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

[bobc2]

.

Thus, the rod is independent of its X coordinates.

This conception of the rod is completely independent of time, for the simple reason that time does not exist.

Greg,

It occurs to me that your description of the situation would be consistent with a concept of the 4-dimensional universe with a 4-dimensional rod existing somewhere in that universe. I'm not trying to propose a cosmology here (and certainly don't have a metric in mind for it), but am just trying to present a picture to see if you identify with it some way.

With this picture I think you could make a case for the length of the rod in the context you seem to be communicating, i.e, a 4-dimensional rod with definite dimensions that are independent of any observers (although an observer would have to be comoving with the rod to be able to observe the actual 4-dimensional "length" (of course a more significant length would be the really long length along X4).

I thought of this when I noticed your mention of "...independent of time."

I'm probably way off base again, but it's the only thing I could imagine that would be consistent with your description. Perhaps you would agree with the picture mathematically but would reject the reality of the 4-dimensional world while retaining the mathematical implication of a "true" rod length (nevertheless, within the context of the picture).

Maybe I've just thrown in too much meaningless double talk here.

 

[Dale]

If we now introduce time into the world, the rod is not changed; it is still fully described by its length. If we wish to represent the rod on a two dimensional graph, one axis for distance and the other for time, the rod must be drawn parallel to the X axis of the graph, because by definition the rod is defined by its length only.

No. Once you introduce time the rod is characterized by duration as well as length.

 

[JesseM]

Or, it could be that you are so sure you are right that you are not listening to me.

I doubt that's it, because instead of explaining that I misunderstood something you said earlier, in this post you simply introduce a new argument that you hadn't made before. It's also pretty clear that you are not listening to me when you completely ignore the substance of my criticisms as you did in your previous post #81, and are now doing again, failing to address any of the things I said and just making a vague psychological accusation about my being too sure I am right. And I also made a request in that previous post, you either didn't see it because you skimmed my post again without reading carefully, or just chose to ignore it:

You also blithely ignored my counterexample, illustrating why your assumptions are circular:

Why can't I define "the rod" to be the 3-dimensional cross-section taken using a surface of simultaneity in the frame F where the rod is moving at 0.99c, and therefore say that only when we draw a spacetime diagram from the perspective of frame F will "the rod" be a horizontal line parallel to the x-axis?

Do you deny that if I define "the rod" in this way, then it follows from my definition that "In frame F (the one where the rod is moving at 0.99c), the time is the same at all points on the rod"? And likewise if we adopt this definition of "the rod", then the rod's rest frame is not seeing "the rod" at a single instant, but rather a collection of points on "the rod" at different instants? Please don't bother responding unless you have actually thought about this alternate definition of "the rod" and can explain in a non-circular way why anything other than aesthetic preferences should lead us to see your definition as more valid than this one.

If you aren't willing to address this alternate definition of "the rod" and tell me specifically what's wrong with it, then again please don't bother responding at all, I'm only interested in a genuine back-and-forth discussion here.

I'll try to develop my concept of what "the rod" is from scratch. This will, I believe, demonstrate that my starting point is fully independent of my conclusion.

In defining the rod, I begin with a minimal world. In this world there is only one parameter, distance. There is just one dimension of distance, which is represented in the usual way by an axis, X. By definition, the rod is fully described by its length. Thus, the rod is independent of its X coordinates.

This conception of the rod is completely independent of time, for the simple reason that time does not exist.

If we now introduce time into the world, the rod is not changed; it is still fully described by its length. If we wish to represent the rod on a two dimensional graph, one axis for distance and the other for time, the rod must be drawn parallel to the X axis of the graph, because by definition the rod is defined by its length only. We can draw the rod at various times, but in every case the rod must be represented by a line drawn parallel to the X axis.

But you can't just assume that when you "introduce time", the rod is at rest! That is exactly equivalent to assuming "the rod" is a 3D cross-section of the 4D world-tube using the rest frame's definition of simultaneity, and assuming that is just another way of assuming what you are trying to prove, i.e. yet another variation on the same circular argument. If you start with "only space" and a rod of length L aligned parallel to the X-axis in one instant, why shouldn't it be possible that when you "introduce time", you now find that rod of length L is moving at 0.99c along the X axis, so that L is not the rest length but rather the length in frame F where the rod is moving at 0.99c?

 

[phyti]

But time and distance both persist, you only think they don't because you are making an invalid comparison of a clock to a ruler. Time dilation does not directly affect the time on a clock, it directly affects the tick rate of a clock and then the clock integrates (or counts) the ticks to keep track of elapsed time.
...

Greg is correct on this.
Upon return by the space traveler, his clock is behind the Earth clock, and he is younger, a permanent change.
Their measurement of distance to the the turn around is the same, i.e.,what it was before he left.
The odometer reading is only a temporary record.

 

[GregAshmore]

Fascinating point of departure, Greg. Now, what do you have to say about someone else's perception of that same rod (someone moving at relativistic speed with respect to your rod's rest system)? Does the rod in its rest system have more claim to reality than that rod existing in the other coordinate system?

I'm working on the assumption that there is one reality which encompasses all coordinate systems, and in that reality there is just one rod. Your question seems to imply that there is a separate reality within each coordinate system.

Does the length of the rod in the moving system not represent a fundamental property of the rod (as perceived by a moving observer--notwithstanding the practical difficulty of actually observing it directly)?

I think this question boils down to, "Is perception reality?" In principle I would say no. The OP attempts to explain how the perception of the rod differs from the real rod.

By the way, there is of course something special about the world line length of the rod as well, i.e., c x (proper time) = millions of miles long (considering the time at the creation of the rod and the time that it disintegrates). Do you consider that a fundamental property of the rod as well?

I'm not sure what you mean by c x with reference to proper time. The definition of proper time is the change in time in a frame in which there is no change in x. Can you explain?

 

[GregAshmore]

I doubt that's it, because instead of explaining that I misunderstood something you said earlier, in this post you simply introduce a new argument that you hadn't made before.

No, not a new argument. The OP is based on this conception of the rod.

If you aren't willing to address this alternate definition of "the rod" and tell me specifically what's wrong with it, then again please don't bother responding at all, I'm only interested in a genuine back-and-forth discussion here.

I've been trying to address your proposition. It seems to me that this definition of the rod is fundamentally different than the one I am using, because time is an integral component of it. That's what I meant when I said that time and distance are not the same thing. The rod, for the purposes of this discussion, is entirely defined by distance--its length. (I say, "for the purposes of this discussion", because there are only two parameters in the LT, time and distance.)

But you can't just assume that when you "introduce time", the rod is at rest!

Whether the rod is at rest or not has nothing to do with the definition of the rod. The rod is entirely defined by its length. Movement involves time, which by definition is not a component of the rod. In other words, the rod is unaffected by its motion. Or, more to the point in a discussion of relativity, the rod never moves; it is always at rest.

That is exactly equivalent to assuming "the rod" is a 3D cross-section of the 4D world-tube.

No, it is not, for the reason given above. The 4D world-tube is not the rod; it is the rod plus time.

 

[GregAshmore]

No. Once you introduce time the rod is characterized by duration as well as length.

Well, here is where we disagree. By my definition, the rod exists (or in principle could exist) apart from time. Both definitions are rational.

btw, my definition does not clear up the murky water of "reality". It solves a particular problem in the interpretation of the spacetime diagram. I suspect that if I pursue it to its logical limit, I will run into trouble in some other area.

 

[bobc2]

I'm not sure what you mean by c x with reference to proper time. The definition of proper time is the change in time in a frame in which there is no change in x. Can you explain?

Sorry. That was a typo error. It was intended to be cT (speed of light times rod time duration--time from manufacture of rod until disentigration at the end of its life). I was wondering if you considered it to have a 4th dimension (a static 4-dimensional existence), even without the flow of time?

 

[Dale]

By my definition, the rod exists (or in principle could exist) apart from time.

How so? In your 2D spacetime universe in what way does "the rod exist apart from time"? From your description it seems to me that the number of dimensions in which the rod exists depends on the space, not the rod. Look at your justification for why it exists independent of time: "the rod is completely independent of time, for the simple reason that time does not exist". That justification is a property of the space, not the rod.

The only way you can say that it could exist apart from time is by considering a "toy universe" without time. In any universe with time, in what sense does "the rod exist apart from time"?

 

[bobc2]

How so? In your 2D spacetime universe in what way does "the rod exist apart from time"? ...
...The only way you can say that it could exist apart from time is by considering a "toy universe" without time. In any universe with time, in what sense does "the rod exist apart from time"?

What if Greg prepares a toy universe in his 3-D world with time added? It's like a 16' long 2x4 transparent beam with various bundles of fine fibers (optical fibers maybe?) extending from one end to the other, configured in such a way as to present a geometric causality and having 2-D cross-sections that are consitent with some sort of physics (as you look along the length of the beam with emerging fiber patterns in a continuous sequence of cross-section views). But the 2x4 is just sitting there static in his 3-D space. It exists as a 3-D structure, independent of time as it were. He arranges one collection of nano optical fibers in a configuration that presents itself as a sub-sub miniature 3-D rod inside of his toy universe--in this 3-D configuration it has a rectangular cross-section (maybe 5 nanoinch x 50 nanoinch) extruded along the 3rd dimension, perhaps for a distance along the 3rd dimension of .01 inch (not long compared to the 16' length of his universe.

Now he passes a light sheet (they use them a lot in manufacturing inspection systems these days) along the length of the beam at constant velocity, starting at one end and finishing at the other end. He smiles as he sees a plane of light penetrating his universe beam, lighting up a new cross-section at each new instant (could the excitation of electrons within special fiber groups give rise to consciousness--but I digress). He identifies that with the flow of time. He could even use some nanotechnology to engrave subminiature pictures of clocks spaced along the beam with time readings that correspond to the position along the beam divided by the light sheet velocity. So, now he can tell you the proper time at any point along the beam (which is really his own elapsed time in his own hyperspace 3-D world). His beam just sits there as a 3-D structure, independent of the passing of his simulated flow of time. The rod inside of his toy universe exists with unique cross-section dimensions and an extruded length along the 3rd dimension. Could the long side of a cross-section of the sub-sub-miniature beam be Greg's absolutness of rod length?

He could have several light sheets, each slanted at different angles wrt the length of the beam, all sheets starting together and moving at the same rate along the beam, simulating different observer cross-section views. If his optical fibers have been arranged sufficiently cleverly, he might even have something similar to Lorentz invariance--he's got to be clever enough to produce a pattern that yields groups of transformations with physically interesting invariances.

But, does this picture change the argument in any way? The universe beam maintains the absolute 2"x4"x16' dimensions but still has cross-section views. Likewise for the embedded sub-sub-miniature beam. Does the perpendicular cross-section view of the miniature embedded beam have more claim to a 2-D cross-section reality than the slanted views of that beam? However, all observers correponding to the different light sheet movements could develop an understanding that there existed a 2-D cross-section sub-sub-miniature beam extruded along the 3rd dimension. Or could they? But would they all agree about the "true" length of the beam?

Sitting out here with our birds eye view of it all, do we even agree about the "true" length of that subminiature 2-D beam at some instant of time?

 

[ghwellsjr]

But time and distance both persist, you only think they don't because you are making an invalid comparison of a clock to a ruler. Time dilation does not directly affect the time on a clock, it directly affects the tick rate of a clock and then the clock integrates (or counts) the ticks to keep track of elapsed time.
...

Greg is correct on this.
Upon return by the space traveler, his clock is behind the Earth clock, and he is younger, a permanent change.
Their measurement of distance to the the turn around is the same, i.e.,what it was before he left.
The odometer reading is only a temporary record.

Are you saying that an odometer taken on a trip loses its reading after it returns?

 

[JesseM]

No, not a new argument. The OP is based on this conception of the rod.

The OP didn't say anything about considering a single frozen snapshot of the rod with no time like you did in your last post.

I've been trying to address your proposition.

But you haven't explained why, when we "introduce time", we might not find that the rod is moving along the X-axis, and thus that the length L that we saw when we just looked at at a single frozen snapshot was not the rest length, but rather the length in a different frame.

Whether the rod is at rest or not has nothing to do with the definition of the rod. The rod is entirely defined by its length.

Yes, and that length needn't be the rest length. If we take a snapshot of a single instant in frame F, this snapshot includes no time, only 3D spatial extension, and in this snapshot the rod's length is shorter than what we'd see if we took a snapshot of a single instant in the rod's rest frame. You have not given a single non-circular reason why this snapshot in frame F isn't just as valid as the snapshot in the rest frame.

Movement involves time, which by definition is not a component of the rod. In other words, the rod is unaffected by its motion.

A velocity of 0 is every bit as much a "state of motion" as a velocity of 0.99c, both involve considering how the position varies with time (a state of being at rest over time cannot be conflated with the fact that the rod shows no motion if you just take a snapshot of space a single instant and ignore time; in such a timeless snapshot, obviously the rod shows no motion regardless of whether the snapshot was taken from the rod's rest frame or the frame where it moves at 0.99c). If we define the "true length" of "the rod" to be the length in frame F where the rod is moving at 0.99c, then "the rod is unaffected by its motion" just means the true length doesn't change when we pick a frame where the rod has a different state of motion than it does in frame F, like the frame where the rod's position stays the same over time.

That is exactly equivalent to assuming "the rod" is a 3D cross-section of the 4D world-tube.

No, it is not, for the reason given above. The 4D world-tube is not the rod; it is the rod plus time.

Er, I didn't say "the 4D world-tube is the rod", I said you are assuming the rod is a 3D cross-section of that world-tube, i.e. a frozen snapshot that includes only space but no time. Anytime you consider how things are arranged in space in a single frozen snapshot without time, the set of events in that snapshot by definition constitute a 3D cross-section of the whole 4D spacetime.

 

[PAllen]

The two are not exactly analogous, since the proper time along a timelike worldline is frame-invariant (and that's the time a clock moving along the worldline would measure), while there is no frame-invariant notion of the distance traveled along a timelike worldline (though along a spacelike worldline there is a frame-invariant proper distance). When you say the odometer integrates "distance traveled", distance of what in what frame? It can't be the distance traveled by the vehicle itself in the vehicle's rest frame, since of course the vehicle is stationary in that frame and doesn't travel at all! To make your comment about the odometer more precise, I guess we could imagine that the vehicle is traveling along some surface (rather than traveling through empty space), and if we draw closely-spaced dots on the surface along the path of the vehicle, then for any two nearby dots on the surface that the vehicle passes in sequence, the odometer will increase by the same amount as the distance between the dots in the vehicle's rest frame as it passes between them (and if the separation between nearby dots is infinitesimal we don't have to worry about the vehicle's velocity relative to the dots changing during the time it's passing between them).

Since I like ghwellsjr's idea, I'll try to give another definition. I agree there are not invariant ways of defining it. So, 'natural', with big quotes, will have to suffice.

Consider that we are talking about an observer taking a trip (that's what an odometer is all about). Let's specify, further, a round trip. You pick a reference object stationary in your start/end frame. During your journey, you use some standard coordinate convention (e.g. Fermi-normal, allowing inertial and accelerated motion). Your odometer simply integrates total spatial coordinate movement of the reference object in your coordinates.

This captures the idea that when you travel to/from star in a short time (relative to your Earth twin), your odometer will show a short distance as well (when you return).

[Edit: most natural reference object? Your start/end point. ]

 

[GregAshmore]

How so? In your 2D spacetime universe in what way does "the rod exist apart from time"? From your description it seems to me that the number of dimensions in which the rod exists depends on the space, not the rod. Look at your justification for why it exists independent of time: "the rod is completely independent of time, for the simple reason that time does not exist". That justification is a property of the space, not the rod.

I'm having trouble following your logic here. In the "minimal world" scenario, there is no such thing as time. How then can the existence of the rod be dependent on time? Isn't that like saying that the existence of the rod is dependent on unicorns?

The only way you can say that it could exist apart from time is by considering a "toy universe" without time. In any universe with time, in what sense does "the rod exist apart from time"?

The illustration was intended to make clear the idea that the rod in and of itself, being characterized solely by its length, is entirely independent of time. The rod exists in time, but does not "mix with" time. This was in response to the charge that my premise is somehow dependent on my conclusion. If the existence of the rod (which is defined as its length) is not tied to, or affected by time, then there is no circularity in my reasoning.

 

[JesseM]

The illustration was intended to make clear the idea that the rod in and of itself, being characterized solely by its length, is entirely independent of time. The rod exists in time, but does not "mix with" time. This was in response to the charge that my premise is somehow dependent on my conclusion. If the existence of the rod (which is defined as its length) is not tied to, or affected by time, then there is no circularity in my reasoning.

Of course it's circular, because for no good reason you just assume that the rod's "length" in this minimal timeless world is equal to its length in a single moment of the inertial frame where the rod has a velocity of 0. You could just as consistently assume that the length in the timeless snapshot is equal to its length in a single moment of the inertial frame where the rod has a velocity of 0.99c, and thus that the rest frame has a "distorted view" of its "true length".

 

[GregAshmore]

Of course it's circular, because for no good reason you just assume that the rod's "length" in this minimal timeless world is equal to its length in a single moment of the inertial frame where the rod has a velocity of 0. You could just as consistently assume that the length in the timeless snapshot is equal to its length in a single moment of the inertial frame where the rod has a velocity of 0.99c, and thus that the rest frame has a "distorted view" of its "true length".

Please explain the concept of velocity in a world without time.

 

[bobc2]

I'm trying to get into Greg's head here. It seems rather straight forward in a way. Here is a simple beam you might pick up at the lumber yard. You've had it laying on the floor for weeks, and it seems to have definite geometric properties that would define its existence in a significant sense. Would we all agree that its true geometric properties are apparent, notwithstanding the fact that I could take a saw and cut through with a diagonal cut and expose an angular cross-section?

These properties appear rather time independent in that a very large number of seconds have ticked off over days and weeks without any changes.

By the way, let's say we're all standing around monitoring this beam for a few weeks, and there are no other observers moving at relativistic speeds observing.

 

[GrayGhost]

Please explain the concept of velocity in a world without time.

Besides the fact that nothing would exist (as we know it) in a world without time, what process would you use to take a measurement to determine length?

GrayGhost

 

[JesseM]

Please explain the concept of velocity in a world without time.

A world without time is just a snapshot of what a world with time looks like at a particular instant, no? If so, then you can take a snapshot using the definition of simultaneity in some frame where the rod has a nonzero velocity. You might say that the rod has no "velocity" in the snapshot itself since the snapshot doesn't include time, but the snapshot was based on a coordinate system in a world with time, and that determines the length of objects in this frozen snapshot. If you define the "snapshot" using a notion of simultaneity different than the one used in rod's rest frame, then the rod's length in the snapshot will be different than its rest length (and remember what I said before, you can't conflate "the rod cannot be said to have a 'velocity' in a world without time" and "the rod has a velocity of 0", the two notions are completely different because a velocity of 0 still involves considering how the rod's position varies with time).

If the "world without time" is not just a snapshot of the world-with-time at a particular instant, then please say so explicitly. And if it's not such a snapshot, what is it? A pure fantasy with not based on anything observable or measurable? If a rod has a length of 10 meters in the frame where it has a velocity of 0, and it has a length of 6 meters in a frame where it is moving at 0.8c, what is your exact argument for why I would be wrong to assert "in the world without time, I will suppose the rod has a length of 6 meters"? What are the rules governing the relation between the static positions things occupy in this "world without time" and the changing positions they occupy in our world with time, if it's not just a snapshot of a particular instant of time?

As an example of why it would seem somewhat meaningless to define the "world without time" as anything other than a snapshot of the real time-based world a particular instant, consider the following question: aside from including the length of each object, does the "world without time" also include information about how multiple objects are arranged in space, like their orientations relative to one another and the distances between any point on one object and a point on another object? If so what would these distances be based on, if they aren't just a snapshot of the distances between the objects at a single moment in time in some inertial frame in our world with time?

 

[Dale]

I'm having trouble following your logic here. In the "minimal world" scenario, there is no such thing as time. How then can the existence of the rod be dependent on time?

It is obviously not dependent on time in a toy universe without time. But again, that is a property of the space, as indicated by your justification. It is not a property of the rod, and the justification doesn't work in a 4D spacetime.

Can you offer an argument in support of your idea that the rod exists independently of time which is applicable in this universe?

 

[phyti]

Are you saying that an odometer taken on a trip loses its reading after it returns?

No, more precisely it's a record of a temporary event.
More precisely it's a temporary record of a temporary event. It's irrelevant .
The key factors are: the time dilation is permanent, the apparent distance contraction is only for the duration of the trip.
This agrees with what he said.

 

[phyti]

GrayGhost;
Besides the fact that nothing would exist (as we know it) in a world without time, what process would you use to take a measurement to determine length?

...and how do they take all these snapshots?

 

[bobc2]

It is obviously not dependent on time in a toy universe without time. But again, that is a property of the space, as indicated by your justification. It is not a property of the rod, and the justification doesn't work in a 4D spacetime.

For the toy universe it is a property of the space, including the rod.

Can you offer an argument in support of your idea that the rod exists independently of time which is applicable in this universe?

Maybe Greg is picturing a 4-D universe that is analagous to the toy universe. The physics arises from the geometry of spacetime and the 4-D objects. The clocks are themselves of course 4-D objects providing a useful parameter in the equations representing the geometry of the space and the objects.

I think the point you are missing in what Greg is trying to convey is has to do with the metaphysical aspect of time as opposed to time values used in a parametric way. The metaphysical aspect of time could be removed from the universe model without taking away from the physics. The metaphysical and philosophical use of the time concept should perhaps be outlawed from this thread, allowing the discussion to focus on the physics of the 4-D structure. In that sense you can have a universe that can be described in terms of geometry and patterns of world lines.

Further, you don't lose the different observer 3-D cross-section views by removing metaphysical time, because those views are expressed in the mathematical description of the geometies, world line derivatives with respect to spatial parameters, etc. The interval, S, is really a spatial concept, not a metaphysical time concept.

 

[PAllen]

No, more precisely it's a record of a temporary event.
More precisely it's a temporary record of a temporary event. It's irrelevant .
The key factors are: the time dilation is permanent, the apparent distance contraction is only for the duration of the trip.
This agrees with what he said.

I agree with ghwellsjr here. I see the analogy as follows:

During travel, each twin observes the other's seconds as short, and meters short. Once back together, they, they find their seconds and meters are the same again.

However, the elapsed age of one twin is, e.g. 1 year versus 10 for the other. Similarly, the 'earth twin' thinks the other twin has traveled a number of light years, while the traveling twin thinks they traveled less than one light year. It is true that only the proper times are invariants; it is also true that neither twin loses their perception of their travel distance.

I make the common sense definition that the traveling twin knows they are traveling, and defines their travel distance by the distance the Earth moved over the course of their trip, as they measure it.

 

[atyy]

As far as I can tell, no one disagrees substantively that a "true length" can be defined, and is the "proper length". In science you can use any terminology you want, so that's fine.

Reading JesseM's responses, it seems that the part that is wrong is the assertion that only the rest frame can determine the true length. In fact, all frames can determine the true length - which is in the first place, why we believe in a true length - all frames agree on it.

 

[JesseM]

As far as I can tell, no one disagrees substantively that a "true length" can be defined, and is the "proper length". In science you can use any terminology you want, so that's fine.

Reading JesseM's responses, it seems that the part that is wrong is the assertion that only the rest frame can determine the true length. In fact, all frames can determine the true length - which is in the first place, why we believe in a true length - all frames agree on it.

"Proper length" means nothing more and nothing more than "length in the frame where the object is at rest" (it isn't really analogous to proper time, a frame-invariant quantity that is calculated along timelike worldlines; the analogy to proper time would be "proper distance" along a spacelike worldline). Why does this deserve to be called "true length", any more than "length in the frame where the object is moving at 0.99c"? If it were simply a matter of arbitrary definition, that Greg was choosing to define "true length" as rest length rather than length in some other frame, but acknowledging that this was just an aesthetic preference, then there would be no problem. The problem is that he asserts that length in the rest frame is really more "true" than length in other frames in some sense that's based on the ordinary English understanding of the word "true", and is not just an arbitrary label. But every argument he makes for why we are compelled to see the length in the rest frame as more fundamental ends up being totally circular, with his arguments implicitly assuming what he is trying to prove.

 

[atyy]

"Proper length" means nothing more and nothing more than "length in the frame where the object is at rest" (it isn't really analogous to proper time, a frame-invariant quantity that is calculated along timelike worldlines; the analogy to proper time would be "proper distance" along a spacelike worldline). Why does this deserve to be called "true length", any more than "length in the frame where the object is moving at 0.99c"? If it were simply a matter of arbitrary definition, that Greg was choosing to define "true length" as rest length rather than length in some other frame, but acknowledging that this was just an aesthetic preference, then there would be no problem. The problem is that he asserts that length in the rest frame is really more "true" than length in other frames in some sense that's based on the ordinary English understanding of the word "true", and is not just an arbitrary label. But every argument he makes for why we are compelled to see the length in the rest frame as more fundamental ends up being totally circular, with his arguments implicitly assuming what he is trying to prove.

Could it work if a rod is defined as a family of inertial time like wordlines? In which case that would pick out a unique orthogonal spatial hypersurface.

 

[GrayGhost]

PAllen,

Let's say twin B has a standard accelerometer and nav system. It determines his instantaneous speed each inch the way wrt his inertial starting frame, and does not use any relativistic calculations in the nav software. Wouldn't this be his odometer?

Per twin A, B's clock slows down. Per B himself, his clock rate never seems to change, and the separation between Earth and the turnabout point contracts with increased acceleration. Seems to me that that nav system would tell him the distance he traveled (at any point), which could be compared wrt Twin A's measure of that distance using light signals.

no?

GrayGhost

 

[Dale]

As far as I can tell, no one disagrees substantively that a "true length" can be defined, and is the "proper length". In science you can use any terminology you want, so that's fine.

I wouldn't say it quite that way. I would say, "no one disagrees substantively that a 'true length' can be defined, and [STRIKE]is[/STRIKE] it could be defined to be the 'proper length'". The "is" makes it sound as though there is no possible alternative definition of "true length" which there certainly could be. E.g. the length in the aether frame, or the length in the CMBR frame, or the length in JesseM's .99c frame, etc. All of the alternative definitions of "true length" would be equally valid if we used them instead.

 

[Dale]

Maybe Greg is picturing ...

I prefer to let Greg tell me what he is picturing. If you have an argument of your own I will be glad to discuss it, but I won't address your interpretations of someone else's argument. Whenever I have tried that in the past it became very confusing very quickly.