Confusion over length contraction

[hunter31]

I understand why an object in a moving frame would be longer, but not why one in a stationary frame would be shorter. Wouldn't one in a stationary frame be the same as the object would be in the rest one? I understand the time dilation behind it, it's just that the rest of the reasoning makes no sense to me.

 

[ghwellsjr]

I understand why an object in a moving frame would be longer, but not why one in a stationary frame would be shorter. Wouldn't one in a stationary frame be the same as the object would be in the rest one? I understand the time dilation behind it, it's just that the rest of the reasoning makes no sense to me.

Hi hunter31, welcome to PF.

Let me see if I can explain it in a way that will make sense to you. Let's start with a frame with a stationary object in it that is 5 feet long. We'll make a spacetime diagram for it (I'm using 1 foot per nanosecond as the speed of light):

The blue line represents one end of the object and the red line represents the other end. The dots represent 1-nanosecond increments of time at each end of the object. Since the object is at rest in this frame, time for it progresses the same as the Coordinate Time.

Now we want to use the Lorentz Transformation process to create another frame moving at 0.6c with respect to the first frame. I'm using 1 foot per nanosecond as the speed of light so this would be 0.6 feet per nsec or 3 feet in 5 nsec:

Can you see that the object is now moving to the left at 3 feet in 5 nsec? That's because this frame is moving to the right at 0.6c relative to the first frame.

And can you see that the object is 4 feet long in this frame? It's important when determining distances that you do it at the same Coordinate Time for both ends so at the Coordinate Time of zero, the blue end is at zero feet while the red end is at 4 feet.

Does this now make sense to you?

 

[hunter31]

Yes, thank you.

 

[andromeda]

Graph 1

Graph2

...
Can you see that the object is now moving to the left at 3 feet in 5 nsec? That's because this frame is moving to the right at 0.6c relative to the first frame.

And can you see that the object is 4 feet long in this frame? It's important when determining distances that you do it at the same Coordinate Time for both ends so at the Coordinate Time of zero, the blue end is at zero feet while the red end is at 4 feet.

Does this now make sense to you?

Sorry ghwellsjr, but it does not make sense to me so far. Something may be broken in the time machine.

On graph 1 it is OK two ends are at the same distance for each time coordinate

Let us assume two points to be the ends of "Born-rigid" rod.

Assuming the maths for graph 2 is correct and coordinate time of blue taken as reference:

a) two ends do not exist together for the same coordinate time from roughly above 1 s onwards
meaning the blue end moves alone most of the time
b) Most of the red end lives in the past of the blue end even before the motion of the blue has started.

If that makes sense, what does not?

I believe Einstein has seen the purpose of synchronisation of clocks such that one coordinate time value ("common time" as he puts it) could be used for the moving observer to determine what is simultaneous in his frame.

Now we see two ends of the rod that coexists in the stationary system do not always coexist in the moving one. The corresponding positions of the two ends are clearly seen as successive dots though, so I conclude from this graph that corresponding points of the trajectory are real positions of the existing rod happen to have different clock times which could be achieved in the stationary system if you have used time based on the position of the Sun while moving West.
In the area where a single coordinate time intersects two ends, these positions are in different moments of history not simultaneous. Then (exagerating) the distance between ends is between
the blue end right now and the red end yesterday (given sufficient v)

These kind of things are the reason I am skeptic about the meaning of the relative simultaneity discussed in another thread "Is Relative Simultaneity Real" in which I am not making much progress as yet.

And don't get me wrong, I am not attacking Special Relativity. I argue that the meaning of time as it results from the theory is not well understood.

Perhaps this is only me who does not understand a thing.

 

[Nugatory]

Now we see two ends of the rod that coexists in the stationary system do not always coexist in the moving one.

Yes they do - you just have to extend the lines. Ghwellsjr only calculated and plotted the points that correspond to a four nanosecond period in the rest frame, but the red and blue lines both extend out to infinity in both directions in both graphs.

Let us assume two points to be the ends of "Born-rigid" rod.

That's no problem, but be aware that Born rigidity is a red herring here because there's no acceleration involved - this is a constant-speed comparison between two inertial frames.

The corresponding positions of the two ends are clearly seen as successive dots though, so I conclude from this graph that corresponding points of the trajectory are real positions of the existing rod happen to have different clock times

depending on whose clock you use, yes. But...

which could be achieved in the stationary system if you have used time based on the position of the Sun while moving West.

Yes, you could indeed achieve that effect, and that's one of the reasons why you'll sometimes hear us saying that "simultaneity is a convention". However, the problems with using the position of the sun in the sky as a clock have been understood for centuries now, and we resolve them by using Greenwich time or some other standard as our convention, and then transform to local time as necessary.

The problem with relativity of simultaneity is much deeper than that. Using any simultaneity convention that does not agree with Einstein clock synchronization in an inertial frame defies both common sense and our intuitive notion of what "at the same time" means... and even using that irreproachable (You want to reproach it? Suggest a better alternative ) convention simultaneity turns out to different for observers moving at different speeds relative to one another.

Have you tried the exercise that I suggested in https://www.physicsforums.com/showpost.php?p=4728067&postcount=28 yet?

 

[ghwellsjr]

Sorry ghwellsjr, but it does not make sense to me so far. Something may be broken in the time machine.

On graph 1 it is OK two ends are at the same distance for each time coordinate

Let us assume two points to be the ends of "Born-rigid" rod.

The rod does not have to be rigid. It could be made out of rubber. Neither the rod nor its ends ever move. The rod is inertial.

Assuming the maths for graph 2 is correct and coordinate time of blue taken as reference:

The Coordinate Time does not apply just to one end of the rod. It applies universally everywhere within the frame which extends out to infinity in all directions, including time.

a) two ends do not exist together for the same coordinate time from roughly above 1 s onwards
meaning the blue end moves alone most of the time
b) Most of the red end lives in the past of the blue end even before the motion of the blue has started.

If that makes sense, what does not?

Sorry, I did not mean to imply that the rod only existed at the points where you see the blue and red lines or that it had a change in motion. Here's another pair of diagrams that show them going off the ends of the graphs (out to plus and minus infinity):

Graph 1:

Graph 2:

Does that help?

I believe Einstein has seen the purpose of synchronisation of clocks such that one coordinate time value ("common time" as he puts it) could be used for the moving observer to determine what is simultaneous in his frame.

Einstein used imaginary observers and clocks and rulers to explain what a frame is with its Coordinate Time and Coordinate Distance but once we understand the concept, we don't need to think in terms that they actually exist in any frame or scenario we want to consider. I didn't specify any observers, clocks or rulers in my explanation.

Now we see two ends of the rod that coexists in the stationary system do not always coexist in the moving one. The corresponding positions of the two ends are clearly seen as successive dots though, so I conclude from this graph that corresponding points of the trajectory are real positions of the existing rod happen to have different clock times which could be achieved in the stationary system if you have used time based on the position of the Sun while moving West.

You shouldn't think of positions as being "real". That's a very elusive concept. The Coordinate Positions or Distances are a result of very precise definitions.

There are two different "times" defined in Special Relativity, Proper Time and Coordinate Time. I didn't focus on them because the OP said he understood Time Dilation which is the relationship between the two. But if you're going to bring up "coexistence", it sounds like you really mean "simultaneity" which has to do only with the Coordinate Time defined according to a particular frame. At every instance of Coordinate Time, there is an event on the blue line and an event on the red line. There is no example of an event on the blue line that doesn't have a simultaneous event on the red line and vice versa. I apologize for making the original diagrams look like that may have been the case but hopefully the new diagrams will eliminate that concern.

In the area where a single coordinate time intersects two ends, these positions are in different moments of history not simultaneous.

Since they have the same Coordinate Time, then they are simultaneous. That's the definition of simultaneity. It doesn't have anything to do with "different moments of history" or even with Proper Time.

Then (exagerating) the distance between ends is between the blue end right now and the red end yesterday (given sufficient v)

Simultaneity is frame dependent. Or to put it another way, it's different in different frames. What was simultaneous (the bottoms ends of the two worldlines) in the first of my original diagrams is not simultaneous in the second of my original diagrams.

These kind of things are the reason I am skeptic about the meaning of the relative simultaneity discussed in another thread "Is Relative Simultaneity Real" in which I am not making much progress as yet.

And don't get me wrong, I am not attacking Special Relativity. I argue that the meaning of time as it results from the theory is not well understood.

That's not an argument you're going to win. Instead, you should try to learn Special Relativity. It really is very simple. Einstein said so himself, and he ought to know.

Perhaps this is only me who does not understand a thing.

If you are unwilling to abide by the definitions used in Special Relativity, then you'll never understand it.

 

[Fredrik]

I understand why an object in a moving frame would be longer, but not why one in a stationary frame would be shorter. Wouldn't one in a stationary frame be the same as the object would be in the rest one? I understand the time dilation behind it, it's just that the rest of the reasoning makes no sense to me.

I don't quite understand the question. If L is longer than L', doesn't that imply that L' is shorter than L?

I calculated the length of a rod in a comoving inertial frame and a not comoving inertial frame this post. pervect and vanhees71 made similar calculations in this thread.

 

[andromeda]

I do not want to be l’enfant terrible here, but the purpose of this forum is to resolve reasonable doubts about description of physical world which and rational human being is entitled to have. Despite a few clarifications some reasonable doubts still exist.

The rod does not have to be rigid. It could be made out of rubber. Neither the rod nor its ends ever move. The rod is inertial.

Thank you for the detailed response.
You are right about the rigid. I only invoked that to focus attention on the fact that there can be a physical object of extended length between two points not just two abstract events in space-time.

The Coordinate Time does not apply just to one end of the rod. It applies universally everywhere within the frame which extends out to infinity in all directions, including time.

Fully agree, as this is the way to use coordinates of any kind.
On the other hand there may be a physical clock in each of the discussed points and once both are “Einstein synchronised” and not moving relative to their established 3d coordinate system origin.
Under this constraint the indicated time on each clock is in agreement with time coordinate at its location in the abstract 4d representation of the stationary system.

But if you're going to bring up "coexistence", it sounds like you really mean "simultaneity" which has to do only with the Coordinate Time defined according to a particular frame. At every instance of Coordinate Time, there is an event on the blue line and an event on the red line. There is no example of an event on the blue line that doesn't have a simultaneous event on the red line and vice versa. I apologize for making the original diagrams look like that may have been the case but hopefully the new diagrams will eliminate that concern.

Since they have the same Coordinate Time, then they are simultaneous. That's the definition of simultaneity. It doesn't have anything to do with "different moments of history" or even with Proper Time.

I am glad you have linked coexistence with "simultaneity". The concept of simultaneity now is almost exclusively defined "at the same time" and this is not a surprise it is made equivalent to two the same indications of two synchronised clocks. You insist on this notion an consistently provide correct arguments of frame dependent relative nature of it. That is why I prefer term "temporal coincidence" which is rarely used and considered sometimes as a synonym of simultaneity. It feels bad to say coexistence is frame dependent and I believe there is no proof of that.

Next paragraphs of this post will touch the subject of coexisting states which has some "existential" flavour. Paraphrasing your statement, I believe this is also true that:

There is no example of a state of an object on the blue line that doesn't have a coexisting state of an object on the red line and vice versa. These states however have different clock times, but I believe they are temporally coincident. I hope it will become clearer later.

Sorry, I did not mean to imply that the rod only existed at the points where you see the blue and red lines or that it had a change in motion. Here's another pair of diagrams that show them going off the ends of the graphs (out to plus and minus infinity):

It is true one can extend lines to infinity but what does it mean?
In my opinion the problem is not solved by extending lines but it is swept under the carpet which I hope I can present below.
Let us assume that at both ends of a physical rod there are computerised devices which by definition have high precision clocks and ability to display or physically engrave in metal some part of their internal state at selected equal time intervals generated by the clocks.
State of matter and then that of material objects is a valid physical concept heavily used in thermodynamics or in quantum theory.
From the logical and theoretical point of view both devices at two ends of the rod start operating simultaneously in their common rest frame.
A programmed experiment described in the stationary frame may look as on the graph below:

At time 0 “0000” is engraved, then every nanosecond we have “0010”, “0210”,
At third nanosecond “3210” is engraved but soon after that time the red end computer programmed to self-destruct havs exploded.

Note that such short times make such defined real physical experiment with ordinary devices not practical.

When interpreting this “experiment” from the abstract representation of the moving system after Lorentz transformation we can see the following:

To achieve a 3 second lag between ends we could have roughly v=0.999999c and the distance of 1272.7 km between them. In these conditions the 5 second experiment would last about 2828 s. The nanosecond graph still remains representative for the concept of simultaneous events.

Applying the logic that simultaneous events are those for the same coordinate time we see:
State of the blue end at t'=0 is "0000", simultaneous with the 3210-the last alive state of the red end, and the state "0010" and "0210" are simultaneous with non-existing exploded red device. How does that stand for real?

The simple explanation is that while a physical motion in the stationary system can be described in the moving system using Lorentz transformed equations of motion and expect to find exact experimental time/position relationships for any point, the judgement of what are corresponding coexisting states cannot be made by equal time coordinates.
On the graph above the simultaneity line is the line joining corresponding coexisting states in stationary frame while horizontal pink lines show equal conventional coordinate times for non-coexisting states. The meaning of coexisting is simple from the above. The non-existing end cannot coexist with the existing one.
The other interesting observation from above which is closely related to the first question on this thread is that for the same Lorentz calculated times the rod length in the moving system units is gamma times shorter than that known in the stationary system, while if you compare the corresponding states it would be gamma times longer which is consistent with the claims that the system that has moved and continues moving is shorter hence it “sees” the stationary as larger as long as it moves.

 

[ghwellsjr]

You seem to understand simultaneity issues perfectly well but I get the impression you don't like them and so you feel compelled to invent a new concept called "corresponding coexisting states". Let me see if I can adequately discuss it.

Let's consider that at both ends of the rod there are clocks synchronized as you have stated and explosive devices that upon detonating, send a signal to the other one at the speed of light to detonate it also. And let's say the one on the red end detonates at its time of 3 nsecs. From then on the blue and red devices no longer coexist according to your concept (I think that's what you mean).

However, after the 3 nsec time on the blue device, it has no awareness of the fact that it no longer coexists with the red device even though in their mutual rest frame that is true. Instead, the blue device has to wait until its time of 8 nsecs to be able to see and become aware of the non-existence of the red device. But, of course, this causes the non-existence of the blue device and so from then on there are "corresponding non-existing states", correct?

Here is a spacetime diagram to show what I have just described:

Now the question is, what does this look like when we transform to another frame? First, as we did previously in this thread, we go to a frame moving at 0.6c relative to the rest frame of the rod:

Even though in this frame, the non-existence of the red device is simultaneous with the clock on the blue device reading 0, the blue device still has no awareness of this until its time of 8 nsec, just like in the rest frame. So the only legitimate thing you can say about the timing of "corresponding coexisting states" is that from the time of 3 nsecs on the red clock until the time of 8 nsecs on the blue clock, there are "unbalanced existing states" between the two devices.

What about other frames? Let's try 0.6c in the other direction:

Here we see that when the blue clock reads 3 nsecs, instead of the red device exploding, its time is 0 nsecs. The explosion doesn't occur until 6 nsecs on the blue clock and 2 nsecs later, it becomes aware of the explosion.

Let's try one more, a frame moving at -0.9 with respect to the original rest frame:

Now we see that the red device explodes just a half nanosecond before the blue device explodes according to the blue clock. In fact, we can narrow this margin as much as we want by increasing the speed. We can get to the point where they both explode at almost the same time according to the blue clock.

But still, in any frame we want to transform to, it can be said that from the time of 3 nsecs on the red clock until the time of 8 nsecs on the blue clock, there are "unbalanced existing states" between the two devices. After 8 nsecs on the blue clock, there are "corresponding non-existing states" and before 3 nsecs on the red clock there are "corresponding coexisting states".

Have I adequately summarized your concept and shown the ramifications of it? I think you should be aware that it has nothing to do with simultaneity and is nothing more than just another example of how different frames have no bearing on what any observer can see or measure. That is, in fact, the main point of transforming to different frames, they all contain exactly the same information. That is why none of them is preferred. They are all equally valid. That is also why simultaneity cannot be a real physical issue. It's just an arbitrary coordinate effect.

 

[andromeda]

You seem to understand simultaneity issues perfectly well but I get the impression you don't like them and so you feel compelled to invent a new concept called "corresponding coexisting states". Let me see if I can adequately discuss it.

...

Have I adequately summarized your concept and shown the ramifications of it? I think you should be aware that it has nothing to do with simultaneity and is nothing more than just another example of how different frames have no bearing on what any observer can see or measure. That is, in fact, the main point of transforming to different frames, they all contain exactly the same information. That is why none of them is preferred. They are all equally valid. That is also why simultaneity cannot be a real physical issue. It's just an arbitrary coordinate effect.

It is not a matter of liking. I do not look at physics from aesthetic point of view. It's a matter of conflicting information I am trying to resolve.
Your understanding of my problem is correct and your presentation is high quality and clear. I would like my next response to match yours but it may take a few days to achieve this.

 

[ghwellsjr]

It is not a matter of liking. I do not look at physics from aesthetic point of view. It's a matter of conflicting information I am trying to resolve.
Your understanding of my problem is correct and your presentation is high quality and clear. I would like my next response to match yours but it may take a few days to achieve this.

Thank you.

If you are trying to resolve conflicting information, it would be most helpful if you would state the source (a link would be great) to the first piece of information and then state the source or link to the second piece of information so that we can understand where and what the conflict is.

 

[andromeda]

it would be most helpful if you would state the source (a link would be great) to the first piece of information and then state the source or link to the second piece of information so that we can understand where and what the conflict is.

It is not that simple as two links. It is the totality of information coming to me from multiple sources that appears to produce conflicting views of the same thing. It is really a good idea to capture some of them which are relevant to this particular thread. I will try to single them out while preparing my response which may take a while.

 

[Dale]

Next paragraphs of this post will touch the subject of coexisting states which has some "existential" flavour.

Do you have a scientific (I.e. not a philosophical reference) for this concept which distinguishes it in any way from the standard concept of "simultaneous"?

While I appreciate the difficulty you are having coming to grip with your philosophical questions, that isn't the purpose of this forum.