# Relativity Paradox – RoS: Trains, Tunnels & Guillotines

1. May 4, 2013

Professor Mike Merrifield from the University of Nottingham has made this excellent video for Sixty Symbols, explaining the apparent paradox of Relativity of Simultaneity:

* Relativity Paradox - Sixty Symbols *

After this video, it should be perfectly clear to everyone what happens. What looks illogical becomes perfectly consistent in the light of Relativity.

Now, my question is:

– What happens if we place two cameras at front & end of the train, directed towards the observer, recording the trip through the tunnel together with the time-code from two synchronized atomic clocks (at front & end), and also having the observer recording the trip through the tunnel. What will we see when we play the 3 videos? It can’t possibly show the observer filming a completely disappeared train, while the observer is also simultaneously being filmed from the front & end of the same disappeared train??

My guess is that the following will happen:

• The camera in the front of the train will record the observer, and then it gets black and then shortly after, the observer reappears when the front is out of the tunnel.

• The camera in the back of the train will record the observer during all the time above, and then it gets black and then shortly after, the observer reappears when the back is out of the tunnel.

• The observer will for a moment record the train disappearing completely in the tunnel.

• Now, the only way to reconcile this paradox is disagreements on timing/events, and this time the disagreement must also occur onboard the train within the same frame of reference...?
Is this correct? Or am I missing something??

P.S: I’m willing to alter the camera/atomic clock recording setup to a ‘centralized unit’ in the middle of the train... in case the answer is a trivial no-brainer...

Last edited by a moderator: Sep 25, 2014
2. May 4, 2013

### Staff: Mentor

What do you mean by "the observer"?

3. May 4, 2013

### bahamagreen

I think he means the observer at rest wrt the tunnel...

The two cameras at each end of the train are in the same inertial frame. Both are capturing images of the outside observer as the train approaches and exits the tunnel.

He is suggesting/asking; if the outside observer sees the train fully disappear from his view, then likewise the two cameras running in synchrony in their frame should record a period where neither camera can capture an image of the outside observer... but. if from within the train frame the tunnel is shortened so that one of the ends of the train (and therefore at least one of the cameras) sticks out from the tunnel, there would be no period during which the two synchronized cameras don't capture the image of the outside observer.
So, like the video said - the train either gets cut of it doesn't - he is suggesting/asking - do the films of the cameras upon examination indicate that the cameras both lose sight of the observer or that at least one of the cameras always had the observer in sight?

Now, although the two cameras are in synch wrt the train frame, all that means is that their clocks show the same rate and the two individual film frames may be lined up and examined to determine what they captured. But, it does not mean that they are capturing light images that left the observer at the same time - one camera is closer to the observer than the other before entering the tunnel, that relationship is reversed when exiting the tunnel.

The length of the train (distance between the cameras) and the lateral distance of the observer from the tunnel, a choice of frame from which to measure these, and some SR math with some trigonometry may show that the apparent paradox of the camera images of the observer will be accounted for.

At a high level, clearly you can see that when the first camera at the front of the train enters the tunnel and loses the image of the observer, the image going to the back camera that corresponds to the same time of emission from the observer is still in route to the back camera even when the length contraction of the distance to the observer is included; the front camera is recording an earlier image of the observer "at the same time" as the back camera... and this reverses when the cameras exit the tunnel.

The timing of when the observer sees the cameras does not correspond to what the cameras captured of the observer... you can't infer what was recorded on the cameras of the observer based on when and where the observer saw the cameras.

Maybe someone can demonstrate this formally.

4. May 4, 2013

Sorry Doc, my fault. In the video Mike is talking about “the trainspotter”. This is my “observer”, i.e. a stationary person filming the event from a distance. Here’s a picture with the 3 cameras (front/back is filming “the observer”):

Note: This is not what the trainspotter/observer will see according to the video.

EDIT: bahamagreen got it right, thanks!

5. May 4, 2013

Yep, exactly!

And there should be (or at least according to “my speculations”) a period when the “train cameras” are filming from both sides of the tunnel...

True, and we all know why the guillotines misses the train – because they are controlled by “the observer” and he is not in sync with the events at the tunnel/train. But the cameras change the game slightly...

Now we’re getting close... real close. My only 'rebuttal' would be – does it really matter? There should be a point when both “train cameras” is filming from both sides of the tunnel... no matter what photons hit the CCD... or...?

This is probably the answer. The trainspotter/observer will see the train contracted, and the guys on the train will see the tunnel contracted. So, the cameras will see... ouch this doesn’t compute in my gray clump... :uhh:

Yes, but what about my earlier 'rebuttal' on filming from both sides of the tunnel...

This is perfectly okay with me.

That would be great and very interesting!

Thanks
DA

6. May 4, 2013

### pervect

Staff Emeritus

Who synchronized the clocks. (And for extra credit, what experimental procedure did they use?)

The last may be a bit demanding, since I don't think the professor covered it in his lecture.

The first question is very important and you need to think about it and answer it.

If you recall, events that are simultaneous in one frame are not simultaneous in another. Therefore, clocks that are synchronized in the train frame will not be synchronized in the obsever frame, and vice-versa.

The second part of the question is really designed to make you think about the first part, really. If we say that two observers differ in their concept of synchronized clocks, it is helpful to expound on this in enough detail so that we know what experimental procedures are used to determine when clocks are synchronized or not.

Unfortunately, when we don't proceed in such detail, people tend to assume that synchronziation should be "universal" - because that's what they are used to. Of course, the point that the professor is trying to make is that it isn't. Understanding the details of how synchronization is done helps one understand why it isn't universal.

Saying this, though, doesn't seem to be enough - people sit through a video telling them that simultaneity is relative, then, like you, they start talking about "synchronized" clocks without telling us in what frame they are synchronized!

I'm not sure what references you hae at your disposal, but for the moment I'll leave it up to you to try and find some good references about how , exactly, clocks are syncrhonized, since I think you can find this out, and that you will learn better if you do.

Looks good so far

right

I don't see why you say that? There is a disagreement over the clocks being synchronized - this is what the relativity of simultaneity means.

But it only occurs between observers in different frames of reference, and I don't understand the logic that led you to believe otherwise.

I think you're missing something.

7. May 4, 2013

It was not that smart of me to bring in the clocks, it only complicates things. The reason was that I (in my ignorance) was so sure of some ‘drift’ onboard the train (to solve this) that I wanted to know what happened to the clocks, but that was obviously wrong. We can forget about the clocks, the important thing is the cameras and what they are recording.

If you look at post #3, #4 and #5 the problem should hopefully be clearer.

8. May 5, 2013

### pervect

Staff Emeritus
My turn to be dense, becaue I don't see what you're getting at. So I'll talk about what you asked about and hope that some of it helps.

Before I do that, I'll try to emphasize again what I see as the message that the professor of the film clip is trying to get across.

(BTW, I'd say that it's not a bad effort to address a confusing subject, but I woluldn't say that it makes relativity clear to anyone who watches it :-) otherwise we wouldn't be having this thread).

This primary message is that the concept of "now" is different for the moving observer and the stationary observer.

That's really all there is too it, it's basically that simple.

As far as what the cameras record - it will be different from the gods-eye view that we've been referring to informally as "seeing", because of light speed propagation delays.

On the train, the main effect will be to tend to crowd the images on the film towards the "front" of the train. This phenomenon is given the name "relativisic abberation".

It'd be hard to draw the images for this case, so I'll settle for images that people have already drawn that show what happens to a uniform field of stars when a spaceship passes through it.

http://www.exo.net/~pauld/stars/PD_images_relativ.html

The math for it is also discussed in the wiki article http://en.wikipedia.org/wiki/Relativistic_aberration

The observer's camera, being a lot further ways away, will have significant lightspeed delays, so there will be large propagation delays. There will be some aberration effects additionally, they may or may not be important to your question (I don't quite understand what the question is). Since the basic principle is the same, I won't go into the details.

There will also be significant light speed delays to the observer, being a distance away from the tunnel, so that by the time the images of the train reaching the tunnel are recorded, the train (according to the observer's notion of time) is already past the tunnel.

The mathematics of what will actually be seen is described by Terrel Rotation, there's a wiki article at http://en.wikipedia.org/w/index.php?title=Terrell_rotation&oldid=543889005

and a sci.physics.faq article at http://math.ucr.edu/home/baez/physics/Relativity/SR/barn_pole.html

Their article on the "barn and the pole" paradox might also be of interest.
http://math.ucr.edu/home/baez/physics/Relativity/SR/barn_pole.html

One other thing that might be of interest are relativistic visualizations. Unfortunately nobody actually did this particular case - it might be interesting if they did. See for example http://www.anu.edu.au/physics/Searle/ [Broken] for some movies that illustrate the visual effect for driving down a highway at high speed.

Hopefully, the differences between the mental image (which involves a concept of now) and the raw image (what the camera actualy photographs, which includes lightspeed delays and aberration effects) will help you answer your question, whatever it is.

Last edited by a moderator: May 6, 2017
9. May 5, 2013

Thank you pervect! This is just great! :!!)

I haven’t had the time to study your links thoroughly, but clearly relativistic aberration must be a part of the answer to this question, since it enables speeding objects to “see around corners”, so to speak.

I’ll have to read and think, and hopefully get back with a coherent picture.

Thanks!
DA

Last edited by a moderator: Sep 25, 2014
10. May 5, 2013

Well, if anyone is dense in this place it’s me, and only me. My question was poorly formulated...

The distortion of objects in Special Relativity is a visual phenomenon (naturally). Neither the train, nor the tunnel, is changing its ‘real’ physical properties (naturally). It only looks that way at speeds close to c. And different observers will see different distortions, in their frame of reference. If the guy running the train hits the break real hard and stop to measure the tunnel, it will have same the length as trainspotter perceives it, i.e. in the rest frame. The trainspotter and the folks on the train will also agree on the length of train when standing still (naturally).

The train and the tunnel are of equal length in the rest frame.

Now, my (not so clever) attempt was to show that these ‘real’ physical properties are there, even when things are moving fast, by ‘rigging a trap’ for divine Einstein... which of course failed catastrophically.

But, I’m probably one of the most stubborn people (still) alive... and I’ll give it another try!

Let’s forget completely about the trainspotter/observer, to get rid of significant light speed delays, etc. It only complicates things (and my gray clump can’t handle complicated things ;). We only want to answer this very simple question:

- Does the length of the train and/or tunnel really change?

Of course we already know the answer to this question, but let’s show why this is a not the smartest inquiry in the universe, by using only two (2) cameras rigged at the end & start of the tunnel.

The camera at the end of the tunnel is rigged to the most sophisticated photodetector on earth, and is programmed to take a picture as soon as something is sticking out of the tunnel, and also send a laser signal through an optical fiber to trigger the camera at the start of the tunnel to also take a picture.

- What happens??

Well, the laser signal (traveling at c) has to travel the length of the tunnel (+ electronic delays), back to the start, to trigger the camera back there, and by then the end of the train has already slipped inside the tunnel! Bingo!!

= It’s impossible to capture any ‘proof’ of disagreements on length/distortions in Special Relativity because the measurements are always performed at < c.

Albert Einstein was no fool.

(Please tell me I got it right this time?)

Last edited: May 6, 2013
11. May 6, 2013

### ghwellsjr

Since you didn't provide what you think is the obvious answer to your question, how can we tell if you got it right this time?

Length Contraction is dependent on the Inertial Reference Frame (IRF) in which the scenario is described. The observations that people or cameras make are not dependent on any IRF. Therefore people or cameras cannot detect Length Contraction.

Of course nothing really changes just because we describe things differently in a different IRF but you didn't say if that was the question you were asking. In other words, you didn't ask: Does the length of the train and/or tunnel really change just because we used different IRF's? The answer to this is "no".

But earlier you said:
Here again you weren't clear. But the video was even less clear. It just said they were the same length without any mention of any frame. What they should have said is that before the train started down the track while it was at rest in the tunnel's rest frame, they both had the same length. Then while the train accelerated to its high speed, its length really changed. This is a true statement in all IRF's. In the tunnel's rest frame, the train became Length Contracted after it accelerated. But in what is called the train's frame in the video, the train started out Length Contracted when it was traveling at the same high speed as the tunnel and then it "decelerated" to a stop and so became it "normal" length while the tunnel remained Length Contracted.

And there are other IRF's to consider. For example, the IRF in which near the end, both the train and the tunnel are approaching each other at the same speed and therefore have the same Length Contraction but in which near the beginning, prior to acceleration, the train had the same speed in the opposite direction (the same as the tunnel) and so it again had the same Length Contraction as at the end, but during its change in speed, it first decelerated to a stop, became normal length, and then accelerated to it final speed and it final Length Contraction.

In all cases (all IRF's), the train's length really changes and the tunnel's length never changes but since there is no preferred IRF, we cannot say uncategorically that the train is the only one that experienced Length Contraction or that it went from normal to contracted length.

Last edited: May 6, 2013
12. May 6, 2013

### pervect

Staff Emeritus
There is a fairly simple experiment, originally due to Einstein, that shows that the length of the train, in some sense, "really" does vary.

What you do is you set up a control box, midway down the tunnel. This sends light signals to cameras at each end - simultaneously, in the "observer" frame. This activates two flash cameras that take pictures of the train. The cameras are mounted half an inch away from the train so that the light speed delays aren't too important to what they observe.

Now we need to get the control box to trigger at the right time.

We set up a movable sensor.This sends a signal up to the control box at lightspeed, the control box which causes the two flash cameras to flash.

The train is 20 meters long, the tunnel is 10 meters long, and the gamma factor for the train's veloicity is 2:1

We need to move the sensor around until we find the "perfect spot", but when we finally do, it flashes the tunnel exit lamp at just the right time to catch the front of the train exiting it.

Succes! When we get the position right, we then look at the entrance cam. And we see that it caught the end of the train!

xxx(end of train)=========================(front of train)
(tunnel entrance)=========================(tunnel exit)

So, we conclude that the train is 10 feet long in our frame.

What happens in the train frame? The observer on the train agrees that the flash at the tunnel exit went off just as the front of the train was at the exit, and that the flash at the tunnel entrance went off just as the end of the train was there. But he denies that the two flashes occured at the same time.

Neither person is wrong. In the train frame, the flashes did not occur at the same time, and the length of the train was 20 meters.

In the camera frame, the flashed did occur at the same time (simultaneity is relative!), and the length of the train was 10 meters.

13. May 6, 2013

### Mentz114

In fact what this setup does is measure the length of the moving train, so it confirms that a change of coordinates does result in a smaller number for the length.

Is this experiment direct evidence for the validity of the LT ?

14. May 6, 2013

### pervect

Staff Emeritus
I haven't really thought about it. It's basically just a recasting of Einstein's train. In the original, "two simultaneous lightening bolts" just happen to hit in the station frame at just the right time.

Einstein didn't specify any mechanism for making this happen. I've read that students find the "it just happens" explanation confusing, so I attempted to avoid this by sepcifying a mechanism which would MAKE the flashes simultaneous. And a timing means for making them happen at the desired time.

THis makes the t hought experiment a bit more compelling, as it is something you can envision actually doing.

15. May 6, 2013

ghwellsjr, thank you for taking the time. However, either I’m totally-complete-ultra-dense, or there might be some other ‘glitch’ in the ‘communication system’ – because I don’t understand what you are talking about...

Okay, but who does detect length contraction? Trains & 'spacemites'??

Okay, but I thought it was pretty obvious that this whole thread was about a train moving close to c through a tunnel... they (the train & tunnel) can’t possibly be in the same inertial frame...

I’m no expert, but to me this was 100% clear when watching the video for the first time. They match the train & tunnel, standing still, at the same place, in the rest frame, and same inertial frame – by driving the train into the tunnel, stop and check proper length. Pretty obvious, if you ask me...

Eh... I’m lost... “traveling at the same high speed as the tunnel and then it "decelerated" to a stop and so became it "normal" length”... have no idea what you’re talking about.

But... there is no acceleration/deceleration/stop in the video... or, are you talking about “my stop” to do the measurement check??

This doesn’t sound compatible with what Professor Mike Merrifield says in the video. Length contraction should be symmetrical for both observers/objects (i.e. train & tunnel).

I’m completely lost...

16. May 6, 2013

Thank you pervect. This is the solution to all my questions.
(I actually had an idea of a ‘centralized unit’ in my OP to solve this puzzle)

Relativity of Simultaneity is the solution and explanation why a 20 meter train could fit in a 10 meter tunnel. Piece of cake!

Simultaneity - Albert Einstein and the Theory of Relativity

In fact, the Ladder paradox is (almost) identical to the problem we discuss here, and the Minkowski diagram of ladder paradox will also work for the “Train paradox”:

“In the garage frame, the front of the ladder will hit the back of the garage at event A on the diagram. All lines parallel to the garage x axis will be simultaneous according to the garage observer, so the dark blue line AB will be what the garage observer sees as the ladder at the time of event A. The ladder is contained inside the garage. However, from the point of view of the observer on the ladder, the dark red line AC is what the ladder observer sees as the ladder. The back of the ladder is outside the garage.”

The “reality question” seems to be a ‘subtle’ matter in RoS. According to Wikipedia on the Reality of Lorentz contraction, Einstein replied to Vladimir Varićak’s claim that length contraction is “real” according to Lorentz, while it is “apparent or subjective” according to Einstein:

Physics is breathtaking!

I think all this also explains my original question: Yes – the cameras in the front & end of the train will record the observer/trainspotter from both sides of the tunnel, and the observer will record the train completely disappeared in the tunnel – but they can never agree on whether the recording was simultaneous, and this is what ‘saves’ the paradox.

Thanks
DA

Last edited by a moderator: Sep 25, 2014
17. May 6, 2013

### ghwellsjr

Well, this is the way to overcome communication glitches--lots of back and forth dialog until we understand each other.

Observers can make assumptions, take measurements, and do calculations to determine Length Contraction but it is not something they can observe directly. It's easiest if they do it for their own rest frame and that makes it deceptively seem like they are actually observing a real Length Contraction so it often needs to be pointed out what they are actually doing.

Of course they can be in the same IRF. They just aren't at rest in the same IRF while the train is going through the tunnel. There's no preferred IRF, not even the one in which an observer or object is at rest.

Where is this sequence? I never saw it. At 3:30 into the video, when the professor is talking about the train being exactly the same length as the tunnel, they don't show the train and the full length of the tunnel for comparison. And every time thereafter, whenever the train is inside the tunnel they always show them with different lengths and yet the professor states that they have the same Proper Length. So I'm asking you - when (what time on the video) is it pretty obvious?

It's only when the train and the tunnel are in relative rest that they have the same length. It doesn't matter if they are both at rest in an IRF or both traveling at the same velocity in an IRF. I mentioned earlier that if they wanted to illustrate or make the point that the train and the tunnel are exactly the same length, they should have started with the train at rest with respect to the tunnel. They could have driven the train into the tunnel and stopped it to show that they were the same length. Then they could have backed the train out of the tunnel and stopped it again, pointing out that it is still the same length as the tunnel. Then they could have accelerated the train towards the tunnel and pointed out that it is now length contracted in the tunnel's rest frame.

If you understand the foregoing, then you can consider what the professor is calling the rest frame of the train. In this frame, while the tunnel is approaching the train, the tunnel is Length Contracted and the train is its normal length. But the tunnel wasn't always approaching the train. If we go back far enough in time, we'll come to a point before the train accelerated when the train and the tunnel were the same distance apart (just like I'm suggesting they should have shown it in the rest frame of the tunnel) and both traveling at the same speed. In this situation, they are both Length Contracted. So the train really changes its length from being Length Contracted to being normal length while it is accelerating in this particular IRF.

I was talking about a modification to the sequence in the video to show that the Proper Length of the train and the tunnel are the same by having them start out in mutual rest and then showing three different IRFs where the train changes its length while accelerating from its position of mutual rest to relative motion. I was making the point that only the train experiences real change in length, because you asked:

Yes, the professor repeatedly states that both observers can see the Length Contraction of the other objects which is not true. He should say that in the rest frame of the train, the tunnel is Length Contracted and in the rest frame of the tunnel, the train is Length Contracted. He also talks about the observer on the ground having a lever which he can pull to make the two guillotines operate at the same time without ever explaining what that is all about and leaving the impression (maybe) that the train is really shorter than the tunnel.

But suppose we change the scenario such that there are two missiles on the train, one at the front and one at the rear and an observer on the train fires both missiles when the train is in the center of the tunnel but because the train is longer than the tunnel, both missiles go straight up into the air without damaging the tunnel. That is just as reasonable as his scenario and (maybe) leaves the impression that the train is really longer than the tunnel.

But do you notice the glee that he expresses when asked how this goes down with his students and he states they love it and that it's conceptually mind-blowing and he says if you get a stupid answer, you've probably done it right? I would rather teach students how to understand Special Relativity in a way that makes sense and doesn't seem stupid or mind-blowing.

That's right. I said that during acceleration, the train changes its length but we cannot say without qualification that it went from normal to contracted or contracted to normal. Maybe it doesn't make sense now but I think it will make sense after you learn SR more. I think the biggest problem is that there are so many false teachers out there that it's hard to sort through the conflicting ideas to get to the truth.

18. May 6, 2013

George, I’ll get back to you tomorrow. It’s getting late over here.

19. May 6, 2013

### ghwellsjr

This is a terrible video. It was discussed at great length in this thread. I finally realized how bad it was at post #139 (page 9) and analyzed it thoroughly starting at post #170 (page 10) and continuing through a great many posts. I hoped it would never be referenced again.

I also found an excellent video made by yuiop and mentioned it at post #235 (page 14). It was analyzed by cepheid in post #337 (page 20) and shapshots taken by me starting at post #340. If you want a correct understanding of the train scenario, study yuiop's video with my commentary.

Where did you get this from? I would help to understand it.

This is another terrible video with a bogus explanation. Briefly, just before the two minute mark, they show two observers at rest in the same frame to explain Relativity of Simultaneity and attribute it exclusively to the light travel time. Totally wrong. If you rely on videos like these, you're going to think you understand when you don't.

Last edited by a moderator: Sep 25, 2014