Which Clock Accumulates More Time in the Train Paradox of Relativity?

[CrtY]

Its my first post here so i say hello to you.

I have a 'problem' with understanding relativity. Here is the thing:

We have a train and a station. Station and train have same lengths measured in rest (v=0).
Station has attached two detectors, one at rear and one at front. They have built in clocks which start running when detector go off. Front detector goes off when front of train passes it. Rear goes off when back of train pass it. I hope you can imagine this situation.

Now here is the question: which clock will accumulate more time? There is no acceleration involved, train is traveling at constant speed, 0.9c.

Option A: front, because from train point of view station is length contracted, so front detector will go first
Option B: rear, because from station point of view train is length contracted, and read fill go first
Option C: none, because from another train traveling at 0.45c in parallel both station and train have the same lengths.

Obviously all options can't be right. How would you explain this? If i would perform this experiment in reality, what would be the state of clocks on station?

 

[Penn.6-5000]

This question is usually discussed as the train paradox: lightning strikes the front and back of a train simultaneously, but not all observers think it's simultaneous. The resolution is to just leave it unresolved. If you observe events taking place simultaneously at two different points, some other jerk will observe one of the events happening before the other and a third jerk will see the events happening in the opposite order. Only when events happen at the same time *and in the same place* does everyone have to agree.

Your question is cleverer than that. You don't ask whether the detectors fire at the same time because that's observer-dependent. But you are comparing clocks. How do you know whether one clock is ahead of the other? "Oh, just look at the two clocks at the same time and see if one has a higher number than the other." Are you sure? Look at the clocks at the same time? That's not relativistically invariant because the clocks are separated by a significant distance.

An observer on the train will look at the two clocks and at any particular "instant in time" he'll see that the front clock has a bigger number on it than the rear clock, for the reason you gave in option A. An observer on the platform will look at two clocks "at the same time" and he'll see that the rear clock has a bigger number on it, for the reason you gave in option B. An observer on the half-speed train will pick option C. These observers are allowed to disagree because you can't compare two clocks that are far apart from each other.

OK, fine, let's step out onto the darned platform, pick up one of the clocks, and set it down next to the other clock. Now we can compare them, right? Yes, but the results are tainted. You had to move the clock across the platform, and while it was moving it experienced time dilation, and it will report an earlier time than it "should." That's not a good way to compare clocks.

Let's get around that problem by picking up each clock and moving them to the halfway point. Is that fair? Do all the observers agree on where the halfway point is? Yes, they do. Great. You're standing on the platform and the clocks start off at rest with respect to you. So you pick up the rear clock and move it forward to the halfway point with a constant velocity of +v, then put it down. Then you pick up the front clock and move it backwards with a constant velocity of -v, and then put it down next to other first clock. You compare the two and, sure enough, the rear clock has a bigger number on it. Problem solved. Right?

No! I was watching you from the train. Before you picked up the clocks, I saw them both moving backwards (relative to me) at -0.9c. When you grabbed the forward clock and started moving it backwards, the train observer didn't see it moving at -0.9c-v like you'd expect... The relativistic velocity-addition formula gives a total speed of (-0.9c - v) / (1 + 0.9v/c). The forward-moving clock, likewise, doesn't now appear to be moving at -0.9c+v; it actually seems to be moving at (-0.9c + v) / (1 - 0.9v/c). These expressions are pretty different and kind of awkward to appreciate intuitively. But if we plug in a test number, like v = 0.05c, we get that the front clock moved backwards at -0.909c and the rear clock moves forwards at -0.89c. So I think you cheated! The front clock was moving faster, so it ticked more slowly while you were transporting it. I'm not surprised that the rear clock shows a bigger number. But it's not because I agree with you that the rear end of the train passed you before the front end of the train did, it's because of an asymmetry in the way you moved the clocks to the center of the platform!

So, long story short, you get bitten by the fact that simultaneity doesn't exist in relativity. But this was probably the most fun question I answered all evening.

 

[CrtY]

Observer standing in the middle of station will see train length-contracted triggering first rear then forward detectors, right? He will see rear clock more advanced than forward. Now he gets a call from a train, and is asked which clock has more time on it. He answers the rear one, right?Passenger in train sees station length contracted. Forward detector is triggered first as station is moving throughout length of train. He then calls person on station and asks which clock is more advanced. What answer will he get? If there is a symmetry, he should get answer that rear clock is more advanced. Will he? I don't think so. No matter what, station was length contracted (and it is still) so the rear clock must have less time on it.You introduced moving clocks and acceleration. I wanted to avoid it.Tell me a very simple thing. If you were on this train, and your friend on station, what would he tell you about the clocks if youd call him after passing station?

 

[Penn.6-5000]

An observer at the station will think the rear clock started first.

But other observers will give different answers.

 

[CrtY]

But other observers will give different answers.

so, for one observer rear clock has more time, and for other forward has more?
if I am in train and know relativity, i know for sure that forward clock will have more time. if i call person on station, he must confirm it.
if I am on station and know relativity, i know that rear clock will have more time. if i get call from person in train, i must contradict him.

something is REALLY wrong with this.

 

[jtbell]

No, there's nothing wrong with it. It's an integral part of relativity, called "relativity of simultaneity." It's right up there with length contraction and time dilation, and is required in order to make length contraction and time dilation consistent with each other. I think of the three phenomena as the "three-legged stool" of relativity.

There's a formula for it: If two clocks are synchronized in one inertial reference frame, and separated by a distance \Delta x in that reference frame, then in another reference frame moving at speed v along the direction from one clock to the other, the two clocks are not synchronized, and the difference in their readings (in that frame) is

\Delta t = \frac{v \Delta x} {c^2}

 

[Penn.6-5000]

Some measurements don't have a consistent answer between observers. That's one of the rules you had to sign when you first agreed to live in a relativistic world. A person on the train *will* observe things that directly contradict an observer on the platform, and that's just life. It turns out that in Einsteinian relativity, the phrase "at the same time" is relative---different observers will disagree on what that means.

Wikipedia has an article on a related simultaneity paradox called the "Pole-Barn Paradox," though the article uses a different name for it:
http://en.wikipedia.org/wiki/Ladder_paradox

It's one of the standard issues that you just have to wrap your head around before you can be comfortable with relativity. But that really is how things work.

 

[CrtY]

I am comfortable with relativity, i understand it but when i read your posts i don't know if you contradict me, or just going around without resolving paradox.

Latter scenario is also perfect. But evidently when garage is contracted, forward doors must close first.
This destroy symmetry, because observer from ladder will expect forward door closed first, while observer from garage would expect both doors closed at same time.
To be more straightforward, i will not use term 'at the same time', but put a clock on both doors and compare them. Their state will be different.

 

[MikeLizzi]

I believe jtbell has misunderstood your issue. Please consider the following calculations:


Given
Train and Station both of Proper Length .133 light seconds.
Relative velocity between the two is .866c (gamma = 2).
Detectors (with clocks) at the rear and front of the station.

With respect to the Station:

Front of Train passes Forward Detector
Forward Clock = 0.194 Arbitrary Start Condition
Rear Clock = 0.194 Synchronized to Forward Clock

Rear of Train passes Rear Detector
Forward Clock = 0.04 Train is 50% length contracted.
Rear Clock = 0.04 Synchronized to forward clock

Both clocks indicate .154 seconds elapsed.



With respect to the Train:

Front of Train passes Forward Detector
Forward Clock = 0.156 Arbitrary Start Condition
Rear Clock = 0.040 Station is length contracted and traveling to the LEFT.

Rear of Train passes Rear Detector
Forward Clock = 0.194
Rear Clock = 0.078

Both clocks indicate .038 seconds elapsed.

(sorry for the many edits)

 

[Penn.6-5000]

To be more straightforward, i will not use term 'at the same time', but put a clock on both doors and compare them. Their state will be different.

That's clever, but it isn't enough. How do you compare two clocks that are in different places? There are two ways:
1. Look at them "at the same time" and compare the numbers on the front. But that means different observers will get different answers.
2. Move the clocks to the same position and compare them there. Now the clocks are directly comparable but the clocks had to move and so time dilation skews the answer.

 

[CrtY]

How do you compare two clocks that are in different places? There are two ways:

they both send light to person in the middle. This person is the first to know their state, and then communicate it to the other one inside train. Result should always indicate that rear clock has more time.
Thats the fact, you are on station, train is shorter. Light takes exactly same amount of time to reach you from both sides.

Now if you are in train, in order to keep things right you should hear in your phone that rear clock has more time. But when you passed station, it was contracted. Ok, Light traveled from forward clock to station observer slower, and from rear faster, probably compensating for this, but as soon as train stops station observer will notice the difference, forward clock is ahead.
This isn't the case if you are all the time on station, you just see train stopped, while rear clock still is ahead.

 

[King Wildog]

How can observed simulaneity have anything to do with actual simultaneity? Observation cannot affect an event of this nature (it is not film exposure or anything). Also, when should we assume clocks measure time anyways? Clocks don't measure anything. They are mechanical repeating devices.

 

[Janus]

they both send light to person in the middle. This person is the first to know their state, and then communicate it to the other one inside train. Result should always indicate that rear clock has more time.
Thats the fact, you are on station, train is shorter. Light takes exactly same amount of time to reach you from both sides.

Now if you are in train, in order to keep things right you should hear in your phone that rear clock has more time. But when you passed station, it was contracted. Ok, Light traveled from forward clock to station observer slower, and from rear faster, probably compensating for this, but as soon as train stops station observer will notice the difference, forward clock is ahead.
This isn't the case if you are all the time on station, you just see train stopped, while rear clock still is ahead.

If the train stops, it has to undergo an acceleration and this acceleration will, according to a person in the train cause the clock in the front of the train to run slower by a factor that depends on the length of the train and the value of the acceleration.

Also, for someone at the station, when the train stops, it must "un-contract" as it does so. But this means that the front of the train and back of the train must move different distances from the time one end stops to when the other end stops. In other words, the station observer will also see the time on the two clocks change at different rates. These two effects are such that by the time the train comes to a rest, both observers will agree as to what each clock reads.

So even though while the train and station will disagree as to which clock is ahead of the other while they are the have a relative motion with respect to each other, when you bring both back into the same frame, they end up agreeing as to what the times on the clocks are.

 

[Janus]

How can observed simulaneity have anything to do with actual simultaneity? Observation cannot affect an event of this nature (it is not film exposure or anything). Also, when should we assume clocks measure time anyways? Clocks don't measure anything. They are mechanical repeating devices.

If you are halfway between the locations of two events and observe them simultaneously, you can conclude that events took place simultaneously, because the light took an equal amount of time to reach you from each event.

Conversely, if you are halfway between the locations of two events and see them at different times, you can conclude that the two event did not take place simultaneously, again because the light from each event takes a equal amount of time to reach you.

What happens with the Relativity of Simultaneity is that you have two observers, each "halfway between the locations of two events" but moving with respect to each other and one sees the events as being simultaneous and the other doesn't.

So according to one observer this is what happened:

http://home.earthlink.net/~jparvey/sitebuildercontent/sitebuilderpictures/trainsimul1.gif

And for the other, this is what happens:

http://home.earthlink.net/~jparvey/sitebuildercontent/sitebuilderpictures/trainsimul2.gif