Rest Length, Coordinate Length, and an argument for 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.

 

[atyy]

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.

Definitely.

 

[ghwellsjr]

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.

Actually, each twin observes the other's seconds as long (time dilation) not short which means the other one's clock is running slower since it takes longer to accumulate the same number of seconds.

Are you sure that "it is true that only the proper times are invariants"? Have you worked out the details to see whether or not the distance measured on an odometer is invariant or to see whether or not the speed measured on a speedometer is invariant?

 

[ghwellsjr]

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

I would say yes. An inertial navigation system is one way a traveler could keep track of his time, speed and distance traveled with respect to his inertial starting frame and the measurements on his odometer, speedometer and clock are all invariants.

 

[bobc2]

I'm still trying to get into Greg's head. I don't know if these thoughts add anything to the discussion or not.

 

[PAllen]

I would say yes. An inertial navigation system is one way a traveler could keep track of his time, speed and distance traveled with respect to his inertial starting frame and the measurements on his odometer, speedometer and clock are all invariants.

I don't see how to define invariant odometer reading. Can you propose something? For a timelike path, I don't know of any spacelike invariant. That was Jessem's original point, and if he knows of no such invariant, I certainly don't.

I don't think it is critical to the odometer/clock vs ruler/ticker argument that the odometer reading is not invariant.

 

[PAllen]

Actually, each twin observes the other's seconds as long (time dilation) not short which means the other one's clock is running slower since it takes longer to accumulate the same number of seconds.

Right, long seconds, short rulers.

 

[GrayGhost]

I don't see how to define invariant odometer reading. Can you propose something? For a timelike path, I don't know of any spacelike invariant. That was Jessem's original point, and if he knows of no such invariant, I certainly don't.

I don't think it is critical to the odometer/clock vs ruler/ticker argument that the odometer reading is not invariant.

Well, the spacetime interval is the distance traveled thru 4-space by he who resides at both events.

Wrt the odometer deal here, we're only talking about the perceptable 3-space separation between 2 events. Given 2 synchronised clocks, one at each event (and thus they reside in the same inertial frame), they define the largest possible separation between the events, and that would be invariant. Basically, the proper separation. He who resides at both events must record a smaller (integrated traversal) length than the proper separation, since moving separations length contract. Assuming the rate of proper time is the same for all, and each of the 2 frames record the same velocity between each another, this is consistent with the lesser aging for he who resides at both events.

GrayGhost

 

[harrylin]

In an earlier thread, I asserted that a rod has one true length, its rest length. If so, then the shorter coordinate length which is measured in some other frame must be somehow untrue. In this thread I argue that the coordinate length is a distorted view of the true length.
[..]

The interpretation proposed here is that the integration describes the compressive shifting of the individual snapshots in frame S. The coordinate length of the rod in frame S is thus a distorted view of the rod, while the rod itself is completely unaffected. The rest length of the rod is therefore its one true length.

Of course, the measured coordinate length is the same regardless of the interpretation of the result.

Please excuse me for not having read the whole thread. But what do you mean with "true" in this context?

More specifically - and I think this has been mentioned in other ways - if you mean with "true" that the measured (or, until now always, calculated) change of length of an accelerated object is somehow an illusion and that its length is completely unaffected - just as unaffected as that of identical object that stayed in rest - then that can't be right.
For this would mean that, for example:

1. also the clock frequency of a co-moving clock is "truly" unaffected[1] (and note that resonance frequency is a property like length). In that case a fast moved clock should not lag behind when it is brought back - but we know as fact that over the time that it was in travel, it made less ticks than the rest clocks.
and
2. also the light bouncing off a moving prism should reflect under an angle that corresponds to an uncontracted prism - but that would violate either the laws of optics or relativity.[2]

So, if that's not what you meant with "true", then what did you mean?

1. http://en.wikipedia.org/wiki/Kennedy–Thorndike_experiment
2. http://adsabs.harvard.edu/abs/2004AmJPh..72.1316G

 

[PAllen]

Well, the spacetime interval is the distance traveled thru 4-space by he who resides at both events.


Wrt the odometer deal here, we're only talking about the perceptable 3-space separation between 2 events. Given 2 synchronised clocks, one at each event (and thus they reside in the same inertial frame), they define the largest possible separation between the events, and that would be invariant. Basically, the proper separation. He who resides at both events must record an integrated separation smaller than the proper separation, since moving separations length contract. Assuming the rate of proper time is the same for all, and each of the 2 frames record the same velocity between each another, this is consistent with the lesser aging for he who resides at both events.

GrayGhost

This doesn't help me in any way. Spacetime interval between events someone can reside at always has the characgter of time, not distance.

What is an invariant definition of perceptible 3-space separation between events with timelike connection? The issue is I've never heard of one, and you don't give one (also, JesseM suggests there is none). Proper separation is between spacelike separated events. The issue here is defining distance traveled along world line in some invariant way. I don't see how to do this, and you've provided no definition.

Please note, I see particularly natural coordinate definitions of odometer readings; I don't see how to define any invariant definition.

Try to define some quantities in terms of the metric (including, e.g. vector dot products, covariant derivatives, etc.) that define an invariant odometer.

The best I've come up with is 'natural' coordinate definition. I can make it 'invariant sounding' as follows:

twin B integrates the projection of twin A's instantaneous 4 velocity onto a spatial hypersurface orthogonal (4-space sense) to B's world line (at that instant). Even if each is undergoing different periods of acceleration, this gives a well defined definition of how each thinks they have traveled relative to the other. Unfortunately, it is 'fake invariant' in that it is just a disguised way of doing a coordinate dependent calculation in Fermi-Normal coordinates.

 

[phyti]

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.

gray ghost #68
Since they are at rest with each other, there can be no bodily length contractions, because their relative v = 0. So the length contractions that existed prior, no longer exist on reunion. Also, clock "rate" differentials no longer exist after return, and for the same reason. However the differential in "proper time experienced" (ie relative aging) is always captured, because the time readout (and date) of any clock is the result of its own ticking over the prior period, ie over the defined interval. So the accrued proper-time of either clock is not lost, and the clocks may be compared for relative aging.

greg #82
The point I was trying to make is that time and distance do not behave the same way in SR. The fact that the time difference persists while the length difference does not (quite aside from how it happens "physically") underscores that difference.

phyti #111
The key factors are: the time dilation is permanent, the apparent distance contraction is only for the duration of the trip.

All three posts have the same conclusion.

 

[JesseM]

It's not right to say this alone proves that "time and distance do not behave the same way" though. Elapsed time along a timelike worldline is not analogous to length, but it is analogous to proper distance along a spacelike worldline; if you had two entities moving FTL (they needn't be tachyons, could just be something like the spots of two lasers), and you recorded the proper distance each one accrued along their trip, then if they started out from the same point and later reunited, one may have accrued a greater total proper distance, analogous to one clock accruing a greater amount of total proper time in the twin paradox.

 

[GrayGhost]

PAllen,

Although it would be nice, I don't think a coordinate independent calculation is possible. The proper length (or proper separation) is invariant only because it is assumed not to change over time. The spacetime interval is invariant only because the rate of proper time equates to the speed of light (which is invariant), which also equates to the rate at which one travels thru the 4-space. I don't see anything that could allow for an invariant for 3-space traversal.

However, although observers of differing v will disagree on the separation between the 2 events, they can all successfully predict how he who travels between the 2 events will measure it. Since they all agree, I don't see that a coordinate independent calculation is needed.

GrayGhost

 

[GrayGhost]

It's not right to say this alone proves that "time and distance do not behave the same way" though.

True.

By "time and distance", I assume he means "space and time". The clock integrates the proper time experienced. The ruler does not. As ghWellsJr pointed out though, an odometer would. Only problem ... there's no road for the rubber in vacu. Doesn't matter though, because the clock is the odometer in free space if we all travel thru the medium at c. The one sticking point IMO is this ... time flows. That's the behavioral difference. Although the Minkowski model presents time as more space, there is still the matter of "time flows" that persists. However, it may be more accurate to say "spacetime flows".

The problem I had with Greg's statement wasn't so much in what he said, but that he seemed to think it supported his argument that rest length is the only real length. In that I disagree.

GrayGhost

 

[bobc2]

 

[GregAshmore]

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

My proposition is that the rod's length is independent of time. I emphasized the point by postulating a world without time. If I define the rod's length to be independent of time, then there can be no time-related circularity in the argument which follows, even in a world with time.

Why would I propose that the rod's length is independent of time? Because, as has been observed by experts in the field (Taylor-Wheeler among them), the rod is not affected by the movement of some other body.

The OP provides a rational explanation for the contradictory length which is observed in other frames. [Note to those who are bristling at the word 'contradiction': When two observers measure different lengths for one and the same rod, that, as a simple matter of fact, is a contradiction. This does not imply an error in the theory of SR.]

Can I prove the assertion that the rod's length is independent of time? Of course not; that point was implicitly conceded in the last sentence of the OP, and explicitly soon afterward. Then again, neither can it be proved that it isn't. Nor can it be proved that the rod is "really" shorter in other frames, rather than "apparently" shorter.

I'm now working seven days a week, and will be for several more weeks. I'll be off the forum for a while.

The question as to whether true length implies true time (which I have not postulated) is very interesting. I'll think about it as I have time.

 

[JesseM]

My proposition is that the rod's length is independent of time. I emphasized the point by postulating a world without time. If I define the rod's length to be independent of time, then there can be no time-related circularity in the argument which follows, even in a world with time.

But you didn't define what the relation between the "world without time" and our time-based world is supposed to be, or give a non-circular reason as to why we should believe "length of rod in world without time" = "length of rod in our time-based world seen the perspective of the inertial frame where it has a velocity of 0". My simple question from post #109 (and I hope you will address that post in its entirety):

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"?

 

[PAllen]

I propose a thought experiment that I hope will resolve some
differences between phyti and myself and ghwellsjr, and at the same
time make length contraction hard to ignore as merely an 'optical
illusion'.

Consider two twins (twin H, home inertial twin, twin T, traveling
twin, does some flavor of long relativistic round trip starting and
ending coincident with H) attached to light year long born rigid rulers,
constructed as follows: a movie camera and motive source are attached
to a 1 meter rigid ruler that maintains contact at all times with its
neighbor, however the neighbor moves. Say twin T moves a light year to
the right and back, according to twin H. Then have H's ruler extend to
the right, and T's extend to the left. T an H also have clocks, of
course.

After the trip, the twins can gather all information from the cameras
and compare notes. They find:

1) After they are back together, their rulers are the same length
again, and seconds are the same length. No difference between time
dilation and length contraction. Both are relative, transitory
phenomena.

2) Of course, their elapsed times are very different - twin T is much
younger. Similarly, twin H's camera from one light year along his
ruler shows an image of T turning around; while twin T's cameras show
twin H turning around at e.g. 1/10 light year. Thus, their preserved
measure of distance traveled (the set of cameras showing the other
twin) remains different, on re-uniting, in the same way as elapsed time.

Some conclusions:

1) Contrary to phyti, and in agreement with ghwellsjr, there is
perfect symmetry between time dilation and length contraction (both
transitory and relative); and between elapsed time and distance
traveled (both summed measures are persistent on re-uniting). That
distance traveled is not conveniently definable as an invariant does
not stop it from being measured in some reasonable way.

2) Separate from metaphysical debates about 'true' and 'real', it
seems hard to avoid attaching some empirical reality to length
contraction given the preserved differences in camera records in this
thought experiment. Further (assuming twin T has some long coasting
phase), each twin watching the other's ruler go by will see meter
markings go by at a rate that if they believe they are 'true' meters
(in the ruler's rest frame), implies the other twin is moving highly
superluminally. Meanwhile, any direct measures of velocity
(e.g. Doppler shift) have each twin seeing the other moving at the
same speed less than c.

 

[GrayGhost]

My proposition is that the rod's length is independent of time. I emphasized the point by postulating a world without time. If I define the rod's length to be independent of time, then there can be no time-related circularity in the argument which follows, even in a world with time.

Well, that's like emphasizing darkness in a world w/o light. Technically, it's meaningless.

Why would I propose that the rod's length is independent of time? Because, as has been observed by experts in the field (Taylor-Wheeler among them), the rod is not affected by the movement of some other body.

A shortsighted view IMO Greg. I do not believe that your assumption is consistent with what Taylor-Wheeler meant.

When two observers measure different lengths for one and the same rod, that, as a simple matter of fact, is a contradiction. This does not imply an error in the theory of SR.

Given your position, I understand why you would believe as such.

It would be a contradiction if each observer predicted the moving-other should measure values he does not. That's not the case though, as all observers agree on their disagreements, as the reason for their disagreement in known and accepted.

Can I prove the assertion that the rod's length is independent of time? Of course not; that point was implicitly conceded in the last sentence of the OP, and explicitly soon afterward. Then again, neither can it be proved that it isn't.

The theory shows that time is meaningless w/o designating a location in space, and that space is meaningless w/o designating a location in time. IOWs, they are 2 apsects of a fused spacetime continuum. Therefore, independent of time does not mean devoid of time.

EDIT: The rod's length may be considered independent of time "if it does not change over time". This does not lead that time does not exist. A moving contracted length may be as independent of time as a proper length, if the measured length is constant over time.

Nor can it be proved that the rod is "really" shorter in other frames, rather than "apparently" shorter.

Untrue. Lasers and processing systems can confirm such. A beam is broken when the moving vessel arrives (of known proper length), and is unbroken after it passes. Nothing to it. The timing of "laser beam continuity changes" tells the story.

The measurement is that it is, with the result being frame dependent. When a moving vessel is measured as length-contracted, this does not lead that the vessel has ever changed in and of itself. Everyone knows it has not. Yet, the measurement is not illusionary effect.

The question as to whether true length implies true time (which I have not postulated) is very interesting. I'll think about it as I have time.

A fascinating subject indeed.

GrayGhost

 

[GregAshmore]

It's not right to say this alone proves that "time and distance do not behave the same way" though. Elapsed time along a timelike worldline is not analogous to length, but it is analogous to proper distance along a spacelike worldline; if you had two entities moving FTL (they needn't be tachyons, could just be something like the spots of two lasers), and you recorded the proper distance each one accrued along their trip, then if they started out from the same point and later reunited, one may have accrued a greater total proper distance, analogous to one clock accruing a greater amount of total proper time in the twin paradox.

Good point. I'll have to think about what this means in relation to the OP.

 

[GregAshmore]

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?

The "world without time" is causing more trouble than it's worth, so I'll retract it. Instead, I'll simply define that the rod's length is independent of and therefore unaffected by time. With that definition in mind, I don't see any circularity in the argument of the OP.

That leaves the question of aesthetics. Is the selection of the rest length as the defined length of the rod objectively better than selecting the length of the rod as observed from a frame moving at 0.6c relative to the rod? I think so, given the assumption that the rod's length is independent of time. In the rest frame, all locations on the rod are at the same time. When observed from the moving frame, each location on the rod is at a different time. Imprecisely put, the moving frame doesn't get a good reading of the rod.

I'm afraid I can't do any better than that right now. I'm late for work. This will have to be my last post for a while. I don't get enough sleep if I spend time here. I get sleepy at work, can't think properly, and have to clock out and take a nap. Not good.