Ladder Paradox: Understanding the Contradictory Length Contraction Phenomenon

[durant]

Hi, I've red a lot these days about the ladder paradox and I have a question about it, but firste let me quote wikipedia so I can describe you the problem:
"In the context of the paradox, when the ladder enters the garage and is contained within it, it must either continue out the back or come to a complete stop. When the ladder comes to a complete stop, it accelerates into the reference frame of the garage. From the reference frame of the garage, all parts of the ladder come to a complete stop simultaneously, and thus all parts must accelerate simultaneously.
From the reference frame of the ladder, it is the garage that is moving, and so in order to be stopped with respect to the garage, the ladder must accelerate into the reference frame of the garage. All parts of the ladder cannot accelerate simultaneously because of relative simultaneity. What happens is that each part of the ladder accelerates sequentially, front to back, until finally the back end of the ladder accelerates when it is within the garage, the result of which is that, from the reference frame of the ladder, the front parts undergo length contraction sequentially until the entire ladder fits into the garage."

How can the last sentence be possible? Shouldn't the ladder it the rest frame of itself have its proper length and not get contracted, while the garage should be contracted from the ladder point of view?

 

[ghwellsjr]

That is a little confusing but what they are pointing out is that if you actually stop the ladder by simultaneously (in the garage frame) forcing each portion of the ladder to stop and stay stopped, then it will end up being shortened. Of course, as they point out later in the article, there's no such thing as a truly rigid ladder and so we have to imagine that this ladder can deform by whatever mechanism caused it to stop.

Maybe it would help to think of a large clamp or braking system that is almost ten feet long that comes together on both sides of the ladder and clamps it in place. This clamp must be stronger than the ladder so that it keeps it from expanding back to its Proper Length.

 

[durant]

That is a little confusing but what they are pointing out is that if you actually stop the ladder by simultaneously (in the garage frame) forcing each portion of the ladder to stop and stay stopped, then it will end up being shortened. Of course, as they point out later in the article, there's no such thing as a truly rigid ladder and so we have to imagine that this ladder can deform by whatever mechanism caused it to stop.

Maybe it would help to think of a large clamp or braking system that is almost ten feet long that comes together on both sides of the ladder and clamps it in place. This clamp must be stronger than the ladder so that it keeps it from expanding back to its Proper Length.

So they actually must be referring to the garage frame as the frame in which the ladder is contracted? I agree that it's confusing :/

 

[HallsofIvy]

So they actually must be referring to the garage frame as the frame in which the ladder is contracted? I agree that it's confusing :/

Yes, they say that explicitely:
"When the ladder comes to a complete stop, it accelerates into the reference frame of the garage. From the reference frame of the garage, all parts of the ladder come to a complete stop simultaneously, and thus all parts must accelerate simultaneously."

 

[ghwellsjr]

So they actually must be referring to the garage frame as the frame in which the ladder is contracted? I agree that it's confusing :/

That's not the part I thought was confusing. While the ladder is moving in the garage frame, it is contracted as shown in Figure 4. What I thought was confusing is the way they twice expressed that the ladder accelerates "into the reference frame of the garage". It's always confusing to think of objects changing reference frames as a result of their own acceleration and it's completely unnecessary and in my opinion wrong. They did express it correctly when they said, "From the reference frame of the garage, all parts of the ladder come to a complete stop simultaneously, and thus all parts must accelerate simultaneously", except I would have said "decelerate" instead of "accelerate", but that's a minor point. Then they should have pointed out that the way to express that scenario, which is simply and clearly defined in the garage frame, into any other frame is to use the Lorentz Transformation process. With the Lorentz Transformation, you don't have to know the "resolution" to the "paradox" ahead of time, you let the Lorentz Transformation "solve" the problem for you.

I have lots of other complaints about the article:

1) In the very first paragraph where they supposedly state the problem, they never mention the doors and how they are defined to close and open, which is all-important to the problem.

2) A couple paragraphs later they say, "both the ladder and garage occupy their own inertial reference frames, and thus both frames are equally valid frames to view the problem" as if no other frames are equally valid frames. Whatever happened to the idea that no frame is preferred? Instead, this statement implies that only the rest frame of an object is not just preferred frame but valid. Terrible.

3) In the section entitled "Relative simultaneity" where they discuss the solution, they finally bring up the doors, "A clear way of seeing this is to consider a garage with two doors that swing shut to contain the ladder and then open again to let the ladder out the other side". This implies that the doors are part of the solution instead of part of the problem. It's the statement that it is in the garage frame where the doors are simultaneously closed and then reopened that is part of the problem. If they had instead said something to the effect that it was in the rest frame of the ladder that two guns at either end of the ladder were fired straight up and the garage was unharmed, we'd have just as valid a "solution" to the paradox but it would be a different problem. What a mess.

4) Their Minkowski diagrams are very confusing. I doubt that they communicate anything to anyone other than someone who could have drawn the diagrams. I think it is far better to draw separate diagrams for each frame with the Lorentz Transform being the mechanism to get from the frame in which the problem is stated to any other frame, including the initial rest frame of the ladder.

5) In their explanation for Figure 8 they repeatedly use the words "sees", "see" and "seen" and apply it to a person, which is incorrect. No person anywhere in any scenario can see what is depicted in a spacetime diagram unless light signals are also drawn in. They did not draw them in any of their diagrams. If you do drawn them in, it won't matter which frame you use, they all will show the same thing for what any observer sees.

Later on, if I get the time, and I'm still motivated, I'll draw some diagrams that correctly communicate the problem of the Ladder Paradox and its solution.

 

[durant]

That's not the part I thought was confusing. While the ladder is moving in the garage frame, it is contracted as shown in Figure 4. What I thought was confusing is the way they twice expressed that the ladder accelerates "into the reference frame of the garage". It's always confusing to think of objects changing reference frames as a result of their own acceleration and it's completely unnecessary and in my opinion wrong. They did express it correctly when they said, "From the reference frame of the garage, all parts of the ladder come to a complete stop simultaneously, and thus all parts must accelerate simultaneously", except I would have said "decelerate" instead of "accelerate", but that's a minor point. Then they should have pointed out that the way to express that scenario, which is simply and clearly defined in the garage frame, into any other frame is to use the Lorentz Transformation process. With the Lorentz Transformation, you don't have to know the "resolution" to the "paradox" ahead of time, you let the Lorentz Transformation "solve" the problem for you.

I have lots of other complaints about the article:

1) In the very first paragraph where they supposedly state the problem, they never mention the doors and how they are defined to close and open, which is all-important to the problem.

2) A couple paragraphs later they say, "both the ladder and garage occupy their own inertial reference frames, and thus both frames are equally valid frames to view the problem" as if no other frames are equally valid frames. Whatever happened to the idea that no frame is preferred? Instead, this statement implies that only the rest frame of an object is not just preferred frame but valid. Terrible.

3) In the section entitled "Relative simultaneity" where they discuss the solution, they finally bring up the doors, "A clear way of seeing this is to consider a garage with two doors that swing shut to contain the ladder and then open again to let the ladder out the other side". This implies that the doors are part of the solution instead of part of the problem. It's the statement that it is in the garage frame where the doors are simultaneously closed and then reopened that is part of the problem. If they had instead said something to the effect that it was in the rest frame of the ladder that two guns at either end of the ladder were fired straight up and the garage was unharmed, we'd have just as valid a "solution" to the paradox but it would be a different problem. What a mess.

4) Their Minkowski diagrams are very confusing. I doubt that they communicate anything to anyone other than someone who could have drawn the diagrams. I think it is far better to draw separate diagrams for each frame with the Lorentz Transform being the mechanism to get from the frame in which the problem is stated to any other frame, including the initial rest frame of the ladder.

5) In their explanation for Figure 8 they repeated use the words "sees", "see" and "seen" and apply it to a person, which is incorrect. No person anywhere in any scenario can see what is depicted in a spacetime diagram unless light signals are also drawn in. They did not draw them in any of their diagrams. If you do drawn them in, it won't matter which frame you use, they all will show the same thing for what any observer sees.

Later on, if I get the time, and I'm still motivated, I'll draw some diagrams that correctly communicate the problem of the Ladder Paradox and its solution.

We all know that wikipedian articles are oftenly BS. Anyway, thanks for your post and opinion about the article.

 

[Meir Achuz]

Once again, Wikipedia is all wrong. Any time the ladder is stopped, it has its original length.
A description of what happens is in http://arxiv.org/pdf/1105.3899.pdf
It is a bit complicated, but explains what happens when a rigid body stops moving.

 

[PAllen]

Once again, Wikipedia is all wrong. Any time the ladder is stopped, it has its original length.
A description of what happens is in http://arxiv.org/pdf/1105.3899.pdf
It is a bit complicated, but explains what happens when a rigid body stops moving.

Well, there are multiple cases: the ladder is rigid, and various assumptions about more realistic material behavior. All are interesting. Assume both doors close and remain closed, in the barn rest frame (until something happens). Then, assuming rigidity, you cannot assume the doors are impervious - you must accept that the ladder ruptures the door it reaches second. If you assume the doors are impervious, you must assume the ladder compresses or fragments. For realistic materials, the ladder material will mostly go through the door, and the door will initially have a fairly clean hole; however the energy released may soon vaporize both.

 

[Fredrik]

Once again, Wikipedia is all wrong. Any time the ladder is stopped, it has its original length.

Only if it's allowed to restore its original length. What they're describing is essentially to grab hold of each component part of the ladder simultaneously (in the coordinate system comoving with the garage) and just prevent it from moving away from the position where it was grabbed. This would not only stop the ladder, but also prevent it from restoring its original length.

Not exactly a realistic scenario, but it's not forbidden by SR alone.

 

[durant]

Once again, Wikipedia is all wrong. Any time the ladder is stopped, it has its original length.
A description of what happens is in http://arxiv.org/pdf/1105.3899.pdf
It is a bit complicated, but explains what happens when a rigid body stops moving.

Wasn't it known even before special relativity that it's impossible for an object to be 'rigid' in the sense that when moving one part another will move simultaneously?

 

[Fredrik]

Wasn't it known even before special relativity that it's impossible for an object to be 'rigid' in the sense that when moving one part another will move simultaneously?

I'm sure that physicists understood that e.g. that if you hit a rod at one end, you are really only pushing the first layer of its component parts, which then pushes the next layer, and so on, and that this disturbance will propagate at a finite speed.

But this is a practical issue. Pre-relativistic classical mechanics doesn't say that objectively rigid motion is impossible. SR on the other hand, says that if you give every component part of an object a velocity boost at the same time in one inertial coordinate system, the component parts will change their velocities at different times in another inertial coordinate system. So rigid motion in one inertial coordinate system is consistent with SR, but rigid motion in all inertial coordinate systems is not.

 

[mananvpanchal]

I'm sure that physicists understood that e.g. that if you hit a rod at one end, you are really only pushing the first layer of its component parts, which then pushes the next layer, and so on, and that this disturbance will propagate at a finite speed.

But this is a practical issue. Pre-relativistic classical mechanics doesn't say that objectively rigid motion is impossible. SR on the other hand, says that if you give every component part of an object a velocity boost at the same time in one inertial coordinate system, the component parts will change their velocities at different times in another inertial coordinate system. So rigid motion in one inertial coordinate system is consistent with SR, but rigid motion in all inertial coordinate systems is not.

Perfect.

But, I have a question. If we give a velocity boost on every component part of an object at the same time in one inertial coordinate system then do the object become length contracted in the same inertial coordinate system after changing its velocity?

 

[Fredrik]

Perfect.

But, I have a question. If we give a velocity boost on every component part of an object at the same time in one inertial coordinate system then do the object become length contracted in the same inertial coordinate system after changing its velocity?

Immediately after this (unrealistic but logically possible) boost, the rod will have the same length as before the boost. If it was a large boost, it will simply tear the rod apart. If it was a small boost, so that the rod doesn't break, then the internal forces in the rod will pull it together, so that it eventually settles down to the length predicted by the Lorentz contraction formula.

 

[mananvpanchal]

Immediately after this (unrealistic but logically possible) boost, the rod will have the same length as before the boost. If it was a large boost, it will simply tear the rod apart. If it was a small boost, so that the rod doesn't break, then the internal forces in the rod will pull it together, so that it eventually settles down to the length predicted by the Lorentz contraction formula.

Can you please explain me little more. If it was small boost then will the length be contracted suddenly or will it be a slow process by the traveling time?

 

[Dale]

First, it is important to understand that length contraction is NOT due to acceleration! Length contraction is a disagreement between two inertial frame about the length of an object at one point in time, not a change in the length of an object in a single inertial frame over time.

If you have an object that undergoes some change in length during the course of acceleration that change in length is not length contraction. Before the change different frames will disagree about is length, that is length contraction. After the change different frames will disagree about its length, that is also length contraction. Before and after the change a single frame will disagree about its length, that is NOT length contraction.

 

[Fredrik]

Can you please explain me little more. If it was small boost then will the length be contracted suddenly or will it be a slow process by the traveling time?

A "boost" of the type we're discussing (when we force each component part to change its velocity by the same amount at the same time in the coordinate system where the rod was at rest) will by definition ensure that the length of the rod doesn't change (in the inertial coordinate system where it was at rest). It's natural length however, i.e. the length it would have if the internal forces had been allowed to do their thing, is the length given by the Lorentz contraction formula. So the natural length it's shorter by a factor of $\gamma$. This means that the "boost" has forcefully stretched the rod. So what will happen is roughly the same thing that will happen if you just stretch the rod without changing its velocity. The rod will contract-expand-contract until all the energy that the stretch added to the rod has been converted to heat.

 

[DrGreg]

First, it is important to understand that length contraction is NOT due to acceleration! Length contraction is a disagreement between two inertial frame about the length of an object at one point in time, not a change in the length of an object in a single inertial frame over time.

If you have an object that undergoes some change in length during the course of acceleration that change in length is not length contraction. Before the change different frames will disagree about is length, that is length contraction. After the change different frames will disagree about its length, that is also length contraction. Before and after the change a single frame will disagree about its length, that is NOT length contraction.

To illustrate this, here's something I said several years ago: https://www.physicsforums.com/showpost.php?p=2443229&postcount=38

I think it is perhaps worth pointing out that some people have a false impression about what Lorentz contraction is. They may think that "when something accelerates it gets shorter". Or to be a bit more precise, if Alice measures (=x) something at rest (relative to Alice) and then later measures (=y) the same thing in motion, the length contracts. There may then be some debate over whether or not the "things" this applies to are just solid objects, or gaps between objects, or "space itself".

The above description of Lorentz contraction is wrong.

In many circumstances, what I said above is true, but reason it is true is not simply Lorentz contraction alone; it is Lorentz contraction plus some other reason combined.

A more accurate description of Lorentz contraction is that when inertial observer Bob measures the length z between two things both at rest relative to Bob, and another inertial observer Alice in relative motion measures the length y between the same two things at the same time, Alice measures a shorter distance than Bob.

So, the situation I described in the first paragraph will arise if there is a reason why Alice's initial "rest distance" x between the two things beforehand is the same as the Bob's final "rest distance" z. For example if the the two things are the two ends of a rigid object that doesn't break into pieces as a result of the acceleration.

The attached illustration emphasises my point. The transformation of x to y is not Lorentz contraction. The transformation of z to y is Lorentz contraction. If there is a reason why x = z, then the transformation of x to y will be a contraction. But if there's no reason, then contraction need not occur.

 

[mananvpanchal]

First, it is important to understand that length contraction is NOT due to acceleration! Length contraction is a disagreement between two inertial frame about the length of an object at one point in time, not a change in the length of an object in a single inertial frame over time.

If you have an object that undergoes some change in length during the course of acceleration that change in length is not length contraction. Before the change different frames will disagree about is length, that is length contraction. After the change different frames will disagree about its length, that is also length contraction. Before and after the change a single frame will disagree about its length, that is NOT length contraction.

In this scenario length is not going to alter by force applied on the object. I am guessing that force applied at all points of the component so it will not be alter. So, here no issue about alteration.

A "boost" of the type we're discussing (when we force each component part to change its velocity by the same amount at the same time in the coordinate system where the rod was at rest) will by definition ensure that the length of the rod doesn't change (in the inertial coordinate system where it was at rest). It's natural length however, i.e. the length it would have if the internal forces had been allowed to do their thing, is the length given by the Lorentz contraction formula. So the natural length it's shorter by a factor of $\gamma$. This means that the "boost" has forcefully stretched the rod. So what will happen is roughly the same thing that will happen if you just stretch the rod without changing its velocity. The rod will contract-expand-contract until all the energy that the stretch added to the rod has been converted to heat.

What I understand from your quote that component is actually stretched by applied force??! Actually we are supposing a logical situation here that what will happen when force is not altering the object?

DrGreg

Suppose that in a A frame object is with length L. After sometime the object is not at rest in A frame and length of the object being measured as l. We just know that force hasn't alter the object. Then can this be said as a length contraction?

 

[Fredrik]

What I understand from your quote that component is actually stretched by applied force??! Actually we are supposing a logical situation here that what will happen when force is not altering the object?

Yes to the first question. I don't understand the second one. You asked me specifically about the situation when every component part of the rod is given the same velocity increase simultaneously in the inertial coordinate system in which the rod is originally at rest. This is not a situation where "force is not altering the object".

DrGreg

Suppose that in a A frame object is with length L. After sometime the object is not at rest in A frame and length of the object being measured as l. We just know that force hasn't alter the object. Then can this be said as a length contraction?

I'm inclined to agree with DrGreg's picture, which says that this is not Lorentz contraction. The length l can however be calculated using the Lorentz contraction formula, and the assumption that the rod has the same length L in its new comoving inertial coordinate system, as it did in it's old comoving inertial coordinate system before the acceleration began.

 

[mananvpanchal]

Yes to the first question. I don't understand the second one. You asked me specifically about the situation when every component part of the rod is given the same velocity increase simultaneously in the inertial coordinate system in which the rod is originally at rest. This is not a situation where "force is not altering the object".

I can't get the all. First you said the rod is stretched under force. And second you said

This is not a situation where "force is not altering the object".

.

 

[Fredrik]

There are two nots in that sentence, and they cancel each other out. This is a situation where force is altering the object.

 

[Dale]

In this scenario length is not going to alter by force applied on the object. I am guessing that force applied at all points of the component so it will not be alter. So, here no issue about alteration.

This is neither true nor relevant.

First, strain-free acceleration is called "Born rigid" acceleration, and is well studied. What you propose is not Born rigid acceleration and therefore there will be strain. In particular, the rod will experience tension and be physically stretched by this suggested acceleration profile.

Second, even if it were Born rigid acceleration, it is still simply NOT Lorentz contraction. Lorentz contraction is always a comparison between two reference frames, not in one reference frame at different points in time. What you are discussing is simply not Lorentz contraction, by definition.

 

[harrylin]

[..]
I have lots of other complaints about the article:

It's a wiki. Thus, after a short discussion here you can improve the text (sticking to the literature), which is most constructive.

 

[ghwellsjr]

It's a wiki. Thus, after a short discussion here you can improve the text (sticking to the literature), which is most constructive.

Yeah, and then someone else can come along and change it all back again.

The problem is that there are lots of other articles on wikipedia and other places that I also disagree with but other people think my approach has problems and so they would edit anything I wrote. At least on PF, no one (except their Exellencies the Right Honourable Mentors) can edit my posts.

 

[ghwellsjr]

Later on, if I get the time, and I'm still motivated, I'll draw some diagrams that correctly communicate the problem of the Ladder Paradox and its solution.

As promised, here are the diagrams that depict the scenario in the wikipedia article regarding the stopping of the ladder while it is inside the barn. First, the diagram for the frame in which the scenario is defined. I show the two ends of the barn in red and just for the fun of it, I colored in blue the 1 nanosecond interval when both doors are closed. I think you can see that if the ladder had continued on without being clamped to a stop at the Coordinate Time of 0 nanoseconds, it would have passed out the back of the barn (on the right side) right after that door reopened.

The rungs of the 12-foot long ladder are depicted in black:

Now the diagram for the rest frame of the ladder prior to its being clamped to a stop:

(Note the blue intervals when both doors are closed and how if the ladder had continued on it way, it would have made it without hitting either door.)

But as a result of the ladder being stopped simultaneously in the barn's rest frame, each rung gets stopped at a different time in the ladder's original rest frame resulting in the ladder's length being shortened after it comes to a complete stop. Note also that the Proper Times of all the rungs go out of sync as a result of the ladder enduring the forced stop.