Does Bell's Paradox Suggest String Shouldn't Break Due to Length Contraction?

[Dale]

It must surely also follow that if it doesn't break in one frame then it doesn't break in all frames ;-)

Indeed. If you think that you can show that it doesn't break in any frame, then please post your work and I am sure we can figure out where you made the mistake.

The main point of frame invariance is that if you can show that it breaks in one frame then you should be able to show that it breaks in other frames.

The main point of frame invariance is that you can always choose to work the problem in the easiest frame.

 

[Eli Botkin]

DaleSpam:
I agree "...that you can always choose to work the problem in the easiest frame." However, with this type of problem, where there are questions of interpretation, it can't hurt (and may help) to seek that same solution, from other directions, in other frames.

An even "easier" frame than the ship's co-moving frame would have the observer moving at some constant speed, V, in the direction opposed to the ship's motion. Can your solution be found there?

Bear in mind that I've never claimed that the string doesn't break. My claim is that string breakage has not been proven, not even by Bell himself. Bell did opine that it would break because of the internal EM field distortion (which I find strange since , through V, the distortion is also frame dependent). But that doesn't make a proof.

I must admit, though I have great respect and admiration for Bell's contributions to the understanding of QM, I am puzzled by his claim that breakage is due to Fitzgerald contraction though he knows there is no such contraction of the separation between ships.

If the correct solution is that breakage must take place, then I think the culprit is more likely to be the ships' acceleration, a component of the problem that exists in all frames.

Thanks for "listening."

 

[Austin0]

You are welcome.

That and the fact that the limit is infinite, therefore it will eventually break regardless of the actual values of L or d. It is possible for a function to be monotonically increasing to a finite limit, in which case the breakage would depend on the details of how big L was compared to the limiting value of d'.

Excellent, therefore the string breaks and Bell's paradox is resolved using only SR.

The laws of physics are frame invariant, therefore if it breaks in one frame then it breaks in all frames.

I leave the details of any other frame as an exercise for the interested reader (you). I would be glad to look over your efforts if you get stuck somewhere.

Hi First I will say that I personally have the view that the light speed electromagnetic interactions involved in maintaining physical structure do result in a contraction through the tensile forces. I.e. ; the string will break.
Even so I feel impelled to play Devil's advocate here.
Just looking at the scenario from a target frame of say 0.8c it is clear that the leading ship initiates acceleration before the trailing ship. This not only means increasing the separation but also creating a velocity differential that will persist even after the trailing ship is also accelerating.
Therefore relative to that frame, the distance between the ships must continue to increase without bounds as you calculated with your analysis.
This itself seems problematic. How do you reconcile frame agreement of time and position observations of the ships when the distance remains constant in one frame and is indefinitely expanding in the other frame??
Or conversely: Assume the ships started accelerating from signals that were simultaneous
in the 0.8c frame. Now in that frame the distance remains constant, but in the launch frame the trailing ship fires up first and must eventually intercept and overtake the lead ship.
How do we reconcile these contradictory expectations?

Adopting for the moment a purely kinematic interpretation of SR which has been expressed by many knowledgeable people on this forum in the past.
Assuming no contraction of either the distance or the string. At a velocity of 0.8 c what would be the measurement of that distance?
According to an 0.8c inertial frame the clock in the launch frame would be running ahead at the instantaneous location of the lead ship.Therefore the trailing ship and string would move forward an additional distance before reaching a clock with the same proper time. I.e.; Would be measured as being contracted relative to the initial separation.

As I said I don't necessarily accept this interpretation but I can't dismiss it out of hand.

I might point out that if the physical reality of contraction as implied by the Maxwell maths is actually valid this leads strongly to the logical conclusion that all motion is actual and in a sense absolute, even if it is not determinable or measurable in non-relative quantitative terms. Would you agree?? ;-)

 

[Dale]

I agree "...that you can always choose to work the problem in the easiest frame."

Then you must agree that proving an outcome in anyone frame is sufficient to prove the outcome.

Bear in mind that I've never claimed that the string doesn't break. My claim is that string breakage has not been proven

I proved it above. If you disagree then please point out any mistake I made and I will correct it. If I made no mistakes, then I have proven it.

 

[A.T.]

A.T.:
Let’s try this. Have the string attached to the two ships only loosely. Say each end is firmly in a sleeve such that when you tug the string it will slide in the sleeve.

With the ships always maintaining the same separation why will the string slide out of a sleeve?

Beacuse it will contract, so it cannot span the distance between the sleeves.

And which sleeve, fore or aft?

An ideal mass-less string? From both I guess.

And why is the space between the string atoms being reduced

Because the EM-fields that hold the atoms together are contracting, and are pulling the atoms closer together. If the atoms are forced to span a constant length (like in the original scenario) the bounds between the atoms will break somewhere.

but not the space between the ships?

That is given in the scenario. They accelerate such that the distance between the string attachments stays constant.

 

[Dale]

Therefore relative to that frame, the distance between the ships must continue to increase without bounds as you calculated with your analysis.
This itself seems problematic. How do you reconcile frame agreement of time and position observations of the ships when the distance remains constant in one frame and is indefinitely expanding in the other frame??

Why should this be problematic or need reconciliation? The worldlines of the spaceships are curved. This same effect happens with curved lines in Euclidean geometry.

Take a piece of paper and draw two copies of the same curved shape separated by some fixed horizontal distance. Then draw a set of parallel oblique lines and see how the distances are not fixed on the oblique lines even though they are fixed on the horizontal lines.

Adopting for the moment a purely kinematic interpretation of SR which has been expressed by many knowledgeable people on this forum in the past.
Assuming no contraction of either the distance or the string. At a velocity of 0.8 c what would be the measurement of that distance?

I don't know what you are asking here, nor why you would assume no contraction.

I might point out that if the physical reality of contraction as implied by the Maxwell maths is actually valid this leads strongly to the logical conclusion that all motion is actual and in a sense absolute, even if it is not determinable or measurable in non-relative quantitative terms. Would you agree?? ;-)

I don't know what you mean by this either.

 

[Austin0]

Or conversely: Assume the ships started accelerating from signals that were simultaneous
in the 0.8c frame. Now in that frame the distance remains constant, but in the launch frame the trailing ship fires up first and must eventually intercept and overtake the lead ship.
How do we reconcile these contradictory expectations?

You missed this one.

Adopting for the moment a purely kinematic interpretation of SR which has been expressed by many knowledgeable people on this forum in the past.
Assuming no contraction of either the distance or the string. At a velocity of 0.8 c what would be the measurement of that distance?
According to an 0.8c inertial frame the clock in the launch frame would be running ahead at the instantaneous location of the lead ship.Therefore the trailing ship and string would move forward an additional distance before reaching a clock with the same proper time. I.e.; Would be measured as being contracted relative to the initial separation.

I don't know what you are asking here, nor why you would assume no contraction.

The assumption of no contraction was the kinematic interpretation of SR. That all effects were purely the result of relative motion. Coordinate evaluations without necessary physical imp0lications. As I said this is not my assumption but an assumption made by others who adopt this interpretation.
How many times over the years has someone asked , "is contraction real or what causes it" and the expert response has been that it is not necessarily physical but merely an effect of relative motion?
I merely noted that employing this assumption to clock desynchronization could produce a coordinate contraction without an assumption of actual physical contraction.

I might point out that if the physical reality of contraction as implied by the Maxwell maths is actually valid this leads strongly to the logical conclusion that all motion is actual and in a sense absolute, even if it is not determinable or measurable in non-relative quantitative terms. Would you agree?? ;-)

I don't know what you mean by this either.

Likewise, this is a perennial question, "Is motion real " and the majority response has been a resounding NO , it is purely relative with no physical actuality.
But if one accepts the validity of the physical contraction implied by Maxwell then logically it should be said that, yes it is real, with actual physical consequences but is simply undetectable and unquantifiable. So I will ask you , in your view is motion real or purely relative??

 

[Dale]

You missed this one.

It appears to be the same question as the other one. Similarly with the question below.

The assumption of no contraction was the kinematic interpretation of SR. That all effects were purely the result of relative motion. Coordinate evaluations without necessary physical imp0lications. As I said this is not my assumption but an assumption made by others who adopt this interpretation.
How many times over the years has someone asked , "is contraction real or what causes it" and the expert response has been that it is not necessarily physical but merely an effect of relative motion?
I merely noted that employing this assumption to clock desynchronization could produce a coordinate contraction without an assumption of actual physical contraction.

My usual response to such questions, and to this one, is to ask the person to define "real" and "actual" and "physical". Those terms are quite ambiguous and without a solid definition of them the question cannot be answered.

If you have a scenario which is described completely in one inertial frame, then you can use the Lorentz transform to obtain the equivalent scenario in any other inertial frame and be assured that the same laws of physics explain the scenario in both frames.

 

[A.T.]

How many times over the years has someone asked , "is contraction real or what causes it" and the expert response has been that it is not necessarily physical but merely an effect of relative motion?

Is death from rifle bullets physical? Nah, merely an effect of relative motion.

 

[Eli Botkin]

DaleSpam:
I've reviewed your math (your earlier reply #43) and find no error. What you've shown for certain is that for a sequence of instantaneously co-moving observers the ships' separation is some d' > d. I'm familiar with the math, having done this myself. And of course your conclusion favoring string breakage is pre-ordained since you've also stipulated that the string must remain L (= d) at any d'.

But why does every co-moving observer "see" d expand to a value d' > d but not see L undergo a proportionate expansion to an L' > L? I think that would be an essential point to address so others couldn't claim that you have, in effect, proved what you assumed.

By selecting the frames of the co-moving observers (who always deal with a d' > d) you have avoided the problem of finding a solution for frames of observers for whom d' < d (and, as you know, there are many such frames.)

I note that you don't claim string contraction, as many others (including Bell) do. Is that because you've selected a massless string (no atoms to contract)? ;-)

 

[Dale]

I've reviewed your math (your earlier reply #43) and find no error. What you've shown for certain is that for a sequence of instantaneously co-moving observers the ships' separation is some d' > d. I'm familiar with the math, having done this myself. And of course your conclusion favoring string breakage is pre-ordained since you've also stipulated that the string must remain L (= d) at any d'.

Good, so now you have seen it proven using SR.

But why does every co-moving observer "see" d expand to a value d' > d but not see L undergo a proportionate expansion to an L' > L? I think that would be an essential point to address so others couldn't claim that you have, in effect, proved what you assumed.

We did assume it. We assumed the string was stiff. If we had instead assumed it was elastic then it could have stretched, but then we would have been working a different problem, one requiring a relativistic version of Hookes law and the elasticity of the string material.

By selecting the frames of the co-moving observers (who always deal with a d' > d) you have avoided the problem of finding a solution for frames of observers for whom d' < d (and, as you know, there are many such frames.)

Yes, that is why I chose the MCIF.

 

[Eli Botkin]

DaleSpam:
Saying "Yes, that is why I chose the MCIF" sounds like an admission that you knew you couldn't get that SR conclusion in the other frames.

Calling the string "stiff" is just another way of saying "all L' = L, regardless of the frame.." I think that SR should decide that. You should be aware that SR tells us that even "proper lengths" of so-called stiff objects can be altered for a particular inertial observer if the object undergoes a history of acceleration. If this is unfamiliar territory, let me know so I would then describe how and why. But that would have to wait until next weekend since I will be sans computer until then.

 

[Dale]

Saying "Yes, that is why I chose the MCIF" sounds like an admission that you knew you couldn't get that SR conclusion in the other frames.

Wow! You sound paranoid.

There are an infinite number of equally valid ways of doing a problem, and we agree that we can choose to do it the easy way. There is also an obvious difficulty with some of those ways. So I pick a way which avoids the obvious difficulty precisely because it avoids the obvious difficulty (which you agree is valid to do).

To me the obvious conclusion is that I am lazy and don't want to do things the hard way, but what comes to your mind is instead that I am trying to hide the fact that it can't be done the other way. Sounds like you think I am some sinister agent of a cover up.

The problem can be worked in an infinite number of frames, and because of the principle of relativity we know that the answer must be the same. I welcome you to do it the hard way if that interests you, but I am lazy and will stick with the easy way. If you get stuck (as often happens when doing things the hard way) then post your work, and I have already offered to help get you unstuck.

 

[Eli Botkin]

QDaleSpam:
Since no paranoia has been detected in my family it's not likely that I carry that trait, nor do I think that you are lazy :-)

I think you should also address the issues in my 2nd paragraph, they are important for understanding what SR is saying.

 

[Dale]

Calling the string "stiff" is just another way of saying "all L' = L, regardless of the frame.."

No, stiff means that the proper length is always L, regardless of the forces on the string. In frames other than the MCIF it is certainly possible that L≠L'.

 

[Eli Botkin]

DaleSpam:
Returned from vacation, willing to continue discussion of our differences.

You say “… stiff means that the proper length is always L, regardless of the forces on the string.” I would modify that to: stiff means that the length is an unchanging value (regardless of forces applied) in any selected frame. In a different frame it will still be unchanging, but at a different value. SR’s transformation equations require that.

As for the term “proper length,” care must be taken not to think of it as a synonym for “true length.” “Proper length” is only shorthand for “the length measured in a frame wherein the body is at rest.” In SR that’s just another frame like any other.

So, if a log (or string) has “proper length” L in frame A, then, after acceleration to a velocity V, it will have a “proper length” >L in a frame B that has a velocity V relative to frame A.

Therefore it cannot be correct to assume that the stiff string maintains the same length L (or proper length L) throughout its acceleration interval.

 

[Dale]

I would modify that to: stiff means that the length is an unchanging value (regardless of forces applied) in any selected frame.

This is only true if the object and the frame are both inertial, which isn't the case here.

So, if a log (or string) has “proper length” L in frame A

Proper length is frame invariant. If it has proper length L then it has proper length L in frame A, B, C, D, ... Proper length only equals coordinate length in the rest frame.

 

[Eli Botkin]

DaleSpam:
You say "This is only true if the object and the frame are both inertial,..."

I presume that you would define a non-inertial object as an object under acceleration. But it is reasonable to consider such an object to be "jumping" from one instantaneously co-moving inertial frame to the next such frame. That's what acceleration is, its a transfer from one inertial frame to another. SR deals with coordinate transformations between inertial frames.

I don't know what you mean by "Proper length is frame invariant. If it has proper length L then it has proper length L in frame A, B, C, D, ... " Please tell me what SR says about proper length when it isn't related to the object's rest frame.

 

[yuiop]

EDIT: Oops, I had only read the first page of this thread when I posted this so completely missed 4 pages of the discussion. Hope it is still relevant.

This also makes sense-- but why doesn't the distance between the ships shrink along with the string? Shouldn't it all be length contracted?

It does and it doesn't ...bear with me and I will try and explain. Imagine we have two spaceships parked on the ground that 1 kilometre apart. They are joined by a 1km string. This string is designed to snap when stretched to twice its rest length. In the ground frame both rockets take off simultaneously and accelerate equally, such that they maintain a spatial separation of 1 km at all times (as measured in the ground frame). When the rockets reach a velocity V, which corresponds to a gamma factor of 2, the length of the string in the ground frame should be 1/2km due to length contraction and is stretched over a space of 1km so it snaps. In the ground frame, there is no length contraction of the space between the rockets. Length contraction requires we have a velocity and we cannot assign a velocity to the space (vacuum) between the rockets.

Now let us have a look from the point of view of the rockets. Initially they see the separation as 1 km. As they accelerate the space between them appears to increase and as they arrive at the critical velocity V they measure the space between themselves as 2km. The string is approximately at rest in there reference frame because it is co-moving with the rockets so it should have a length of 1km but it is stretched over a distance of 2km so it also snaps from their point of view. (note that I am using a loose definition of rest frame for the rocket observers, because from their point of view they are are not exactly at rest with respect to each other, but it is a reasonable approximation for our purposes).

Note that according tot he rocket observers the distance between the rockets is 2km and according to the ground observers the distance is 1 km, so there is a sort of "length contraction" of the space between the rockets because different observers disagree on the length, but at no time does the spatial separation between the rockets contract according to any observer. In fact the distance expands according to the rocket observers and remains constant according to the ground observers.

The string on the other hand can be assigned a velocity relative to the ground frame and so it really does physically contract according to the ground observers.

In summary, according to the ground based observers the string contracts, but the space does not and according to the rocket based observers the space expands (because they consider themselves to be moving apart from each other) and the theoretical length of the string remains constant. The space between the rockets according to the ground based observers is 1/2 the distance measured by the rocket based observers, so the Lorentz transformation of space is still satisfied, even though there is no actual "contraction" of the separation space according to any observer.

You cannot accelerate space (vacuum) to make it contract, but your perception of the space between two markers can vary with your velocity relative to the markers.

If we do a variation of the paradox, whereby the rocket captains are instructed to maintain a constant distance of 1km between their rockets (by their own measurements) as they accelerate, then at the critical velocity V, the ground based observers will measure the space between the rockets to be 1/2km and so any string between the rockets will not snap, because this time the string and the separation distance, length contract at the same rate according to the ground based observers, while the rocket based observers say the string length and separation space remain constant so they also agree that the string does not snap.

 

[A.T.]

I don't know what you mean by "Proper length is frame invariant. If it has proper length L then it has proper length L in frame A, B, C, D, ... " Please tell me what SR says about proper length when it isn't related to the object's rest frame.

What do you mean by "proper length when it isn't related to the object's rest frame". Proper length is per definition the length measured in the the object's rest frame. From any other frame, the proper length of a moving object is measured by a co-moving ruler.

 

[Dale]

SR deals with coordinate transformations between inertial frames.

See http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html

I don't know what you mean by "Proper length is frame invariant. If it has proper length L then it has proper length L in frame A, B, C, D, ... " Please tell me what SR says about proper length when it isn't related to the object's rest frame.

The proper length is
$$\int_P \sqrt{g_{\mu \nu} dx^{\mu} dx^{\nu}}$$
where P is the space like path consisting of the intersection of the objects worldsheet with a hyperplane orthogonal to the tangent vector.

Eli, you seem to be grasping for straws now. As I said, the fact that the string breaks can be proven with SR. With this current line of questioning you are straining at very minor details that are already well established in the SR literature. How to handle acceleration and the definition of proper length are well known. The fact that you don't know about them is not a flaw in the proof I gave.

I am willing to continue the conversation in the context of improving your education, but not in the context of defending the proof above. It is a valid proof that the string breaks. Is that acceptable to you?

 

[Eli Botkin]

DaleSpam:
Thanks for your reply. I am always willing to continue my education, as we all should.

It seems to me that in your replies there has been an avoidance of a certain SR question, which is not a minor detail.

Why would an inertial observer, who is traveling in a direction opposed to the ships', predict that the string will break? In his frame the ships approach each other. Your own prediction of breakage is based on frames wherein the ships increase their separation while (you say) the string retains its "proper" length (reply #43). After all, a principal SR teaching is that the physics should be coordinate-free.

I would truly appreciate your view of this issue.

 

[Dale]

Do you accept the proof as valid?

 

[Eli Botkin]

What I accept is that under all your assumptions your mathematical deduction is correct.

You've correctly shown that for frames in which the ships are momentarily at rest, successive frames show an increasing ship separation. One of your assumptions is that the string length, L, stays constant from such frame to frame. Therefore you can correctly conclude that the string will break.

Now to my reply #82, would you be willing to address that? Thanks.

 

[Dale]

What I accept is that under all your assumptions your mathematical deduction is correct.

Fair enough.

Do you think the assumptions I made are the standard ones relevant to Bells spaceships? Do you think they are correct assumptions?

 

[Dale]

Why would an inertial observer, who is traveling in a direction opposed to the ships', predict that the string will break? In his frame the ships approach each other.

The string length contracts more than the distance between the ships decreases. I would encourage you to work this out for yourself.

 

[Eli Botkin]

DaleSpan:

You've taken me aback. I can hardly imagine that someone who exhibits such SR expertise would not be aware of the inertial frames within which the ships are approaching each other.

Say the ships accelerate to the right in the (rest) frame where they started at the same time. In that frame their separation remains constant.

An inertial observer moving (at some speed V) to the right will say that the lead ship started to accelerate earlier than the aft ship, leading to a continual increase in separation.

Conversely, an observer moving (at speed -V) to the left will say that the aft ship started to accelerate earlier than the lead ship, leading to a continual reduction in separation, without ever overtaking it.

I would encourage you to work this out for yourself ;-)

 

[Dale]

I would encourage you to work this out for yourself ;-)

I did, you apparently missed my update.

That is precisely why working things out for yourself is so important and why I encourage others to do so also. Did you think that I don't follow my own advice?

 

[Eli Botkin]

DaleSpan:
Your reply#88 puzzles me. What update did I miss? My reply #87 was an answer to your #86.

Are you still maintaining that there are no inertial frames wherein the ships' separation decreases? Can you show that?

 

[Dale]

Re-read 86.