Two trains. Two equivalents. Two outcomes. Can anyone explain?

[JesuisSean]

Let us think of an observation that is similar to that seen as the outcome of the Hafele–Keating experiment. https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment

 

Imagine you had two very very very very long trains, trains A and B, and that they are positioned closely side by side. Located at each window of each section of each train, is a clock. All the clocks have been synchronized. Those inside train A can look out a window and see clocks that are located in train B, and vice versa.

 

Now imagine that one of the two trains, train A, rapidly accelerated up to a high velocity and then maintains that very same high velocity for a significant time period. During that lengthy time period, from train B's Point of view, the clocks on board train A appeared to be ticking at a slower rate than were the clocks on board train B itself.

 

Eventually, at a later period, train A slows down and stops, and thus it is again at rest in respect to train B. Those on board each train had recorded the time in which train A had maintained its high velocity. They then compared the recorded time periods and quickly noticed that the clocks on board train A had measured a shorter time period had passed by when compared to the time period recorded using the clocks that are located on board the stationary train B. However, during the time that train A was moving at a high velocity, those on board train A also noticed that the clocks on board train B each seemed to be ticking at a slower rate than were the clocks ticking on board train A itself.

 

But it is to be noted that when special relativity is understood properly, you understand why these 2 instances of observing the ticking of opposing train clocks each "other" appearing to be ticking at a slower rate.

By understanding SR, you Immediately understand why and how this will still occur despite the fact that the clocks had actually only truly slowed down on board Train A.

 

It gets complicated. For instance, let's say you have 3 trains positioned side by side, rather than have just 2. The train velocities are relative to the train tracks, tracks which are all mounted on the same ground.

 

Train 1) [--------] [--------][--------][--------][--------][--------][--------] Velocity = 0 km/s

Train 2) [--------] [--------][--------][--------][--------][--------][--------] Velocity = 260,000 km/s

Train 3) [--------] [--------][--------][--------][--------][--------][--------] Velocity = 297,000 km/s

 

Those on board train #1, would see that the clocks on board train #2 seem to be ticking at only half speed.

 

Those on board train #2, would see that the clocks on board train #1 seem to be ticking at only half speed.

Those on board train #2, would see that the clocks on board train #3 seem to be ticking at only half speed.

 

Those on board train #3, would see that the clocks on board train #2 seem to be ticking at only half speed.

 

Can you figure it out ?

[Strange]

Figure what out?

[Tim88]

Let us think of an observation that is similar to that seen as the outcome of the Hafele–Keating experiment. https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment

 

Imagine you had two very very very very long trains, trains A and B, and that they are positioned closely side by side. Located at each window of each section of each train, is a clock. All the clocks have been synchronized. Those inside train A can look out a window and see clocks that are located in train B, and vice versa.

 

Now imagine that one of the two trains, train A, rapidly accelerated up to a high velocity and then maintains that very same high velocity for a significant time period. During that lengthy time period, from train B's Point of view, the clocks on board train A appeared to be ticking at a slower rate than were the clocks on board train B itself.

 

Eventually, at a later period, train A slows down and stops, and thus it is again at rest in respect to train B. Those on board each train had recorded the time in which train A had maintained its high velocity. They then compared the recorded time periods and quickly noticed that the clocks on board train A had measured a shorter time period had passed by when compared to the time period recorded using the clocks that are located on board the stationary train B. However, during the time that train A was moving at a high velocity, those on board train A also noticed that the clocks on board train B each seemed to be ticking at a slower rate than were the clocks ticking on board train A itself.

[...]

 

That's where it goes wrong: You forgot to re-synchronize the clocks in the train to the "moving frame". Without that, those on board train A would not have noticed that the clocks on board train B each seemed to be ticking at a slower rate.

[Janus]

Let us think of an observation that is similar to that seen as the outcome of the Hafele–Keating experiment. https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment

 

Imagine you had two very very very very long trains, trains A and B, and that they are positioned closely side by side. Located at each window of each section of each train, is a clock. All the clocks have been synchronized. Those inside train A can look out a window and see clocks that are located in train B, and vice versa.

 

Now imagine that one of the two trains, train A, rapidly accelerated up to a high velocity and then maintains that very same high velocity for a significant time period. During that lengthy time period, from train B's Point of view, the clocks on board train A appeared to be ticking at a slower rate than were the clocks on board train B itself.

 

Eventually, at a later period, train A slows down and stops, and thus it is again at rest in respect to train B. Those on board each train had recorded the time in which train A had maintained its high velocity. They then compared the recorded time periods and quickly noticed that the clocks on board train A had measured a shorter time period had passed by when compared to the time period recorded using the clocks that are located on board the stationary train B. However, during the time that train A was moving at a high velocity, those on board train A also noticed that the clocks on board train B each seemed to be ticking at a slower rate than were the clocks ticking on board train A itself.

 

But it is to be noted that when special relativity is understood properly, you understand why these 2 instances of observing the ticking of opposing train clocks each "other" appearing to be ticking at a slower rate.

By understanding SR, you Immediately understand why and how this will still occur despite the fact that the clocks had actually only truly slowed down on board Train A.

 

It gets complicated. For instance, let's say you have 3 trains positioned side by side, rather than have just 2. The train velocities are relative to the train tracks, tracks which are all mounted on the same ground.

 

Train 1) [--------] [--------][--------][--------][--------][--------][--------] Velocity = 0 km/s

Train 2) [--------] [--------][--------][--------][--------][--------][--------] Velocity = 260,000 km/s

Train 3) [--------] [--------][--------][--------][--------][--------][--------] Velocity = 297,000 km/s

 

Those on board train #1, would see that the clocks on board train #2 seem to be ticking at only half speed.

 

Those on board train #2, would see that the clocks on board train #1 seem to be ticking at only half speed.

Those on board train #2, would see that the clocks on board train #3 seem to be ticking at only half speed.

 

Those on board train #3, would see that the clocks on board train #2 seem to be ticking at only half speed.

 

Can you figure it out ?

The first thing I need to point out is that in your first example you accelerate train A after the clocks are synced. This open a whole new can of Worms as train A during that period becomes an accelerated frame, and the clocks in the train will not remain synced to each other, even to someone riding in the train. Also, even if you synced them after acceleration, they would still not be in sync according to train B. Put another way, the situation is much more complicated than just saying that Train A sees the clocks on Train B running slow and vice-versa.

 

In your example with three trains, we will assume that the clocks in each train are synced according to someone in that train. We still end up with the situation where occupants of each train will say that the clocks in the other trains are not in sync with each other( relativity of simultaneity). Also, the occupants of each train will note that the other trains will be length contracted.

 

So let's say that in each train, the occupants measure the distance between clock as being 1 meter. This means that Train three measures the distance between the clocks of train 2 as being 1/2 meter and vice-versa. You get the same between Train 1 and 2. Train 1 will measure the clocks on train 3 as being ~1/7 meter apart and vice-versa.

 

If we number the clock on each train, we can now compare what any two clocks on any two trains read when they pass each other. For instance, when clock 4 on train 1 passes clock 7 on train 2, we can note the reading on those two clocks. The point is that all observers, no matter which train they are on, will agree as to the respective readings on those two clocks when they pass each other, and this holds for any pair of clocks passing each other. Combination of time dilation, length contraction, and the relativity of simultaneity insure this.

 

This is something turns out to be a stumbling block for many when they try to work out such scenarios; they focus on just one thing, like time dilation, and fail to take into account that you need all three for things to make sense.