "actually" shorter for the platform observers. But Doc you seem to think this violates SR.

I don't understand why you think I would think that. The train is "actually" shorter for platform observers! It's not an optical illusion.

To make things crystal clear, please restate your questions using my notation.

 

[Doc Al]

D = distance between the triggers
D1 = distance between gun powder

D and D1 equal 20ft at rest.

If D = D', then neither frame observes the explosions happen at the same time. Since they both see the other's distance as shorter, the only way that one of them can see (meaning: measure) the explosions happen at the same time is if the other's distance is greater. If the train distance (D') is greater than the platform distance (D) by just the right amount, then the platform observers will observe the train just fit perfectly so that TN & PN and TS & PS line up at exactly the same time. This of course is not symmetric--since D' > D. (Note that D' and D are the distances as measured in their own frames.)

 

[Dale]

After consideration, in my first explanation of length contraction, I seem to have forgotten about time dilation. It's difficult to put them all together at the same time.

No it isn't. Look at the diagram I provided. Length contraction, time dilation, and relativity of simultaneity. And all together at the same time.

 

[Joanna Dark]

Right.

So if that is true then the observed length contraction actually occurs. The train "actually" is shorter from the platform. The platform "actually" is shorter for the train observers.

Now we have a problem because you objected so often to me saying the the moving clocks "actually" ran slow and that this violates SR. Why can I say that length contraction actually occurs for the moving train, if I'm on the platform, but I can't say that about time dilation? In my understanding both should actually occur.

Hmm Doc. What's up here?

 

[Doc Al]

Right.

So if that is true then the observed length contraction actually occurs. The train "actually" is shorter from the platform. The platform "actually" is shorter for the train observers.

Right.

Now we have a problem because you objected so often to me saying the the moving clocks "actually" ran slow and that this violates SR. Why can I say that length contraction actually occurs for the moving train, if I'm on the platform, but I can't say that about time dilation? In my understanding both should actually occur.

We're getting tangled up in semantics a bit. What I object to is specifying a clock rate without specifying as measured by what frame. To say that a clock is slow (or fast) is meaningless without specifying who is doing the measuring. It would be just as meaningless to say that the train is a certain length without specifying according to who.

Does that help?

 

[Joanna Dark]

Well I'm not so sure about the semantics but, as you can understand, this helps a lot.

So after our little word mix-up, and after analysing length contraction, I'm back to: A moving train is "actually" shorter and its clock "actually" runs slow for the platform observer. The platform is "actually" shorter and its clock "actually" runs slow for the moving train observer. Got it after a few days, and now you can't change my mind on this thankfully . Now I can run my experiments properly.

I have to figure out how this would be possible.

Let's head back to Sarah's example for a moment. Light travels at a constant speed for all observers. In this example the stationary observer sees a light beam travel 10ft at c and the moving observer sees the same light beam travel 20ft at c.

If the train observer measures the light beam speed by observing the platform clock what would the result be? Hmmm...?

The light beam travels 20ft at c on the passenger's and they now time the light beam on the station clock. The stationary observer times the light traveling 10ft at c. The distance is doubled for the passenger. So I would need to slow down the station clock by half, (for the perspective of the passenger in this example) to see the same time as the platform observer. Unless simultaneity affects my ability to do this. Yes? The stations's clock should be running fairly slow but could I say it's half the speed ? Let's correct for the movement of the train by using two passengers on the train to measure the speed of the stationary clock (just like the first experiment I set up). I'm not sure but am guessing I'm right.

 

[Joanna Dark]

I'll reset the experiment. Two observers on the train with synchronized clocks 20ft apart. The train is traveling at a velocity where the Platform observer sees the light beam travel only 10ft.

At the exact time the front observer (Fr) passes the platform observer (P) he also shines a light beam towards the back observer (B). When the (B) sees the light hit his sensor would it be the exact same time he passes (P).

 

[Doc Al]

I'll reset the experiment. Two observers on the train with synchronized clocks 20ft apart. The train is traveling at a velocity where the Platform observer sees the light beam travel only 10ft.

OK. At some point TN sends a light beam towards TS. In the train frame, the distance between TN and TS is 20ft, so in the train frame the light travels 20ft. But the speed of the train is such that as seen by the platform observers the light only travels 10ft.

At the exact time the front observer (Fr) passes the platform observer (P) he also shines a light beam towards the back observer (B). When the (B) sees the light hit his sensor would it be the exact same time he passes (P).

No. In the platform frame the light travels 10ft before hitting the back train observer (TS). So when TS "sees the light", he must be at a point that is 10ft south of P according to the platform observers. (TS will agree that he is south of P when he sees the light, but will disagree that he is 10 ft south of P.)

 

[Joanna Dark]

That couldn't be. P is perpendicular to TN when the light flashes. P sees the light travel 10ft. Then by the time he sees the light hit the back of the train, TS is 10ft south. That means the train is ten feet short and stationary.

Doc this train is traveling at a tremendous speed.

Are you ok? Or just a little tired?

 

[Doc Al]

That couldn't be. P is perpendicular to TN when the light flashes. P sees the light travel 10ft.

Right. So where is the light when it reaches its target?

Then by the time he sees the light hit the back of the train, TS is 10ft south.

Exactly! (According to the platform observers, TS is 10ft south of P when the light hits him.)

That means the train is ten feet short and stationary.

How do you figure that?

Doc this train is traveling at a tremendous speed.

Indeed it is!

Are you ok? Or just a little tired?

I'm fine. Thanks for asking.

Hey, you are the one who said the light travels 10ft according to the platform! Platform observers see the light begin at point P and travel south for 10 ft. That's what "travels 10ft" means.

 

[Joanna Dark]

Oh no I had the train traveling 180000 times faster than the speed of light.

 

[Doc Al]

Oh no I had the train traveling 180000 times faster than the speed of light.

Huh?

FYI: To meet the conditions you described, the train needs to travel at 3/5 the speed of light. (Assuming I did the calculation correctly.)

 

[Joanna Dark]

The problem as I see it is I am trying to run an experiment I'm not sure is even possible. It's extremely complicated, however the result should explain roughly what is happening, so I'll go with it anyway. It is difficult to follow I know, but is based on the premise that if I could see a moving clock I could have a better understanding of what is happening in relativity. It includes time-dilation, length contraction and simultaneity in the one experiment. Read it through before commenting and ignore the errors please.

On the train there are two observers, train north (TN) and train south (TS), who are 20ft apart and traveling north.

The platform has two observers of it's own, platform north (PN) and platform south (PS), who are 10ft apart on the platform.

When TN passes PN he shines a light (L1) south towards TS, and the train is traveling at such a velocity where, TS's sensor sees the light as it is perpendicular to PS.

When PS passes TS he shines a light (L2) nouth towards PN.

I have two points on each side with which to time the speed of the train. PS and PN record the time it takes for TN to travel between them and TN does the same but records a lesser value (just like my first experiment). TN and TS record the time that it takes PS to travel between them and PS does the same but also shows a lesser result.

So that is the set up and now we will try to use it.

L1 travels 20ft at c for observers on the train and 10ft at c for observers on the platform, which is half the distance in half the time. L2 travels 10ft at c for platform observers and 5ft at c (I'm guessing here, but is not overly relevant) for train observers, which is half the distance at half the time.

They are timing the same light beam, so I'm thinking hypothetically that the platform observers should see the train observers clocks operating at the same speed as their own clocks, thus timing the light travel 10ft at c.

I'm not sure this part is even possible, but bear with me for a minute and let's pretend we could. In the end it seems to confirm what SR is saying.

The platform observers time L1 on the TN's clock traveling 10ft at C. This would mean that the train observers clocks are going half the speed of the platform clock in their own frame but don't recognise their clocks are running slow.

If I reversed the situation then the same thing should occur.

When PS, PN, and TN time the journey between PS and PN they all agree on the location of each observer but the distance between the two is shorter for TN. So in my hypothetical above I'd assume that PS_PN and TN would agree, but they don't. I'm guessing that TN's time should be closer to half of PS_PN but he also believes he traveled half the distance. TN thinks he only traveled 5 ft. So from the platform while they should see TN's clock timing L1 travel 10ft at c TN says his clock only elapsed about half that time.

So from this I could possibly make the assumption: It wouldn't matter whether the platform observers see the light travel 10ft at c (on the train's clock) and be correct or c/2 and be correct, because, for the train observers the light has only traveled 5ft. Thus making sense and alleviating the contradiction I saw previously.

But that is to say that the light traveling between the two points doesn't occur simultaneously. PN sees the light arrive early and PS calculates the light leave late. TS and PN agree with TN (when he shines the light) but TS doesn't agree with PN on the arrival time.



So when I put it altogether I can see how relativity works without necessarily contradicting itself. But, and this is a huge but, it does require accepting something that doesn't follow any logical sense. If my clocks are traveling slower than yours and yours are traveling slower than mine, then you could look at it as our clocks are simultaneously running normally and slowly at the same time. How do I deal with that?

If experiments into SR really work, and are empirical, then I would need to accept it without the necessity of logic. Extraordinary claims require extraordinary evidence, so I now need to study this evidence.

Thanks so much Doc for you patience.

 

[Dale]

So when I put it altogether I can see how relativity works without necessarily contradicting itself. But, and this is a huge but, it does require accepting something that doesn't follow any logical sense. If my clocks are traveling slower than yours and yours are traveling slower than mine, then you could look at it as our clocks are simultaneously running normally and slowly at the same time. How do I deal with that?

There is nothing at all illogical about this! Consider a spatial parallel.

Let's say two surveyors are marking out coordinates (meters North, meters East) of different landmarks from the same reference point. One is using magnetic north and the other is using celestial north which differ by e.g. 20º. Each will find that the other's "meters North" is short by a factor of cos(20º). They will each also find that the other's "meters East" is short by the same factor of cos(20º). Go ahead and work out the geometry for yourself to confirm that I am correct.

What is illogical about a rotation?

 

[jtbell]

So when I put it altogether I can see how relativity works without necessarily contradicting itself.

Therefore relativity is logical. It follows from its postulates, and is self-consistent.

But, and this is a huge but, it does require accepting something that doesn't follow any logical sense.

Relativity is counterintuitive for people who have not had practice in thinking in relativistic terms. This is not the same thing as being illogical.

If experiments into SR really work, and are empirical, then I would need to accept it without the necessity of logic. Extraordinary claims require extraordinary evidence, so I now need to study this evidence.

Here is a large collection of references to experimental evidence:

Experimental Basis of Special Relativity

 

[Joanna Dark]

Well my understanding is premature because I have disregarded 3 factors: aceleration, deceleration and total time dilation.

Someone on this thread stated that the train clocks needed to be sychronized and stopped before any change in velocity occurs. The train's clocks are out of sync for the platform observer. So a ref sync'ed the clocks when the train was stationary and rechecked at a constant speed they would have changed. I'm going to take a stab in the dark and suggest the south clock is traveling faster than the north clock.

I don't know whether decelleration should begin to bring the clocks closer to sync again, but that would be my best guess. The same should occur for the stationary clocks when viewed from the train frame.

Now for total time dilation. I am led to believe that if a plane flys around the equator for a long enough time it's clock won't match airport clocks. Same with satellites. A trip to Neptune and back would cause a great time difference suggesting that a person taking this trip would hypothetically actually be younger than a person on earth.

This could affect this new symmetry I've come to understand as one clock is actually going slower than a stationary clock. This I think is where my confusion arose in the first place.

Is this total time dilation caused by movement in general or only by aceleration and decelleration? My guess would be even a constant velocity would cause time dilation. Well I wonder if a plane traveling at a constant speed compared to a plane that acelerates and decellerates so both plane's average speed is the same, whether they will both experience the same total time dilation at the end of the journey.

I don't know how this effects my illogical/counter-intuitive understanding of special relativity.

 

[Doc Al]

The problem as I see it is I am trying to run an experiment I'm not sure is even possible. It's extremely complicated, however the result should explain roughly what is happening, so I'll go with it anyway. It is difficult to follow I know, but is based on the premise that if I could see a moving clock I could have a better understanding of what is happening in relativity. It includes time-dilation, length contraction and simultaneity in the one experiment. Read it through before commenting and ignore the errors please.

FYI: To fully understand just about any relativistic thought experiment requires all three: time dilation, length contraction, and simultaneity.

On the train there are two observers, train north (TN) and train south (TS), who are 20ft apart and traveling north.

OK.

The platform has two observers of it's own, platform north (PN) and platform south (PS), who are 10ft apart on the platform.

OK.

When TN passes PN he shines a light (L1) south towards TS, and the train is traveling at such a velocity where, TS's sensor sees the light as it is perpendicular to PS.

OK. This defines the relative speed of the train and platform to be 3/5 c.

When PS passes TS he shines a light (L2) nouth towards PN.

OK.

I have two points on each side with which to time the speed of the train. PS and PN record the time it takes for TN to travel between them and TN does the same but records a lesser value (just like my first experiment). TN and TS record the time that it takes PS to travel between them and PS does the same but also shows a lesser result.

They both measure each other's speed to be the same value: 3/5 c. And they both measure each other's clocks as running slow (compared to their own) by the same factor. (That factor = 1.25, by the way.)

So that is the set up and now we will try to use it.

L1 travels 20ft at c for observers on the train and 10ft at c for observers on the platform, which is half the distance in half the time. L2 travels 10ft at c for platform observers and 5ft at c (I'm guessing here, but is not overly relevant) for train observers, which is half the distance at half the time.

OK.

They are timing the same light beam, so I'm thinking hypothetically that the platform observers should see the train observers clocks operating at the same speed as their own clocks, thus timing the light travel 10ft at c.

Nope. Platform observers "see" (really, measure) the train clocks as running slow. And vice versa.

I'm not sure this part is even possible, but bear with me for a minute and let's pretend we could. In the end it seems to confirm what SR is saying.

The platform observers time L1 on the TN's clock traveling 10ft at C. This would mean that the train observers clocks are going half the speed of the platform clock in their own frame but don't recognise their clocks are running slow.

Sorry, not sure how platform observers would time something using train clocks. You'd have to define exactly what you mean by that. And the speed of light is measured to be c, but only if you use your own clocks and lengths in the usual manner to measure the speed of light with respect to you.

If I reversed the situation then the same thing should occur.

When PS, PN, and TN time the journey between PS and PN they all agree on the location of each observer but the distance between the two is shorter for TN.

They agree on the definition of the two events: (1) TN passes PS, (2) TN passes PN. But they don't agree on the location, distance, or times. (On the train, TN doesn't even move--it's the platform that moves.)

So in my hypothetical above I'd assume that PS_PN and TN would agree, but they don't. I'm guessing that TN's time should be closer to half of PS_PN but he also believes he traveled half the distance. TN thinks he only traveled 5 ft. So from the platform while they should see TN's clock timing L1 travel 10ft at c TN says his clock only elapsed about half that time.

The analysis of how the platform measures the light to travel 10 ft while the train measures it to travel 20 ft is more complicated than that. It involves length contraction as well as time dilation. (And to understand it from both points of view also requires simultaneity.)

So from this I could possibly make the assumption: It wouldn't matter whether the platform observers see the light travel 10ft at c (on the train's clock) and be correct or c/2 and be correct, because, for the train observers the light has only traveled 5ft. Thus making sense and alleviating the contradiction I saw previously.

You lost me a bit. Now you're talking about L2, not L1. Per your setup, L2 travels 10 ft on the platform, but only 5 ft according to the train.

But that is to say that the light traveling between the two points doesn't occur simultaneously. PN sees the light arrive early and PS calculates the light leave late. TS and PN agree with TN (when he shines the light) but TS doesn't agree with PN on the arrival time.

Not sure what you're talking about here. But it's certainly true that each frame sees the other's clocks as being out of synch. (Which helps explain how they can both measure the speed of the same beam of light to be c.)

So when I put it altogether I can see how relativity works without necessarily contradicting itself. But, and this is a huge but, it does require accepting something that doesn't follow any logical sense. If my clocks are traveling slower than yours and yours are traveling slower than mine, then you could look at it as our clocks are simultaneously running normally and slowly at the same time. How do I deal with that?

You still seem unable to shake the idea that clock rates depend on the frame in which they are observed.

If experiments into SR really work, and are empirical, then I would need to accept it without the necessity of logic.

It's not logic that needs to be abandoned, but certain of your premises about how the world works.

Extraordinary claims require extraordinary evidence, so I now need to study this evidence.

Always a good idea.

 

[Joanna Dark]

I wanted to see what would happen if the platform observers recorded L1 on the train's clock and wondered if they would agree it is traveling at c. Someone in my class asked the question why the speed of light is constant but we use two clocks operating at different speeds to measure it? That made me curious.

In the end I worked out that the trains clock is running slower but for the train observers they had traveled an equally less distance. The platform observers agree on the time L1 left TN and when it hit TS's sensor. But the train observers disagree.

TN thinks that PS saw the light arrive in double the time he claims. TS saw TN flash his light earlier than PN claims he did. Or another way to look at it, TN and PN agree that they were perpendicular when TN flashed his light. TS doesn't agree they were perpendicular, he believes that PN is much closer and therefore PN's clock is slow. The same in reverse if I conducted the experiment regarding L2.

How could I observe the speed of L1 using the train's clock? In my first experiment I set up 3 observers to record the "time elapsed" between two points (F and S) from the trains perspective (T) and the tracks perspective (F,S). The string of sensors I set up parallel to the tracks between F and S, I didn't use, because in the process I realized it would achieve the same result as T on the train, but the only difference is that it records each second of the T's clock from multiple stationary positions. In theory, using a computer and a string of sensors, I could watch a moving clock next to a stationary clock and therefore time the speed of light on the moving clock.

What I expected would occur didn't (that for the platform observers both clocks would see light travel at c) but in SR that is to be expected.

So to answer the original question: If light is constant why do we use 2 clocks speeds to time it?

We don't. The light travels only one distance between two points and the clocks are operating at the same speed in their own frames. The only reason there is a difference of opinion is because from the platforms perspective, meters and seconds are smaller and out of sync. But otherwise both agree that L1 traveled between point A and Point B at v=c.

The way I have explained the situation sounds really simple though, when it's not. If a person snaps their fingers on both hands, did they do it simultaneously? For some, traveling at a certain velocity and direction, they did, for some the right one snapped first and others still saw the left one snap first. Those who saw the right hand snap first will not all agree on the delay before the left hand snaps. If you are to accept SR then you need to understand that all answers are simultaneously correct and there is no way to know when each hand snapped, unless you are basing this on a single set of frames.

I would still like to know how aceleration and deceleration and total time dilation affect my understanding as per my last post.