Understanding Length Contraction in Relativity: Explained and Analyzed

[durant]

As I've just recently started studying relativity, I need some help with these basic concepts.

So this thread is about length contraction. Can somebody explain the effects of it, since I've already found out some things, like the existence of the invariant proper lenght, the fact that contraction is in the direction of the motion of the object. Is the length of the object the only property that is affected by length contraction? For instance, does its proper mass stay invariant, its composition and so on? Thanks in advance.

 

[Simon Bridge]

Not heard of an invariant length before.
The space-time interval is invariant ... that what you mean?

The effect of the length contraction is exactly what you'd expect ... moving objects get shorter.
Strictly, an observer stationary with respect to an object, and an observer moving with respect to the object will disagree about it's length, with the moving observer recording the shorter measurement. The will also each claim that the other's "meter ruler" is shorter than their own. It can get confusing.

Length contraction is part of the general Lorentz transformation - which has far-reaching effects in space and time. It is probably not a good idea to get ahead of yourself though.

Proper measurements are those taken with an instrument which is at rest with respect to the thing being measured.

Proper mass, therefore, is the rest mass ... which is always the mass measured at rest, so it is invariant pretty much by definition. The picture of a mass increase has fallen out of favor but you will see it in some older texts and discussions. However, all the Newtonian physics you learned should work as long as all the distances and times are measured in the same reference frame. All the Lorentz transformation does is relate measurements of the same thing made in different reference frames.

The main thing to watch as you learn is to make sure you are clear about who measures what.

The FTL and Relativity FAQ has an accessible crash-course on the Lorentz transformations as well as discussions on some of the subtler and potentially more confusing consequences, if you want a look ahead.

 

[durant]

Not heard of an invariant length before.
The space-time interval is invariant ... that what you mean?

The effect of the length contraction is exactly what you'd expect ... moving objects get shorter.
Strictly, an observer stationary with respect to an object, and an observer moving with respect to the object will disagree about it's length, with the moving observer recording the shorter measurement. The will also each claim that the other's "meter ruler" is shorter than their own. It can get confusing.

Length contraction is part of the general Lorentz transformation - which has far-reaching effects in space and time. It is probably not a good idea to get ahead of yourself though.

Proper measurements are those taken with an instrument which is at rest with respect to the thing being measured.

Proper mass, therefore, is the rest mass ... which is always the mass measured at rest, so it is invariant pretty much by definition. The picture of a mass increase has fallen out of favor but you will see it in some older texts and discussions. However, all the Newtonian physics you learned should work as long as all the distances and times are measured in the same reference frame. All the Lorentz transformation does is relate measurements of the same thing made in different reference frames.

The main thing to watch as you learn is to make sure you are clear about who measures what.

The FTL and Relativity FAQ has an accessible crash-course on the Lorentz transformations as well as discussions on some of the subtler and potentially more confusing consequences, if you want a look ahead.

Yes, I referred to the proper length of the object.

Same goes, for proper time, right? It defines sequence of evens which all observers will agree upon, if I got it right?

So in the rest frame of the object the events that happen to the object are fixed, all observers will agree after comparing their observations what the proper events that are in the objects rest frame are, just like they will with the objects proper length?

 

[Simon Bridge]

Yes, I referred to the proper length of the object.

Cool.

Same goes, for proper time, right? It defines sequence of evens which all observers will agree upon, if I got it right?

It is the time as measured on a clock at rest with respect to the observer.

You are thinking that all observers know relativity and so will be able to calculate the same proper time and length? However, the same is true for measurements in any reference frame so this would be a trivial example of all observers agreeing.

There are non-trivial invariants - the most famous one is the speed of light in a vacuum.
The kind of invariant that matters is "Lorentz Invariant". (look it up)

So in the rest frame of the object the events that happen to the object are fixed, all observers will agree after comparing their observations what the proper events that are in the objects rest frame are, just like they will with the objects proper length?

It's a definition of "proper" - but no special importance should be attached to the proper lengths and times. There is no preferred reference frame.

Have you had a look through that link I gave you?
It covers quite a lot of this.

 

[durant]

It's a definition of "proper" - but no special importance should be attached to the proper lengths and times. There is no preferred reference frame.

Have you had a look through that link I gave you?
It covers quite a lot of this.

I know that they are not priviliged, but they are nonetheless necessary for calculations and all observers (with knowledge of physics) will agree on them.

 

[Simon Bridge]

All observers with a knowledge of physics will calculate the same results for any reference frame - you are not saying anything special.

If you still want to stay with this idea, then you need to be aware that when a text says that a number of observers agree on something, they usually mean that the observers measure the same amount: their own clocks and rulers agree on the measurement. They don't mean the observers calculate the same thing.

The only concession is that the observers compensate for the time light takes, on their clocks, to reach them from an event.

Enjoy.

 

[Mentz114]

All observers with a knowledge of physics will calculate the same results for any reference frame - you are not saying anything special.

If you still want to stay with this idea, then you need to be aware that when a text says that a number of observers agree on something, they usually mean that the observers measure the same amount: their own clocks and rulers agree on the measurement. They don't mean the observers calculate the same thing.

The only concession is that the observers compensate for the time light takes, on their clocks, to reach them from an event.

Enjoy.

In the case of proper-time, all observers will get the same value for any worldline between the same two events. I think Durant was trying to make this point. One ought to distinguish between invariant and coordinate dependent quantities.

 

[VantagePoint72]

In the case of proper-time, all observers will get the same value for any worldline between the same two events. I think Durant was trying to make this point. One ought to distinguish between invariant and coordinate dependent quantities.

And Simon's point is that this is a trivial statement because only the observer at rest with respect to the clock in question will measure it on her own clock. Proper time is not an "invariant" in the usual relativistic sense, it's just the time between two events as measured by a clock at rest with respect to them. Of course everyone will agree on its value, because it's been defined with respect to a particular reference frame. Everyone else can calculate it using their own coordinate time and the Lorentz transformations, but they're not measuring it.

Contrast this with something like the speed of light, which is an invariant—meaning that its value is measured to be the same in all frames.

 

[Mentz114]

And Simon's point is that this is a trivial statement because only the observer at rest with respect to the clock in question will measure it on her own clock. Proper time is not an "invariant" in the usual relativistic sense, it's just the time between two events as measured by a clock at rest with respect to them. Of course everyone will agree on its value, because it's been defined with respect to a particular reference frame. Everyone else can calculate it using their own coordinate time and the Lorentz transformations, but they're not measuring it.

Contrast this with something like the speed of light, which is an invariant—meaning that its value is measured to be the same in all frames.

Proper time is a geometric invariant of the space-time. It is coordinate independent. I don't think this is trivial.

 

[Sugdub]

The effect of the length contraction is exactly what you'd expect ... moving objects get shorter.
Strictly, an observer stationary with respect to an object, and an observer moving with respect to the object will disagree about it's length, with the moving observer recording the shorter measurement. The will also each claim that the other's "meter ruler" is shorter than their own. It can get confusing.

I'm not a physicist and I must admit I get more and more confused. I can understand the second sentence in the quote above. However the wording "moving objects get shorter" does not seem to fit with it. According to the second sentence, the length of the object is not effected. It "looks" shorter, it "appears" shorter to an observer in relative motion. Can you confirm (or disagree) that the "apparent length" is shorter, i.e. that whilst SR deals with real outcomes of actual measurements, these reflect the appearance of physical objects and not changes to their proper dimensions?
The same comment holds for the third sentence: the ruler would "appear" shorter for an observer in relative motion to it as compared to the measured length of a similar ruler at rest in respect to him/her. Can you confirm this as well?
In the third sentence we only deal with one observer and two rulers, whereas the second sentence deals with two observers and one target object, but these are equivalent paradigms isn't it?
Thanks in advance to anyone proposing an unambiguous wording.

 

[VantagePoint72]

I'm not a physicist and I must admit I get more and more confused. I can understand the second sentence in the quote above. However the wording "moving objects get shorter" does not seem to fit with it. According to the second sentence, the length of the object is not effected. It "looks" shorter, it "appears" shorter to an observer in relative motion. Can you confirm (or disagree) that the "apparent length" is shorter, i.e. that whilst SR deals with real outcomes of actual measurements, these reflect the appearance of physical objects and not changes to their proper dimensions?

No, none of this has anything to do with how the ruler looks. You can compute that too, but it's a separate question. In the coordinates of the frame in which the ruler is moving, it is shorter. It's not some kind of optical illusion. As far as an observer in that frame is concerned, it is a real effect with real consequences. "Proper" length just means the length of the ruler in its rest frame. You are using the word as if it means "real, as opposed to illusory" and that is not right.

This is the hardest thing for people new to relativity to wrap their head around, but there is no such thing as "the length" of the ruler. It has one length in its rest frame and another length in frames in motion relative to the ruler. Each length is the correct length of the ruler in the appropriate frame of reference.

Take an example: you're standing next to a barn with doors on both ends while a man carrying a ladder runs (very fast) towards the barn. Beforehand, when the ladder was at rest on the driveway, you observed that it was longer than the barn. Now as the man runs towards you, the ladder is length contracted. If he's running fast enough (an appreciable fraction of the speed of light), the ladder is contracted enough that it would actually fit inside the barn as he runs through it. You can confirm this: as the man and ladder pass through the barn, you simultaneously slam both garage doors shut and then open them. So, for at least an instant, the ladder was completely inside the barn. Clearly this would not be possible if the length contraction of the ladder was just an illusion.

However, bear in mind that the description of events depends on your frame of reference. In the frame of the man and ladder, the ladder's length stays the same but the barn contracts. At first glance, this seems problematic: now it's even more certain the ladder won't fit in the barn. However, special relativity also tells us that simultaneity is relative. Remember I said that you slam both doors shut simultaneously? Well in the ladders frame, it's not simultaneous: in its frame, you slam the rear door before the front end of the ladder has passed the threshold and you slam the front door after the rear end of the ladder has already passed the rear threshold. So, all observers agree that the ladder makes it through the barn unscathed. So, the description of a series of events in terms of length contraction, etc., is coordinate dependent—but is not illusory.

 

[Sugdub]

...It's not some kind of optical illusion. As far as an observer in that frame is concerned, it is a real effect with real consequences. ... This is the hardest thing for people new to relativity to wrap their head around, but there is no such thing as "the length" of the ruler. It has one length in its rest frame and another length in frames in motion relative to the ruler. Each length is the correct length of the ruler in the appropriate frame of reference.

I must admit I was far from this "understanding", however I think your wording does not remove the initial ambiguity I quoted. I do understand that the outcome of the length measurement process is different for both observers, i.e. there is indeed a genuine effect of their difference of relative velocity in respect to the target object. But still this does not imply that the length of the object is different.
In the classical, low speed, Doppler effect the frequency of the sound you receive/perceive is actually different depending on your relative velocity in respect to the source, but the frequency of the source itself is not effected by your own relative motion. It remains the same for all observers. One must distinguish between the frequency of the source and the frequency of the received sound. Both are real... So I can understand that my initial wording "appearance" is not appropriate ...
Can we then state that the "measured length" of the object is different for both observers?
Or do you actually mean that the object has a different length in different frames of reference irrespective of an observation process taking place in that frame?

 

[VantagePoint72]

Or do you actually mean that the object has a different length in different frames of reference irrespective of an observation process taking place in that frame?

Yes.

 

[Ookke]

Can we then state that the "measured length" of the object is different for both observers?
Or do you actually mean that the object has a different length in different frames of reference irrespective of an observation process taking place in that frame?

A common misconception, or at least something that I have tried hard to unlearn, is to imagine some global frame where we can actually see (or measure real-time) everything that happens. So we could see a length-contracted ladder enter into barn, be inside the barn both doors closed, and so on.

Actually, each observer can see/measure only the things in immediate vicinity. So an observer at one end of the barn has no way to real-time check whether the other door is closed or not. The opinion whether the other door is closed "right now" is relative and depends on clock sync.

In a sense, length contraction can be viewed as a consequence of relative simultaneity. Measuring length is not as straightforward as it might seem and always involves simultaneity, which is relative. When we have one end of the ladder at hand, it depends on the clock sync where the other end of the ladder is "right now", and there is no absolute answer for that.

 

[PAllen]

A common misconception, or at least something that I have tried hard to unlearn, is to imagine some global frame where we can actually see (or measure real-time) everything that happens. So we could see a length-contracted ladder enter into barn, be inside the barn both doors closed, and so on.

Actually, each observer can see/measure only the things in immediate vicinity. So an observer at one end of the barn has no way to real-time check whether the other door is closed or not. The opinion whether the other door is closed "right now" is relative and depends on clock sync.

In a sense, length contraction can be viewed as a consequence of relative simultaneity. Measuring length is not as straightforward as it might seem and always involves simultaneity, which is relative. When we have one end of the ladder at hand, it depends on the clock sync where the other end of the ladder is "right now", and there is no absolute answer for that.

I wish I could find it, but there was a thread here a while back where the question was literally whether anyone could visually see the latter in the barn with both doors closed (e.g. with perfect eyes or camera). The answer is the with a perfect fisheye lens set close enough to the center of the barn (but not hit by the ladder), the answer is yes. You could get a photograph including all parts of ladder and both doors closed.

 

[Nugatory]

A common misconception, or at least something that I have tried hard to unlearn, is to imagine some global frame where we can actually see (or measure real-time) everything that happens. So we could see a length-contracted ladder enter into barn, be inside the barn both doors closed, and so on.

The closest you can come to that non-existent global frame is to imagine that the universe is full of observers, all at rest relative to each other and all carrying synchronized clocks. Each observer writes down on his piece of paper exactly what happens under his nose, and exactly when. Then we can, at our leisure, collect all the pieces of paper and construct the complete history of what happened. If this flock of observers at rest relative to one another happens to be at rest relative to the barn, you'll get a history that describes a length-contracted rod fitting in the barn with the doors shut: You'll have piece of paper from the guy stationed at the far end of the barn that says "3:00 PM, door shut, tip of pole hasn't hit door yet; 3:51 PM, tip of pole hit door" and another one from the guy at the near end saying "2:59 PM, back of pole passed through the door; 3:00 PM, door shut". From this, you can conclude that both doors were shut with the pole inside at 3:00 PM.

But a different flock of observers at rest relative to one another but not at rest relative to the barn may give you slips of paper that lead to a different chronology.

 

[Mentz114]

I wish I could find it, but there was a thread here a while back where the question was literally whether anyone could visually see the latter in the barn with both doors closed (e.g. with perfect eyes or camera). The answer is the with a perfect fisheye lens set close enough to the center of the barn (but not hit by the ladder), the answer is yes. You could get a photograph including all parts of ladder and both doors closed.

I remember it well. What helped me come to terms with it is that the setup you describe here constitutes the measurement of the length of a rod that is in motion, and so will always give the contracted answer. In the camera frame the light from the endpoints of the rod were emitted simultaneously, but in the rod frame they were not.

 

[Sugdub]

I remember it well. What helped me come to terms with it is that the setup you describe here constitutes the measurement of the length of a rod that is in motion, and so will always give the contracted answer. In the camera frame the light from the endpoints of the rod were emitted simultaneously, but in the rod frame they were not.

Very interesting, however I can't see where this discussion is leading us. I'm looking for a non-ambiguous presentation of what physicists generally agree upon concerning the about-ness of SR and what it "says". May be such a common understanding does not exist at all, ... Many physicists seem to believe that SR deals with phenomena, i.e. with the outcome of experimental measurements. Conversely "LastOneStanding" tells us that SR is about a set of equivalent descriptions of the world, irrespective of any measurement process. This is a major difference I'm afraid. I have no intention to take position on this, but the alternative should not remain open.
On the one hand, I can understand that an "SR theory" would propose a continuous family of mutually exclusive descriptions of the world, the Lorentz transformation allowing to convert one into the other, the difference being enshrined into a change of the Galilean reference frame used for the respective descriptions. No opportunity referring to any "observers" in relative motion, to any "measurement protocol", to any "measured length". Each object may have a different "length" in each description. However this change in description is purely formal, it takes place in the brain of a theoretician, as if one changes the orientation of the coordinate system. It is not a physics theory: nothing has changed in the world and it does not deal with such changes. This is the way I read the "Yes" by "LastOneStanding" to my question.
On the other hand, I can also understand that observers exercising the same measurement protocol against a remote target might obtain different results if their operating conditions are objectively different, e.g. because their relative speed in respect to the target is different. In this case, the Lorentz transformation will convert one experimental result into the other. Please note that in this case the difference cannot be enshrined into a change of Galilean reference frame since the relative speed between the observer and the target changes from "nil" to "v" when swapping from one experiment to the next (a change of reference frame cannot effect the relative speed between any pair of physical objects). This second "SR" paradigm is indeed a physics theory: something different happened in the way observers interacted with the external world, leading to an objective difference in measurement outcomes.
I hope physicists contributing to this forum will be able to explain whether there is actually a shared vision within their community on the about-ness of the SR theory.