# Take on Length Contraction at relativistic speeds

• B
• Simi
In summary: Earth as being 0.9c shorter than it would on their home planet, even though the rod on Earth is the same length as the one on their home planet.
m4r35n357 said:
Just to add for the benefit of the OP, the "contracted length" consists of a "smeared over time" perspective of the moving object, not a snapshot!
What do you mean by "smeared over time"? To determine the contracted length in some frame F, you use the positions of the object's ends at the same time according to F.

@A.T. , I'm thinking that height contraction would still allow the object to fit through the narrow hole.

@jbriggs444 , so you say it's not a physical modification. Here I was a bit confused since reading some of the posts I thought it implied an actual physical change. An idea which I'm not very comfortable with, either. If I understand correctly you say that this implies a mathematical transformation, a change of basis from one coordinate system to another. So, you are talking about a mathematical model to described the phenomena.

Somehow I get the sense that you and @PeroK don't agree on this either or is just my fracture here?

@m4r35n357 in that case, considering the following experiment I stated above:

If I take @Nugatory detectors array setup (garage door example) further, to perform the measurements:
So, the object (traveling at 0.9c relative to the sensors array) has detectors on both ends. In that case, we measure the distance between two sensors which detect both object's ends at the same time. In this case, should I expect the distance between those two sensors to be, the proper length of the object ?
- because, this would be more of a snapshot than a "smear over time"? (length contraction detectors field 1.3.png)

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Simi said:
@A.T. , I'm thinking that height contraction would still allow the object to fit through the narrow hole.
What about the frame of the object, where the hole is contracted, not the object?

Simi said:
- because, this would be more of a snapshot than a "smear over time"? (length contraction detectors field 1.3.png)
If it is not a "smear over time", then there is no "length contraction".

[EDIT] and that is as good a definition of "loss of simultaneity" as I can give!

@jbriggs444 , so you say it's not a physical modification. Here I was a bit confused since reading some of the posts I thought it implied an actual physical change. An idea which I'm not very comfortable with, either. If I understand correctly you say that this implies a mathematical transformation, a change of basis from one coordinate system to another. So, you are talking about a mathematical model to described the phenomena.

Somehow I get the sense that you and @PeroK don't agree on this either or is just my fracture here?
I do not think that @PeroK and myself are in disagreement about this. The problem is with the word "modification". Nothing physical is being modified. We are using a different coordinate system and are measuring different aspects of the same thing. Similarly, there is no physical modification to a ruler whose width is measured on a diagonal. The width that is measured is really, truly physical. But it is not "physically modified".

PeroK
@jbriggs444 got it, nothing physical is being modified.

@m4r35n357 I would like to know how you settle the "smeared over time" matter with @A.T. since to me, it seems that you have different understandings of the matter.

@PeroK , @A.T. or anyone else for that matter, what is your take on this variation of the experiment, what would be the measured length?

If I take @Nugatory detectors array setup (garage door example) further, to perform the measurements:
So, the object (traveling at 0.9c relative to the sensors array) has detectors on both ends. In that case, we measure the distance between two sensors which detect both object's ends at the same time. In this case, should I expect the distance between those two sensors to be, the proper length of the object ?

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Simi said:
@m4r35n357 I would like to know how you settle the "smeared over time" matter with @A.T. since to me, it seems that you have different understandings of the matter.
No idea if it's different because "smeared over time" is completely vague, and might mean anything.

Simi said:
@PeroK , @A.T. or anyone else for that matter, what is your take on this variation of the experiment, what would be the measured length?
Do you understand why you cannot have length contraction perpendicular to velocity (see post #73)?

A.T. said:
No idea if it's different because "smeared over time" is completely vague, and might mean anything.

It is my attempt to translate "not simultaneous" into a B-level reply.

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Simi said:

[USER=493650]@PeroK
, @A.T. or anyone else for that matter, what is your take on this variation of the experiment, what would be the measured length?[/USER]

Let's go back to square 1. Let's assume we do not know whether SR is right or wrong. We may be living in a Newtonian universe, in which lengths are absolute. Or, we may be living in a relativistic universe. We don't know. So, we do an experiment. But, first, we need a definition:

Length: the distance between the measured position of the front of an object and the rear when these position measurements are simultaneous.

So, in order to measure the length of a moving object we need a valid measurement process. We must ensure that we take position measurements at the same time.

Let's assume that we have such a measurement process. We can discuss the details later. That measurement process is valid regardless of which universe we are in.

We measure the length of a moving object:

a) If this measurement returns the rest length, then we shown SR is wrong.
b) If the measurement returns the contracted length, then we have confirmed SR (to some extent at least).

Also, either a) all valid measurements of length (however they are done) must return the rest length; or b) all valid measurements of length must return the contracted length.

You don't get one or the other depending on your setup.

In short, you have the following:

SR predicts a contracted length measurement for moving objects
If you carry out such a measurement you get the contracted length.
The measured length is not an optical illusion of the specific measurement process. It's a valid measurement of the length of an object in that frame of reference.

m4r35n357 said:
It is my attempt to translate "not simultaneous" into a B-level reply.
Ok, it sounded like some sort of motion blur effect. I think we agree that the contracted length in the detector frame is based on the end positions, taken at the same time according to detectors frame simultaneity.

@Simi in addition, you cannot deduce length contraction by considering the way you measure the length of a moving object. The measurement will tell you whether the object is contracted or not, but you cannot deduce that length contraction exists simply by this single measurement.

You can't "prove" SR simply by considering a single measurement of length. You have to actually do the measurement and see what you get.

But, you can deduce length contraction by considering light signals in two frames of reference and using the invariant speed of light to deduce that length measurements in the two frames are different. But, that assumes that the speed of light is invariant. You could then carry out real measurements in both frames of reference to test whether your assumption about light speed was correct.

If you measure the cross-section of a cylinder, it is a circle with some diameter d. If you tip the cylinder over by an angle ##\theta##, the horizontal cross-section becomes an ellipse with minor diameter d and major diameter ##d/\cos\theta##.

This is closely analogous to length contraction. What we call a 3d object is a 3d section through a 4d worldtube. In a frame where the object is at rest we've defined its worldtube as perpendicular to what we're calling "space". In a frame where the object is moving, we've defined its worldtube as non-perpendicular to what we're now calling space. So the purely spatial cross-section of the object has changed. Just like the cross-section of the cylinder, nothing has happened to the worldtube. We've just picked a different cross-section to call "the 3d object" (but hyperbolic geometry means that you get length contraction, not expansion as in the Euclidean analogy of the cylinder).

So nothing happens to the object. Length contraction is a misnomer in this view - rather, we're changing what we call "the length of the object".

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nitsuj, Simi, jbriggs444 and 2 others
Ibix said:
If you measure the cross-section of a cylinder, it is a circle with some diameter d. If you tip the cylinder over by an angle ##\theta##, the horizontal cross-section becomes an ellipse with minor diameter d and major diameter ##d/\cos\theta##.

This is closely analogous to length contraction. What we call a 3d object is a 3d section through a 4d worldtube. In a frame where the object is at rest we've defined its worldtube as perpendicular to what we're calling "space". In a frame where the object is moving, we've defined its worldtube as non-perpendicular to what we're now calling space. So the purely spatial cross-section of the object has changed. Just like the cross-section of the cylinder, nothing has happened to the worldtube. We've just picked a different cross-section to call "the 3d object" (but hyperbolic geometry means that you get length contraction, not expansion as in the Euclidean analogy of the cylinder).

So nothing happens to the object. Length contraction is a misnomer in this view - rather, we're changing what we call "the length of the object".
This is an astonishingly eye-opening explanation of length contraction!

Thanks @PeroK . Going once more thorough the comments, I saw @Nugatory explanation:
Nugatory said:
A reference frame is modeled as as an infinite lattice of detectors with synchronized clocks, each recording only what happens at the point where they are. Measurements are carried out by collecting these recordings at our leisure and then analyzing them after the fact; there are no light travel delays to correct for.
Nugatory said:
we need multiple detectors, one at the event "nose was here at at time T" and the other at the event "tail was here at that time T". Now we don't have a picture but we do have two points where the two ends of the object were at the same; the distance between these points is the ocntracted length.
But, doesn't this contradict with the bellow statement?
jbriggs444 said:
Nothing physical is being modified.
I'm having troubles seeing those statements as non-mutually exclusive.

Well, but I guess @Ibix's last comment cleared things out for me.
Ibix said:
nothing has happened to the worldtube. We've just picked a different cross-section to call "the 3d object".
So nothing happens to the object. Length contraction is a misnomer in this view - rather, we're changing what we call "the length of the object".
This is what I was not understanding actually, the physical change of the object length. No I get it, it's not an actual physical length change but a measurement of length in a different system of coordinates. A transformation applied to the mathematical model which reads different values from a different points of view.

That's great, so the physical properties of the object remain unchanged no matter what the speed. This is perfectly acceptable.

Still, reverting back to what @Nugatory said earlier, shouldn't a real world array of detectors like the ones that he described, register the proper length for the moving object?
At least, I see no reason as to why the detectors would measure anything else but the proper length in any situation (since the physical properties of the object remain unchanged no matter what the speed).

Simi said:
shouldn't a real world array of detectors like the ones that he described, register the proper length for the moving object?
No. Since the geometric picture seems to work for you...

The worldlines of the detectors are a set of straight parallel lines. They are triggered when the object touches them (or passes through the gap between light emitter and light sensor, or whatever). The length measured is (approximately) the number of detectors simultaneously triggered. That is just the number of worldlines that overlap the worldtube at a given time, which is proportional to the length of the purely-spatial slice through the worldtube - i.e. the length-contracted length.

Simi said:
...it's not an actual physical length change but a measurement of length in a different system of coordinates

...I see no reason as to why the detectors would measure anything else but the proper length ...
The detectors don't measure length directly. You need synchronized clocks at each detector to compute the length. The contracted length comes from using clocks synchronized in rest frame of the detectors.

1977ub and Ibix
Simi said:
Still, reverting back to what @Nugatory said earlier, shouldn't a real world array of detectors like the ones that he described, register the proper length for the moving object?
At least, I see no reason as to why the detectors would measure anything else but the proper length in any situation (since the physical properties of the object remain unchanged no matter what the speed).

There is a quotation from Alexander Pope:

Nature and Nature's laws lay hid in night:
God said, "Let Newton be!" and all was light

Epitaph intended for Sir Isaac Newton.

It did not last: the devil, shouting "Ho.
Let Einstein be," restored the status quo.

The problem is that if you lose length contraction, you lose time dilation and all of SR, then you lose GR and all modern cosmology is gone too. You are back in the 19th Century, with all the problems that Newtonian physics could not solve. Not least, the invariant speed of light to explain.

So, actually, it's the other way round. Without SR we'd still be in the dark!

Simi said:
physical properties of the object remain unchanged no matter what the speed.
Depends what you mean by "physical properties". If you accelerate a chain, while keeping its length constant, it will break, because its contracted links cannot span the original length any more.

See:

Simi said:
Thanks @PeroKStill, reverting back to what @Nugatory said earlier, shouldn't a real world array of detectors like the ones that he described, register the proper length for the moving object?
At least, I see no reason as to why the detectors would measure anything else but the proper length in any situation (since the physical properties of the object remain unchanged no matter what the speed).
You are forgetting to take the relativity of simultaneity into account.

Detector A triggers when the front of the moving object reaches it.
Detector B triggers when the rear of the moving object clears it.
A and B are positioned so that, using the frame in which they are at rest and the object is moving, they both trigger at the same time. Because they both trigger at the same time we conclude that the length of the ship using this frame is equal to the distance between the detectors.

However, these two events do not happen at the same time when we use the frame in which the object is at rest and the detectors are moving. Using this frame the timestamps the detectors are recording do not agree with a clock that is at rest relative to the object and moving relative to the detectors, and when we use this clock we find that detector B triggers some time after detector A. During the time between these two events both detectors' beams are blocked by the ship, and we conclude that when we're using this frame the length of the ship is greater than the distance between the detectors. (Note also that when we use this frame the distance between the detectors is less than when we use the frame in which the detectors are at rest).

So no "physical change" but we still find different lengths.

A.T. said:
Depends what you mean by "physical properties". If you accelerate a chain, while keeping its length constant, it will break, because its contracted links cannot span the original length any more.

See:
Relativity of simultaneity is rather cleverly hidden in this problem; find where it's hidden and the resolution of the paradox will become clear.

(@A.T. already knows this of course - this comment is for people following this thread and encountering Bell's spaceship for the first time).

DJ_Juggernaut said:
My position is that to measure the length of an object you need your detector to be on the object you intend to measure. I am still thinking about his technique. I will respond to it later.
Simi said:
Thanks @PeroK . Going once more thorough the comments, I saw @Nugatory explanation:But, doesn't this contradict with the bellow statement?

I'm having troubles seeing those statements as non-mutually exclusive.

Well, but I guess @Ibix's last comment cleared things out for me.

This is what I was not understanding actually, the physical change of the object length. No I get it, it's not an actual physical length change but a measurement of length in a different system of coordinates. A transformation applied to the mathematical model which reads different values from a different points of view.

That's great, so the physical properties of the object remain unchanged no matter what the speed. This is perfectly acceptable.

Still, reverting back to what @Nugatory said earlier, shouldn't a real world array of detectors like the ones that he described, register the proper length for the moving object?
At least, I see no reason as to why the detectors would measure anything else but the proper length in any situation (since the physical properties of the object remain unchanged no matter what the speed).
Consider the following diagrams. The gap between the black objects is the distance between our "detectors" and the red line the object whose length you are measuring.
The top image is when the object is at rest with respect to the detectors. The detectors measure its length by noting how it lines up with the gap (the green dotted lines). It is obviously longer than the gap
The second image is for when the object has a significant speed relative to the detectors. It has It's proper length has not changed, But it is rotated such that its ends now fit between the green lines. By the detector's measurement of "length" is is shorter.
The trick here to realize that "up: and "down" in this diagram represents time and not spatial distance. A separation along the vertical direction is a measurement of a time interval. So if you think of length contraction as a rotation, it is a rotation in space-time and not just space. The only "length" the detectors can measure is the in the horizontal direction, to measure in the vertical direction, you would need to use clocks.

Think of it like trying to fit an object through a doorway. In the top image both ends try to go through at the same time and it won't fit. In the second image, it is rotated so that one end goes through the door first, and then the other, and it does fit through the doorway.

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Very nice explanations.

So, considering all that you have said, if there's no physical change of the object's properties/length in the real world, this means that Length Contraction represents object properties translated from the 3D model into the 4 dimensional space-time coordinates system, where you indeed will have different measurements.
In this case, having an array of detectors in the Atacama desert at a distance of 1 mm away from one-another, and an object passes over the detectors with 0.9c (the object has a switch at the end and one at front) activating with both of the switches, simultaneously (in the array frame of reference) two detectors, the distance between those two detectors will be the proper length of the object. Though, if translated in the 4D space-time model, this will measure a Length Contraction.
So you never lose Length Contraction, you don't lose SR, you just have to translate the experiment in the mathematical form of a space-time model to have different readings, which makes sense in my mind. I mean, it was never meant to convince that length changes at relativistic speeds rather it is a different reading in a 4D space.

Simi said:
In this case, having an array of detectors in the Atacama desert at a distance of 1 mm away from one-another, and an object passes over the detectors with 0.9c (the object has a switch at the end and one at front) activating with both of the switches, simultaneously (in the array frame of reference) two detectors, the distance between those two detectors will be the proper length of the object.
Let me get this straight.

You have an array of detectors. You have an object with a front end and a back end. The object is in motion. We are taking the array of detectors to be at rest.

You have a switch at each end of the object. I take it that these switches are attached to high powered lasers. When switched on, the lasers trigger the adjacent detector.

You trigger both switches, simultaneously in the rest frame of the detectors. The result is to trigger two detectors simultaneously, again in the rest frame of the detectors. The distance between the two triggered detectors is the length measurement.

This procedure measures contracted length, not proper length.

Simi said:
In this case, having an array of detectors in the Atacama desert at a distance of 1 mm away from one-another, and an object passes over the detectors with 0.9c (the object has a switch at the end and one at front) activating with both of the switches, simultaneously (in the array frame of reference) two detectors, the distance between those two detectors will be the proper length of the object.
You failed to specify in which frame you measure the distance between those two detectors. If you meant "distance between those two detectors in the rest frame of the detectors", then you are wrong: it will be the contracted distance of the object.

This has been explained many times in this thread.

I know that length contraction at relativistic speeds has been proven in the lab, experimentally, using subatomic particles and I guess nobody can argue the results. I for one, don't.
But, did we had any attempts of proving length contraction, experimentally, using macro objects (not quantum objects)? Did we actually observe this or it's an extrapolation from the quantum world and a deduction?

Simi said:
I know that length contraction at relativistic speeds has been proven in the lab, experimentally, using subatomic particles and I guess nobody can argue the results. I for one, don't.
But, did we had any attempts of proving length contraction, experimentally, using macro objects (not quantum objects)? Did we actually observe this or it's an extrapolation from the quantum world and a deduction?

The people who we consider to be moving along with the object find there to be no contraction. We find their clocks to not be synchronized with each other.

Simi said:
But, did we had any attempts of proving length contraction, experimentally, using macro objects (not quantum objects)?
Consider the cosmic ray muons. From our perspective they survive to reach sea level because their time is dilated. From their perspective there must be a reason that they can get through the atmosphere - its rest depth is too deep for them to cross given their short lifetimes. So what's happened?

Basically you have two options. The first is that the atmosphere is length contracted in this frame. The second is that the principle of relativity is wrong, and the muons are time dilated in an absolute sense, and see Earthbound clocks ticking fast.

But we test the principle of relativity every day, and there is no evidence of it being wrong. If you don't want to believe length contraction you need to come up with a coherent explanation for gross violation of relativity at high speeds while approximating it absurdly well at low speed.

1977ub
Simi said:
But, did we had any attempts of proving length contraction, experimentally, using macro objects (not quantum objects)? Did we actually observe this or it's an extrapolation from the quantum world and a deduction?
It's not practical to measure length contraction with macroscopic objects. To see why, try calculating the kinetic energy of a 1 kilogram object moving fast enough to contract by 1% and compare with total annual worldwide energy production.
(Time dilation in macroscopic clocks is much easier to observe because we can work with slower speeds over a longer time and because our best clocks are many orders of magnitude more accurate than our best instruments for measuring the length of moving objects.)

However, we can and do measure length contraction of distances: measurements of the distance traveled by subatomic particles before they decay show that, in the frame in which the particle is at rest and the detector is moving towards the particle, the distance between the particle source and the detector is length contracted compared with the distance in the frame in which the particle is moving and the lab, particle source, and particle detector are all at rest.

In any case time dilation, relativity of simultaneity, and length contraction all follow from the same proven-correct equations (the Lorentz transforms) so any consistent theory has to include all three. If any two of them are experimentally confirmed then the third must be present even if we lack the tools to measure it directly.

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Ibix and PeroK
Simi said:
Very nice explanations.

So, considering all that you have said, if there's no physical change of the object's properties/length in the real world, this means that Length Contraction represents object properties translated from the 3D model into the 4 dimensional space-time coordinates system, where you indeed will have different measurements.
It's more the other way around. The 4D model is actually a better representation of the "real world" object, and it is the projection of that object onto our 3D perception of the universe that results in our measuring length contraction. Our three 3D world is just a "shadow play" of the 4D reality. We can only directly measure the shadows cast, and must infer the 4D model from that.
In this case, having an array of detectors in the Atacama desert at a distance of 1 mm away from one-another, and an object passes over the detectors with 0.9c (the object has a switch at the end and one at front) activating with both of the switches, simultaneously (in the array frame of reference) two detectors, the distance between those two detectors will be the proper length of the object. Though, if translated in the 4D space-time model, this will measure a Length Contraction.
So you never lose Length Contraction, you don't lose SR, you just have to translate the experiment in the mathematical form of a space-time model to have different readings, which makes sense in my mind. I mean, it was never meant to convince that length changes at relativistic speeds rather it is a different reading in a 4D space.
With the red arrows being the detectors and the green line the object with the switches at the ends, then if the switches of the object are detected simultaneously in rest frame of the desert, then the top image shows the length measurement you get for the object in the desert frame. (object shown moving left to right) This will be the length contracted length of the object.
However, in the rest frame of the object, it is the detectors that are moving (right to left) and are length contracted while the object is its proper length. Thus you get the two bottom images with the object triggering one detector and then the other.

So, you will directly measure the contracted length, and if you know the relative speed you can use this to work out what the proper length is.

I

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1977ub, Ibix and PeroK
Simi said:
I know that length contraction at relativistic speeds has been proven in the lab, experimentally, using subatomic particle...But, did we had any attempts of proving length contraction, experimentally, using macro objects
Yes, as noted above, the observations imply a contracted macroscopic environment in the frame of the particles.

Also, the macroscopic objects are made of subatomic particles. Accepting the bricks being contracted, but not the whole house makes no sense.

nitsuj and Dale
Janus said:
Our three 3D world is just a "shadow play" of the 4D reality. We can only directly measure the shadows cast, and must infer the 4D model from that.

I don't understand your 'shadow'. 4D world contains far more events than one 3D world. Hence a 3D world cannot be a projection nor shadow of such a vast 4D world. A 3D world is a section through 4D world.

Ebeb said:
I don't understand your 'shadow'. 4D world contains far more events than one 3D world.

Ebeb said:
Hence a 3D world cannot be a projection nor shadow of such a vast 4D world.
You are taking his words too literally.

Simi said:
… Length Contraction at the relativistic speed of object O? Is it a real length contraction which can be perceived and measured from two distinct frame of references...
Not sure about the "two distinct frames of reference" part, but length contraction is real -- even at non-relativistic speeds -- just as time dilation is real. Things will happen more slowly (from your standpoint) for objects in motion with respect to you.

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