The case for True Length = Rest Length

Mentz114

Underlying all these objections is a confusion between length, the property, and it's value.

The 'relative quantities' you speak of are the result of relative velocity on measuring instruments, and unless they are corrected for the effects of the velocity, they are not measurements of the length.

I'm disappointed by your reply, where you've thrown questions rather than reply to what I explained very simply.

rjbeery

What I'm doing is trivially assigning the physical manifestation of the quantity that is invariant under Lorentz tranforms in SR to be considered a "spacetime velocity". This quantity, by definition, will remain constant for all frames. I've "lumped together" the spatial vector components for simplicity because when discussing SR length contraction and time dilation we don't frankly care which way the object is moving, only that it's moving spatially.
I guess I'll start over....
The diagram below is completely different by design - I don't want people trying to analyze it with traditional Minkowski spacetime diagram prejudices which I think may have been part of my problem. The Y direction represents an object's speed through space (specifically, what we traditionally think of when we say "speed"); the -X direction represents an object's "speed" through time; the needle has a fixed length of C, representing the Lorentz-invariant quantity in SR that we are physically representing here; finally, the speedometer's numbers very roughly signify the object's spacetime velocity's percentage of "rotation" through the space and time diagram. (Ideally, 100% would be directly at the top, coinciding with an object moving through space at C, but it's late and I'm tired!)

The above diagram shows an object at rest. It's Y component is zero, signifying zero spatial-velocity. It's -X component is C, signifying a temporal-velocity of C. What does it mean to be moving at 1 second per second? It means that we observe that the object is experiencing no time dilation.


Now, this object is moving such that it's spatial and temporal component vectors are equal in magnitude. As you can see, its Y component would be .707, as would its -X component. This coincides precisely with what SR calculates as the time dilation factor of an object moving in such a manner.

Presuming the above explanations make sense, the parallax-induced (aka "foreshortening") length contraction analogy is simple (it may help to consider the speedometer to be sitting flat on a table for this): Physically replace the needle with the Rubik's Cube face, as the "true" face width (let's call it W) is invariant to rotation; next, consider the -X direction to be the "apparent face width" and the Y direction to be depth; lastly, consider the speedometer reading to be the percentage of rotation of the Rubik's Cube face from being sitting squarely in front of us through being completely inline with our vision such that it's apparent width is zero.

Above is a Rubik's Cube with blue face width of W.


Above is the same Rubik's Cube, rotated 50% (or 45 degrees) through the depth dimension Y. It's "apparent", parallax-induced, foreshortened blue-face width is now .707W, which means that the foreshortening factor is exactly what we calculated the Lorentzian time-dilation and length-contraction factor to be above.

The point is that if we consider foreshortening to be illusory, and I presume we all do, then I maintain that Lorentzian length contraction should also be considered illusory.

...Whew! Does this help?

DaleSpam

Instead of expressing your disappointment perhaps you should actually answer the questions. "Objective reality" is not a term I have ever used and I don't know how you are using it. I am particularly interested in your response to this question:

Is your idea of "objective reality" fundamentally incompatible with relative quantities?

Mentz114

For the purposes of this discussion, all I ask is this:

If an object is transported from one laboratory to another that is moving relative to the first laboratory, then if its length is measured in that lab, the outcome will be the same as the identical procedure that was carried out earlier in the first lab. So there is some property of the object that was unaffected by being moved between the labs. Sort of like "the laws of physics are the same in all inertial frames".

Allowed this premise, I assert that relativistic effects cause miscalibrated measurements to give wrong answers. However if the instruments are made so they can take into account these effects, then all inertial observers will actually be measuring the length, and agreeing. Using miscalibrated measurement procedures, one is not measuring anything.

A question for you, if there is no objective reality, what exactly is the nature of thing you call length, and is there any point in measuring it ?

I have to leave now, but I'll check in again in about eight hours.

PAllen

So, to be clear, you would call the following a mis-calibrated mearurement:

A rocket with rest length of 100 meters is moving by at relativistic speed. I have a pair of super fast barriers I can raise and lower simultaneously (as I see it). I can momentarily contain the rest length 100 meter rocket between my barriers set 10 meters apart. My conclusion that the length is less than 10 meters is mis-calibrated, and inferior in some way to the rocket's own perception that what happened is that a barrier went up and down in front of the rocket, then both barriers moved, then the other barrier went up and down behind the rocket.

rjbeery

First of all, I'd like to announce that I realize this issue is 100% subjective and ultimately "undecidable". I just started this thread to issue my take on it.

The scenario above, though, has an analogy in my Rubik's Cube mentioned a couple of posts back...relative motion is analogous to a differing amount of physical rotation, so your 100m rocket would indeed fit between two barriers set 10m apart if it was twisted to the side. My position is that to say that the twisted rocket is now "truly" less than 10m long is a fallacy.

JesseM

Yes, I understood that you were just talking about the total magnitude of the velocity vector rather than its individual components, that's why I said:
If you plot speed vs. [tex]d\tau/dt[/tex] (which is just 1/gamma) you get the graph that ghwellsjr posted in this thread:



Just a segment of a circle, with each point having the same distance from the origin. That's what you're representing in your speedometer drawings too. But the argument about what this implies about length contraction is still unclear to me. You say:
First of all, it seems a little confusing to have the cube rotate in the "depth" direction (into the computer monitor) when the vector representing "speed through spacetime" (the needle on the speedometer) never rotates in that direction (always stays in the plane of the monitor). I suppose you could say that we were just changing our perspective on the speedometer so we were looking up at the needle from "below", sitting far down along the y-axis (our line of sight being in the same plane as the speedometer, as if we were flatlanders), in this case as the needle rotated, the needle itself would appear visually foreshortened in exactly the same way as the side of the Rubik's cube, and if we were a far distance away so that our lines of sight to each end were essentially parallel, we would see it foreshortened by exactly the same amount as needle's length component on the x-axis. Since the length contraction factor and [tex]d\tau/dt[/tex] are both equal to 1/gamma, this foreshortening would be proportional to the length contraction factor. In fact we might improve the analogy if instead of having the axes be speed vs. [tex]d\tau/dt[/tex], we had the y-axis be (speed*rest length/c) and the x-axis be moving length...in that case the length of the "needle" could always be equal to the rest length, while the component of the needle that lies along the x-axis would be the moving length, and that would be the foreshortened apparent visual length seen by an observer a great distance away along the y-axis.

But it seems to me this is just a happenstance fact about mathematical similarities between length contraction and rotated needles seen at a great distance, it doesn't make sense to me to say that because the numbers work out, that means length contraction "really is" just a consequence of "viewing" some vector in an abstract space of (speed*rest length/c) vs. (moving length) at an angle--after all, we measure length in real space and time, not in this abstract space! Also, wouldn't it be just as much a consequence of your argument that you should say the "true" value of the rate a clock ticks is always 1 second/second, and that any apparent change is an "illusion" caused by the foreshortening of a needle in an abstract space of (speed/c) vs. (rate of ticking in frame where clock is moving)?

rjbeery

Agreed! The perspective with the cube and the speedometer isn't perfect, which is why I wrote
I'm actually a bit frustrated because I stayed up quite late crafting my speedometer post, which is why I had to cut some corners, but apparently I never hit "Submit" in my sleepy state so I had a late night for naught!
Anyway, I'm asserting a correspondence between Lorentz transforms and foreshortening. When you view SR in the manner that I described, both phenomena involve rotating an invariant between 2 perpendicular dimensions. In the former case we are claiming that something is "actually changing" while in the latter case we all acknowledge that it's simply an illusion. All are welcome to continue to hold their own views, obviously, but I find the analogy to be convincing.

Also, consider Wiki's explanation of the bar-and-ring paradox:
In other words, the bar and ring problem is resolved by one party disagreeing with the other on the amount of relative rotation between the two objects! If a simple rotation can explain how we fit a 2 meter pole through a 1 meter ring in that scenario, doesn't it support the analogy between Lorentz length contraction and a physical rotation?

DaleSpam

No, it won't, unless the object is Born-rigidly accelerated such that the object's velocity relative to the second lab is the same as the velocity relative to the first lab.

I am sorry Mentz114, but your stance here is completely contrary to SR. In SR length is a relative quantity, meaning that it depends on the frame of reference. It is not a property of the object itself.

I have told you 3 times now that I don't know what you mean by "objective reality". You should stop being evasive about the meaning of your terms. In that vein, I will gladly define "the thing I call length":

The length of an object is the distance between the two ends of an object at the same time in some specified reference frame.

I would now appreciate an answer to the question which you have avoided twice now: Is your idea of "objective reality" fundamentally incompatible with relative quantities?

DaleSpam

Yes. That is why it was discussed at length in the other threads.

rjbeery

DaleSpam, presuming that you consider foreshortened lengths to be illusory, your definition of "distance" probably involves correcting for foreshortening effects, correct? Why is it that you would correct for foreshortening effects but not for relativistic effects if the math and procedure is similar?

DaleSpam

No, I don't consider projected lengths (what you call "foreshortened lengths") to be illusory. They are simply projections from a higher dimensional space onto a lower dimensional space.

e.g. I do not consider the length of a shadow to be an illusion; the shadow actually has some length. Also, I would not confuse the length of a shadow with some property of the object casting the shadow, and I would recognize that if the shadow were cast from a different light source that the result could be different without there being any paradox or contradiction.

Matterwave

I think there is an issue here with rigid bodies and how they are, in fact, not allowed in special relativity. Special relativity is violated by rigid bodies since, they transmit cause and effect at infinite speed between 2 end points. We should only consider point particles in special relativity. In this sense, the fact that the space "contracts" between 2 points when you are moving between those 2 points can easily be seen in the following experiment (taking time-dilation for granted):

Suppose particle A and particle B is 1 light-second apart in my (at rest w.r.t. A and B) frame of reference. There is a spaceship, traveling at velocity = .5c from particle A to particle B. At t=0 me, particle A, and the spaceship all coincide in position and the spaceship and me synchronize our watches.

From my POV, the space-ship is traveling at .5c from A to B, from the space-ship's POV, point B is moving towards him at .5c and point A is moving away from him at .5c. So, for me the spaceship will get from A to B in 2 seconds. For the space-ship though, B will arrive at where he is in (I am unprimed, space-ship is primed)

[tex]t'=\frac{t}{\gamma} = 1.732s[/tex] (The space-ship is the one measuring proper time)

Therefore he will say that the distance that point B has traveled from the time that A coincided with him to be:

d'=.5c*t'=.866 light-seconds.

Therefore, the space-ship must have measured a contracted distance between A and B than I did because his time has been dilated and I must measure the same velocity that the spaceship has that the spaceship measure me to have (otherwise, we would have some paradox like he is traveling at .5c away from me, but in his perspective, I am traveling at v not equal to .5c away from him).

If we then ask "what is the ACTUAL or TRUE distance between A or B", then there is no good answer because if you take the length I measure to be some "true distance", then the space-ship which is moving relative to me can never MEASURE this "true" distance (the people can write some equations and try to figure it out, I suppose).

Since there are no rigid bodies in SR, I cannot somehow put a perfectly rigid rod between A and B, and allow the space-ship to measure the distance I would measure by transporting that rod to the space-ship.

Whether you really want to call my measurement the "true" distance and the space-ship's measurement somehow an "untrue" distance, I think is more up to philosophy than actual physics.

rjbeery

Ahh, exactly! The shadow has some length, but when you say
...you are referring to the distance between the ends of the projected object, NOT the shadow itself. When you observe a foreshortened object you do 1 of 3 things:

1) Turn the foreshortened object squarely with yourself and measure it.
2) Turn your measuring device to match the angle of the foreshortened dimension you're trying to measure.
3) Measure the foreshortened length (or, if you prefer, the length of the shadow), and mathematically calculate what the "true length" of the object is.

Each one of these actions has an SR analogue. When you observe a Lorentz-contracted object you (could) do 1 of 3 things:

1) Bring the object under consideration into your rest frame.
2) Send your measuring device into the moving frame of the object.
3) Measure it's contracted length and correct your answer taking into consideration the Lorentz transform determined by your relative velocity.

In my opinion it's the SAME THING. By claiming that a shadow has a definite length which is separate from the object's "true length" you are merely reasserting my claim that a length-contracted object has a definite length which is separate from that object's "true length".

rjbeery

Yes, in the end this is undeniable of course. However, would you also argue that the foreshortened width of the Rubik's Cube a few posts back is just as "true" a width as if we were to measure the width squarely?

DaleSpam

The measurement of length is a projection of a 4D object onto a 3D space called a hyperplane of simultaneity. In that sense, it is in fact a "shadow".

Or 4) measure the length of the projection and recognize that it is a projection.

The point is that the distance between two ends of an object at some instant in a given reference frame is some number. That number is named "length". You may not like the fact that that number is called "length" but your opinion is not relevant and does not change the facts.

Matterwave

I would hope that you would read more of my post than just the last statement.

Since option 1 and 2 that you posted, are not possible, and 3 is just manipulating some formulas, then I would say that there is no better reason to call the distance I measure to be "true" than to call, say, the rest-energy of a particle as it's "true" energy.