How can the Pole&Barn Paradox be solved?

[Sagar_C]

I have studied in some details pole&barn paradox and I thought I could solve the following paradox, but sadly I couldn't! Please help me with it so that I can understand special relativity better. Thanks in advance for any help.

The paradox: There are two trains T1 and T2 of equal proper length "L" (say) running on two parallel tracks in opposite direction with a relative velocity V such that due to length contraction one appears of length L/2 w.r.t. the other. Train T1 has a gun right at the "back end" which can shoot perpendicularly right towards the track of train T2. Suppose, it has been arranged for the the gun to shoot "as soon as" (has to be defined properly, I guess) the front end of T1 coincides with the back end of train T2. Now one can see that in the frame of T1, T2 will appear contracted (to L/2) so that T2's front wouldn't have crossed the back-end of T1 when gun shoots, and thus gunshot will not hit T2. But in the frame of reference of T2, T1 will appear contracted (to L/2) and thus, the gunshot will hit T2. So is T2 hit or not hit?

 

[ghwellsjr]

I have studied in some details pole&barn paradox and I thought I could solve the following paradox, but sadly I couldn't! Please help me with it so that I can understand special relativity better. Thanks in advance for any help.

The paradox: There are two trains T1 and T2 of equal proper length "L" (say) running on two parallel tracks in opposite direction with a relative velocity V such that due to length contraction one appears of length L/2 w.r.t. the other. Train T1 has a gun right at the "back end" which can shoot perpendicularly right towards the track of train T2. Suppose, it has been arranged for the the gun to shoot "as soon as" (has to be defined properly, I guess) the front end of T1 coincides with the back end of train T2. Now one can see that in the frame of T1, T2 will appear contracted (to L/2) so that T2's front wouldn't have crossed the back-end of T1 when gun shoots, and thus gunshot will not hit T2. But in the frame of reference of T2, T1 will appear contracted (to L/2) and thus, the gunshot will hit T2. So is T2 hit or not hit?

You guessed right, except that there is no "proper" way to define "as soon as". Which ever frame you choose to define "as soon as" in determines whether the gunshot hits T2, as you have already pointed out.

 

[GAsahi]

I have studied in some details pole&barn paradox and I thought I could solve the following paradox, but sadly I couldn't! Please help me with it so that I can understand special relativity better. Thanks in advance for any help.

The paradox: There are two trains T1 and T2 of equal proper length "L" (say) running on two parallel tracks in opposite direction with a relative velocity V such that due to length contraction one appears of length L/2 w.r.t. the other. Train T1 has a gun right at the "back end" which can shoot perpendicularly right towards the track of train T2. Suppose, it has been arranged for the the gun to shoot "as soon as" (has to be defined properly, I guess) the front end of T1 coincides with the back end of train T2. Now one can see that in the frame of T1, T2 will appear contracted (to L/2) so that T2's front wouldn't have crossed the back-end of T1 when gun shoots, and thus gunshot will not hit T2. But in the frame of reference of T2, T1 will appear contracted (to L/2) and thus, the gunshot will hit T2. So is T2 hit or not hit?

1. You can't have different outcomes for the same event, all observers need to agree that the projectile hits (or doesn't hit) the other train. In this case, they will agree that it hits (reasons are explained below)

2. It is a very bad idea to use length contraction in solving relativity problems. Length contraction is a consequence of the full-fledged Lorentz transforms and it is applicable ONLY when one marks BOTH ends of an object SIMULTANEOUSLY as viewed from a SINGLE frame of reference. This is not the case in your example.

3. The projectile is shot perpendicular to the frame attached to T1, i.e. it has zero speed component along the train T1. This is no longer true when judged from the frame of the track (or from the frame of train T2), the projectile has a component along train T2. In other words, the projectile is "angled" in the direction of motion of train T2. This is true in Galilean relativity just the same as it is true in SR. Since the projectile is angled in the direction of the incoming train (T2), it WILL hit it.

 

[Sagar_C]

3. The projectile is shot perpendicular to the frame attached to T1, i.e. it has zero speed component along the train T1. This is no longer true when judged from the frame of the track (or from the frame of train T2), the projectile has a component along train T2. In other words, the projectile is "angled" in the direction of motion of train T2. This is true in Galilean relativity just the same as it is true in SR. Since the projectile is angled in the direction of the incoming train (T2), it WILL hit it.

Thanks. Can I get rid of issue of angle by saying that the trains are almost touching each other?

 

[Warp]

The paradox: There are two trains T1 and T2 of equal proper length "L" (say) running on two parallel tracks in opposite direction with a relative velocity V such that due to length contraction one appears of length L/2 w.r.t. the other. Train T1 has a gun right at the "back end" which can shoot perpendicularly right towards the track of train T2. Suppose, it has been arranged for the the gun to shoot "as soon as" (has to be defined properly, I guess) the front end of T1 coincides with the back end of train T2. Now one can see that in the frame of T1, T2 will appear contracted (to L/2) so that T2's front wouldn't have crossed the back-end of T1 when gun shoots, and thus gunshot will not hit T2. But in the frame of reference of T2, T1 will appear contracted (to L/2) and thus, the gunshot will hit T2. So is T2 hit or not hit?

As far as I can see, this is basically the exact same problem as the pole-in-barn one. The solution to that one is, as I assume you know, that from the barn's perspective its doors close and open at the same time, but from the pole's perspective the back door closes and opens first, and then after the pole has advanced enough, the front door will do the same.

In your case the problem is that the back end of T1 disagrees about the moment when the ends meet, compared to the front end of T1. (In other words, the front end of T1 sees the event to happen at a different time than the back end of T1 does.)

Or if we state it a bit differently, if you were to command T1 to shoot at the right moment from an external point of view (where it looks like the trains are moving at the same speed), from the back end of T1 it would look like you are commanding it "too late".

 

[GAsahi]

Thanks. Can I get rid of issue of angle by saying that the trains are almost touching each other?

It will not help you. You still can't use length contraction to solve the problem. Here is a simple reason why you cannot: from the point of view of an observer on the track, both trains appear length contracted (to the same extent), yet both ends coincide when the trains pass each other. One more time : length contraction is a BAD idea when trying to solve relativity problems. My advice for you is to forget about it.

 

[Sagar_C]

It will not help you. You still can't use length contraction to solve the problem. Here is a simple reason why you cannot: from the point of view of an observer on the track, both trains appear length contracted (to the same extent), yet both ends coincide when the trains pass each other. One more time : length contraction is a BAD idea when trying to solve relativity problems. My advice for you is to forget about it.

While I agree with you regarding the "bad" idea, have a look at the article "Beyond the pole-barn paradox: How the pole is caught" (just google). There length contraction has been put to very good use to address the paradox. Is there any flaw there? Thanks again.

 

[ghwellsjr]

1. You can't have different outcomes for the same event, all observers need to agree that the projectile hits (or doesn't hit) the other train. In this case, they will agree that it hits (reasons are explained below)

2. It is a very bad idea to use length contraction in solving relativity problems. Length contraction is a consequence of the full-fledged Lorentz transforms and it is applicable ONLY when one marks BOTH ends of an object SIMULTANEOUSLY as viewed from a SINGLE frame of reference. This is not the case in your example.

3. The projectile is shot perpendicular to the frame attached to T1, i.e. it has zero speed component along the train T1. This is no longer true when judged from the frame of the track (or from the frame of train T2), the projectile has a component along train T2. In other words, the projectile is "angled" in the direction of motion of train T2. This is true in Galilean relativity just the same as it is true in SR. Since the projectile is angled in the direction of the incoming train (T2), it WILL hit it.

Concerning 1: Yes, all observers (and by that I think you mean all reference frames because all observers are in all reference frames) will agree on the outcome. However, you cannot say, "in this case" because no case was stated as I pointed out in my previous post.

Concerning 2: When the length of an object is viewed from a frame in which the both ends are simultaneous, you have just specified the rest frame of the object in which there is no length contraction, it has what is called its Proper Length. Length contraction is a perfectly valid explanation but it must always be stated what frame of reference is being used if it is a different one from the one in which its length is defined.

Concerning 3: In problems involving parallel trains, it is always assumed that we disregard the perpendicular travel times of bullets or light, especially when the distance between the train tracks is not specified. But even if we do consider this extra detail, the problem has still not been defined adequately to provide an answer.

 

[ghwellsjr]

Thanks. Can I get rid of issue of angle by saying that the trains are almost touching each other?

It will not help you. You still can't use length contraction to solve the problem. Here is a simple reason why you cannot: from the point of view of an observer on the track, both trains appear length contracted (to the same extent), yet both ends coincide when the trains pass each other. One more time : length contraction is a BAD idea when trying to solve relativity problems. My advice for you is to forget about it.

No, you don't want to forget about length contraction, you want to understand it.

You can get rid of the angle issue simply by ignoring it (just like we ignore gravity issues in thought problems and a host of other issues that would make our thought problems too cumbersome to analyze), we all know what you meant and if someone brings it up, just say you are ignoring it.

 

[Nugatory]

Thanks. Can I get rid of issue of angle by saying that the trains are almost touching each other?

You can. That's a good simplification of the thought experiment, takes the angle complications away without losing the important aspects of the thought experiment.

(If you really wanted to be mathematically rigorous about this, you'd say that the tracks are close enough that you can ignore the small corrections from the angle effects. This procedure is valid because you can make the angle effect as small as you want by moving the tracks close enough together, so no matter how small your tolerance for error, you can find a separation that brings the angle effect in under that tolerance).

 

[Sagar_C]

Please allow me to ask an intermediate step. Is it valid to assume that as T1's tip touches T2's tail, T1's tip sends a (fastest possible) signal as light pulse towards its tail to command the gun to shoot? I am under the impression that this mode of signal (being the speed of light) will have same speed in both the train's frame. However, in T1's frame this signal travels L distance to reach the gun, but in T2's frame this signal travels only L/2 to reach the gun. Is this concept right?

I am so frustrated over this problem and am losing my sleep over it. :-(

 

[Doc Al]

Is it valid to assume that as T1's tip touches T2's tail, T1's tip sends a (fastest possible) signal as light pulse towards its tail to command the gun to shoot?

There's nothing wrong with setting up such a situation. It's perfectly unambiguous. (But it's different than your original set up.)

I am under the impression that this mode of signal (being the speed of light) will have same speed in both the train's frame.

Sure. Both frames will see the light pulse move at speed c.

However, in T1's frame this signal travels L distance to reach the gun, but in T2's frame this signal travels only L/2 to reach the gun. Is this concept right?

Almost. Don't forget that in T2's frame, the rear of T1 is moving towards the oncoming light pulse, so the light pulse has even less distance to cover to reach the rear of T1.

 

[Sagar_C]

(But it's different than your original set up.)

Thanks for the reply. I couldn't grasp how this is different form the original set up?

 

[Doc Al]

I couldn't grasp how this is different form the original set up?

Your original set up had the gun at the rear of T1 fire 'as soon as' the front of T1 passed the rear of T2. That needed a bit more definition, which could have been done by simply adding that the gun would fire at the same time that the front of T1 passed the rear of T2 according to T1 clocks. (You'd have to arrange for that in advance, of course, but it's certainly doable.)

But in your new scenario there will be a delay for the signal to pass from front to rear of T1 no matter whose frame you view things from.

 

[Sagar_C]

Your original set up had the gun at the rear of T1 fire 'as soon as' the front of T1 passed the rear of T2. That needed a bit more definition, which could have been done by simply adding that the gun would fire at the same time that the front of T1 passed the rear of T2 according to T1 clocks. (You'd have to arrange for that in advance, of course, but it's certainly doable.)

Oh! I never thought that was doable! I thought at best there would be time difference needed for light to travel the length. I think I have a wrong concept in my head. Would appreciate a little more clarification.

 

[Doc Al]

Oh! I never thought that was doable! I thought at best there would be time difference needed for light to travel the length. I think I have a wrong concept in my head. Would appreciate a little more clarification.

All you'd need to do is have synchronized clocks at the front and rear of T1. Arrange things so that the front of T1 passed the rear of T2 when the T1 front clock showed 1pm (for example). Easily done (in a thought experiment, at least). Then just have the gun at the rear of T1 fire when the T1 rear clock also shows 1pm.

 

[Dale]

Oh! I never thought that was doable! I thought at best there would be time difference needed for light to travel the length. I think I have a wrong concept in my head. Would appreciate a little more clarification.

What isn't doable is to send information from point A to point B faster than light. But you can do something at A and something at B at some times which are too close together for light to go from A to B as long as information doesn't have to go between them. E.g. you could get information to come from some 3rd point C at light speed to both A and B.

 

[Sagar_C]

What isn't doable is to send information from point A to point B faster than light. But you can do something at A and something at B at some times which are too close together for light to go from A to B as long as information doesn't have to go between them. E.g. you could get information to come from some 3rd point C at light speed to both A and B.

This was helpful. Is there anywhere this paradox has been solved? I found it as a problem in a (youtube) lecture on Relativity by Prof. R. Shankar of Yale. I have almost given up! I think my basics are too weak to address this.

 

[Doc Al]

This was helpful. Is there anywhere this paradox has been solved?

Where exactly is the paradox? First you have to agree on a setup. Then you can unambiguously figure out if the bullets will hit train T2 or not. Then the trick is to see if you can understand that result from both frames. (Both frames must agree that the bullets either hit or missed T2.)

The stumbling block for most is that they forget about the relativity of simultaneity. If two things are simultaneous according to T1, then they will not necessarily be simultaneous for T2.

 

[Dale]

This was helpful. Is there anywhere this paradox has been solved?

In keeping with Doc Al's comment, any time you are ever presented with a paradox in special relativity, chances are better than two to one that the key is the relativity of simultaneity. That is almost always not correctly specified in the problem setup (as was done here) or it was neglected in the analysis.

The relativity of simultaneity is the single most difficult concept of SR.

 

[GAsahi]

Concerning 1: Yes, all observers (and by that I think you mean all reference frames because all observers are in all reference frames) will agree on the outcome. However, you cannot say, "in this case" because no case was stated as I pointed out in my previous post.

Sure there was, it is the case expressed by the OP.

Concerning 2: When the length of an object is viewed from a frame in which the both ends are simultaneous, you have just specified the rest frame of the object in which there is no length contraction,

This is outright incorrect, there is an infinity of frames , all DIFFERENT from the proper frame of the object, where both ends of the object can be marked simultaneously without any difficulty.

it has what is called its Proper Length. Length contraction is a perfectly valid explanation but it must always be stated what frame of reference is being used if it is a different one from the one in which its length is defined.

No one claimed that length contraction is invalid, all I told the OP is that attempting to use length contraction instead of using the full-fledged Lorentz transforms often leads to incorrect solution, as it happened in the case of his (false) "paradox".

Concerning 3: In problems involving parallel trains, it is always assumed that we disregard the perpendicular travel times of bullets or light, especially when the distance between the train tracks is not specified.

Says who?

 

[GAsahi]

While I agree with you regarding the "bad" idea, have a look at the article "Beyond the pole-barn paradox: How the pole is caught" (just google). There length contraction has been put to very good use to address the paradox. Is there any flaw there? Thanks again.

OK,

Words rarely convey the precise meaning, so we'll resort to the actual language employed by physics, math.

In the frame of the track, at time \tau_1 the rear of train T1 coincides with the front of train T2:

x_{T1,rear}=x_{T2,front}

In the frame attached to T1 the positions of the rear of train T1 and the front of train T2 are:

X_{T1,rear}=\gamma(v_1)(x_{T1,rear}-v_1 \tau_1)

and

X_{T2,front}=\gamma(v_1)(x_{T2,front}-v_1 \tau_1)

So, contrary to your intuition about how length contraction plays into the paradox:

X_{T1,rear}=X_{T2,front}

You need to use the full-fledged Lorentz transforms in order to solve the problem correctly.

 

[ghwellsjr]

Concerning 1: Yes, all observers (and by that I think you mean all reference frames because all observers are in all reference frames) will agree on the outcome. However, you cannot say, "in this case" because no case was stated as I pointed out in my previous post.

Sure there was, it is the case expressed by the OP.

The OP never said that both trains were traveling at the same speed in the track frame, only that they were going in opposite directions at a specific relative speed. He then mentioned frames for both trains but he didn't say which one defines "as soon as". As a matter of fact, from all the details he gave, the trains could be traveling at almost any speed as long as their relative speed was 0.866c.

Concerning 2: When the length of an object is viewed from a frame in which the both ends are simultaneous, you have just specified the rest frame of the object in which there is no length contraction,

This is outright incorrect, there is an infinity of frames , all DIFFERENT from the proper frame of the object, where both ends of the object can be marked simultaneously without any difficulty.

Now you're talking about an infinity of frames whereas before you said there was a single frame:

Length contraction is a consequence of the full-fledged Lorentz transforms and it is applicable ONLY when one marks BOTH ends of an object SIMULTANEOUSLY as viewed from a SINGLE frame of reference.

Why did you say "a SINGLE frame"?

it has what is called its Proper Length. Length contraction is a perfectly valid explanation but it must always be stated what frame of reference is being used if it is a different one from the one in which its length is defined.

No one claimed that length contraction is invalid, all I told the OP is that attempting to use length contraction instead of using the full-fledged Lorentz transforms often leads to incorrect solution, as it happened in the case of his (false) "paradox".

If he had defined the frame in which "as soon as" was applicable, then no other frame would have been necessary to solve the problem, he could have simply used the knowledge of length contraction to know whether or not the other train was hit. The only reason to use the Lorentz transform is to see how the same solution "looks" in other frames.

Concerning 3: In problems involving parallel trains, it is always assumed that we disregard the perpendicular travel times of bullets or light, especially when the distance between the train tracks is not specified. But even if we do consider this extra detail, the problem has still not been defined adequately to provide an answer.

Says who?

Einstein set the precedent in his 1920 book involving observers on trains and observers on embankments, never once concerned with the issue that they were not in the same vertical plane.

 

[Dale]

No one claimed that length contraction is invalid, all I told the OP is that attempting to use length contraction instead of using the full-fledged Lorentz transforms often leads to incorrect solution, as it happened in the case of his (false) "paradox".

I agree, and the same with the time dilation formula. I think that students should always be recommended to use the full Lorentz transform. In any case where the simplified length contraction or time dilation formulas are appropriate then they will automatically fall out Lorentz transform, but using the full transform will avoid situations where the simplified formulas are used inappropriately.

 

[GAsahi]

The OP never said that both trains were traveling at the same speed in the track frame, only that they were going in opposite directions at a specific relative speed. He then mentioned frames for both trains but he didn't say which one defines "as soon as". As a matter of fact, from all the details he gave, the trains could be traveling at almost any speed as long as their relative speed was 0.866c.

Now you're talking about an infinity of frames whereas before you said there was a single frame:

Why did you say "a SINGLE frame"?

You are splitting hairs while you are perpetrating outright mistakes.

If he had defined the frame in which "as soon as" was applicable, then no other frame would have been necessary to solve the problem, he could have simply used the knowledge of length contraction to know whether or not the other train was hit.

No, he couldn't, I just disproved this misconception using detailed math. Why do you persist in pushing fringe ideas?

 

[GAsahi]

I agree, and the same with the time dilation formula. I think that students should always be recommended to use the full Lorentz transform. In any case where the simplified length contraction or time dilation formulas are appropriate then they will automatically fall out Lorentz transform, but using the full transform will avoid situations where the simplified formulas are used inappropriately.

Yes, I agree 100% with the above. If you think about it, the "paradoxes" are created by the naive application of length contraction/time dilation in the first place. The "paradoxes" are invariably explained away by the proper application of the full-fledged Lorentz transforms.

 

[Doc Al]

If he had defined the frame in which "as soon as" was applicable, then no other frame would have been necessary to solve the problem, he could have simply used the knowledge of length contraction to know whether or not the other train was hit.

No, he couldn't, I just disproved this misconception using detailed math. Why do you persist in pushing fringe ideas?

Fringe ideas? And you proved no such thing. (It's not clear what scenario you were analyzing in your post.)

If the original scenario was defined so that "as soon as" was with respect to T1, as I suggested in post #14, then it is trivial to use only length contraction to answer the question.

 

[Dale]

No, he couldn't, I just disproved this misconception using detailed math. Why do you persist in pushing fringe ideas?

Hold on. There is nothing fringe about that statement, and it was completely correct. For pedagogical reasons I discourage the use of length contraction, but it isn't "fringe".

 

[GAsahi]

Woah there! This is out of line. There is nothing fringe about that statement, and it was completely correct. For pedagogical reasons I discourage the use of length contraction, but it isn't "fringe" and your math was unrelated to his statement.

Sure it is, it shows how the application of the full-fledged Lorentz transform proves that , from the frame of reference of T1, the projectile hits T2 DESPITE the false impression that it doesn't created by the naive application of length contraction. Applying length contraction is what created the "paradox" in first place.
I challenge any of you to solve the paradox using length contraction ONLY. Let's see the math.

 

[Doc Al]

Sure it is, it shows how the application of the full-fledged Lorentz transform proves that , from the frame of reference of T1, the projectile hits T2 DESPITE the false impression that it doesn't created by the naive application of length contraction. Applying length contraction is what created the "paradox" in first place.
I challenge any of you to solve the paradox using length contraction ONLY. Let's see the math.

Please demonstrate that claim using "the full-fledged Lorentz transform". Your previous post did nothing of the kind. (Hint: You cannot, since your conclusion is incorrect.)

First start by defining the scenario as you understand it. (I believe you are mixing it up with a different scenario.) Be sure to go back to the original post.

 

[GAsahi]

Please demonstrate that claim using "the full-fledged Lorentz transform". Your previous post did nothing of the kind. (Hint: You cannot, since your conclusion is incorrect.)

First start by defining the scenario as you understand it. (I believe you are mixing it up with a different scenario.) Be sure to go back to the original post.

I challenged you first, solve the problem using length contraction ONLY. Please show your math, words don't count.

 

[ghwellsjr]

Sure it is, it shows how the application of the full-fledged Lorentz transform proves that , from the frame of reference of T1, the projectile hits T2 DESPITE the false impression that it doesn't created by the naive application of length contraction. Applying length contraction is what created the "paradox" in first place.
I challenge any of you to solve the paradox using length contraction ONLY. Let's see the math.

There was no paradox stated. The OP correctly described two different scenarios and how the outcome would be different in each of those two, neither of which was the one that you claim he was describing.

I never stated that the full-fledged Lorentz transform shouldn't be used. I only said that you don't need it to describe a scenario because a scenario should be described from only one frame. In fact, describing a scenario using two or more frames can itself present problems. That's also a common mistake when presenting "paradoxes". But until the presenter states clearly what the scenario is in one frame, you can't even use the Lorentz transform. And once you "solve" a problem in one frame, the solution applies to all frames and so it is not even necessary to use the Lorentz Transform except to satisfy your curiosity that all frames yield the same answer.

 

[GAsahi]

OK,

Words rarely convey the precise meaning, so we'll resort to the actual language employed by physics, math.

In the frame of the track, at time \tau_1 the rear of train T1 coincides with the front of train T2:

x_{T1,rear}=x_{T2,front}

In the frame attached to T1 the positions of the rear of train T1 and the front of train T2 are:

X_{T1,rear}=\gamma(v_1)(x_{T1,rear}-v_1 \tau_1)

and

X_{T2,front}=\gamma(v_1)(x_{T2,front}-v_1 \tau_1)

So, contrary to your intuition about how length contraction plays into the paradox:

X_{T1,rear}=X_{T2,front}

You need to use the full-fledged Lorentz transforms in order to solve the problem correctly.

By using the same math, you can easily prove that:X_{T1,front}=X_{T2,rear}

So, the trains line up perfectly as measured from T1. You can repeat the same exact reasoning and you'll get the same result from T2.

 

[GAsahi]

There was no paradox stated. The OP correctly described two different scenarios and how the outcome would be different in each of those two, neither of which was the one that you claim he was describing.

I never stated that the full-fledged Lorentz transform shouldn't be used.

What you said is that you could solve the "paradox" using length contraction ONLY. Can you please post the math that supports your point?

I only said that you don't need it to describe a scenario because a scenario should be described from only one frame.

...which is precisely what I did by solving the problem from the perspective of the track. Math included.

In fact, describing a scenario using two or more frames can itself present problems.

Not at all. I solved the problem in the frame of the track and AFTER that, I transformed it in the frame of T1. You can also transform into the frame of T2 just the same.

 

[Doc Al]

By using the same math, you can easily prove that:


X_{T1,front}=X_{T2,rear}

So, the trains line up perfectly.

All you did was quote your earlier post. What's the point of this 'conclusion'? What do you mean by 'the trains line up perfectly'?

Please show how that tells you that the bullets from T1 hit T2.

 

[ghwellsjr]

By using the same math, you can easily prove that:


X_{T1,front}=X_{T2,rear}

So, the trains line up perfectly as measured from T1. You can repeat the same exact reasoning and you'll get the same result from T2.

Only in the scenario that you decided the OP had in mind where the two trains are traveling at the same speed in opposite directions in the track frame where "as soon as" is defined but for which there is no clue that the OP actually had that in mind.

 

[Doc Al]

I challenged you first, solve the problem using length contraction ONLY. Please show your math, words don't count.

It's trivial!

At the same moment--according to the T1 frame--that the front of T1 passes the rear of T2, the gun at the rear of T1 fires. In the T1 frame the distance (at that instant) between the front and rear of T2 is L/γ, yet the distance from the gun and the rear of T2 is L. The bullets miss the train. Done!

 

[GAsahi]

It's trivial!

At the same moment--according to the T1 frame--that the front of T1 passes the rear of T2, the gun at the rear of T1 fires. In the T1 frame the distance (at that instant) between the front and rear of T2 is L/γ, yet the distance from the gun and the rear of T2 is L. The bullets miss the train. Done!

It might be trivial but it is wrong. The two trains match up perfectly in all frames of reference.
The bullet has a component along the direction of the trains. EVEN IF the perpendicular distance between the trains goes to 0 in the limit, the bullet will still hit the other train. ALWAYS. This is what naive application of length contraction does to you.

 

[GAsahi]

Only in the scenario that you decided the OP had in mind where the two trains are traveling at the same speed

Incorrect. Doesn't have to be the same speed, the path I posted works for different speeds just the same.

in opposite directions in the track frame where "as soon as" is defined but for which there is no clue that the OP actually had that in mind.

He was quite precise. The bullet is fired when the tail of one train coincides with the front of the other. The OP did not specify the frame, I specified it to be the frame of the track. Now, can you please post your solution?

 

[Doc Al]

It might be trivial but it is wrong. The two trains match up perfectly in all frames of reference.

Please define clearly the scenario you are analyzing.

The bullet has a component along the direction of the trains. EVEN IF the perpendicular distance between the trains goes to 0 in the limit, the bullet will still hit the other train. ALWAYS. This is what naive application of length contraction does to you.

Applying the principle of charity, I will assume you are discussing a different scenario that I am. Please define what you are talking about. (That bit about the bullet's having a component along the direction of the trains is irrelevant.)

Certainly there's nothing 'naive' about applying length contraction to the scenario that I am discussing.

 

[Doc Al]

He was quite precise. The bullet is fired when the tail of one train coincides with the front of the other. Now, can you please post your solution?

When according to whom?

The frame was not defined in the OP. I took (as an example) the frame of T1, which is common in these kinds of problem. What frame are you taking?

 

[GAsahi]

Please define clearly the scenario you are analyzing.

Applying the principle of charity, I will assume you are discussing a different scenario that I am. Please define what you are talking about.

See the post above yours. The bullet is fired when the tail of T1 coincides with the front of T2 as measured in the frame of reference of the track. Post 22 gives you the precise math, please go back and read it.

(That bit about the bullet's having a component along the direction of the trains is irrelevant.)

Actually, it isn't. Neglecting it combined with your naive application of length contraction is what led you to your incorrect resolution.

Certainly there's nothing 'naive' about applying length contraction to the scenario that I am discussing.

You got the wrong answer. I suggest that you go over the math and you think some more. I can see that you are a "PF Mentor", so you certainly must know about solving problems correctly.

 

[Doc Al]

See the post above yours. The bullet is fired when the tail of T1 coincides with the front of T2 as measured in the frame of reference of the track. Post 22 gives you the precise math, please go back and read it.

OK, now you've defined your scenario. And this, like the other, is perfectly analyzable using only length contraction. In this case, of course, the trains contract by the same amount, thus both ends line up at the same time. Do you seriously need "the full Lorentz transform" to analyze this trivial situation?

Actually, it isn't. Neglecting it combined with your naive application of length contraction is what led you to your incorrect resolution.




You got the wrong answer. I suggest that you go over the math and you think some more. I can see that you are a "PF Mentor", so you certainly must know about solving problems correctly.

Next time you might like to read the thread that you are posting in. I clearly defined my scenario. And, given that scenario, my answer is correct. Nothing naive about it!

 

[ghwellsjr]

What you said is that you could solve the "paradox" using length contraction ONLY. Can you please post the math that supports your point?

Where did I ever say that? I never said that. I said there was no paradox stated. I said the OP described two different scenarios using length contraction correctly in both of them. If he chooses to define "as soon as" in the frame of T1, his description is the "solution". Nothing more needs to be solved. If, on the other hand, he chooses to define "as soon as" in the frame of T2, then that different definition of the different scenario produces a different "solution" which again leaves nothing more to be solved. But you defined "as soon as" in the track frame with the added and unwarranted assumption that the two trains were traveling at the same speed in opposite directions (as speed that you could have numerically calculated precisely, by the way).

...which is precisely what I did by solving the problem from the perspective of the track. Math included.

You invented your own scenario that was not defined by the OP and you declared the "solution" without using any math, just like the OP did for his two scenarios.

Not at all. I solved the problem in the frame of the track and AFTER that, I transformed it in the frame of T1. You can also transform into the frame of T2 just the same.

That's what I said. Once you define a scenario in one frame, nothing further needs to be solved, the "solution" does not need a second frame or the Lorentz transform. But if you want, just for the fun of it, you can transform the coordinates from the frame of definition into any other frame to show that the "solution" is the same.

 

[ghwellsjr]

See the post above yours. The bullet is fired when the tail of T1 coincides with the front of T2 as measured in the frame of reference of the track. Post 22 gives you the precise math, please go back and read it.

OK, now you've defined your scenario. And this, like the other, is perfectly analyzable using only length contraction. In this case, of course, the trains contract by the same amount, thus both ends line up at the same time. Do you seriously need "the full Lorentz transform" to analyze this trivial situation?

Saying that he measures the time in the track frame is not enough, he also assumed that the two trains were traveling in opposite directions at the same speed in that frame, something that was not stated by the OP.

And then he did exactly what he says we shouldn't do: he used the fact that the two trains were length contracted by the same amount and declared that the other ends of the two trains coincided as his solution, without using any math or the LT. Then he used the Lorentz transform to show that the same solution applied in one of the train's rest frame.

 

[Doc Al]

Saying that he measures the time in the track frame is not enough, he also assumed that the two trains were traveling in opposite directions at the same speed in that frame, something that was not stated by the OP.

You are correct, of course. Again, I was giving what I thought was the most charitable interpretation of what he was doing.

And then he did exactly what he says we shouldn't do: he used the fact that the two trains were length contracted by the same amount and declared that the other ends of the two trains coincided as his solution, without using any math or the LT. Then he used the Lorentz transform to show that the same solution applied in one of the train's rest frame.

Yep.

 

[ghwellsjr]

GAsahi, do you not agree that we could have done exactly what you did using train T1's rest frame to define "as soon as" and used the fact that only T2 would have been contracted and the bullet would not hit T2 (just like the OP said) and then transformed that scenario into the rest frame of T2 to show again that the bullet wouldn't hit T2?

And then do you not agree that we could have done the same thing again using train T2's rest frame to define "as soon as" and used the fact that only T1 would have been contracted and the bullet would hit T2 (just like the OP said) and then transformed that scenario into the rest frame of T1 to show again that the bullet would hit T2?

 

[GAsahi]

OK, now you've defined your scenario. And this, like the other, is perfectly analyzable using only length contraction.

I very much doubt it. You will get another incomplete result and another incorrect conclusion.

In this case, of course, the trains contract by the same amount, thus both ends line up at the same time. Do you seriously need "the full Lorentz transform" to analyze this trivial situation?

Wrong again, the trains do not have the same speed, my solution works just as well.

Next time you might like to read the thread that you are posting in. I clearly defined my scenario. And, given that scenario, my answer is correct. Nothing naive about it!

Your scenario may be correct, it is your conclusion that is wrong. You are missing some very important components in your naive application of length contraction. It is precisely this insistence on applying length contraction that produces the "paradoxes".

 

[GAsahi]

It's trivial!

At the same moment--according to the T1 frame--that the front of T1 passes the rear of T2, the gun at the rear of T1 fires. In the T1 frame the distance (at that instant) between the front and rear of T2 is L/γ, yet the distance from the gun and the rear of T2 is L. The bullets miss the train. Done!

You are missing a couple of components in your solution, this is why you arrive to the wrong conclusion.
For you to apply length contraction in order to claim that "the length of T2 is L/γ", you need to mark both ends of T2 simultaneously in the frame of T1. But the order to "fire" needs to travel from the front of T1 to its rear and you well know that such a signal takes some time. So, in frame 1, you may be marking both ends of T2 simultaneously but you fire AT SOME LATER TIME. While the signal to fire travels from the front of T1 to its rear, T2 CONTINUES MOVING IN THE SAME DIRECTION. If you add the missing pieces to your solution, you will find out that you were wrong all along, contrary to your claims, the bullet HITS T2.
Look at it another way, in the frame of the track, the bullet hits T2, so you need to arrive (via proper computation) to the SAME conclusion in ALL frames. Otherwise, your attempt to apply length contraction only is plain wrong.

 

[Doc Al]

Your scenario may be correct, it is your conclusion that is wrong. You are missing some very important components in your naive application of length contraction. It is precisely this insistence on applying length contraction that produces the "paradoxes".

In my scenario, length contraction allows you to instantly get 'the answer' by viewing things from one frame. (The frame in which the defining events are simultaneous.) The apparent paradox is generally obtained from naively applying length contraction in a different frame without also taking into account the relativity of simultaneity (as I pointed out in post #19).