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

GregAshmore

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

In the graphic below (fig. 1), there are two rods, each with a rest length of four units. The horizontal rod is at rest in frame S; the other rod is at rest in frame M. Frame M is moving at 0.6c relative to frame S. In this discussion, the focus will be on the rod in frame M [red], and its coordinate length in frame S [purple].



[1] The rod which is at rest in frame M will always be parallel to the XM axis, no matter at what time the rod is drawn. Thus each instance of the rod in the graphic is parallel to the XM axis.

[2] The view of the rod from frame S (its coordinate length) is horizontal. Therefore, the view is not one view, but a composite of many views. The coordinate length is composed of many snapshots, each snapshot showing a specific point on the rod at a specific time in frame M. In the graphic, four instances of the rod are marked as they cross time TS = 6.68. The four marked points of the rod are at times TM = 5.35, 6.01, 6.67, and 7.33, respectively. The fifth mark, at TM = 7.75, is where the trailing end of the rod crosses TS = 6.68; no instance of the rod is drawn at that point.

[3] As one moves along the view of the rod in frame S, from leading end to trailing end, the time in frame M increases. This means that, in the view as seen from frame S, the trailing end of the rod has traveled farther than the leading end of the rod. This explains, qualitatively, the contracted coordinate length in frame S.

[4] When the velocity is reversed, the effect is the same, as shown in this graphic (fig. 2):



[5] The time differential from the leading end to the trailing end is equal to the relative velocity of the two frames multiplied by the rest length of the rod. In this example, (7.75 - 5.35) = 2.4 = 0.6 * 4, where T = ct and V = v/c. This can be better seen in figure 3:



Thus, the apparent contraction of the rod is directly related to the relative velocity of the frames. Taylor and Wheeler show that the contracted length can be developed by integrating from relative velocity 0 to V. (See exercise L-14 in Spacetime Physics.) Their interpretation is that the trailing end of the rod, as seen in frame S, begins to move before the leading end, thus contracting the rod in frame S. (The same is true for each differential segment of the rod. The differential time at each segment is smaller than at the trailing end, thus leading to a contraction proportional to the length.)

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

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

John232

In relativity the length of an object doesn't change due to an outside observer traveling at relativistic speeds as it is anyways. For example, a ship traveling 0.6c relative to Earth couldn't make us earthlings observe ourselves to be distorted in the ships direction of motion. I don't think it actually proves that there is no length contraction, but the contraction is only observed differently by different observers.

I think it is key to remember that the length contraction is used to only maintain the constant speed of light. In time dialation, the amount of time you observe another object to expereince has to be a value that makes you see them measure the speed of light to be the same as you do. If they expereince the dialation themselves this wouldn't happen because they measure light to still travel the same 300,000 km/s or so faster than them. So then two observers couldn't see themselves as their measuring rods as being a different length because the measured speed of light is always the same speed faster for both of the two, otherwise the theory would be useless because they would measure different speeds of light with their distorted measureing rods.

bcrowell

Please define "true."

GrayGhost

GregAshmore,

You are thorough, I'll give you that.

There is the "proper length" of a body, observable by only those who reside at rest in the proper frame of said body.

There is the "contracted length", observable by only those in luminal motion wrt (the proper frame of) the body.

Relative motion allows us to witness the body as it exists "not in its own instant of time", but rather as it exists "across a duration of its own time". Technically, as it exists "over a length of its own worldline" thru spacetime. Temporally, and wrt its own sense of time, FWD points of the moving body lag AFTward points.

That said, its not about "true or real" vs "untrue or apparent". It's about "proper vs non-proper". There is a "proper view" of the body, which can only be obtained while at rest with the body. There also exists a "non-proper (rotated) view" of the body, when seen in motion. Both views are verifiable via measurement (in theory) by the observer. Neither is less real, or less true, than the other. It's all about POV.

There is one point worth mentioning IMO. None of us ever experience ourselves except in the moment. If we do exist in the continuum as luminally moving others record of us, we are unware of this in our own experience. Yet, this alone does not require that the rotated view does not exist, or that it is any less true or real. The mathematics of the theory require it.

Good work though ! Believe me, everyone who understands relativity theory eventually came across your discovery here at one time or another, and questioned it at length. So, you are in good company.

GrayGhost

ghwellsjr

Greg, when you think about two rods with a relative motion between them, do you think that the true speed of both of them is zero? Or do you think that the true speed of both of them is whatever their relative speed is? Or do you think that the true speed of both of them is some smaller identical value but in opposite directions?

I doubt it. I'm going to guess that you have no problem with the concept of relative speed and you realize that even though each one views the other one as traveling in the opposite direction at the same speed, you understand that you cannot then say that the true difference in speed is double their relative speed.

It is a fact that when two rods are in relative motion, you cannot say that both are stationary at the same time and for the same reason, you cannot say that both their true lengths are their rest lengths at the same time. Special Relativity is all about picking a single frame of reference from which to assign locations, dimensions, and times to everything. It is not possible to pick a frame of reference in which both rods will be their rest length.

If we could say that the length of one rod was its true length, then we would also be saying that we have identified the absolute ether rest frame and the true length of the second moving rod would be a contracted length. When you rotate the first rod, since it is stationary in the ether, its true length will remain the same but when you rotate the second moving rod, its true length would be changing, even though its speed is not changing.

So the bottom line is that your effort to attribute "trueness" to a rod's length is no different than an effort to promote an absolute ether rest frame. Is that really what you want to do?

DaleSpam

Hi GregAshmore, none of what you presented here is new, surprising, or unusual. It does not change the way that the term "length" is defined nor does it make anything about that definition invalid.

You are certainly free to adopt the arbitrary personal definition that the term "true length" means the same as the common term "proper length". In fact, you don't need any of the above justification to do so, you can simply assert "I am defining the term 'true length' to mean the proper length". However, since a common term already exists for the concept why not use it? All you are going to accomplish with this approach is to make communication barriers and encourage confusion.

ghwellsjr

But, DaleSpam, would you also say that Greg is free to call a contracted "measured coordinate length" the "untrue length" or "a distorted view of the true length" and that rods are "completely unaffected" by their accelerations?

I'm saying that when two rods are in relative motion, Greg needs to understand that they cannot both be at rest at the same time, and so it is not a true statement that their "true lengths" are the same as their "rest lengths".

PAllen

Dalespam,

I have only seen proper length described as length measured between two spacetime points in an inertial frame in which they are simultaneous (SR; more complex in GR). It is invariant given the points (and the implied geodesic between them). Thus one can talk about the proper length of a moving ruler, whose end events will not be simultaneous in the frame of the ruler, but are simultaneous in the frame of the observer seeing the moving ruler.

OP is defining proper length in the frame in which the ruler is at rest, sort of analgous to rest mass. I haven't seen such a definition in any of my books (or in Wikipedia def. of proper length). Seems a somewhat interesting idea.

Can you clarify this terminology?

clem

Hi Greg,
I agree with your assertion that "a rod has one true length, its rest length.",
but I find your pictures confusing.
You might want to look at <http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1919v3.pdf>.

GregAshmore

In relativity--both special and general--every observer is always at rest. As a given observer sees it, all other objects are moving; he himself is stationary. What, then, is the meaning of "their accelerations"? And, how can a rod be affected by acceleration if it never accelerates?

Each rod is at rest, so far as its resident observer is concerned. Each observer says that it is the other rod which is in motion. No one can prove either one of them wrong.

GregAshmore

I didn't expect that I would be the first to interpret length contraction this way. However, I have not seen this interpretation in the books I have read.

The choice to accept or reject the coordinate length as a valid length of the object will be driven by one's philosophy of reality. There is no disagreement as to the result of the measurement.

For me, the issue is the meaning of the measurements. Such questions cannot be resolved by the adoption of a definition.

I'm not trying to change the world. I have no problem with using the accepted terms in all discussions of practical matters.

GregAshmore

Thanks for the link. I've started reading the paper; I'll finish it this evening.

I found the opening paragraph very interesting. I'll bet it gives DaleSpam fits. (I appreciate your point of view, DaleSpam. There is little practical value in a philosophical discussion of "reality". That's why such discussions must be limited to the appropriate times and places.)

DaleSpam

You are correct. According to my understanding the term "proper length" (or better "proper distance") refers to the invariant interval between two spacelike separated events. Whereas the term "rest length" refers to the length of an object in the reference frame where it is at rest. For massive inertial objects they are equivalent so the distinction is somewhat hazy at times. This is similar to "invariant mass" vs "rest mass".

DaleSpam

One's philosophy of reality is not relevant here. The coordinate length is the length of the object in the given reference frame by definition. There is no question of validity, it is defined as such.

Although you are free to introduce new terms and their definitions you are not free to un-define already defined terms.

Then I don't know what you hope to accomplish by defining your new term "true length".

IMO, the entire discussion is purely semantic and will remain purely semantic while you focus on introducing new words for already defined concepts. You would be better off to simply adopt the existing terminology "coordinate length" and "proper length" and discuss the physics.

DaleSpam

I wouldn't say it this way. I would say that for every observer there exists a valid frame where that observer is at rest. However, the observer is no more constrained to use that frame than any other frame.

Acceleration is not relative, particularly not proper acceleration.

bcrowell

GregAshmore, please listen to what DaleSpam is trying to tell you. You're not accomplishing anything meaningful here.

PAllen

Actually, the part I don't quite agree with is that proper length of a massive body is equivalent to rest length as you've (and Greg and the paper linked by Clem) have defined it. Proper length is defined for the object in motion, and is then less than rest length. That is, an observer seeing the massive object in motion perceives a different slice of its world tube as simultaneous than the object's rest frame, and proper length along a cut of this slice will be shorter than proper length along a cut of the simultaneity slice of the object's rest frame. So proper length of a massive object is observer dependent (even though it is invariant *given* a particular simultaneity slice), while rest length is defined as observer independent by the specification that is always computed in the object's rest frame.