On the interpretation of a spacetime diagram

[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 theorem (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.

 

[GrayGhost]

You are correct that Lorentz considered the Fitzgerald contraction a physical deformation of the electron field about the atom. They assumed the body contacted in an aether that does not.

Einstein's length contraction is completely different, yes. If an aether exists, it doesn't matter at all per SR. Consider 2 bodies of the same frame, one in front of the other separated by some proper distance s. Now, assume yourself in motion wrt said bodies, with velocity vector parallel wrt the center-line connecting the 2 bodies. While viewed in motion, not only do the moving bodies length-contract, but all the space between them (and within them) does as well. IOWs, it's not that the moving body contracts in a space that does not ... but rather that the way in which space and time are measured changes with a change in one's own state of motion. This is why you remain unaffected when (say) 10 moving observers measure your length differently. Yet, their measurements are correct, and quite real. Theoretically, with sophisticated enough measuring systems using lasers, light signals must reveal what the math of SR predicts for any of said 10 moving measurers.

GrayGhost

 

[bobc2]

You are correct that Lorentz considered the Fitzgerald contraction a physical deformation of the electron field about the atom. They assumed the body contacted in an aether that does not.

Einstein's length contraction is completely different, yes. If an aether exists, it doesn't matter at all per SR. Consider 2 bodies of the same frame, one in front of the other separated by some proper distance s. Now, assume yourself in motion wrt said bodies, with velocity vector parallel wrt the center-line connecting the 2 bodies. While viewed in motion, not only do the moving bodies length-contract, but all the space between them (and within them) does as well. IOWs, it's not that the moving body contracts in a space that does not ... but rather that the way in which space and time are measured changes with a change in one's own state of motion. This is why you remain unaffected when (say) 10 moving observers measure your length differently. Yet, their measurements are correct, and quite real. Theoretically, with sophisticated enough measuring systems using lasers, light signals must reveal what the math of SR predicts for any of said 10 moving measurers.

GrayGhost

Nice summary of the situation, GrayGhost.