Rest Length, Coordinate Length, and an argument for True Length

[phyti]

Begin with an object at rest in frame A.
Claude passes the object at speed b in direction x.
Maude passes the object at speed c (with c > b) in direction x.
Both pass the first end simultaneously.
Their length measurements of the object will differ.

How can the object, in a state of equilibrium, with no forces acting on it, simultaneously have two different lengths?

Why does the measured length only depend on the speed of the passing observer?

Are the length measurements 'real', or are they just calculations?

 

[PAllen]

Begin with an object at rest in frame A.
Claude passes the object at speed b in direction x.
Maude passes the object at speed c (with c > b) in direction x.
Both pass the first end simultaneously.
Their length measurements of the object will differ.

How can the object, in a state of equilibrium, with no forces acting on it, simultaneously have two different lengths?

Why does the measured length only depend on the speed of the passing observer?

Are the length measurements 'real', or are they just calculations?

It is measured to have two different lengths. The object has a length in its own rest frame, and other lengths measured by different observers moving relative to it. I refuse to discuss real. The measured length depending only on speed follows from the Lorentz transform. You can consult any book on SR for a derivation of Lorentz transform.

How real are the measurements? Well, each observer can whip a ring past the object without hitting it, if the ring is a little bigger than the measured length (in theory). You decide whether this constitutes reality.

How can this be? Well how can two observers measure the same clock going at two different rates? How can a particle diffract or not passing through two slits depending on the type of measurement made *after* it has passed through the slits? Ultimately, you have to accept that that is just the way it is.

 

[GrayGhost]

How can the object, in a state of equilibrium, with no forces acting on it, simultaneously have two different lengths?

Because the 2 differing lengths are the result of 2 differing POVs, not a change in and of the body itself. Each differing POV measures space and time differently. Also, the Lorentz transformations are kinematic, so forces play no role.

Why does the measured length only depend on the speed of the passing observer?

If the object radiated a beam of light after both Claude and Maude passed it by, they would each record the light to pass them at speed c even though they both move relatively at v. This can only happen if Claude measures time differently than Maude does. Since x/t = c = X/T, then they also each measure space differently. Because they do, they both measure a different contracted length of the moving object. Deriving the Lorentz tranformations will answer your question best though.

Are the length measurements 'real', or are they just calculations?

The calculations are predictions, and they will match what is measured. If you believe that what is measured is real, then they are real.

GrayGhost

 

[GrayGhost]

PAllen,

I gave it further thought. I know you do not wish to discuss "real", so no response is necessary. I agree that the 3d analogs are analogous to the SR 4d case. However, regarding the matter as to whether the moving contracted lengths should be labeled apparent vs real, I maintain this ...

The contracted length is the real length of the desynchronised body in motion, as it's measurable. The proper length is the real length of the synchroised body as stationary, as it's measurable. Anywho, that's how I see it.

I suppose there's a valid argument that the contraction be labeled apparent, since the proper length of the body never changes. However, I figure it best to say "moving length and proper length" vs "apparent length and true length".

GrayGhost

 

[bobc2]

In this discussion there appears to be a misunderstanding that is due to Minkowski. Based on his presentation of SR, some people here think that length contraction (and as a consequence, time dilation) "just relates to differences in cross-section views" so that "the rod itself is completely unaffected".

I seems as though someone in the example you've cited has been applying an operator rather than performing a coordinate transformation. The differences in cross-section views we've been talking about, by definition, are in fact just the components of a fixed vector after a coordinate transformation. The vector does not change.

If you wish to have a 4-D rod that has physically changed (the 4-D vector has changed), then you apply the appropriate operator.

 

[PAllen]

PAllen,

I gave it further thought. I know you do not wish to discuss "real", so no response is necessary. I agree that the 3d analogs are analogous to the SR 4d case. However, regarding the matter as to whether the moving contracted lengths should be labeled apparent vs real, I maintain this ...

The contracted length is the real length of the desynchronised body in motion, as it's measurable. The proper length is the real length of the synchroised body as stationary, as it's measurable. Anywho, that's how I see it.

I suppose there's a valid argument that the contraction be labeled apparent, since the proper length of the body never changes. However, I figure it best to say "moving length and proper length" vs "apparent length and true length".

GrayGhost

I think I could agree with this. I would perhaps call it moving length and rest frame length, because of overloading of proper (I have checked 5 relativity texts, including MTW, and none use proper length to mean rest frame length of a rigid object). I also perceive rest frame length as having a unique invariant definition, at least for a Born rigid object.

There are many subtleties I haven't worked out to my own satisfaction. These revolve around what would be 'seen' in some of the cases under discussion due to Terrell-Penrose rotation.

 

[harrylin]

harrylin,

Interesting, this topic of real vs apparent. It's the debate between ... (1) the proper POV presents what is real, versus (2) what is measurable is real, proper or not.

Let's say you and another fellow are standing on a train track, and a wonder train is whizzing toward you at v=0.866c. He who jumps first "is chicken", and loses the $1 bet. You ask yourself, what is the last moment I should jump off the track to win the bet and avoid being splattered? So you get your handy dandy calculator out, punch in some numbers right quick like, obtain the solution and the plan. You run your figures based on the contracted length, and the other fellow runs his based on the proper length. The only data you have is the train velocity, and the location of the train's center at any instant.

I'd bet $2 you win and he's chicken.

GrayGhost

That's a nice variant of the "Einstein (dead or alive) cat" example in the paper that I referred too. And easier to calculate.

However, it's incompatible with the PoR to claim that your POV is correct (so that the train is really contracted by gamma), as than all other POV's incl. that of the train frame would be wrong.

 

[GrayGhost]

That's a nice variant of the "Einstein (dead or alive) cat" example in the paper that I referred too. And easier to calculate.

However, it's incompatible with the PoR to claim that your POV is correct (so that the train is really contracted by gamma), as than all other POV's incl. that of the train frame would be wrong.

harrylin,

If you reread my stated scenario, you will likely realize that both betters are of the same frame, that the loser of the bet fails because he makes mis-assumptions of the train's length, while the winner runs the LTs and learns how to win. Passengers of the wonder train will agree with everyone else, that you win the bet, and he was chicken. That said, the PoR was not violated. Quite the opposite, as it was upheld.

GrayGhost

 

[harrylin]

harrylin,

If you reread my stated scenario, you will likely realize that both betters are of the same frame, that the loser of the bet fails because he makes mis-assumptions of the train's length, while the winner runs the LTs and learns how to win. Passengers of the wonder train will agree with everyone else, that you win the bet, and he was chicken. That said, the PoR was not violated. Quite the opposite, as it was upheld.

GrayGhost

GrayGhost, passengers of the train will tell you that they made the same bet as you while they hold that the train had its proper length. That's the PoR. The PoR would be violated if it could be shown that their POV was "less real", and that you could claim that the train was "truly contracted" by the Lorentz factor.

 

[PAllen]

GrayGhost, passengers of the train will tell you that they made the same bet as you while they hold that the train had its proper length. That's the PoR. The PoR would be violated if it could be shown that their POV was "less real", and that you could claim that the train was "truly contracted".

But that has nothing to do with GrayGhost's example. In the train, the you win if you bet the train has 'rest frame length'. On the platform, you win if you bet the train has 'moving length'. That is correct, and is what GrayGhost was saying (as I understood him). What is 'real' (e.g. wins bet) is frame dependent.

 

[GrayGhost]

GrayGhost, passengers of the train will tell you that they made the same bet as you while they hold that the train had its proper length. That's the PoR. The PoR would be violated if it could be shown that their POV was "less real", and that you could claim that the train was "truly contracted".

PAllen summed it up pretty well there. Train passengers will indeed measure their train length just as usual, its proper length. They'll see the betters moving and length contracted. Train passengers will witness the chicken to jump off the track first, then you second (just missing the train). Everyone agrees as to what happened, and everyone can predict what the other experiences via the LTs. You still disagree?

GrayGhost

 

[harrylin]

But that has nothing to do with GrayGhost's example. In the train, the you win if you bet the train has 'rest frame length'. On the platform, you win if you bet the train has 'moving length'. That is correct, and is what GrayGhost was saying (as I understood him). What is 'real' (e.g. wins bet) is frame dependent.

I understood GreyGhost differently, it sounded as if the view of the stationary observer is more real than the view of the train observer. And who wins the bet is certainly real; but that has nothing to do with "True Length"!
The topic of this thread is that some people claim that a certain inertial POV is "true", so that a conflicting inertial POV is not true or not real but "distorted". That is in conflict with the PoR, according to which none of the observers can say that his view is "the true view".

 

[PAllen]

I understood GreyGhost differently, it sounded as if the view of the stationary observer is more real than the view of the train observer. And who wins the bet is certainly real; but that has nothing to do with "True Length"!
The topic of this thread is that some people claim that a certain inertial POV is "true", so that a conflicting inertial POV is not true or not real but "distorted". That is in conflict with the PoR, according to which none of the observers can say that his view is "the true view".

There is a philosophic choice available that is not (in my view) not inconsistent with PoR. You can choose to say the rest length of a rigid object is an intrinsic feature of the object, without saying there is anything preferred about a frame or observer in which the object is at rest (or that measurements made in other frames are less real measurements). Make analogy to rest mass or invariant mass - its existence and usefulness does not give special preference to any particular inertial frame. The additional fact that rest mass is fundamental for elementary particles is not really part of SR as a classical theory.

Similarly, I have argued that one can (if desired) define a unique, invariant rest length for a Born rigid object, based on the fact that a flat spacelike hypersurface 4-orthogonal to one of its world lines is also orthogonal to all of them; and further, any point along the world tube you take an orthogonal slice, you get the same result (even if the object is moving non-inertially). These statements are true only for a Born rigid object.

[EDIT: To be clear, I am aware that you cannot generalize all desirable properties of Born rigidity to GR; and this is related to the inability, for example, to have strict Born rigidity in a rotating object in SR]

 

[harrylin]

There is a philosophic choice available that is not (in my view) not inconsistent with PoR. You can choose to say the rest length of a rigid object is an intrinsic feature of the object, without saying there is anything preferred about a frame or observer in which the object is at rest (or that measurements made in other frames are less real measurements). Make analogy to rest mass or invariant mass - its existence and usefulness does not give special preference to any particular inertial frame. The additional fact that rest mass is fundamental for elementary particles is not really part of SR as a classical theory. [..]

Intrinsic is again another poorly defined word..

But I fully agree that proper measurements are special, if that is what you mean.
In a certain sense they are absolutes: everyone agrees on it, and even everyone agrees with you when you compare a rod with your standard, co-moving ruler and you find that the rod is shorter than your ruler. In that sense, it is truly shorter!

 

[bobc2]

In a certain sense they are absolutes: everyone agrees on it, and even everyone agrees with you when you compare a rod with your standard, co-moving ruler and you find that the rod is shorter than your ruler. In that sense, it is truly shorter!

harrylin, my take on this is that you must decide whether you wish to talk about a 4-D rod or a 3-D rod. But, you must make it clear what physical object you are referring to.

If you are talking about observers making measurements on a 3-D rod, then you must recognize that the two observers are not even looking at the same rod. So it would not be correct to say that a given 3-D rod has two different measurements that are unequal. They are definitely not even the same 3-D rod. When you are viewing a different cross-section of a 4-D rod you are not looking at the same 3-D object--thus, no confusion. One rod has one length and the other has another.

If you are talking about measurements of a 4-D rod, then you emphasize that the cross-section views are different--therefore no surprize or puzzle about getting different measurements.

 

[GrayGhost]

harrylin,

I think you read more into my prior thought experiment than I actually said. I for one believe that contractions and desynchronised bodies are real. By real, I mean that "it must exist as the math requires it". If not, then as you said, the PoR is violated. In theory it should be measurable, although light transit time (and effects) would need negated.

GrayGhost

 

[Dale]

I for one believe that contractions and desynchronised bodies are real. By real, I mean that "it must exist as the math requires it".

That is very refreshing to use the word real and define what you mean by it. Thank you.

 

[harrylin]

harrylin, my take on this is that you must decide whether you wish to talk about a 4-D rod or a 3-D rod. But, you must make it clear what physical object you are referring to.

One measures the dimensions of objects with rulers... Length, width and height. Thus 3D, obviously!

[/QUOTE]If you are talking about observers making measurements on a 3-D rod, then you must recognize that the two observers are not even looking at the same rod. So it would not be correct to say that a given 3-D rod has two different measurements that are unequal. [/QUOTE]

You lost me there. Do you disagree with special relativity? By definition it is the same object:

A rigid body which, measured in a state of rest, has the form of a sphere, has in a state of motion (viewed from the stationary system) the form of an ellipsoid.
- http://www.fourmilab.ch/etexts/einstein/specrel/www/

 

[harrylin]

harrylin,

I think you read more into my prior thought experiment than I actually said. I for one believe that contractions and desynchronised bodies are real. By real, I mean that "it must exist as the math requires it". If not, then as you said, the PoR is violated. In theory it should be measurable, although light transit time (and effects) would need negated.

GrayGhost

That is what I read in what you said; and I tried to make clear that, if what you say is not a self contradiction, then it is in conflict with the PoR. But perhaps I misunderstand what you mean with "exist"... has "exist" perhaps for you the same meaning as "appear" for Einstein? Or do you hold that A>B AND B>A (from which follows that A not>B) is not a self contradiction, so that for a certain event A>B and B>A can both be true?

 

[GrayGhost]

But perhaps I misunderstand what you mean with "exist"... has "exist" perhaps for you the same meaning as "appear" for Einstein?

I'm not certain precisely what Einstein was thinking when he said "appears" in OEMB, however I'm rather sure it encompassed what I stated wrt "exists". I'm not sure to what extent Einstein thought it thru wrt light transit effects (in depth) for the observational process.

Or do you hold that A>B AND B>A (from which follows that A not>B) is not a self contradiction, so that for a certain event A>B and B>A can both be true?

Wrt moving body lengths, and for 2 bodies (A & B) of the same proper length in relative motion colinearly at luminal v ... I hold that A>B per A AND B>A per B, and that this is not a contradiction because A & B will each agree that the other records it differently (and as such). Both are correct.

GrayGhost

 

[harrylin]

[..]
Wrt moving body lengths, and for 2 bodies (A & B) of the same proper length in relative motion colinearly at luminal v ... I hold that A>B per A AND B>A per B, and that this is not a contradiction because A & B will each agree that the other records it differently (and as such). Both are correct.
GrayGhost

Sure - and that is not relevant for this thread! Sorry if my question was not clear. There is no question that different views exist, but this thread is certainly not about the reality that different views exist. It is about the "truth" value of certain views.

According to most people and math books, A>B AND B>A (not "per A" and "per B" but "in truth" (that is, according to a single, absolute view) is a contradiction. Except for a small number of people who think that there can be contrary truths, so that for example Obama is the president of the USA AND Obama is not the president of the USA.

In an case, it is contrary to the PoR to claim that one of two random views is preferred or "the true view", corresponding to a reality that "exists", so that the other view is distorted and does not correspond to the existing reality.

 

[JesseM]

One measures the dimensions of objects with rulers... Length, width and height. Thus 3D, obviously!

If you are talking about observers making measurements on a 3-D rod, then you must recognize that the two observers are not even looking at the same rod. So it would not be correct to say that a given 3-D rod has two different measurements that are unequal.

You lost me there. Do you disagree with special relativity? By definition it is the same object

I don't know what you mean by "object", but if we're talking about the 3D rod at a particular moment rather than the entire 4D world-tube of the rod, then obviously different frames disagree about simultaneity so the set of points in spacetime they're referring to when they say "the 3D rod at a particular moment" must be different.

 

[Samshorn]

There is no question that different views exist, but this thread is certainly not about the reality that different views exist. It is about the "truth" value of certain views. According to most people and math books, A>B AND B>A (not "per A" and "per B" but "in truth" (that is, according to a single, absolute view) is a contradiction.

Bob is standing to Alice's left, and Alice is standing to Bob's left. This is not a contradiction, and both statements are absolutely true. You might prefer to say that two people can't both REALLY be to the left of each other, because if A is to the left of B then B must be to the right of A. However, such a statement would apply only to some third person's left and right, which is no more absolutely correct than Bob's or Alice's senses of left and right.

The concepts of leftness and rightness have no meaning without specifying the reference system, so one cannot talk about absolute left, or absolute right, free of any specified system of reference.

Exactly the same applies to spatial "length". It has no meaning without specifying a system of reference. So the only statements whose truth value we can even consider must specify the system of reference.

 

[Dale]

Hi Samshorn, welcome to PF, and well said!

 

[Dale]

One measures the dimensions of objects with rulers... Length, width and height. Thus 3D, obviously!

And duration, thus 4D.

 

[GrayGhost]

Sure ... it is contrary to the PoR to claim that one of two random views is preferred or "the true view", corresponding to a reality that "exists", so that the other view is distorted and does not correspond to the existing reality.

Well, I do not (and have never) disagreed with these statements. My scenario surely did not. Therefore, your issue seems to be in that you believe both observers cannot concurrently record contractions of the other, which you say is a contradiction and a violation of the PoR. That's incorrect, here's why ...

The PoR says that all observers use the very same mechanics, and thus the laws of physics are invariant under rotation. Both observers use the very same LTs, and per the LTs, each must record the other contracted when viewed from a relative state of motion. If both observers did not record the contractions of the other, then the PoR would be violated. Each their POVs are as valid as the other's, and no more preferred than the other's.

Samshorn's example is a good one. Here's another ... 2 ships in calm waters drift apart in the ocean. Eventually, each claims the other is dropping under the horizon. It would be incorrect for either to assume that they themselves are not dropping under the horizon per the other, even though no one ever believes himself to drop under the horizon. It's all about reference and POV.

GrayGhost

 

[harrylin]

Bob is standing to Alice's left, and Alice is standing to Bob's left. This is not a contradiction, and both statements are absolutely true. You might prefer to say that two people can't both REALLY be to the left of each other, because if A is to the left of B then B must be to the right of A. However, such a statement would apply only to some third person's left and right, which is no more absolutely correct than Bob's or Alice's senses of left and right.

The concepts of leftness and rightness have no meaning without specifying the reference system, so one cannot talk about absolute left, or absolute right, free of any specified system of reference.

Exactly the same applies to spatial "length". It has no meaning without specifying a system of reference. So the only statements whose truth value we can even consider must specify the system of reference.

Yes indeed: according to the PoR, an observer cannot claim to observe the "True Length" in the same way that an observer cannot claim "Absolute Left".
Note: funny enough this turned out to be not true for all kinds of left and right, in view of broken symmetry - but that's another topic.

 

[harrylin]

Well, I do not (and have never) disagreed with these statements. My scenario surely did not. Therefore, your issue seems to be in that you believe both observers cannot concurrently record contractions of the other, which you say is a contradiction and a violation of the PoR. [..]

I say exactly the contrary: there is no question about the reality that different views exist and I explained how that works with the train example. We fully agree on that. I disagree with calling measured contractions "real", as it suggests "true contraction", corresponding to "True Length" (there can be no doubt that a real measurement was done; thus the precision "real" cannot refer to the reality of doing measurements!).

Again: the topic of this thread is the claim that one observer has the right to say to be observing the "True Length". That is incompatible with the PoV, as I tried to explain, and now Samshorn added a nice illustration about "True Left".

 

[harrylin]

And duration, thus 4D.

As this is not the topic I won't come back to this here, but how many dimensions does a cube have? Usually one understands dimensions of objects to mean "spatial dimensions" - thus a cube is 3D. If one does not limit it in that way then the right answer is undefined. For example 5 physical "dimensions": height, width, depth, temperature, time.
http://en.wikipedia.org/wiki/Dimensional_analysis

Note: one can measure the length of an object with duration if one replaces a ruler by a laser ray, a stopwatch and the speed of light; and then one still measures its 3 spatial dimensions.

 

[Dale]

As this is not the topic I won't come back to this here, but how many dimensions does a cube have? Usually one understands dimensions of objects to mean "spatial dimensions" - thus a cube is 3D. If one does not limit it in that way then the right answer is undefined. For example 5 physical "dimensions": height, width, depth, temperature, time.

Sure, but if you are measuring the extent of an object it takes 4 numbers: length, height, width, and duration.

Temperature certainly can be a dimension in some space, e.g. in state space, but it does not measure the extent of an object. Additionaly, from a mathematical perspective, temperature is not a dimension in the same mathematical space as space and time. Similar arguments cannot be made for excluding duration.