On the interpretation of a spacetime diagram

[lugita15]

Can velocity be measured at one time?

Not directly, but by determining how much the length of someone's ruler has Lorentz contracted, you can calculate their velocity relative to you.

 

[bobc2]

Can velocity be measured at one time?

Not by us mere mortals. But, some super hyperdimensional guy with the big hyperdimensional view of the 4-dimensional poles can just measure the orientations of the objects in hyperdimensional space and tell you the velocities. Of course the 4-D poles are just static objects and don't have any motion at all, so he just uses the orientations to come up with numbers that are compatible with the inferior impressions of the 3-dimensional mortals.

 

[GregAshmore]

How do you know it's 4 units long? What do you mean by "it's length"?

In your thought experiment, nobody made any claims about the length of a rod

No, the two rods were made to be identical, each four units long. One never left the lab; the other was set in motion and observed from the lab frame.

 

[Dale]

Can velocity be measured at one time?

I was talking about measuring length, not velocity. Your reply is rather irrelevant.

 

[bobc2]

Not by us mere mortals. But, some super hyperdimensional guy with the big hyperdimensional view of the 4-dimensional poles can just measure the orientations of the objects in hyperdimensional space and tell you the velocities. Of course the 4-D poles are just static objects and don't have any motion at all, so he just uses the orientations to come up with numbers that are compatible with the inferior impressions of the 3-dimensional mortals.

By the way, I really am not a troll. I just meant this as a pedagogical device.

 

[GregAshmore]

I was talking about measuring length, not velocity. Your reply is rather irrelevant.

I disagree. The projected length by itself is not particularly meaningful.

 

[Dale]

If you disagree then please expain how a reply about measuring velocity is relevant to a comment about measuring length?

Btw, I sympathize with your position that coordinate length is less meaningful than rest length, I was only pointing out that the way you described measuring length is incorrect. There is nothing different about the process of measuring the length of an object in different frames.

 

[GregAshmore]

If you disagree then please expain how a reply about measuring velocity is relevant to a comment about measuring length?

Btw, I sympathize with your position that coordinate length is less meaningful than rest length, I was only pointing out that the way you described measuring length is incorrect. There is nothing different about the process of measuring the length of an object in different frames.

The measurement you describe does not measure the length of the object; it measures the length of the object's projection in another inertial frame. You pointed out earlier in this thread that the projection is analogous to the shadow of a tilted rod. Just as the length of the shadow on the table is not the length of the rod, the projected length in the moving frame is not the length of the rod.

Suppose we have two observers of the one rod on the lab bench, moving at different speeds relative to the bench. There will, of course, be two distinct measurements of length--yet there is only one rod, with one length. The only way to resolve the contradiction is for each observer to account for the rod's velocity relative to his frame.

To emphasize the point, suppose that each observer is bidding to purchase the rod, for use in his own laboratory. How much room must be cleared for the rod on the observer's bench? That question cannot be answered correctly without taking the velocity into account.

 

[JesseM]

The measurement you describe does not measure the length of the object; it measures the length of the object's projection in another inertial frame. You pointed out earlier in this thread that the projection is analogous to the shadow of a tilted rod. Just as the length of the shadow on the table is not the length of the rod, the projected length in the moving frame is not the length of the rod.

But all lengths are really cross-sections of the 4D world-tube with some plane of simultaneity; the rest length is just "special" in the sense that it's the maximum possible length, and also that in this case the plane of simultaneity is orthogonal to the worldline of any point on the object.

Suppose we have two observers of the one rod on the lab bench, moving at different speeds relative to the bench. There will, of course, be two distinct measurements of length--yet there is only one rod, with one length.

Why should one rod imply "one length"? Do you think the rod must also have "one velocity" or "one x-coordinate"? I see nothing wrong with saying "length" is an intrinsically frame-dependent quantity and therefore that it has no one "correct" frame-independent value. Of course you are free to define the "true length" as the "rest length", but this is just an arbitrary convention you've invented--one could equally well define the "true length" as "length in the frame where the rod is moving at 0.5c", there's no physical reason compelling us to adopt one definition or the other, it's purely a matter of aesthetic preference which definition we use, if indeed we want to use a silly phrase like "true length". Do you disagree with any of the above? If so, what, and why?

To emphasize the point, suppose that each observer is bidding to purchase the rod, for use in his own laboratory. How much room must be cleared for the rod on the observer's bench? That question cannot be answered correctly without taking the velocity into account.

Here you seem to assume that the rod will be brought to rest on the work bench, in which case of course it's the rest length we'll be interested in, but that doesn't make it the "true length". You could equally well ask, "suppose each observer wants to use the rod in an experiment involving a toy train car moving at 0.5c relative to his lab, which has a length of 1 meter in the observer's frame--will the rod fit in the train car?" In this case it would be "length in the frame where the rod is moving at 0.5c" that would be relevant to each observer.

 

[Dale]

The measurement you describe does not measure the length of the object

Yes, it does. This is not a matter of debate, it is a matter of definition.

 

[ghwellsjr]

Suppose we have two observers of the one rod on the lab bench, moving at different speeds relative to the bench. There will, of course, be two distinct measurements of length--yet there is only one rod, with one length. The only way to resolve the contradiction is for each observer to account for the rod's velocity relative to his frame.

To emphasize the point, suppose that each observer is bidding to purchase the rod, for use in his own laboratory. How much room must be cleared for the rod on the observer's bench? That question cannot be answered correctly without taking the velocity into account.

Don't forget that a rod moving in a frame doesn't have just one length if it is rotated in a particular orientation. Its length will be changing between its rest length and its most contracted length.

So if the seller of the rod wants to make sure the two bidders observe the same length of the rod they want to purchase (and they're too stupid to make the correct calculation) he can reposition the rod so that its length is perpendicular to both bidders' directions of motion. Problem solved.

Or he could just tell them its length.

 

[GregAshmore]

Yes, it does. This is not a matter of debate, it is a matter of definition.

Definitions should not allow for contradictory statements. The rod is a physical object, with its own length that is unaffected by the relative motion of other objects. If the definition of "length of an object" allows for multiple lengths, the definition is bad.

It seems that we are debating the definition of length, rather than the result of any measurement in the lab.

 

[JesseM]

Definitions should not allow for contradictory statements.

Where is the "contradiction" in allowing "length" to be defined in a frame-dependent way? It seems that all you really mean is that it conflicts with your personal aesthetic intuitions about what the word "length" should imply, you haven't stated any more rational objection so far.

The rod is a physical object, with its own length that is unaffected by the relative motion of other objects. If the definition of "length of an object" allows for multiple lengths, the definition is bad.

Do you think a 3D physical object should have a single 2D "cross-sectional area", or would you agree it has multiple cross-sectional areas depending on the orientation of the 2D plane used to take the cross-section? If you accept multiple "cross-sectional areas" in that case, why can't you accept multiple 3D cross-sections (with different 'lengths' along a given spatial axis, not to mention different spatial volumes) of a rod's 4D world-tube?

 

[GregAshmore]

Don't forget that a rod moving in a frame doesn't have just one length if it is rotated in a particular orientation. Its length will be changing between its rest length and its most contracted length.

So if the seller of the rod wants to make sure the two bidders observe the same length of the rod they want to purchase (and they're too stupid to make the correct calculation) he can reposition the rod so that its length is perpendicular to both bidders' directions of motion. Problem solved.

Then they will argue about the thickness of the rod. (I think there are other problems with getting an accurate measurement of length of an oblique object, but they are not central to this post, so I'll not investigate the issue further.)

Or he could just tell them its length.

I would say, "It's true length."

 

[GregAshmore]

Where is the "contradiction" in allowing "length" to be defined in a frame-dependent way? It seems that all you really mean is that it conflicts with your personal aesthetic intuitions about what the word "length" should imply, you haven't stated any more rational objection so far.

Do you think a 3D physical object should have a single 2D "cross-sectional area", or would you agree it has multiple cross-sectional areas depending on the orientation of the 2D plane used to take the cross-section? If you accept multiple "cross-sectional areas" in that case, why can't you accept multiple 3D cross-sections (with different 'lengths' along a given spatial axis, not to mention different spatial volumes) of a rod's 4D world-tube?

If I place a block of aluminum on the table, everyone in the room will agree on the absolute value of its physical extents, label them however you wish. Therefore, all will agree on the volume of the block. Any observer, moving or not, who measures a different set of extents, and thus a different volume, is wrong.

Here you seem to assume that the rod will be brought to rest on the work bench, in which case of course it's the rest length we'll be interested in, but that doesn't make it the "true length". You could equally well ask, "suppose each observer wants to use the rod in an experiment involving a toy train car moving at 0.5c relative to his lab, which has a length of 1 meter in the observer's frame--will the rod fit in the train car?" In this case it would be "length in the frame where the rod is moving at 0.5c" that would be relevant to each observer.

It is not possible to demonstrate that the rod will fit in the car. The rod will smash through the ends of the car because it is moving relative to the car.

 

[JesseM]

If I place a block of aluminum on the table, everyone in the room will agree on the absolute value of its physical extents, label them however you wish.

Not if they use a ruler moving relative to the room to measure it, which is perfectly possible.

Therefore, all will agree on the volume of the block. Any observer, moving or not, who measures a different set of extents, and thus a different volume, is wrong.

An assertion, not an argument. Why are they wrong?

It is not possible to demonstrate that the rod will fit in the car. The rod will smash through the ends of the car because it is moving relative to the car.

You misunderstood my scenario, I was suggesting that the rod would be used in an experiment where both the rod and the car would be moving at 0.5c relative to the lab. This is no more arbitrary than your suggestion that the rod be at rest relative to the lab.

 

[Dale]

Definitions should not allow for contradictory statements.

I agree, but you seem to believe that a definition is contradictory simply because it is frame-dependent. If you have any physics background then you are already familiar with using quantities that are defined in a frame dependent manner, such as position, velocity, kinetic energy, momentum, and work. Do you wish to get rid of all of these quantities simply because they are frame-dependent? I hope not. Length is no different, it is a very useful quantity despite the fact that it is frame dependent.

The only difference between length and these other frame-variant quantities is that the fact that length is frame-dependent is a new concept. There is nothing wrong or contradictory here, any more than statements about an objects position are wrong or contradictory. You merely have to specify your reference frame when making statements about frame-variant quantities, then your statements are clear and do not lead to contradictions.

The rod is a physical object, with its own length

If by "own length" you mean "rest length" then I agree.

 

[GrayGhost]

Ahh, the ole argument over what is real versus merely apparent (or illusion).

Based on my understanding of the theory, what you measure is real. I might add, what the Lorentz transforms mathematically require, must be real. This of course assumes that the 2 relativity postulates are indeed true, which virtually all great physicists today believe to be the case. For those who haven't studied the theory at length, I usually summarize it this way ...

The proper length of a body is its length as measured by a ruler at rest with it. All observers agree on a body's proper length, even if they witness it moving and contracted wrt said proper length. Moving bodies are length contracted and the contracted length is real as real gets, as they are measurable. The mathematics alone shows that length contractions must be real. However, the theory of relativity also requires that no body ever change in and of itself, that is, it's proper length never changes. That is, a body does not change in its own proper length simply because 10 other folks who move relatively measure it 10 different ways.

Yet, there's more to "moving contracted bodies" than most folks realize, which is where most the confusion arises wrt real vs less-than-real ... No 2 points along a moving contracted length exists in the same moment of said body's time. IOWs, it's rotated relatively in a fused spacetime continuum (or 4-space if you prefer). The rotation is not fully realized by the observer who moves relatively, because space and time are not the very same thing, yet relativistic effects of said rotation do reveal the rotation's existence ... eg length contraction and time dilation, which are measurable (at least in theory). Anyways, that's my 2 cents :)

GrayGhost

 

[GregAshmore]

Not if they use a ruler moving relative to the room to measure it, which is perfectly possible.

If the ruler is moving relative to the room, then it is not in the frame of the room.

An assertion, not an argument. Why are they wrong?

The assertion is based on the notion--which I think we all agree to be true--that the rod itself is not affected by the motion of other objects. In that case, it seems to me that any claim that the rod has a different length is wrong. However, you are correct in saying that I have not presented a logical argument to support my position. I've been working on that since my last post; I'll have something in a day or two, I think. When I got here this morning, I saw that I am following a line of reasoning which is similar to that presented by GrayGhost, in #53.

You misunderstood my scenario, I was suggesting that the rod would be used in an experiment where both the rod and the car would be moving at 0.5c relative to the lab. This is no more arbitrary than your suggestion that the rod be at rest relative to the lab.

While it is not arbitrary, it is a different scenario than the one we are discussing. We are talking about a measuring device in one frame, and the object to be measured in another frame.

btw, in an earlier series of posts, I used the word 'contradiction' inappropriately, thus putting the discussion in a confrontational mode. It seems to me that the word 'silly' tends to do the same thing.

 

[GregAshmore]

I agree, but you seem to believe that a definition is contradictory simply because it is frame-dependent. If you have any physics background then you are already familiar with using quantities that are defined in a frame dependent manner, such as position, velocity, kinetic energy, momentum, and work. Do you wish to get rid of all of these quantities simply because they are frame-dependent? I hope not. Length is no different, it is a very useful quantity despite the fact that it is frame dependent.

The only difference between length and these other frame-variant quantities is that the fact that length is frame-dependent is a new concept. There is nothing wrong or contradictory here, any more than statements about an objects position are wrong or contradictory. You merely have to specify your reference frame when making statements about frame-variant quantities, then your statements are clear and do not lead to contradictions.

I need to think about this more, but I'm not convinced that there is nothing wrong in saying that the length of an object is frame-dependent. The concept is not merely new; it is of a fundamentally different character, because of the relativity of time. I am not contesting the relativity of time, or the result of the length measurement; I am contesting the interpretation of the length measurement.

If by "own length" you mean "rest length" then I agree.

I do mean rest length, but I would say that the rest length is the only length. I would also say that the shorter distance measured in another frame is not the length of the rod, but a measure of its ________. (I've been trying for two hours to find a word to put in that blank, without success. I need to study the spacetime diagram some more. I may confirm the concept I have in mind, and find a word to express it. Or, perhaps not. In the meantime, I see that I am on the same track as GrayGhost, in #51.)

 

[Dale]

I am not contesting the relativity of time, or the result of the length measurement; I am contesting the interpretation of the length measurement.

... I would also say that the shorter distance measured in another frame is not the length of the rod, but a measure of its ________.

The process of measuring the distance between the front and the back of an object at a given instant in time according to two synchronized clocks results in some number. The scientific community has given that number a name: "length". It is a defined term. You may think that it would have been better to pick a different name, but you need to know and use the standard name anyway. Otherwise you will be unable to communicate your concepts to others who use the standard name.

Luckily, there is another term used by the community to denote the concept that you prefer: "rest length". So, another way to communicate clearly is for you to simply always talk about "rest length". You can also use the term "coordinate length" to refer to "length" as defined above and thereby simply avoid ever using the unqualified term "length" yourself at all. I have tried to adopt a similar stance regarding "mass" and try to always use the qualified terms "invariant mass" or "relativistic mass", rather than ever using the unqualified term "mass".

 

[bobc2]

But all lengths are really cross-sections of the 4D world-tube with some plane of simultaneity; the rest length is just "special" in the sense that it's the maximum possible length, and also that in this case the plane of simultaneity is orthogonal to the worldline of any point on the object.

Why should one rod imply "one length"? Do you think the rod must also have "one velocity" or "one x-coordinate"? I see nothing wrong with saying "length" is an intrinsically frame-dependent quantity and therefore that it has no one "correct" frame-independent value. Of course you are free to define the "true length" as the "rest length", but this is just an arbitrary convention you've invented--one could equally well define the "true length" as "length in the frame where the rod is moving at 0.5c", there's no physical reason compelling us to adopt one definition or the other, it's purely a matter of aesthetic preference which definition we use, if indeed we want to use a silly phrase like "true length". Do you disagree with any of the above? If so, what, and why?

Here you seem to assume that the rod will be brought to rest on the work bench, in which case of course it's the rest length we'll be interested in, but that doesn't make it the "true length". You could equally well ask, "suppose each observer wants to use the rod in an experiment involving a toy train car moving at 0.5c relative to his lab, which has a length of 1 meter in the observer's frame--will the rod fit in the train car?" In this case it would be "length in the frame where the rod is moving at 0.5c" that would be relevant to each observer.

Excellent assessment, Jesse. I think it good to refresh your ideas at this point.

 

[JesseM]

If the ruler is moving relative to the room, then it is not in the frame of the room.

It's not at rest in the frame of the room--is that what you mean by "in"? A frame is just a coordinate system, so in another sense any object is "in" the region of spacetime covered by those coordinates, regardless of whether the object is at rest in those coordinates. In any case you didn't say anything about an (arbitrary) requirement that all observers in the room use rulers at rest in that room, you just said:

If I place a block of aluminum on the table, everyone in the room will agree on the absolute value of its physical extents, label them however you wish.

They won't agree if they use rulers not at rest in that room. You can impose the requirement that each observer use a ruler at rest in the room to measure the aluminum block, but this is just an arbitrary convention you've invented, no less arbitrary than the convention "each observer must use a ruler moving at 0.5c relative to the room to measure the aluminum block". Do you disagree, and think there is some non-arbitrary, non-aesthetic reason why your requirement is the correct one to use?

The assertion is based on the notion--which I think we all agree to be true--that the rod itself is not affected by the motion of other objects. In that case, it seems to me that any claim that the rod has a different length is wrong.

But here you are making the implicit assumption that length must be a characteristic of "the rod itself", and not just a characteristic of how the rod looks when described in a particular frame. Do you think velocity or x-coordinate are characteristics of "the rod itself", and therefore the rod can only have one "true" velocity or one "true" x-coordinate? If not, why do you have this strange mental block about seeing "length" the same way you see velocity and x-coordinate? Can you give any rational argument as to why you feel "length" must be such an intrinsic characteristic of the rod, whereas velocity and x-coordinate are not? (assuming you don't think they are--please address this one way or another!)

Speaking of non-intrinsic characteristics like velocity and x-coordinate, you never really gave me a definite answer to the question of whether you agree that a 3D object doesn't need to have a single "true" 2D cross-sectional area, you just changed the subject to talk about an aluminum block on a table...can you please address this section from comment #48, telling me whether you agree that there is no single "true" 2D cross-sectional area of a 3D object?

Do you think a 3D physical object should have a single 2D "cross-sectional area", or would you agree it has multiple cross-sectional areas depending on the orientation of the 2D plane used to take the cross-section? If you accept multiple "cross-sectional areas" in that case, why can't you accept multiple 3D cross-sections (with different 'lengths' along a given spatial axis, not to mention different spatial volumes) of a rod's 4D world-tube?

btw, in an earlier series of posts, I used the word 'contradiction' inappropriately, thus putting the discussion in a confrontational mode. It seems to me that the word 'silly' tends to do the same thing.

Why, did I use the word "silly" at some point? I don't remember. Anyway, to me the problem with your use of "contradiction" is not that it's overly confrontational, but just that it suggests you have some actual logical argument in the form of a specific pair of conclusions that follow from the idea that length is frame-dependent, but which end up contradicting each other. That would allow you to do a proof by contradiction to show that there must be a frame-independent notion of length. However, it seems like you don't actually have any sort of specific contradiction in mind, so my objection to your use of that word is just that it's misleading and makes it sound like your argument has more substance than it actually does.

 

[ghwellsjr]

Not if they use a ruler moving relative to the room to measure it, which is perfectly possible.

If the ruler is moving relative to the room, then it is not in the frame of the room.

In Special Relativity, all objects are in all frames of reference. The whole point of SR is that you define your entire scenario in any arbitrarily selected single frame of reference. So if you choose the lab frame, then rulers moving in that frame aligned along the direction of motion will be length contracted. It is wrong to think that a ruler moving relative to the room is not in the frame of the room.

This is one of the most common misconceptions about Special Relativity, that is, that every object has it own frame of reference in which it is at rest, simultaneously. You started this post with a diagram showing two identically constructed objects in relative motion from the frame of reference in which one of them, which you called the "lab bench", was at rest, and then you asked the question: "Knowing that the stationary rod is four units long, would we not have to conclude that the instruments on the moving rod are incorrect when they report its length to be 3.2 units?"

Do you see how this question mixes up two different frames of reference? The expression, "Knowing that the stationary rod is four units long" implies that you are using the lab frame because that is the only one in which the lab rod has a length of four units. And the rest of your question implies the "moving" frame because in that one the stationary rod is 3.2 units long. So because you have switched between frames within a single question, you think there is a contradiction.

Your question could have been asked: "In the frame of reference in which the 'moving' rod is at rest, would we not have to conclude that its instruments are incorrect when they report the length of the lab rod to be 3.2 units?" And the answer is no. And by the same token, in the frame of reference in which the lab rod is at rest, would we not have to conclude that its instruments are incorrect when they report the length of the 'moving' rod to be 3.2 units? And, again, the answer is no.

We have to make a distinction between the spatial values that a frame of reference assigns to objects and the measurements that those objects make. So in the rest frame of the lab rod, the length of the moving rod is shortened and in the rest frame of the moving rod, the lab rod is shortened. But the amazing thing is that in each of these frames, the "lab" frame and the "moving" frame, or in any other frame, we can view how the instruments on each rod measure the length of the other rod as being contracted.

I described in detail how this is done in the single lab frame in answer to your original question in post #3 and I pointed out how you could transform the entire scenario into the rest frame of the "moving" rod and view how both sets of instruments again correctly measure the other one's rod as length contracted. But I stated that doing this would be rather unexciting because the calculatons are reciprocal.

What would be more exciting would be to use the frame of reference in which both rods are traveling at the same speed but in opposite directions. In this frame, both rods are contracted to the same length, somewhere between 3.2 and 4.0 units, but still, each one measures their own length as 4.0 and the other one's length as 3.2 units. And, even though the common speed in this frame is not 0.3c, they still measure the other one's relative speed as 0.6c.

So the bottom line of what I'm saying here is that if you want to talk about Special Relativity and the lengths that are assigned according to a particular frame of reference, then it is correct to say that one of the rods is length contracted while the other is not. But if you want to talk about what each rod measures of its own length and that of the other one, it doesn't matter which frame you use, they will always measure the same length for their own rod and a shortened length for the moving rod. And all of these definitions of lengths are correct in their own contexts and none are better than the others or more correct.

 

[GregAshmore]

In Special Relativity, all objects are in all frames of reference. The whole point of SR is that you define your entire scenario in any arbitrarily selected single frame of reference. So if you choose the lab frame, then rulers moving in that frame aligned along the direction of motion will be length contracted. It is wrong to think that a ruler moving relative to the room is not in the frame of the room.

I'll try to remember to say "at rest in the same frame" instead of "in the same frame".

You started this post with a diagram showing two identically constructed objects in relative motion from the frame of reference in which one of them, which you called the "lab bench", was at rest, and then you asked the question: "Knowing that the stationary rod is four units long, would we not have to conclude that the instruments on the moving rod are incorrect when they report its length to be 3.2 units?"

Do you see how this question mixes up two different frames of reference? The expression, "Knowing that the stationary rod is four units long" implies that you are using the lab frame because that is the only one in which the lab rod has a length of four units. And the rest of your question implies the "moving" frame because in that one the stationary rod is 3.2 units long. So because you have switched between frames within a single question, you think there is a contradiction.

No, I think there is a contradiction because I know that the rod is 4 units long, and the instruments mounted in the moving rod tell me that it is 3.2 units long.

I understand that the instruments read as they do because they are in motion with respect to the rod which they are measuring. However, that by itself does not mean that the rod is actually shorter. As I mentioned, I do have a rational alternate explanation, but it will take me a few days to prepare it. I'm working overtime for the next month or so; there won't be much time in the evenings.

Your question could have been asked: "In the frame of reference in which the 'moving' rod is at rest, would we not have to conclude that its instruments are incorrect when they report the length of the lab rod to be 3.2 units?" And the answer is no.

You say "no", but I don't see any explanation as to why they are not in error. Again, I understand that the instruments are in perfect working order, and that the results will be the same no matter how many times the experiment is conducted. And I understand the reciprocal nature of the phenomenon. I just don't believe that the rod is shorter; I believe that the moving instruments are unable to see the rod properly, due to the nature of light [in brief, c].

So the bottom line of what I'm saying here is that if you want to talk about Special Relativity and the lengths that are assigned according to a particular frame of reference, then it is correct to say that one of the rods is length contracted while the other is not. But if you want to talk about what each rod measures of its own length and that of the other one, it doesn't matter which frame you use, they will always measure the same length for their own rod and a shortened length for the moving rod. And all of these definitions of lengths are correct in their own contexts and none are better than the others or more correct.

Again, we do not disagree on the result of the measurements; we disagree as to whether the results are correct.

 

[GregAshmore]

The process of measuring the distance between the front and the back of an object at a given instant in time according to two synchronized clocks results in some number. The scientific community has given that number a name: "length". It is a defined term. You may think that it would have been better to pick a different name, but you need to know and use the standard name anyway. Otherwise you will be unable to communicate your concepts to others who use the standard name.

Luckily, there is another term used by the community to denote the concept that you prefer: "rest length". So, another way to communicate clearly is for you to simply always talk about "rest length". You can also use the term "coordinate length" to refer to "length" as defined above and thereby simply avoid ever using the unqualified term "length" yourself at all. I have tried to adopt a similar stance regarding "mass" and try to always use the qualified terms "invariant mass" or "relativistic mass", rather than ever using the unqualified term "mass".

Rest length and coordinate length it is (or, they are).

 

[lugita15]

You say "no", but I don't see any explanation as to why they are not in error. Again, I understand that the instruments are in perfect working order, and that the results will be the same no matter how many times the experiment is conducted. And I understand the reciprocal nature of the phenomenon. I just don't believe that the rod is shorter; I believe that the moving instruments are unable to see the rod properly, due to the nature of light [in brief, c]

Light is not at all relevant to Lorentz contraction. If all the light in the universe disappeared tomorrow, so that the "nature of light" no longer mattered, nothing in relativity would be affected. It is true that in order to derive the Lorentz transformations we make use of the postulate that there exists a speed c that is invariant in all reference frames. However, we do not make use of the fact that there is something that actually travels at this speed.

And the phenomenon of Lorentz contraction is NOT due to the fact that we use light for seeing. The measurement of length can be computed using sound or any other means, and it won't affect the result one bit. (Of course, we have to take into account the time of travel of whatever we're using to make our measurement. If we don't correct for this, results like time dilation and length contraction will come out all wrong.)

 

[GrayGhost]

GregAshmore,

You ask a question that all relativists have asked at one time or another. Is a moving contracted length real, or merely apparent? I must admit, even relativists debate this matter at length, and in most those cases comes down to an argument of semantics vs theory.

An accelerating body rotates in its orientation within spacetime, yet we do not witness this rotation in the same way we witness the rotation of a pencil in 3-space. The full rotation we do not perceive, but fortunately, we do perceive effects of the rotation. The effects are length contraction and time dilation, relativistic effects that arise with relative motion. This is proof that bodies remain at their proper length (per themselves) even when moving per others, even though rulers moving relatively can never measure it as such. That said, the contractions are real, while at the same time, bodies always remain their proper length per themselves even during acceleration.

To understand why this is the case, you should really be asking ... WHY does a moving body shrink in length, as opposed to ... is the measurement data correct or not? Understand why, and the question generally no longer needs asked. If you are the persistent type, you may save months (or years) by pursuing the former question first. A good understanding of Minkowski illustrations helps immensely ... food for thought.

GrayGhost

 

[ghwellsjr]

No, I think there is a contradiction because I know that the rod is 4 units long, and the instruments mounted in the moving rod tell me that it is 3.2 units long.

I understand that the instruments read as they do because they are in motion with respect to the rod which they are measuring. However, that by itself does not mean that the rod is actually shorter. As I mentioned, I do have a rational alternate explanation, but it will take me a few days to prepare it. I'm working overtime for the next month or so; there won't be much time in the evenings.

You say "no", but I don't see any explanation as to why they are not in error. Again, I understand that the instruments are in perfect working order, and that the results will be the same no matter how many times the experiment is conducted. And I understand the reciprocal nature of the phenomenon. I just don't believe that the rod is shorter; I believe that the moving instruments are unable to see the rod properly, due to the nature of light [in brief, c].

Again, we do not disagree on the result of the measurements; we disagree as to whether the results are correct.

Greg, are you aware of the origin of Lorentz contraction? It predated Einstein. It came about because of the null result of the Michelson-Morley Experiment in which a massive slab of marble was believed to be changing its physical dimensions as it was rotated, even though the slab itself was experiencing different speeds at different times of the day and of the seasons.

You have argued, apparently based on logic or common sense, that an object cannot change its dimensions simply because of the speed by which it is viewed, but what about when the object itself is having its speed changed? Do you insist that when you push on an object, it cannot change its dimensions? Or are you willing to understand how those early scientists employed Lorentz contraction to explain the null result of MMX?

 

[JesseM]

I understand that the instruments read as they do because they are in motion with respect to the rod which they are measuring. However, that by itself does not mean that the rod is actually shorter.

What does "actually shorter" mean? The claim is that it is shorter in the coordinates of a given frame, i.e. the difference between the position coordinate of the front end and the position coordinate of the back end at a single time coordinate is shorter. One could also define this in terms of simultaneous measurements in the observer's frame, for example if the rod is 3.2 meters long in my frame, that means if I have some calipers set to 3.2 meters apart, the back end of the rod will be passing the back caliper simultaneously with the front end of the rod passing the front caliper, according to my frame's definition of simultaneity. But no one is claiming that this definition of simultaneity is "correct" in any absolute sense, in relativity there is no absolute simultaneity and thus we can only talk about length relative to a particular simultaneity convention. The advantage of the convention Einstein came up with for inertial frames is just that the laws of physics appear to be invariant under a transformation from one inertial frame to the next (the equations of all known fundamental laws are Lorentz-invariant), which implies that if you do any experiment and describe the results in terms of the coordinates of the apparatus' rest frame, you'll get the same result regardless of which frame the apparatus happens to be at rest in.

You say "no", but I don't see any explanation as to why they are not in error.

"In error" with regards to what? You seem to have this quasi-metaphysical notion of the "true value" of various quantities, but in physics all quantities can only be measured relative to a particular choice of measurement procedure (like assigning coordinates in a given frame), it's completely meaningless to say something like "the measurement procedure itself is wrong" unless this is just a matter of definitions (i.e. if the quantity you're interested in is 'rest length' but you measure it with the procedure for moving length, then you've used the 'wrong procedure' given the usual definition of rest length). You can't ask what the true value of "length" is independent of human definitions of what the word "length" means, unless perhaps you believe that God has a "true" definition of the word "length" and if we use a different one then we are objectively wrong.

I just don't believe that the rod is shorter

See above, you are using metaphysical/theological language again. What does "is shorter" mean, if it doesn't just refer to definitions of "length" used by physicists which are defined in terms of simultaneous measurements of the front and back in a given frame?

I believe that the moving instruments are unable to see the rod properly

Same as above. "Properly" with respect to what, if not the human definition of "length" in a given frame?

due to the nature of light [in brief, c].

The only place light might enter into it would be in the definition of simultaneity, but there are other ways to define simultaneity in a given frame that don't make use of light, like the [post=2937771]slow transport method[/post].

Again, we do not disagree on the result of the measurements; we disagree as to whether the results are correct.

"Correct" with respect to what? Again, there is no place in physics for some metaphysical notion that words have any "true" definition aside from how we choose to define them.

 

[GregAshmore]

Greg, are you aware of the origin of Lorentz contraction? It predated Einstein. It came about because of the null result of the Michelson-Morley Experiment in which a massive slab of marble was believed to be changing its physical dimensions as it was rotated, even though the slab itself was experiencing different speeds at different times of the day and of the seasons.

You have argued, apparently based on logic or common sense, that an object cannot change its dimensions simply because of the speed by which it is viewed, but what about when the object itself is having its speed changed? Do you insist that when you push on an object, it cannot change its dimensions? Or are you willing to understand how those early scientists employed Lorentz contraction to explain the null result of MMX?

In special relativity--inertial frames--there is no acceleration, therefore no pushing at the time of the experiment. If the objects under test have been accelerated to bring them into position for the experiment, we can assume for the purposes of discussion that any deformation was elastic. In that case, yes, I would say that the length of a rod at speed is unchanged from its length at rest. It seems to me that the reciprocal nature of length contraction would be violated otherwise--the rod which was not accelerated would see the other rod as even shorter than the amount predicted by the Lorentz transformation.

At an even more basic level, it seems to me that the length contraction proposed by Fitzgerald, and applied by Lorentz, is different than the length contraction of special relativity. The equation of the transformation is the same, of course. However (it seems to me), the contraction of Fitzgerald and Lorentz must be a physical deformation, because the moving rod is absolutely in motion and the resting rod is absolutely at rest. At any rate, my understanding is that Lorentz believed that the deformation was a physical deformation, in the same sense that compression under load is a physical deformation.

In contrast, the contraction in special relativity is attributed to a projection of the rod's strip on the x, ct plane onto a line of simultaneity in another frame. Thus (says Born), "the contraction is only a consequence of our way of regarding things and is not a change of a physical reality." However one may interpret these words, it is clear that (for Born, at least) a physical deformation of the rod is not implied by special relativity.

Born goes on to say that this view "does away with the notorious controversy as to whether the contraction is "real" or only "apparent". If we slice a cucumber, the slices will be larger the more oblique we cut them. It is meaningless to call the sizes of the various oblique slices "apparent" and call, say, the smallest which we get by slicing perpendicular to the axis the "real" size."

I'm not fully satisfied with Born's explanation. When one slices a cucumber at an oblique angle, the resulting surface is spatial, just as the surface resulting from a perpendicular slice is spatial. Not so with a slice across the rod's strip in the x, ct plane; that slice is a combination of distance and time. Time and distance are interrelated, but they are not convertible one to the other. Therefore, it is very likely a mistake to treat the length of that slice as a simple distance--notwithstanding the fact that the slice is parallel to the x-axis in another frame.

I will spend the next few evenings putting together the case for my argument. I'll sign off until then--or I'll never get it done.

 

[GregAshmore]

GregAshmore,

You ask a question that all relativists have asked at one time or another. Is a moving contracted length real, or merely apparent? I must admit, even relativists debate this matter at length, and in most those cases comes down to an argument of semantics vs theory.

An accelerating body rotates in its orientation within spacetime, yet we do not witness this rotation in the same way we witness the rotation of a pencil in 3-space. The full rotation we do not perceive, but fortunately, we do perceive effects of the rotation. The effects are length contraction and time dilation, relativistic effects that arise with relative motion. This is proof that bodies remain at their proper length (per themselves) even when moving per others, even though rulers moving relatively can never measure it as such. That said, the contractions are real, while at the same time, bodies always remain their proper length per themselves even during acceleration.

To understand why this is the case, you should really be asking ... WHY does a moving body shrink in length, as opposed to ... is the measurement data correct or not? Understand why, and the question generally no longer needs asked. If you are the persistent type, you may save months (or years) by pursuing the former question first. A good understanding of Minkowski illustrations helps immensely ... food for thought.

GrayGhost

I appreciate the advice. I will look into Minkowski illustrations--after I see how my idea works out. I need to do that first, if only because it will help me appreciate Minkowski better.

 

[bobc2]

At an even more basic level, it seems to me that the length contraction proposed by Fitzgerald, and applied by Lorentz, is different than the length contraction of special relativity. The equation of the transformation is the same, of course. However (it seems to me), the contraction of Fitzgerald and Lorentz must be a physical deformation, because the moving rod is absolutely in motion and the resting rod is absolutely at rest. At any rate, my understanding is that Lorentz believed that the deformation was a physical deformation, in the same sense that compression under load is a physical deformation.

I think you have that exactly right, GregAshomre.

In contrast, the contraction in special relativity is attributed to a projection of the rod's strip on the x, ct plane onto a line of simultaneity in another frame. Thus (says Born), "the contraction is only a consequence of our way of regarding things and is not a change of a physical reality." However one may interpret these words, it is clear that (for Born, at least) a physical deformation of the rod is not implied by special relativity.

This is certainly correct.

Born goes on to say that this view "does away with the notorious controversy as to whether the contraction is "real" or only "apparent". If we slice a cucumber, the slices will be larger the more oblique we cut them. It is meaningless to call the sizes of the various oblique slices "apparent" and call, say, the smallest which we get by slicing perpendicular to the axis the "real" size."

I think most any special relativity physicist would agree with Born's characterization.

I'm not fully satisfied with Born's explanation. When one slices a cucumber at an oblique angle, the resulting surface is spatial, just as the surface resulting from a perpendicular slice is spatial. Not so with a slice across the rod's strip in the x, ct plane; that slice is a combination of distance and time. Time and distance are interrelated, but they are not convertible one to the other. Therefore, it is very likely a mistake to treat the length of that slice as a simple distance--notwithstanding the fact that the slice is parallel to the x-axis in another frame.

You've done a nice job of summarizing the situation, but Greg, I think I would not agree with this characterization. However, you probably have some good company among physicists on this point. The thing that is usually cited to bolster your view is that it is common practice to relate the 4th dimension as X4 = ict, where the imaginary number is associated with the 4th dimension. And it is this imaginary number that cautions physicists to hold back from accepting the 4th dimension as a bonafide spatial dimension. After all, how can something with "imaginary" in front of it be considered real (Minkowski, himself, started it)? However, I think this is artificially contrived as I'll try to show in the sketches below where I've derived the Lorentz transformation (rotation only) modeling space as strictly 4-dimensional.

The 4-dimensional space concept is illustrated below with a red rocket and a blue rocket speeding in opposite directions as represented in the black rest coordinate system. This is a symmetric space-time diagram that has the advantage of having straight lines in the red and blue coordinate systems having the same distance calibrations. When each rocket is at his position number nine along his own world line we will call that the "NOW" point in time for each observer. Fundamentally, everything is actually spatial here. All of the objects are actually 4-dimensional, even the physical clocks on board. So, time is just a mathematical parameter in terms of its role in the physics. The distances along X4' (blue) and X4'' (red) are the fundamental aspects of the objective ontological reality. Sure, the blue guy and the red guy can each calculate how far he has moved along the respective 4th dimension by multiplying X4' = ct' and X4'' = ct'', respectively. We could do a side bar on who or what is actually doing the moving, but that's another post. Here all physical objects are 4-dimensional (including the blue guy and the red guy).

So, just regarding the distances in the 4-D space we have the diagram in the upper right sketch that shows you can form a right triangle using X4'', X1', and X4'. You can relate these distances with the Pythorean theorm (hypotenuse squared equals the sum of the two legs squared). The Lorentz transformation follows directly from these purely spatial distances. Calling the 4th dimension "time" is just a habit picked up and continued to this day. Of course the time is associated with the X4' and X4'' because the "observer" (whatever that is, ...consciousness, awareness, or whatever...) moves along X4' for the blue guy and X4'' for the red guy at the speed of light, c. Herman Weyl, one of Einsteins close colleagues, used the phrase, "...the observer crawls along his own world line.).

The lower left sketch emphasizes the effect of length contraction. You can see that, in the blue guy's 3-D cross-section of the universe, the red guy's rocket is definitely shorter than the blue guy's rocket. Also, returning to the upper left sketch, you can see that the blue guy's cross-section of the universe intersects the 4-dimensional red rocket at a much earlier position (red position number 8--and of course corresponding to an earlier red clock time).

 

[GregAshmore]

What does "actually shorter" mean?

...

"In error" with regards to what? You seem to have this quasi-metaphysical notion of the "true value" of various quantities, but in physics all quantities can only be measured relative to a particular choice of measurement procedure...

You assume that all measurements provide an accurate picture of reality. How does this square with the fact (as reported by Taylor and Wheeler--I didn't work out the math myself) that no matter how fast an object is moving away from us, we will never measure its speed as greater than 0.5c?

The only place light might enter into it would be in the definition of simultaneity, but there are other ways to define simultaneity in a given frame that don't make use of light, like the [post=2937771]slow transport method[/post].

The equations of special relativity have the form they do because they start with the observed nature of light--its speed is the same for all observers. The nature of light is thus tightly bound to the characteristics of time and space which flow from the equations.

I must sign off for a few days and see how my ideas work out on the spacetime diagram.

 

[GregAshmore]

I think you have that exactly right, GregAshomre.



This is certainly correct.



I think most any special relativity physicist would agree with Born's characterization.



You've done a nice job of summarizing the situation, but Greg, I think I would not agree with this characterization. However, you probably have some good company among physicists on this point. The thing that is usually cited to bolster your view is that it is common practice to relate the 4th dimension as X4 = ict, where the imaginary number is associated with the 4th dimension. However, I think this is artificially contrived as I'll try to show in the sketches below where I've derived the Lorentz transformation (rotation only) modeling space as strictly 4-dimensional.

I'll have a look at your sketches after I see how my concept works out. See you in a few days.