How can a train appear shorter to an outside observer due to its speed?

[arkain]

Please note: While this post may seem to be me trying to advertise a new theory, I'm actually trying to find a reason to discount/discredit what I'm thinking. I don't actually expect anyone would buy into this notion given the lack of convincing reason to do so.

That being said, is it at all possible that both length-contraction and time-dilation are due to some actual structural relationship between spacetime, energy(EM in particular), and matter? Some time ago, I came to the peculiar conclusion that all of relativity, special or otherwise, the general energy equation (E = sqrt((mc^2)^2 + (pc)^2)), and some of the weird particles that show up in accelerators after a smash could all be explained by spacetime having some kind of structure that gets distorted in various ways due to the presence of sufficient quantities of energy.

Before I even bother to try and explain what I'm thinking, let's try a thought experiment to test out the basic idea. What if we modified the train paradox just a bit... Let's say the train is moving at 0.86666c (makes the numbers nice and easy). Let's get rid of the tunnel and replace it with a series of photo-sensors. Along the track at a spacing of 20m there will be 4 photo-eye sensors set up. The first two are set up so that when the first eye is crossed, a timer is started (one inside the train, and another one outside the train). When the second eye is crossed, the timers are stopped. The 3rd and 4th eyes are used to stop individual timers, starting when the eye is crossed and stopping when the eye is cleared, each having a timer both inside and outside the train. Just as a point of note, the timers inside the train are triggered by sensor panels along the outside of the train so that the signal travel distance between the clock and the sensor panel is negligible.

According to relativity, if the length of the train is L then to an outside observer, the length of the train will be 0.5L. My question is this: since a) the timers are being triggered by photo signals which move at the same speed regardless of reference frame, b) the photo signals are perpendicular to the motion of the train and as such, not subject to translative effects, will we, as a result of simple Euclidean math, register the shortened length for the train from the outside observers standpoint and the actual speed of the train from the conductors standpoint using the timers in their respective reference frames?

The reason this raises a question for me is because the triggering of these timer events happens at the speed of light. Also it's a simultaneousness problem where the events in question occur in two different reference frames. The original version of the train paradox used a problem where the two events both occurred in the same reference frame. As such, they were subject to relativistic skewing. But how does this work when the two events are in 2 different reference frames?

I can't even begin to validate what I've been thinking until I can at least understand this.

 

[Bill_K]

Arkain, The first thing you need to understand is what an "event" is. It's a single point in space and time. An event happens instantaneously, and you can't claim it's either moving or standing still. It's not in any way tied to a particular reference frame.

In your example, event A is "front of train passes eye 1". Event B is "front of train passes eye 2". Special relativity defines an invariant interval between two events, namely (Δx)² - c²(Δt)². Invariant means it has the same value in every reference frame.

(Δx)² - c²(Δt)² = (Δx')² - c²(Δt')²

Let's say that x', t' refer to the train's frame of reference and x, t refer to the station's frame of reference. In the train's frame, both events A and B occur at the same place, namely the front of the train. So Δx' = 0. In the station's frame, A and B occur a distance Δx = v Δt apart. So

(v Δt)² - c² (Δt)² = - c²(Δt')²

Solving for Δt' we get the usual time dilation relation:

Δt' = Δt sqrt(1 - v²/c²)

 

[arkain]

I understand the definition of an event as a coordinate in 4D Minkowski? space. And I was hoping someone would come back with an answer like that because this is were things get fun.

Since both the AB timer in side and the AB timer outside of the train are started simultaneously (ie, they constitute a single event), they should both be measuring the same interval of time. Let's add a quirk now. Suppose the same signal that stops the AB timers (event B) also causes a camera inside the train to take a picture of the timer outside the train so that the captured image represents the time as recorded from outside the train of when the cut-off event happened inside the train. Some time later, when the train has stopped, the conductor compares this picture with the AB timer that was outside the train.

According to relativity, from the conductor's point of view when the train was running, time outside the train was moving slow. So if T is the time from the AB timer inside the train at event B, the photo should show time 0.5T. However, when the conductor looks at the AB timer that was outside the train after stopping, it shows a time of 2T. What did I get wrong here? There's no way that decelerating the train caused a speed-up in time outside the train since there's no way to move slower that rest-speed for a given reference frame.

 

[Doc Al]

You'll need to precisely define your scenario. Where exactly are the timers? How are they turned on and turned off?

Before I even bother to try and explain what I'm thinking, let's try a thought experiment to test out the basic idea. What if we modified the train paradox just a bit... Let's say the train is moving at 0.86666c (makes the numbers nice and easy). Let's get rid of the tunnel and replace it with a series of photo-sensors. Along the track at a spacing of 20m there will be 4 photo-eye sensors set up. The first two are set up so that when the first eye is crossed, a timer is started (one inside the train, and another one outside the train).

One timer at the front of the train? Another timer at some location along the track? There's no problem in having both timers start as they pass each other--but stopping them is another matter.

When the second eye is crossed, the timers are stopped.

The stopping of the timer at the front of the train is clear enough, but the timer on the track is no longer co-located with the front of the train. You'll have to send a signal back to that timer to shut it off. You'll need to take that into account.

Since both the AB timer in side and the AB timer outside of the train are started simultaneously (ie, they constitute a single event), they should both be measuring the same interval of time.

Not really. While they start at the same time (since they are at the same location when they start), they don't stop at the same time. (See comments above.) And the train timer is seen to run slow according to track frame timers--and vice versa.

 

[Mentz114]

This exact scenario was proposed by someone a while back in this forum as a means of measuring 'absolute' velocity. I notice that arkain uses the phrase 'actual speed'. I should point out that we can only measure relative speed, so there's no need for him to disprove his conclusions, unless he believes there is such a thing as 'actual' velocity and his setup is measuring it.

 

[arkain]

Let me try and clarify a few things...

About the timer problem on the train. I knew someone was going to want to throw in a simultaneousness problem due to the length of the train. So here's how we fix this so that is no longer an issue. Put one of the timers at the nose of the train and the other at the tail. Let the signal that triggers the nose timer run to the tail timer. Also use a continuous sensor strip along the outside length of the train. The tail timer will not start unless only the nose timer signal is available. In this way, the nose timer will be started first. At that point, 2 signals will reach the tail timer and it will stay inactive. When the end of the train is reached and the outside sensor no longer reads a signal, the tail timer will start. The measured time will thus be the difference between the nose and tail timers. No relativity issues should be able to interfere with such a system.

When I referred to "actual speed," I meant with respect to the preferred rest frame, i.e. Earth. I've noticed in a lot of discussions on this board that there is difficulty in choosing such a preferred rest frame as GR and SR really do not even begin to attempt to specify one. I say there is no need for relativity to do so since one is defined in basic Newtonian physics, i.e. the frame of the object with the greatest rest mass. Since the mass of the train is insignificant when compared to that of Earth, the preferred frame for this thought experiment is that of the Earth.

 

[arkain]

Not really. While they start at the same time (since they are at the same location when they start), they don't stop at the same time. (See comments above.) And the train timer is seen to run slow according to track frame timers--and vice versa.

I said that "they should be measuring the same interval of time" for a reason, but I'm not yet ready to dig into that reasoning. I've already commented on the apparent time dilation from both perspectives and offered a question: Why did the photo show a time such that it isn't even in line with relativity calculations? Care to posit a response to that? If you think I've made a mistake in the reasoning behind that question, I'd love to hear it. That's the kind of thing I'm looking for.

 

[Mentz114]

Why did the photo show a time such that it isn't even in line with relativity calculations? Care to posit a response to that? If you think I've made a mistake in the reasoning behind that question, I'd love to hear it. That's the kind of thing I'm looking for.

If you did the calculations correctly, then your prediction of what the photo shows is the prediction of the theory.

I'm still not clear about your scenario - can you draw a spacetime diagram of the setup ?

Something like this - A and B are the worldlines of the timers/eyes the blue line is the worldline of the front of the train. The proper interval E1->E2 is the elapsed time on A's clock, the proper interval E1->E3 is the elapsed time on the clock at the font of the train and so on.

 

[Doc Al]

Let me try and clarify a few things...

About the timer problem on the train. I knew someone was going to want to throw in a simultaneousness problem due to the length of the train. So here's how we fix this so that is no longer an issue.

As long as the timers are spatially separated on the train, you'll have to deal with simultaneity. (In typical scenarios, the timers--clocks--would be synchronized in the train frame. Here you are using signals to start/stop the timers; you'll have similar issues to deal with.)

Put one of the timers at the nose of the train and the other at the tail.

You have two timers on the train, separated by the length of the train.

Let the signal that triggers the nose timer run to the tail timer.

This is unclear. What triggers the nose timer to start? Its passing a certain point on the track, right? What triggers it to stop? Realize that any signal sent from nose to tail takes time to travel.

Also use a continuous sensor strip along the outside length of the train.

What is this sensor sensing? Again, there are signals involved that will have to travel to the timers; that travel time must be considered.

The tail timer will not start unless only the nose timer signal is available. In this way, the nose timer will be started first. At that point, 2 signals will reach the tail timer and it will stay inactive. When the end of the train is reached and the outside sensor no longer reads a signal, the tail timer will start. The measured time will thus be the difference between the nose and tail timers.

Sorry, but your scenario is still unclear. Keep it simple. Describe exactly what triggers each timer to start and to stop.

No relativity issues should be able to interfere with such a system.

Sorry, no way to sidestep 'relativity issues' here.

I said that "they should be measuring the same interval of time" for a reason, but I'm not yet ready to dig into that reasoning. I've already commented on the apparent time dilation from both perspectives and offered a question: Why did the photo show a time such that it isn't even in line with relativity calculations? Care to posit a response to that? If you think I've made a mistake in the reasoning behind that question, I'd love to hear it. That's the kind of thing I'm looking for.

There's really no point in attempting to give a detailed response when your setup is still unclear.

 

[arkain]

Doc Al,
I'm not sure what's left unclear here... But let me try one more time.

The reason I introduced a second sensor B is because if I could actually perform this experiment, I'd want multiple data samples. So for now, since this alone seems to be such a point of contention, let's just strip it down to just a single sensor. At some point A along the track is a standard electronic eye type sensor. When the nose of the train crosses the sensor line, the timer outside of the track is triggered. At the same time, that same laser from the electronic eye sensor is now in contact with a sensor panel capable of detecting the presence of that laser. When the laser is detected at the front of the train, the nose timer is triggered along with a continuous signal which is being sent to the timer at the rear of the train from the nose timer. The sensor panel that triggered the nose timer runs along the entire length of the train, continuously detecting the fact that the given sensor is currently occluded by the train. This panel provides a secondary output signal to the timer at the rear of the train. When the train completely passes the given sensor, the secondary signal being provided from the sensor panel stops, leaving only the continuous signal from the nose timer. This state causes the rear timer to activate. At the same, the re-established electronic eye signal causes the outside timer to stop.

So regardless of your FoR, the timer at the nose of the train is started when the nose of the train breaks the electronic eye signal, and the timer at the rear of the train is started when the train no longer occludes the electronic eye. The outside timer measures the amount of time this takes in the FoR of the Earth while the difference between the nose and rear timers measures the amount of time this takes in the FoR of the train.

Since the signals involved propagate faster than the train is moving, and the effective propagation distance to start each timer inside the train is 0, the distance between the two timers inside the train has no relativistic bearing on the amount of time measured.

If you still wish to balk at the setup. I'll make one more modification. Suppose there is just a single electronic eye sensor at the front of the train and another one at the tail such that when the nose of the train crosses the electronic eye, the nose timer starts. No signals are exchanged between the front and rear of the train. Just as the train is about to clear the electronic eye, the emitted beam crosses the tail sensor, activating the timer at the rear of the train. Now you have 2 distinct events with no inhibiting relationship between them and a dx of 0 for each event as all events occur at position A along the track.

Is this setup clear enough?

 

[Doc Al]

Doc Al,
I'm not sure what's left unclear here... But let me try one more time.

The reason I introduced a second sensor B is because if I could actually perform this experiment, I'd want multiple data samples. So for now, since this alone seems to be such a point of contention, let's just strip it down to just a single sensor. At some point A along the track is a standard electronic eye type sensor. When the nose of the train crosses the sensor line, the timer outside of the track is triggered. At the same time, that same laser from the electronic eye sensor is now in contact with a sensor panel capable of detecting the presence of that laser. When the laser is detected at the front of the train, the nose timer is triggered along with a continuous signal which is being sent to the timer at the rear of the train from the nose timer. The sensor panel that triggered the nose timer runs along the entire length of the train, continuously detecting the fact that the given sensor is currently occluded by the train. This panel provides a secondary output signal to the timer at the rear of the train. When the train completely passes the given sensor, the secondary signal being provided from the sensor panel stops, leaving only the continuous signal from the nose timer. This state causes the rear timer to activate. At the same, the re-established electronic eye signal causes the outside timer to stop.

OK. I understand how the track timer at A is started and stopped. And how the train timers are started. (Too bad you don't show how the train timers are stopped. That might clear things up for you.)

So regardless of your FoR, the timer at the nose of the train is started when the nose of the train breaks the electronic eye signal, and the timer at the rear of the train is started when the train no longer occludes the electronic eye.

OK

The outside timer measures the amount of time this takes in the FoR of the Earth

OK.

while the difference between the nose and rear timers measures the amount of time this takes in the FoR of the train.

Since those timers are not co-located, you'll need some procedure (with synchronized clocks, or their equivalent) to determine their difference. But let's say they work that out: The difference between the timer readings will be the travel time for the train to pass point A according to the train frame. OK.

Since the signals involved propagate faster than the train is moving, and the effective propagation distance to start each timer inside the train is 0, the distance between the two timers inside the train has no relativistic bearing on the amount of time measured.

Sure it does. You can't just ignore the distance between the timers--after all, you're using them to measure the speed of the train, I suppose. Of course in your example the distance of 20m is comical at a speed of 0.866c. These would have to be incredibly precise timers! Make the train light-years long and see what happens.

If you still wish to balk at the setup. I'll make one more modification. Suppose there is just a single electronic eye sensor at the front of the train and another one at the tail such that when the nose of the train crosses the electronic eye, the nose timer starts. No signals are exchanged between the front and rear of the train. Just as the train is about to clear the electronic eye, the emitted beam crosses the tail sensor, activating the timer at the rear of the train. Now you have 2 distinct events with no inhibiting relationship between them and a dx of 0 for each event as all events occur at position A along the track.

Is this setup clear enough?

Sounds ok to me. So what? What do you conclude? (Don't try to sweep the distance between train timers under the rug--that's basically ignoring the issue.)

All measurements will prove consistent with relativity.

 

[arkain]

The first thing is that while the train is running, the train's timers are not stopped at all. Some synchronization method (pick one, doesn't matter) is used to stop the timers after the train is brought back into the same FoR as Earth. Since the timers are in locked step with respect to the train's FoR, the time difference between them will not change as a result of the deceleration. So when they are stopped in the Earth's FoR, they should still report the amount of time that passed for the train as it passed the electronic eye.

Yeah, 20m is a bit comical for that ridiculous speed, so I wonder which would be cheaper, 2 custom atomic clocks or a 2 light-second long train. With a train that long, you could use cheap stopwatches for the timers... but I don't think it'd fit linearly on the Earth's surface anymore. :)

Now don't get me wrong. I actually think the timers will show a difference corresponding to the relativity equations. My problem has never been with that. My problem with the model is that it doesn't provide a mechanism to enable all of this weirdness and relies heavily on very peculiar concepts like inconsistent simultaneousness of events. So in this thought experiment, what I've attempted to do is effectively drop both doors of the tunnel at the same time in both reference frames (referring to the original thought experiment).

When the conductor finally gets off the train, I expect that the Earth timer will read some time T. I likewise expect that when the difference between the 2 train timers is read, the result will be some time 0.5T. Let's presume a train length of L instead of the original 20m and try to remove some of the comedy. When the conductor uses this to calculate his speed, he finds a speed of v'=2L/T.

Now we get back to one of the reasons I was using multiple sensors. 2 sensors along the track with a spacing of D between them were used to verify the the train's speed of 0.86666c before it reached sensor A. So now, the conductor uses that speed and the outside timer to measure the length of his train as it appeared from outside the train, that being L'= 0.86666cT. Since the speed v' should also be 0.86666c, L'=2L.

Did you see the mistake I made? I certainly didn't. Where according to relativity, we'd expect to see L'=0.5L, we got the exact inverse. This is why I wanted to remove simultaneousness from the equation. By removing simultaneousness from the equation, there's no improper presumption regarding the synchronization of events.

Since the clocks inside the train were always both in the same FoR, it is impossible (barring mechanical failure) for the time between them to have skewed as would be the case for synchronized events. So what error explains the inverted results? The only possibility I can come up with is that either I royally screwed up this really simple algebra (v=dx/dt) or something is wrong with the model.

 

[Doc Al]

When the conductor finally gets off the train, I expect that the Earth timer will read some time T.

OK.

I likewise expect that when the difference between the 2 train timers is read, the result will be some time 0.5T.

No. Applying time dilation to the track timer--which is a moving clock in the frame of the train--the train timers will show a difference of 2T.

Let's presume a train length of L instead of the original 20m and try to remove some of the comedy.

OK.

When the conductor uses this to calculate his speed, he finds a speed of v'=2L/T.

No. The conductor will calculate his speed as L/2T. Which is exactly what the track observer will calculate for the speed of the train.

Did you see the mistake I made? I certainly didn't. Where according to relativity, we'd expect to see L'=0.5L, we got the exact inverse.

You misapplied time dilation.

This is why I wanted to remove simultaneousness from the equation. By removing simultaneousness from the equation, there's no improper presumption regarding the synchronization of events.

You can't 'remove' the issue of simultaneity.

 

[arkain]

No. Applying time dilation to the track timer--which is a moving clock in the frame of the train--the train timers will show a difference of 2T.

You misapplied time dilation.

Ok... now you've got me a little confused (and I think that's a good thing). By "track timer" I presume you mean the 2 timers inside the train. If so, then aren't they stationary WRT the train? If the frame inside the train is considered the moving frame, then isn't time inside the train running slower than time in Earth's FoR? The twin paradox says that the moving twin experiences less time. Therefore, while time T passed for Earth's FoR as the train crossed the sensor, time 0.5T passed inside the train during the same interval. So why would the train timers show a difference of 2T?

While I agree that your "correction" makes the numbers work, I'm failing to comprehend how this "correction" fits the model as (poorly) explained in so many physics books and classes.

 

[Doc Al]

Ok... now you've got me a little confused (and I think that's a good thing). By "track timer" I presume you mean the 2 timers inside the train.

No. By "track timer" I meant the timer on the track.

If so, then aren't they stationary WRT the train?

No, wrt the train the timer on the track is moving.

If the frame inside the train is considered the moving frame, then isn't time inside the train running slower than time in Earth's FoR?

Sure! The track/earth observer would say that the timers on the train run slow.

The twin paradox says that the moving twin experiences less time. Therefore, while time T passed for Earth's FoR as the train crossed the sensor, time 0.5T passed inside the train during the same interval. So why would the train timers show a difference of 2T?

The difference between the two timers on the train is not the same thing as a time measured on a single clock. You cannot apply time dilation to that time difference. (At least not in a simple way using the time dilation formula.)

The time dilation formula says that when a single moving clock shows an elapsed time of T, then the time according to the observer's clocks will show an elapsed time of 2T (when the speed is 0.866c, as in this example).

Luckily, there is only a single track clock in this scenario, so the train observers can apply the time dilation formula to it.

While I agree that your "correction" makes the numbers work, I'm failing to comprehend how this "correction" fits the model as (poorly) explained in so many physics books and classes.

It's tricky stuff. What books have you studied on this topic?

 

[arkain]

No. By "track timer" I meant the timer on the track.

Ok. So the "track-timer" is the single timer in Earth's FoR. Got it.

The difference between the two timers on the train is not the same thing as a time measured on a single clock. You cannot apply time dilation to that time difference. (At least not in a simple way using the time dilation formula.)

I understood that. This is why I decided to use 2 separate timers. Always keeping them in the same FoR but removing them from Earth's FoR, starting them with independent events, then bringing them both back into Earth's FoR, the difference between their start times should reflect the temporal difference in the other FoR between the events that started them.

The time dilation formula says that when a single moving clock shows an elapsed time of T, then the time according to the observer's clocks will show an elapsed time of 2T (when the speed is 0.866c, as in this example).

Luckily, there is only a single track clock in this scenario, so the train observers can apply the time dilation formula to it.

The reason I can't readily accept your response in this case is the fact that inside the train are 2 single moving clocks. If an observer outside the train takes note of the nose timer at the instant the Earth timer is stopped, that time will read as 1/2 of whatever is on the Earth timer. It should not and does not matter that a second timer was started on the train in that exact same instant.

Let's make things more symmetrical and put a second timer on the Earth a distance L from the first timer. L is still the rest length of the train. Put an eye laser on the train (doesn't matter where), and an eye sensor on each timer so that the timers are only started as the eye laser crosses them. In this way, time intervals outside the train are being measured just like time intervals inside the train. Remember, in this modified scenario, none of the 4 timers are stopped until they are all brought into the same FoR and synchronously stopped. In this scenario, which pair of timers will show the dilation, the Earth pair or the train pair?

Here's the problem with your response in another form. What you said basically claims that for the twin paradox it's equally valid to claim the ship as the preferred FoR and say that the twin left behind on Earth barely aged at all. Here's what I do get. When 2 FoR separate due to some force, one of those 2 frames is going to experience less time than the other. Guaranteed. Here's what I don't get. In the absence of any explanatory mechanics in the model, and given that the model provides no preference for either FoR, how can you predict which FoR experienced less time once the two FoR are brought back in sync by another force?

It's tricky stuff. What books have you studied on this topic?

More books than I can remember the names for. At least 5 different college-level textbooks and some 15 or so books from various libraries. Not to mention several websites.

 

[Mentz114]

In the absence of any explanatory mechanics in the model, and given that the model provides no preference for either FoR, how can you predict which FoR experienced less time once the two FoR are brought back in sync by another force?

(my emphasis)

Surely, somewhere in those 20 books must have been mentioned the proper time, or proper interval along a worldline, and the postulate that this quantity is the same as the elapsed time on a clock traveling along the worldline.

So there exists a straightforward way of calculating exactly what the 'twins' clocks will read when they are re-united. Always.

 

[Doc Al]

I understood that. This is why I decided to use 2 separate timers. Always keeping them in the same FoR but removing them from Earth's FoR, starting them with independent events, then bringing them both back into Earth's FoR, the difference between their start times should reflect the temporal difference in the other FoR between the events that started them.

Yes, the time difference between the two train timers represents the travel time according to the train frame. But not according to Earth observers, who disagree that the timers were stopped at the same time. (Simultaneity again.)

The reason I can't readily accept your response in this case is the fact that inside the train are 2 single moving clocks. If an observer outside the train takes note of the nose timer at the instant the Earth timer is stopped, that time will read as 1/2 of whatever is on the Earth timer.

That's exactly right! According to the Earth frame, at the moment the Earth timer shuts off the nose timer reads 1/2 what the Earth timer reads. But that's not the same as the difference between the two train timers.

According to the Earth frame, the nose timer was not shut off at the same time as the rear timer. According to the Earth frame, the nose timer keeps running for some time after the rear timer has been shut off.

It should not and does not matter that a second timer was started on the train in that exact same instant.

Since you're using the time difference between two timers, of course it matters.

Let's make things more symmetrical and put a second timer on the Earth a distance L from the first timer. L is still the rest length of the train. Put an eye laser on the train (doesn't matter where), and an eye sensor on each timer so that the timers are only started as the eye laser crosses them. In this way, time intervals outside the train are being measured just like time intervals inside the train. Remember, in this modified scenario, none of the 4 timers are stopped until they are all brought into the same FoR and synchronously stopped. In this scenario, which pair of timers will show the dilation, the Earth pair or the train pair?

Sorry, I don't understand this new scenario. Especially the sentence I highlighted. I recommend sticking to your original scenario, which is simpler.

Here's the problem with your response in another form. What you said basically claims that for the twin paradox it's equally valid to claim the ship as the preferred FoR and say that the twin left behind on Earth barely aged at all. Here's what I do get. When 2 FoR separate due to some force, one of those 2 frames is going to experience less time than the other. Guaranteed. Here's what I don't get. In the absence of any explanatory mechanics in the model, and given that the model provides no preference for either FoR, how can you predict which FoR experienced less time once the two FoR are brought back in sync by another force?

Sorry, but I'm not sure what your scenario has to do with the twin paradox. To make a 'twin paradox' situation, have the nose and Earth timers start together (as you have done), let the train keep going for a good distance, then have it turn around and come back to the starting point. Then you can compare the nose timer with the Earth timer. You'll find that the nose timer shows less elapsed time, just as you'd expect.

More books than I can remember the names for. At least 5 different college-level textbooks and some 15 or so books from various libraries. Not to mention several websites.

You need to review the basics of time dilation and clock synchronization.

 

[arkain]

Yes, the time difference between the two train timers represents the travel time according to the train frame. But not according to Earth observers, who disagree that the timers were stopped at the same time. (Simultaneity again.)

Doc, please go back and re-read the thread, the train's timers are not stopped until after the train is brought back into the same FoR as Earth. So according to ALL observers, the 2 timers were indeed stopped simultaneously. The 2 timer method merely starts the timers in response to a particular event. I never said that the nose timer on the train was stopped when the single Earth timer was stopped.

That's exactly right! According to the Earth frame, at the moment the Earth timer shuts off the nose timer reads 1/2 what the Earth timer reads. But that's not the same as the difference between the two train timers.

Key question: If the timer at the rear of the train is started in the same instant as the Earth timer is stopped, which is also the same instant that the nose timer was observed by the Earth observer (which read 0.5T), why would the interval between the 2 train timers be anything other than 0.5T? Put another way, if a pair of cameras were set up further down the track, and somehow setup to take a simultaneous (in Earth's frame) snapshot of the front and rear timers when the train's timers are lined up with the cameras, would the difference between the time recorded in those pictures show anything other than an interval of 0.5T?

According to the Earth frame, the nose timer was not shut off at the same time as the rear timer. According to the Earth frame, the nose timer keeps running for some time after the rear timer has been shut off.

Please re-evaluate the situation as the trains timers are not stopped until after the train returns to Earth's FoR.

Since you're using the time difference between two timers, of course it matters.

What I was trying to say is that the measurement of elapsed time as seen from the train's FoR should be irrespective of the number of timers used to make that measurement.

Sorry, I don't understand this new scenario. Especially the sentence I highlighted. I recommend sticking to your original scenario, which is simpler.

I recommend accepting the new scenario as the extra complexity removes simultaneity considerations. The new scenario merely supplies the Earth FoR with the same timing mechanism as the train FoR. Where in the Earth FoR, an electronic eye fixed to the track triggers the train's timers in succession as the train passes by, in the train FoR, an electronic eye fixed to the train triggers the Earth's timers in succession as the Earth passes by. None of the the timers are ever stopped until the train (Earth) comes to a complete stop with respect to the Earth (train). When all 4 timers are in the same FoR, all 4 timers are stopped synchronously. I can't explain it any easier than this.

Sorry, but I'm not sure what your scenario has to do with the twin paradox. To make a 'twin paradox' situation, have the nose and Earth timers start together (as you have done), let the train keep going for a good distance, then have it turn around and come back to the starting point. Then you can compare the nose timer with the Earth timer. You'll find that the nose timer shows less elapsed time, just as you'd expect.

I brought that up to make a point. The twin on the ship experiences less time because the twin aboard the ship experienced acceleration. The simple fact that the ship's FoR became non-inertial for a period of time caused the expected alteration in the passage of time. Why is this relevant? In the scenarios that I've given, Earth's FoR is always inertial. Only the train experiences acceleration. Therefore only the train will incur a dilation of time.

The "correction" you applied earlier basically said that the Earth FoR incurred time dilation. Since the Earth did not accelerate, this cannot be the case.

 

[Dale]

Arkain, your approach here is unproductive. You are making an elaborately complicated scenario, and then (no surprise) finding that it is confusing. You should instead be looking to simplify your scenario to the simplest form that captures your question.

 

[Mentz114]

In the scenarios that I've given, Earth's FoR is always inertial. Only the train experiences acceleration. Therefore only the train will incur a dilation of time.

The "correction" you applied earlier basically said that the Earth FoR incurred time dilation. Since the Earth did not accelerate, this cannot be the case.

You are mixing up time dilation ( moving clocks run slower) and differential ageing. Time dilation is what moving observers see symmetrically with both observers seeing the others clock running slow. But differential ageing depends on the proper length of the worldline between two events.

 

[arkain]

Arkain, your approach here is unproductive. You are making an elaborately complicated scenario, and then (no surprise) finding that it is confusing. You should instead be looking to simplify your scenario to the simplest form that captures your question.

Actually, in case you hadn't noticed, I'm quite clear as to what happens in my scenario. What I find confusing is the fact that 1) no one else seems to be able to draw the same conclusion as I have, and 2) no one can adequately explain the error in my understanding without reverting to a "Well, that's just how it is" type comment. If someone could just give me a rigorous reason why my conclusions are incorrect, then I'd have no difficulty at all.

What's more is that my scenario is not very elaborate or very complicated. I've actually created timing setups very similar to the one I've described here, but for different purposes. The only thing that complicates this problem is the choice of a preferred FoR. The fact that Doc Al wanted to use the train as the preferred FoR defies rationality to me, especially given that if an accelerometer were placed on the Earth and in the train, only the one in the train would ever register any acceleration.

As such, time dilation is only in effect for the train and my previous calculation of L'=2L stands. The fact that this does not agree with what relativity claims is an oddity that, at least for me, means that either I've screwed up royally or something is horribly incorrect about the model. While I'm prepared to accept the latter as true, I'd much rather believe the former considering that people who's talent, knowledge, and insight far exceed my own (I assume) have been working on this for far longer than I. Hence the series of questions.

In short, I just want someone to show me that I've made a mistake without making one themselves in the process.

 

[Dale]

What's more is that my scenario is not very elaborate or very complicated.

yes, it is, and the description is unclear also. I can't follow it and Doc Al can't either. If someone of his caliber has trouble then the scenario is not well described.

 

[arkain]

You are mixing up time dilation ( moving clocks run slower) and differential ageing. Time dilation is what moving observers see symmetrically with both observers seeing the others clock running slow. But differential ageing depends on the proper length of the worldline between two events.

You might be on to something. This might be where my mistake is. Then again...

Since "world lines" are paths through Minkowski space (4D spacetime) where non-linear portions represent acceleration, it's not simply the length of the world line but rather the difference in the temporal component of the ending event coordinates when mapped from the "moving" frame to the "rest" frame that account for differential aging. Fun fact: the temporal coordinates will not differ unless the world line in the moving frame passed through a different span of the temporal axis than did the world line in the rest frame. I.e. differential aging is a side effect of time dilation.

The time dilation inside the train as seen from the Earth-bound observer is real. Time literally is creeping along inside the train, or rather, the train is crossing temporal coordinates at a decreased rate. Time dilation on Earth as seen from inside the train is merely an "optical illusion" induced/released by the train's acceleration and maintained by the consistent velocity of the train.

 

[arkain]

yes, it is, and the description is unclear also. I can't follow it and Doc Al can't either. If someone of his caliber has trouble then the scenario is not well described.

Try this. Specify exactly what is unclear about the description of the scenario. I still don't get what's so difficult to comprehend about this scenario. I've attached a rather simple image to help with the re-explanation.

The turquoise rectangle is the train moving to the right.
The green circles are the timers.
The blue squares are the electronic eye receivers, wired to start the respective adjacent timers.
The red squares are the electronic eye emitters, which are always on.

Procedure:

1. Reset all timers to 0.
2. Load one timer into the head of the train, wired to start when the receiver at the front of the train senses an emitter.
3. Load a second timer into the tail of the train, wired to start when the tail receiver senses an emitter.
4. Place 2 more timer/receiver pairs further down the track, spacing them by L (the rest length of the train).
5. Place an emitter on the train such that it can cross both previously placed track receivers as the train moves.
6. Place an emitter along the track such that it can cross both receivers mounted on the train as the train moves.
7. Accelerate the train to 0.8666c, ensuring that the train reaches and maintains this speed before the train-mounted emitter reaches the nearest track-mounted receiver.
8. Stop the train after all 4 timers have been started.
9. Bring the 2 timers from the train and the 2 timers from the track to a workstation..
10. Stop all 4 timers simultaneously.
11. Record the interval between the 2 timers from the train.
12. Record the interval between the 2 timers from the track.

This breakdown is akin to a freshman physics experiment. If the setup of this experiment still cannot be understood from this, then you may be right. I may be asking a question too complicated for this forum. Unfortunately, I cannot conceive of a simpler experiment that would not suffer simultaneity issues. To be honest, the 4 timer version is actually simpler to work with than the 3 timer version due to the fact that each of the 4 timers can be activated by its own separate event where with the 3 timer version, 1 timer shares its events with the other 2 timers.

 

[Calrid]

Correct me if I am wrong but in that situation all measurements would be relatively accounted for by the transformations.

Are you saying that there actually is a paradox or that there is an unexpected result?

I agree with what someone said, actual time is a nonsense anyway, seems to me its just another way of saying absolute time which is of course forbidden.

There is a simultaneity issue with the sensors/emitters as well because the clock and the sensors are in different frames of reference.

It might be a good idea to know if you think there is. I have yet to see a "paradox" in this area that actually is one.

Just another angle, sorry if it is not apt.

 

[arkain]

Correct me if I am wrong but in that situation all measurements would be relatively accounted for by the transformations.

Are you saying that there actually is a paradox or that there is an unexpected result?

I agree with what someone said, actual time is a nonsense anyway, seems to me its just another way of saying absolute time which is of course forbidden.

It might be a good idea to know if you think there is. I have yet to see a "paradox" in this area that actually is one.

Just another angle, sorry if it is not apt.

I don't think you're wrong, hence the questions I have. Every time I run this thought experiment, time in the train slows down. So when I stop all timers, the timers on the track give dt=T and the timers from the train give dt=0.5T. What doesn't seem to fit is that this results in L' = 2L where L is the rest length of the train and L' is its length at 0.86666c. I can accept that this means I've done something wrong. I just need someone to explain what that is.

Doc Al posited that if dt from the track timers is T then dt from the train timers should be 2T. However, this implies that time was accelerated for the moving object (the train). Doc's argument favored using the train as the "rest" frame to make the calculations. However, since the train undergoes acceleration to reach the needed speed, its frame is not always inertial and as such is not a valid candidate to use as the "rest" frame.

 

[Doc Al]

Doc, please go back and re-read the thread, the train's timers are not stopped until after the train is brought back into the same FoR as Earth.

Careful here, as you are now introducing acceleration into the mix. If the train manages to slow down in such a way that clock synchronization is preserved in the train, the Earth observers will not agree. There's no way to avoid the issue of simultaneity.

So according to ALL observers, the 2 timers were indeed stopped simultaneously.

That's irrelevant once the train has come to rest. Stick to time measurements made in the original frames!

The 2 timer method merely starts the timers in response to a particular event. I never said that the nose timer on the train was stopped when the single Earth timer was stopped.

Neither did I.

Key question: If the timer at the rear of the train is started in the same instant as the Earth timer is stopped, which is also the same instant that the nose timer was observed by the Earth observer (which read 0.5T), why would the interval between the 2 train timers be anything other than 0.5T?

Because 'at the same instant' is different for different frames. In the train frame, the difference between the timer readings represents the time it takes for the train to pass the Earth timer; the Earth observers disagree.

Put another way, if a pair of cameras were set up further down the track, and somehow setup to take a simultaneous (in Earth's frame) snapshot of the front and rear timers when the train's timers are lined up with the cameras, would the difference between the time recorded in those pictures show anything other than an interval of 0.5T?

Yes, as I explained in an earlier post.

Please re-evaluate the situation as the trains timers are not stopped until after the train returns to Earth's FoR.

As stated above, that just unnecessarily complicates things. All your claims about the time difference between the train timers were made in the frame of the moving train anyway.

I recommend accepting the new scenario as the extra complexity removes simultaneity considerations. The new scenario merely supplies the Earth FoR with the same timing mechanism as the train FoR. Where in the Earth FoR, an electronic eye fixed to the track triggers the train's timers in succession as the train passes by, in the train FoR, an electronic eye fixed to the train triggers the Earth's timers in succession as the Earth passes by. None of the the timers are ever stopped until the train (Earth) comes to a complete stop with respect to the Earth (train). When all 4 timers are in the same FoR, all 4 timers are stopped synchronously. I can't explain it any easier than this.

You'll have to try, as I don't understand it. (I haven't looked at your post #25 yet. One scenario at a time.)

I brought that up to make a point. The twin on the ship experiences less time because the twin aboard the ship experienced acceleration. The simple fact that the ship's FoR became non-inertial for a period of time caused the expected alteration in the passage of time. Why is this relevant? In the scenarios that I've given, Earth's FoR is always inertial. Only the train experiences acceleration. Therefore only the train will incur a dilation of time.

The "correction" you applied earlier basically said that the Earth FoR incurred time dilation. Since the Earth did not accelerate, this cannot be the case.

You have a basic misunderstanding of 'time dilation'. Time dilation is a relationship between frames in relative motion. Both frames see the other's clocks/timers as running slow. Time dilation works both ways!

(As Mentz114 says, you are mixing up time dilation with differential ageing.)

 

[Dale]

Procedure:

1. Reset all timers to 0.
2. Load one timer into the head of the train, wired to start when the receiver at the front of the train senses an emitter.
3. Load a second timer into the tail of the train, wired to start when the tail receiver senses an emitter.
4. Place 2 more timer/receiver pairs further down the track, spacing them by L (the rest length of the train).
5. Place an emitter on the train such that it can cross both previously placed track receivers as the train moves.
6. Place an emitter along the track such that it can cross both receivers mounted on the train as the train moves.
7. Accelerate the train to 0.8666c, ensuring that the train reaches and maintains this speed before the train-mounted emitter reaches the nearest track-mounted receiver.
8. Stop the train after all 4 timers have been started.
9. Bring the 2 timers from the train and the 2 timers from the track to a workstation..
10. Stop all 4 timers simultaneously.
11. Record the interval between the 2 timers from the train.
12. Record the interval between the 2 timers from the track.

Thanks, that helps clarify considerably. In step 7 (Accelerate the train to 0.8666c) you need to specify the acceleration profiles for the front and the back timers, or at a minimum tell whether or not there is any change in the rest length of the train after the acceleration. In step 8 (Stop the train after all 4 timers have been started) you need to specify the deceleration profiles for the front and rear timers as this will affect the results. Please note that the acceleration/deceleration profiles will not in general be the same, particularly if you are considering Born rigid deceleration.

EDIT: I also just noticed that in step 9 you need to specify the motion of all 4 timers from the various locations where they stopped to the workstation since that will also affect the outcome.

I reiterate my advice to find a simpler scenario, the description is much clearer now but the proposed scenario is overly complicated. There is so much going on in this scenario that I doubt you will learn much from the answer. Particularly since it depends on the details of several specific acceleration profiles, so it is not terribly generalizable.

 

[Doc Al]

Doc Al posited that if dt from the track timers is T then dt from the train timers should be 2T.

Note that we are talking about a time interval measured on a single track timer. That is key! According to the train observers, when that track timer shows an elapsed time of T, then the train timers will show a time difference of 2T. Time dilation in action!

Of course, the Earth observers disagree. According to Earth observers, it's the train clocks/timers that are running slow and that are not synchronized.

And since both train and track observers have studied relativity and understand how measurements transform between frames in relative motion, they are not puzzled by this. Everything works exactly as relativity predicts.

However, this implies that time was accelerated for the moving object (the train). Doc's argument favored using the train as the "rest" frame to make the calculations.

Each frame views itself as being the 'rest' frame and the other frame as 'moving'. It's the relative motion between the frames that's the key.

However, since the train undergoes acceleration to reach the needed speed, its frame is not always inertial and as such is not a valid candidate to use as the "rest" frame.

Again, forget about acceleration--you're just adding complications that only serve to hide what's going on. In your scenario the train is already moving at 0.866c. It's a lovely inertial frame. You used it yourself when you described how the difference between the train timers would show the travel time in the train frame. (That's the frame of the train while it's moving with respect to the earth.)

 

[arkain]

Careful here, as you are now introducing acceleration into the mix. If the train manages to slow down in such a way that clock synchronization is preserved in the train, the Earth observers will not agree. There's no way to avoid the issue of simultaneity.

That's irrelevant once the train has come to rest. Stick to time measurements made in the original frames!

Neither did I.


Because 'at the same instant' is different for different frames. In the train frame, the difference between the timer readings represents the time it takes for the train to pass the Earth timer; the Earth observers disagree.

Yes, as I explained in an earlier post.


As stated above, that just unnecessarily complicates things. All your claims about the time difference between the train timers were made in the frame of the moving train anyway.


You'll have to try, as I don't understand it. (I haven't looked at your post #25 yet. One scenario at a time.)


You have a basic misunderstanding of 'time dilation'. Time dilation is a relationship between frames in relative motion. Both frames see the other's clocks/timers as running slow. Time dilation works both ways!

(As Mentz114 says, you are mixing up time dilation with differential ageing.)

Ok. I think we've finally uncovered one of my largest mistakes: Improper use of terminology. Problem: for differential aging to occur, the FoR that experienced acceleration must have traveled less along the time axis than the "rest" frame. Worded differently, differential aging occurred because the train experienced less time than the Earth did between the initial and ending rest (WRT Earth) periods of the train. This "actual" relative decrease in the passage of time for the train is what I've been referring to as time-dilation. If this was an incorrect term usage, then I apologize for the confusion.

What I've been trying to measure is the interval of differential time as seen from each frame. Put another way, I'm trying to measure, from train's perspective how long the train took to pass a fixed point on the Earth (Tt), and from the Earth's perspective how long a length of Earth equal to the rest length of the train took to pass a fixed point on the train (Te). My expectation is that Te = 2 * Tt for a relative velocity of 0.86666c.

Was this a better use of terminology? Did it help relieve the confusion about what I'm trying to do? Is my expectation incorrect? Is there anything improper about the experiment itself?

 

[arkain]

Thanks, that helps clarify considerably. In step 7 (Accelerate the train to 0.8666c) you need to specify the acceleration profiles for the front and the back timers, or at a minimum tell whether or not there is any change in the rest length of the train after the acceleration. In step 8 (Stop the train after all 4 timers have been started) you need to specify the deceleration profiles for the front and rear timers as this will affect the results. Please note that the acceleration/deceleration profiles will not in general be the same, particularly if you are considering Born rigid deceleration.

EDIT: I also just noticed that in step 9 you need to specify the motion of all 4 timers from the various locations where they stopped to the workstation since that will also affect the outcome.

I reiterate my advice to find a simpler scenario, the description is much clearer now but the proposed scenario is overly complicated. There is so much going on in this scenario that I doubt you will learn much from the answer. Particularly since it depends on the details of several specific acceleration profiles, so it is not terribly generalizable.

Yay! More progress! I was beginning to get worried that I'm irrecoverably off my gourd! :)

First thing to note is that in this thought experiment, regardless of the length L of the train, the train itself is to be considered perfectly rigid. There should be no time t at which the Born rigidity should be broken. I know that this is impossible and violates more rules than I'd care to count, but the only way around this is to use 2 separate, identical trains of unimportant but equal length, both with timers in the nose and with an initial separation between the trains of length L. Then you'd have to set up some kind of simultaneous launch of both trains. Both trains would need to have previously synchronized timers used to control the stopping of both trains so that they both shut down at the same time (in their FoR). While such a convoluted thing could be done and doesn't violate any rules, it's way simpler to simply consider 1 perfectly rigid train. It'll have the same effect on the experiment. Put simply, the timers in motion need to experience identical acceleration and deceleration profiles relative to each other. They need to remain in the same frame WRT each other at all points during this experiment.

The motion of the timers to the workstation would be non-relativistic (human walking speed WRT Earth) and should not significantly affect the timers.

 

[Dale]

First thing to note is that in this thought experiment, regardless of the length L of the train, the train itself is to be considered perfectly rigid. ...

Put simply, the timers in motion need to experience identical acceleration and deceleration profiles relative to each other.

These are contradictory requirements. If the ship is undergoing Born-rigid motion then the acceleration profiles will be different for the front and the back. In either case, it is necessary to completely specify the exact details of the acceleration profile because the answer depends on the result.

To determine the time accumulated on a clock in general you use the following approach. First, you write down the position of the clock as a function of time x(t). Then you evaluate the following integral:

$$\int_a^b \sqrt{1-\frac{1}{c^2}\left(\frac{dx}{dt}\right)^2} \, dt$$

Or you can generalize it even further by writing the position and time coordinate as a function of some arbitrary parameter. Then you would evaluate this integral.

$$\int_a^b \sqrt{\left(\frac{dt}{d\lambda}\right)^2-\frac{1}{c^2}\left(\frac{dx}{d\lambda}\right)^2} \, d\lambda$$

Which obviously reduces to the first integral if time is the arbitrary parameter.

So, unless you specify x(t) for each clock, the question cannot be answered other than by pointing to these general formulas. Once you specify x(t) for each clock then the question is simply a matter of computation.

The motion of the timers to the workstation would be non-relativistic (human walking speed WRT Earth) and should not significantly affect the timers.

You can specify this as "slow transport" or "ultra-slow transport".

 

[Doc Al]

What I've been trying to measure is the interval of differential time as seen from each frame.

Despite talk about accelerating and decelerating the train (which complicates things, as DaleSpam points out), you still seem to just want to measure time intervals from the two frames when they are in relative motion. Not really 'differential aging' of anything.

Put another way, I'm trying to measure, from train's perspective how long the train took to pass a fixed point on the Earth (Tt), and from the Earth's perspective how long a length of Earth equal to the rest length of the train took to pass a fixed point on the train (Te).

Is this a new scenario? As I understood it, in your original scenario you wanted to compare:
(1) From the train frame, the time it takes for a point A on the Earth to travel the length of the train.
(2) From the Earth frame, the time it takes for the train to pass point A.

The two events defining this time span are: (E1) The nose of the train passes point A and (E2) The tail of the train passes point A. This is perfectly well-defined.

My expectation is that Te = 2 * Tt for a relative velocity of 0.86666c.

No, as I explained earlier it will be just the opposite. Tt = 2*Te. Time dilation applies from either frame, but only in the Earth frame are we recording the time elapsed on a single clock. Note that both events take place at location A, so the single Earth timer is able to measure that time interval. Not so in the train frame, where multiple clocks are required.

If you mean to change the scenario as you described above, it seems you now want to compare two different time intervals:
(1) From the train frame, the time it takes point A on the Earth to travel the length of the train (which has proper length L).
(2') From the Earth frame, the time it takes the nose of the train to travel a distance L along the tracks.

In this case, both time intervals are the same: Te = Tt. (How could they not be? The situation is perfectly symmetric.) Note that (2') is not the same as (2); (2') requires the use of two clocks/timers in the Earth frame. Note that these time intervals do not correspond to the time between the same two events:

For (1), the events defining the time span are exactly what they were before, (E1) and (E2).
For (2'), the events defining the time span are: (E1) The nose of the train passes point A and (E3) The nose of the train passes point B, where A and B are a distance L apart in the Earth frame.

Was this a better use of terminology? Did it help relieve the confusion about what I'm trying to do? Is my expectation incorrect? Is there anything improper about the experiment itself?

See my comments above.

Note: In my opinion it is essential that you understand everything described above. This is just basic relativity, so any additional complications (such as having the train accelerate and decelerate) will only be that much harder to unravel.

 

[arkain]

Again, forget about acceleration--you're just adding complications that only serve to hide what's going on. In your scenario the train is already moving at 0.866c. It's a lovely inertial frame. You used it yourself when you described how the difference between the train timers would show the travel time in the train frame. (That's the frame of the train while it's moving with respect to the earth.)

I'm not trying to hide anything at all, especially not with undue complexity. What I've been saying in a nutshell is this:

2 identical timers some distance L apart are accelerated to 0.86666c WRT Earth using identical acceleration profiles. The 2 timers maintain this velocity and eventually cross point P WRT Earth, at which point, the timer crossing P is started. At some time after both timers are started, both timers are decelerated to 0c WRT Earth using identical acceleration profiles.

Unless what I'm about to say is another of my mistakes, the following should be true:

1) the 2 timers were always in the same FoR.
2) the timer FoR was non-inertial during the acceleration and deceleration phases.
3) both timers were started while the timer FoR was in an inertial state.
4) before the deceleration phase starts, the difference between the two timers as seen from the timer's FoR represents the amount of time in that FoR between the starting of the 2 timers.
5) (and this seems to be part of the point of contention here) since 1 is true, 4 must also be true after the deceleration phase ends.
6) when this time difference is compared with the time taken for both timers to cross the fixed point P (given that the crossing of the first timer is T=0) in Earth's FoR, the intervals will not be identical, thus indicating differential aging.
7) the timer FoR will show the shorter time interval.