Take on Length Contraction at relativistic speeds

[Ibix]

Would those two different scenarios measure the same length for object O, traveling at .9c, relative to the detectors?

Yes. Setup B is simply a silly setup that may require calculation to derive what its rest frame calls the length of the object.

 

[A.T.]

Wouldn't the distance between the detectors and the traveling object affect the measured length ...

If you place the detectors at different distances from the objects path, you have to correct for that to get the correct length. That has nothing to do with length contraction and SR, as it would happen in classical Newtonian physics too.

No idea why you are making your life so difficult with this impractical setup though.

 

[PeroK]

Wouldn't the distance between the detectors and the traveling object affect the measured length (or the length contraction for that matter)? Like in, the further away the detectors are from the traveling object, the grater the discrepancy between proper length and measured length would get?

That's why I had the clunky but effective setup where the object actually hits a switch.

The fundamental point is this. If you draw a diagram of where an object is in your reference frame, then it is length contracted. How you measure the length of a moving object is up to you. But, you must somehow identify where the front and rear are at the same time.

For example, an object with a rest/proper length of $1m$ traveling at $0.8c$ will be only $0.6m$ long in your frame. If you draw a diagram where (at time $t=0$, say) the rear of the object is at the origin, then the front of the object is at distance of $0.6m$.

There are lots of thought experiments, like a jet of paint is ejected at time $t=0$ (in your frame) at $x=0$ and $x = 1m$. Leaving aside the finite speed of the paint, the jet at $1m$ really does miss the front of the train.

The point is that a contracted length is a contracted length. The front and rear of the object really are closer together in your frame.

 

[Simi]

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 end at the same time. In this case, should I expect the distance between those two sensors to be, the proper length of the object (the length of the object in its own frame of reference)?

 

[Simi]

I'm trying another view, let's say that I am in a medium which alters greatly the speed of light and it is lowered to

c = 100 m/s

I have an object (with the length of one meter) traveling with 99 m/s through that medium. I setup an array of sensors in that medium which are meant to measure the length of the object.
Using Lorentz formula (L = L_0 * Sqrt(1 - v^2/c^2)) to measure the length of the moving object, I get the following measurement: ~0.141 m.
Is it safe to assume that the object's length in this medium is actually ~0.141 m? Or am I missing something and this thought experiment is not valid?

 

[PeroK]

I'm trying another view, let's say that I am in a medium which alters greatly the speed of light and it is lowered to

c = 100 m/s

I have an object (with the length of one meter) traveling with 99 m/s through that medium. I setup an array of sensors in that medium which are meant to measure the length of the object.
Using Lorentz formula (L = L_0 * Sqrt(1 - v^2/c^2)) to measure the length of the moving object, I get the following measurement: ~0.141 m.
Is it safe to assume that the object's length in this medium is actually ~0.141 m? Or am I missing something and this thought experiment is not valid?

$c$ is a universal constant. The speed of light in a medium (other than vacuum) is of no relevance to SR.

Also, the speed of light (or any signal) is not relevant to the measurement result you will get. You could set up an experiment where local observations are communicated to you by carrier pigeon. But, the speed of the carrier pigeon has no relevance to the information than the pigeon is carrying.

SR is a theory about the nature of time and space. The theory predicts certain things. E.g. length contraction. That needs to be confirmed by measurement.

You need, somehow, to get rid of the idea that the travel time of light signals has something to do with SR. It does not. Try to imagine experiments in a darkened room, where all the events are recorded by collisions, switches and time stamps. That information is sent to you electronically, where you can look at it. That way, you take all this irrelevance about the time travel of light signals out of the equation.

 

[Simi]

I see!
In that case, do we know the reason why length contraction (actual physical modification of length) occurs only in the direction of motion? Should not the whole object attempt to undergo the same transformation?
Having the contraction in the direction of motion might be interpreted as an optical / electromagnetic effect along the motion axis due to the fact that information is read by the detectors electromagnetically, at the speed of light.
That was actually the reason why I envisioned the thought experiment in a different medium.

 

[A.T.]

Should not the whole object attempt to undergo the same transformation?

Consider an object flying fast trough a narrow hole, from different frames. All frames must agree whether it fits through or not.

Having the contraction in the direction of motion might be interpreted as an optical / electromagnetic effect along the motion axis due to the fact that information is read by the detectors electromagnetically, at the speed of light.

No, length contraction is what you get after you already accounted for signal delays.

 

[jbriggs444]

actual physical modification of length

It is not a modification of length at all. It is a modification in the coordinate system you are using to assess the length. Just as there is no physical modification of a 1 inch by 12 inch ruler when you measure it crosswise (and measure 1 inch in perpendicular width) and when you then re-measure it (and measure 1.5 inch in diagonal width). The ruler really physically is both 1 inch wide and 1.5 inches wide.

 

[m4r35n357]

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!

 

[A.T.]

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.

 

[Simi]

@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)

 

[A.T.]

@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?

 

[m4r35n357]

- 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]

@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".

 

[Simi]

@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 ?

 

[A.T.]

@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.

@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)?

 

[m4r35n357]

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.

 

[PeroK]

[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.

 

[A.T.]

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.

 

[PeroK]

@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.

 

[Ibix]

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".

 

[DaveC426913]

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!

 

[Simi]

Thanks @PeroK . Going once more thorough the comments, I saw @Nugatory explanation:

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.

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?

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.

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).

 

[Ibix]

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.

 

[A.T.]

...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.

 

[PeroK]

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.

Then someone added:

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!

 

[A.T.]

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:
https://en.wikipedia.org/wiki/Bell's_spaceship_paradox

 

[Nugatory]

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.

 

[Nugatory]

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:
https://en.wikipedia.org/wiki/Bell's_spaceship_paradox

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).