# On the interpretation of a spacetime diagram

Below is a spacetime diagram which depicts the movement of a rod, four units long, relative to another rod of equal length. The red lines are the "moving" rod; the purple lines are the "stationary" rod. The gray lines are the world lines of the unit marks (0, 1, 2, 3, 4) on each rod.

As I attempt to interpret the diagram, I am moved to ask a question which has been asked before on this forum, including by me. There is nothing new in the question itself--only my level of understanding has (hopefully) progressed.

To set up the question, I stipulate that the stationary rod and the moving rod are identical, having been fabricated on the same high-precision equipment. The stationary rod was then fastened to a bench in the lab, and the moving rod was set in motion. As the moving rod passes over the resting rod, it is moving at a constant velocity of 0.6c.

As can be seen in the diagram, instruments mounted on the moving rod will measure the resting rod to be 3.2 units long. Likewise, instruments mounted on the resting rod will measure the moving rod to be 3.2 units long.

My question focuses on the stationary rod, because it has been minding its own business on the lab bench, and its dimensions are known to be the same as when it was fabricated. 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?

My question focuses on the stationary rod, because it has been minding its own business on the lab bench, and its dimensions are known to be the same as when it was fabricated. 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?
The whole point is that there's no meaning in saying that one rod is "really" moving and the other rod is "really" at rest. Think about the situation from the point of view of of the moving rod: "I've just been minding my business sitting in one place, I was fabricated by equipment which started moving at some point, and my dimensions are 4 units long, the same as when I was fabricated, but now a rod attached to a moving lab bench which just passed by me is claiming that I'm only 3.2 units long. But then again, it thinks it is 4 units long, even though I can clearly see that it's only 3.2 units long. So its measurement instruments are incorrect."

ghwellsjr
Gold Member
In the case of a relative speed of 0.6c, which each rod correctly measures, gamma is 1.25, which means that each one sees the other one's times as lengthened by a factor of 1.25 and sees the other's rod as being shortened by a factor 1/1.25 which equals 0.8. Let's say that light takes 1 unit of time to travel 1 unit of length (whatever you want that to be) so c equals 1 unit of length per 1 unit of time.

There are many ways to illustrate how this is happening but one of the easiest, in the case where the two rods pass directly adjacent to each other is to time how long it takes for each rod to pass by the other one.

So let's look at it from the standpoint of the "stationary" rod. It's clock/timer is running at the normal rate and so when the two rod's meet, it starts its timer. How long will it take for the other end of the moving rod to get to the leading edge of the stationary rod? Well, since the rod is 3.2 units long and it is traveling at 0.6 units of length per 1 unit of time, and since time is equal to distance divided by speed, the time will be 3.2 divided by 0.6 which equals 5.3333 units of time. That is what the stationary rod measures of how long it takes for the moving rod to pass a certain point. Now since length equals speed multiplied by time we get 0.6 times 5.3333 which equals 3.2. (Image that!)

Now let's see how the moving rod measures the stationary rod's length. We're still examining this from the lab frame. When the two rods meet, the moving rod starts it's timer. How long will it take for the leading edge of the moving rod to get to the trailing edge of the stationary rod? Well, since the stationary rod is 4.0 units long and the moving rod is traveling at 0.6 units of length per 1 unit of time, and since time is equal to distance divided by speed, the time will be 4.0 divided by 0.6 which equals 6.6666 units of time (according to the stationary rod). Now to see what the moving rod measures for how long it takes for a point on the moving rod to pass the length of the stationary rod we have to divide by gamma so the result is 5.3333. Now since length equals speed multiplied by time, we get 0.6 multiplied by 5.3333 which equals 3.2.

Please note that this is all done in the lab frame. It is also possible to transform the entire scenario into another frame, such as one in which the "moving" rod is at rest but this will result in the exact same calculation as was done for the "stationary" rod so it's rather unexciting.

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JesseM
On this thread I drew up some illustrations of two ruler/clock systems moving past each other, showing how each system measures the other rulers to be shrunk and the other clocks to be slowed-down and out-of-sync. You can see from the diagrams how the situation is completely symmetrical, and if you add to that the fact that all laws of physics work the same way in the coordinate systems defined by each ruler/clock system, you may get a better idea of why there is no basis for preferring one point of view over the other (including the fact that there's no basis for labeling either 'at rest' in an absolute sense).

Dale
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2021 Award
My question focuses on the stationary rod, because it has been minding its own business on the lab bench, and its dimensions are known to be the same as when it was fabricated. 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?
Let me offer you a geometrically similar scenario and see what you think:

Suppose we construct two identical parallel rods, each of unit length, with devices attached to measure projected lengths (e.g. A sliding T). Now we rotate one rod 45 deg and each rod measures the projected length of the other rod. Each rod finds the projection of the other rod to be .7.

The stationary rod has been minding its own business on the lab bench, and its dimensions are known to be the same as when it was fabricated. Knowing that the stationary rod is 1 unit long, would we not have to conclude that the instruments on the rotated rod are incorrect when they report the projection of its length to be .7 units?

Let me offer you a geometrically similar scenario and see what you think:

Suppose we construct two identical parallel rods, each of unit length, with devices attached to measure projected lengths (e.g. A sliding T). Now we rotate one rod 45 deg and each rod measures the projected length of the other rod. Each rod finds the projection of the other rod to be .7.

The stationary rod has been minding its own business on the lab bench, and its dimensions are known to be the same as when it was fabricated. Knowing that the stationary rod is 1 unit long, would we not have to conclude that the instruments on the rotated rod are incorrect when they report the projection of its length to be .7 units?

Well, if we recognize that we are measuring a projected length, then there is no problem with either measurement. But in that case, we understand that the actual (or real) length of the rod is something other than the measured value.

Perhaps this is the experiment which can (in principle) be conducted to determine whether length contraction is physically real.

What about considering the situation with a focus on the 4-dimensional geometry. The situation described below is very similar to what you've set up with the two poles. However, to give the situation a definite physical character to it we use the well-known pole-in-the-barn example. My Special Relativity physics prof gave this one to us as a homework problem. The challenge was to use space-time diagrams to illustrate how a pole that was too long to fit in a barn in one coordinate system could be observed to fit easily in the barn in the view from another system.

I think is helpful to first view the pole and the barn as two 4-dimensional objects. The pole flies toward the barn at relativistic speed. The space-time diagram is sketched as the pole (red 4-D object) moving to the left and the barn (blue 4-D object) moving to the right at the same relativistic speed with respect to a rest system (black coordinates). Again, we assert something more than just coordinate systems: We regard the two objects as literally 4-D objects--you construct the 4-D objects by extruding the easily visualized 3-D objects along their respective 4th dimensions. The impression of speed is a manifestation of the relative rotational orientations of the 4-D objects with respect to each other (and with respect to a photon world line).

Please don't get into the side bar philosophical issue of whether I'm trying to float a literal interpretation of a static 4-D universe here. This is a pedagogical choice. Thus, for the sake of understanding the space-time diagrams, we at least temporarily abandon caution for the sake of getting the hang of cross-section views of 4-D objects.

Once we accept this 4-D object view, we must then adapt another curious aspect of our 4-D world experience. Observers experience the laws of physics only by living in a continuous sequence of instantaneous 3-D cross-sections of the 4-D universe in which the following results: For any rotational angle of a given 4-D object's world line (angle of the world line of the 3-D extrusion along the 4th dimension world line), any instantaneous 3-D cross-section view of that 4-D object (representing a normal spatial axis) will be symmetrically oriented about a world line representing the world line of a photon of light. That is, the angle between the spatial dimension (axis, i.e., X1') and the photon world line is (for any observer) always the same as the angle between the observer's 4th dimension (i.e., world line direction--X4'). Thus, for two observers moving with respect to each other, their 4th dimension world lines will have different angular orientations with respect to a photon world line--and consequently their spatial axes will have different rotational axes as well. This accounts for the different cross-section views of 4-D objects.

In the sketch below we have a red coordinate observer (view from the vantage point of the pole) and a blue observer (view from the viewpoint of the barn). The front and back doors are initially open. When the pole reaches the barn (blue guy's view) the blue guy waits until the pole is inside the barn, then he quickly closes both doors, then opens the back door just in time to let the pole fly on through. But, just for an instant the pole is INSIDE THE BARN WITH BOTH DOORS CLOSED.

However, the red guy (moving with the pole) observes the front and back doors opening at different times, so that, in his cross-section view of the 4-D objects, the pole is never inside the barn with both doors closed (it better not be, since in his world the pole is too long to fit into the barn). Notice, for the red guy, the back door opens before the pole is completely inside the barn. This of course illustrates the length contraction as well as time dilation and differing impressions of simultanaeity.

The point emphasized here is that the blue and red guys are each viewing dimensions of cross-sections views of objects that are in reality 4-dimensional. The 4-D objects are the reality. Our 3-D images are just particular cross-section views that depend on the realtive orientations with respect to photon world lines. It was important to use a SYMMETRIC space-time diagram in order to sketch the 4-D objects to the same scale on the computer screen.

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Hurkyl
Staff Emeritus
Gold Member
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?
Why would we conclude such a thing?

JesseM
Well, if we recognize that we are measuring a projected length, then there is no problem with either measurement. But in that case, we understand that the actual (or real) length of the rod is something other than the measured value.
Geometrically, what would a non "projected", "actual" length be? Just the length measured along the axis parallel to the rod? I suppose you could similarly define the "actual" length or a rod in SR to be the length in the rest frame, and say that all other lengths are "projected". In both cases it's just a matter of definition, and there's no reason we must accept those definitions.
GregAshmore said:
Perhaps this is the experiment which can (in principle) be conducted to determine whether length contraction is physically real.
What does "physically real" mean? I would normally interpret that phrase to mean something like "frame-invariant" but that doesn't really make sense here since obviously length is not frame-invariant.

Dale
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Well, if we recognize that we are measuring a projected length, then there is no problem with either measurement.
Yes. I think that is the important thing, to recognize that any measurement of a length is a projection onto a plane of simultaneity.

Why would we conclude such a thing?

If I am the owner of a rod whose value is determined by its length, and I know that the rod is four units long, you can be sure that I will challenge any claim that the rod is in fact only 3.2 units long.

Geometrically, what would a non "projected", "actual" length be? Just the length measured along the axis parallel to the rod? I suppose you could similarly define the "actual" length or a rod in SR to be the length in the rest frame, and say that all other lengths are "projected". In both cases it's just a matter of definition, and there's no reason we must accept those definitions.
It seems to me that for either definition to be meaningful (or "useful", or "an accurate model of reality"), all parties ought to be able to agree on the length of a particular rod if they adopt one definition or the other and measure accordingly.

What does "physically real" mean? I would normally interpret that phrase to mean something like "frame-invariant" but that doesn't really make sense here since obviously length is not frame-invariant.
If length is not frame invariant, why would time be frame invariant? In the Lorentz transform, distance and time are interdependent and inseparable. The invariant is (of course) the spacetime interval, which is a combination of distance and time.

JesseM
It seems to me that for either definition to be meaningful (or "useful", or "an accurate model of reality"), all parties ought to be able to agree on the length of a particular rod if they adopt one definition or the other and measure accordingly.
Well, rest length is something all parties can agree on, but then so is length in some agreed-upon frame.
GregAshmore said:
If length is not frame invariant, why would time be frame invariant? In the Lorentz transform, distance and time are interdependent and inseparable. The invariant is (of course) the spacetime interval, which is a combination of distance and time.
Proper time along a timelike path through spacetime is frame-invariant, as is proper distance along a spacelike path through spacetime (which could be imagined as the path of a faster-than-light particle). Both are found by integrating the infinitesimal spacetime interval (possible multiplied by or divided by c to make sure the answer has units of time in the first case and distance in the second case) along the path.

If I am the owner of a rod whose value is determined by its length, and I know that the rod is four units long, you can be sure that I will challenge any claim that the rod is in fact only 3.2 units long.
Equally, if you measure the length of the rod moving relative to you as 3.2 units and the observer co-moving with the rod will measure his rod to be 4 units long, so he will challenge your claim his rod is 3.2 units long.

Well, if we recognize that we are measuring a projected length, then there is no problem with either measurement.
While this interpretation sort of solves the problem, it also sweeps the problem under the carpet. Imagine we have a horizontal rod that is one metre long. We place a light above it and its shadow is one metre long. Now we tilt the rod and the shadow is less than one metre long. The shadow is the projected length, but we understand that the length of the rod has not actually changed and is still one metre long. Is length contraction effectively just measuring the shadow of the moving rod? Well consider this thought experiment and tell me what you think. We have a circular railway track that is ten units long and banked like a "wall of death" so that a train can move at high speed on it. On the track we place ten self powered railway carriages each one unit long and connect them to each other all the way around except at one point. The carriages are accelerated to say 0.8c and due to length contraction the total length of the train is 6 units long. With some careful timing, we can now fit another 6.6666 carriages on the track so that we have a total of 16.6666 carriages, each with a proper length of one unit (as measured by an observer on the carriage) circulating on the 10 unit length track comfortably fitting end to end. If the speed is increased, we can fit further carriages on the track and by continuing this process we can fit as many carriages on the track as we like without any change in the proper length of the track or the carriages. Would you agree in this case that we are not just "measuring shadows"?

P.S. Edited to correct the number of extra carriages from 4 to 6.6666

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Equally, if you measure the length of the rod moving relative to you as 3.2 units and the observer co-moving with the rod will measure his rod to be 4 units long, so he will challenge your claim his rod is 3.2 units long.

While this interpretation sort of solves the problem, it also sweeps the problem under the carpet. Imagine we have a horizontal rod that is one metre long. We place a light above it and its shadow is one metre long. Now we tilt the rod and the shadow is less than one metre long. The shadow is the projected length, but we understand that the length of the rod has not actually changed and is still one metre long. Is length contraction effectively just measuring the shadow of the moving rod? Well consider this thought experiment and tell me what you think. We have a circular railway track that is ten units long and banked like a "wall of death" so that a train can move at high speed on it. On the track we place ten self powered railway carriages each one unit long and connect them to each other all the way around except at one point. The carriages are accelerated to say 0.8c and due to length contraction the total length of the train is 6 units long. With some careful timing, we can now fit another 4 carriages on the track so that we have a total of 14 carriages, each with a proper length of one unit (as measured by an observer on the carriage) circulating on the 10 unit length track comfortably fitting end to end. If the speed is increased, we can fit further carriages on the track and by continuing this process we can fit as many carriages on the track as we like without any change in the proper length of the track or the carriages. Would you agree in this case that we are not just "measuring shadows"?

Movement around a circular track does not satisfy the conditions of special relativity. I'm not sure that is a fatal problem, but in any case the experiment can be imagined on a straight track, with a gap between two sets of cars.

On first look, I think this is the same as the pole-in-the-barn scenario. If so, then the answer is that you can't fit more cars in the gap as the speed increases, because (roughly speaking) the gap isn't on the track all at once (that is, at the same time).

Suppose we accept your premise. There are two observers of the train, moving at different speeds relative to the train. The one observer will gauge that he can put two extra cars in the gap; the other will see room for three extra cars. Which one is right?

Well, rest length is something all parties can agree on, but then so is length in some agreed-upon frame.
Yes, but this means that length (and thus time) cannot be measured directly. Which poses a number of practical problems as speeds approach c. Those practical problems leave open the possibility that the theory is inaccurate at speeds very near c.

Dale
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The invariant is (of course) the spacetime interval, which is a combination of distance and time.
Yes, that is why invariants like the spacetime interval are so important and useful in modern physics.

Those practical problems leave open the possibility that the theory is inaccurate at speeds very near c.
That is always a possibility that we cannot exclude. All we can say is that it is accurate at all speeds we have tested so we have no scientific reason to believe that it is inaccurate at higher speeds.

Movement around a circular track does not satisfy the conditions of special relativity. I'm not sure that is a fatal problem, but in any case the experiment can be imagined on a straight track, with a gap between two sets of cars.

On first look, I think this is the same as the pole-in-the-barn scenario. If so, then the answer is that you can't fit more cars in the gap as the speed increases, because (roughly speaking) the gap isn't on the track all at once (that is, at the same time).

Suppose we accept your premise. There are two observers of the train, moving at different speeds relative to the train. The one observer will gauge that he can put two extra cars in the gap; the other will see room for three extra cars. Which one is right?
It does satisfy the conditions of Special Relativity. SR can handle acceleration in flat space which is what we are discussing in the case of the relativistic train on the circular track. All observers will agree on the number of extra cars that can fit on the circular track.

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JesseM
Yes, but this means that length (and thus time) cannot be measured directly.
Why not? What does it mean to measure them "directly"? (why doesn't measurement or proper time along a worldline by an atomic clock moving along that worldline count, for example?) Do you think length and time can be "measured directly" in classical Newtonian physics, and if so what's the difference?

GregAshmore,

I have two questions for you:

1) Are you thinking of your two poles as static 4-dimensional objects? That is, objects that really don't move at all--for those objects it would be hard to define time--they are frozen in place as time flows quite independantly. "But, who or what is doing the moving?", you may ask. Don't worry about it. We just don't talk about that. For pedagogical purposes, you might play like there is some 3-dimensional consciousness that flies along the world line of a 4-dimensional object, "observing" a sequence of 3-dimensional images. Herman Weyl once made a comment something like, "...the observer crawls along his world line..." Actually, it should be more like, "...the observer flies along his world line at 186,000 mi/sec." So, I just wanted to ask if this is the kind of model for your poles that you envision as you pose your question.

2) Are you considering observers moving along the 4th dimension worldlines as having different 3-D cross-section views of 4-D objects?

(Refer to my post #7 on page 1)

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Suppose we accept your premise. There are two observers of the train, moving at different speeds relative to the train. The one observer will gauge that he can put two extra cars in the gap; the other will see room for three extra cars. Which one is right?
I need to stop posting just before I have to leave for work--I don't think my question was properly thought out. On the other hand, it's far enough removed from my original question that I'll let this sub-thread drop, rather than rework the question.

Why not? What does it mean to measure them "directly"? (why doesn't measurement or proper time along a worldline by an atomic clock moving along that worldline count, for example?) Do you think length and time can be "measured directly" in classical Newtonian physics, and if so what's the difference?
When I measure the length of a pin with a calipers, I accept the reading on the dial as the true length of the pin, within the accuracy of the measurement. That is what I mean by a direct measurement of length.

When measuring the length of a pin moving at high speed, the reading on the instrument cannot be accepted as the true length of the pin (unless I happen to be in the frame which has been chosen as the standard frame of reference). The reading on the instrument must be adjusted, using the Lorentz transform, to obtain the length of the rod in the standard reference frame. The measurement is thus indirect. (By the way, I vote for the pin's rest frame as the reference frame.)

I'm pretty sure that measurement of proper time along a worldline is equivalent to agreeing on a standard reference frame--the frame in which the clock is at rest.

JesseM
When I measure the length of a pin with a calipers, I accept the reading on the dial as the true length of the pin, within the accuracy of the measurement. That is what I mean by a direct measurement of length.
That doesn't really tell me what you mean in general by "direct", it just gives an example. But I suppose it'll do for now.
GregAshmore said:
When measuring the length of a pin moving at high speed, the reading on the instrument cannot be accepted as the true length of the pin (unless I happen to be in the frame which has been chosen as the standard frame of reference). The reading on the instrument must be adjusted, using the Lorentz transform, to obtain the length of the rod in the standard reference frame. The measurement is thus indirect. (By the way, I vote for the pin's rest frame as the reference frame.)
Why do you equate the rest length with the "true length of the pin"? Are you just defining the "true" length as the rest length, or do you have some prior notion of "true" which leads you to this as a conclusion?

Anyway, what about just accelerating the calipers so they match speed with the pin, without accelerating yourself? Of course this basically amounts to everyone agreeing on a standard reference frame--the frame of the pin--but it shows that the rest length can be "measured directly" rather than each observer measuring the length using rulers at rest relative to themselves and then applying a Lorentz transform to the measurement.
GregAshmore said:
I'm pretty sure that measurement of proper time along a worldline is equivalent to agreeing on a standard reference frame--the frame in which the clock is at rest.
That would be true if the clock was moving inertially, but proper time is a more general notion that can be applied to non-inertial worldlines as well.

Why do you equate the rest length with the "true length of the pin"? Are you just defining the "true" length as the rest length, or do you have some prior notion of "true" which leads you to this as a conclusion?
Intuitively, the pin's length in its rest frame is its true length. Does intuition hold up under scrutiny? I think it does in this case, because the rest frame is the only frame in which the length can be measured at one time.

In every other frame, the length must be measured by taking readings at a minimum of three different times: two readings of the leading end of the rod to establish velocity, plus a third reading of the trailing end to establish length.

Because of this, I was privately calling the pin's length in its rest frame the "proper length"--its own length--just as the time in that frame is the pin's proper time--its own time. But it seems that "proper length" has been applied to a different condition by physicists.

Dale
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2021 Award
Intuitively, the pin's length in its rest frame is its true length. Does intuition hold up under scrutiny?
If you mean "rest length" then use the term "rest length". It is clearly understood and unambiguous, whereas the term "true length" is not clearly understood. Inventing new words for existing concepts only leads to miscommunication, as demonstrated in this thread.

Btw, length is always measured at one time using two synchronized clocks in any frame. That does not distinguish the rest length from the length in other frames.

Below is a spacetime diagram which depicts the movement of a rod, four units long, relative to another rod of equal length. The red lines are the "moving" rod; the purple lines are the "stationary" rod. The gray lines are the world lines of the unit marks (0, 1, 2, 3, 4) on each rod.

As I attempt to interpret the diagram, I am moved to ask a question which has been asked before on this forum, including by me. There is nothing new in the question itself--only my level of understanding has (hopefully) progressed.

To set up the question, I stipulate that the stationary rod and the moving rod are identical, having been fabricated on the same high-precision equipment. The stationary rod was then fastened to a bench in the lab, and the moving rod was set in motion. As the moving rod passes over the resting rod, it is moving at a constant velocity of 0.6c.

As can be seen in the diagram, instruments mounted on the moving rod will measure the resting rod to be 3.2 units long. Likewise, instruments mounted on the resting rod will measure the moving rod to be 3.2 units long.

My question focuses on the stationary rod, because it has been minding its own business on the lab bench, and its dimensions are known to be the same as when it was fabricated. 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?

Maybe think of it this way:

We have (1) 'stationary' rod, call it Rod A. Now let's say this 'stationary' Rod A is in a lab on earth. Obviously it is moving at some constant velocity because the earth is moving at some constant velocity, let's say equal to 0.2c.

We have (1) 'moving' rod, call it Rod B. This moving rod is moving at a constant velocity, 0.6c.

Now let's introduce a thrid rod, call it Rod C. Let's put Rod C on in a 'stationary' lab on a different planet and say the planet (and thus Rod C) is moving at a constant velocity, 0.1c.

Rod B observes Rod A and notes Rod A is 3.2 units long. This is correct in Rod B's RF.
Rod C observes Rod A and notes Rod A is 4.2 units long. This is correct in Rod C's RF.
Rod A measures itself and notes Rod A is 4.0 units long. This is correct in Rod A's RF.

Moral of the story: There is no preferred RF.

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ghwellsjr
Gold Member
Maybe think of it this way:

We have (1) 'stationary' rod, call it Rod A. Now let's say this 'stationary' Rod A is in a lab on earth. Obviously it is moving at some constant velocity because the earth is moving at some constant velocity, let's say equal to 0.2c.

We have (1) 'moving' rod, call it Rod B. This moving rod is moving at a constant velocity, 0.6c.

Now let's introduce a thrid rod, call it Rod C. Let's put Rod C on in a 'stationary' lab on a different planet and say the planet (and thus Rod C) is moving at a constant velocity, 0.1c.

Rod B observes Rod A and notes Rod A is 3.2 units long. This is correct in Rod B's RF.
Rod C observes Rod A and notes Rod A is 4.2 units long. This is correct in Rod C's RF.
Rod A measures itself and notes Rod A is 4.0 units long. This is correct in Rod A's RF.

Moral of the story: There is no preferred RF.
You have been ambiguous in your statement of your scenario because you haven't stated what Rod C's motion is relative to. However, your conclusion that Rod C observes Rod A as being 4.2 units long can't be correct no matter how any one moves because lengths always contract so if Rod A measures itself to be 4.0 units long, everyone else that is moving with respect to A will observe it as being shorter.

JesseM
Obviously it is moving at some constant velocity because the earth is moving at some constant velocity, let's say equal to 0.2c.
You understand it is meaningless to talk about any object's "velocity" (including that of the Earth) except in relation to some other (arbitrary) choice of object or frame, right? In the Earth's rest frame it has a velocity of 0, in some other frame Earth has a velocity of 0.2c, in some other frame Earth has a velocity of 0.999999999999999c, all are equally valid perspectives.

You have been ambiguous in your statement of your scenario because you haven't stated what Rod C's motion is relative to. However, your conclusion that Rod C observes Rod A as being 4.2 units long can't be correct no matter how any one moves because lengths always contract so if Rod A measures itself to be 4.0 units long, everyone else that is moving with respect to A will observe it as being shorter.

You understand it is meaningless to talk about any object's "velocity" (including that of the Earth) except in relation to some other (arbitrary) choice of object or frame, right? In the Earth's rest frame it has a velocity of 0, in some other frame Earth has a velocity of 0.2c, in some other frame Earth has a velocity of 0.999999999999999c, all are equally valid perspectives.

You are correct. I did miss a part to my explanation: I observed Rod A, B and C from my planet and determined their constant velocities of 0.2c, 0.6c, and 0.1c respectively.

So because Rod C is moving at 0.1c and Rod A is moving at 0.2c (both in constant velocity RF), would Rod C not say 'Rod A is 4.2 units long' (arbitrary number. The point is that it is that Rod A is longer)? I believe so.

JesseM
So because Rod C is moving at 0.1c and Rod A is moving at 0.2c (both in constant velocity RF), would Rod C not say 'Rod A is 4.2 units long' (arbitrary number. The point is that it is that Rod A is longer)? I believe so.
No, you have to figure out how fast Rod A is moving in Rod C's frame--call that velocity vC--then if Rod A's rest length is 4.0, Rod A's length in Rod C's frame is given by the standard length contraction formula, $$4.0 * \sqrt{1 - v_C^2/c^2}$$, which will always give a value smaller than 4.0, never larger. For example, if both Rod A and Rod C are moving in the same direction in your frame, that means Rod A is moving at v=0.2c in your frame, and you are moving at u=-0.1c in the frame of Rod C, so according to the velocity addition formula the speed of Rod A in Rod C's frame would be (u + v)/(1 + uv/c^2) = (-0.1c + 0.2c)/(1 - 0.02) = 0.1c/0.98 = 0.102c. So, in this case the length of Rod A in Rod C's frame would be 4.0*sqrt(1 - 0.102^2) = 3.98.