Length contraction and special relativity

[Karthikeyan]

Special theory of relativity says that, when an object moves with a uniform velocity along its length, the length seems to be less. I believe this is the length measured from the stationary frame of reference. This is justified by Michelson-Morely experiment. What happens when the motion is in perpendicular to the length ??

 

[bernhard.rothenstein]

Physicists consider that when we speak about a physical quantity, we shoulld tell the observer(s) who measures it, when and where the measurement is performed and the measuring devices used.
The proper length of a rod is a well defined physical quantity. The measured lenghts of a rod depends on the measurement procedure which could lead to length contraction or to length dilation as well.
The explanation of the negative result of the Michelson Morley experiment is given in very different ways some of them being based on the Lorentz contraction, but not all of them use that explanation.
The invariance of distances measured perpendicular to the direction of relative motion is a direct consequence of the first postulate its explanation involving no more then paintbrushes. If you are interested I could give you valuable references.

 

[Karthikeyan]

Thanks.Can u refer some materials for this?

 

[finchie_88]

Just as a matter of interest, but if there are two people (A and B let's say). And, A is considered as being the the still, and B is considered as being the moving relative to A, then if they compared the lengths of their metre sticks let's say, who would experience the length contraction relative to A, and will the affect be reversed when the situation is considered from Bs point of view?

 

[bernhard.rothenstein]

consider a rod at rest relative to A and moving with speed v relative to B. observers from A measure the velocity of B detecting simultaneously the ends of the rod. in the experiment described above observers from A measure the proper length of the rod but a non-proper time interval. Observer B can measure alone the speed of rod obtaining a nonproper length but measuring a proper time interval. So we should have L(proper)/t(nonproper)=L(nonproper/t(proper)=v.

 

[finchie_88]

I think I understand, but it seems very strange that it should work like that. I do have two questions to ask though, and that is a)what causes this effect, as in why does it happen like that? b)How are the lengths and times of different points of view related mathematically? Thank you of the previous responses, I still think its a bit strange, but its slowly becoming clearer.

 

[Galileo]

Special theory of relativity says that, when an object moves with a uniform velocity along its length, the length seems to be less. I believe this is the length measured from the stationary frame of reference. This is justified by Michelson-Morely experiment. What happens when the motion is in perpendicular to the length ??

No, the length does not seem to be less, it really IS less. It's not an optical illusion but an actual physical ... thing. Time and space are both relative and are different for things moving relative to each other.
And yes, naturally it's the length measured from the stationary frame. In the frame moving with the object, the object would be standing still.

Note that seeing or seeming is different from measuring. For example, light takes a while to reach your eye and such so you probably see a fast moving object somewhere else than it really is. A measurement takes that in account. All the contraction and dilation in relativity deal with what IS, not with what SEEMS.

Length will only get contracted in the direction of motion. So if a stick is held perpendicular to the motion, it will not get shorter (it will get thinner).

Just as a matter of interest, but if there are two people (A and B let's say). And, A is considered as being the the still, and B is considered as being the moving relative to A, then if they compared the lengths of their metre sticks let's say, who would experience the length contraction relative to A, and will the affect be reversed when the situation is considered from Bs point of view?

Both will see the other one contracted. A sees B moving and B sees A moving (with the same speed, but in the opposite direction). So A's meter-stick is 1 m as seen from A's frame, but it's shorter (say 99 cm) from B's frame. Likewise, B's meter-stick is 1 m as seen from B's frame, but 99 cm as measured from A's frame.
(I spoke of 'seen' above, but I meant 'measured')

I think I understand, but it seems very strange that it should work like that. I do have two questions to ask though, and that is a)what causes this effect, as in why does it happen like that? b)How are the lengths and times of different points of view related mathematically? Thank you of the previous responses, I still think its a bit strange, but its slowly becoming clearer.

a) The effect ultimately comes from the fact that the speed of light is the same for all observers. One of the fundamental postulates of special relativity. Not an in-depth answer, I know.

b) Ofcourse you can calculate everything. If you know how something behaves in a standing frame, you can know how it moves in every other frame moving relative to it. All you need to calculate basically everything in relativity are the Lorentz transformations:

$$x'=\gamma(x-vt)$$
$$y'=y$$
$$z'=z$$
$$t'=\gamma \left(t-\frac{v}{c^2}x\right)$$

Here x,y,z,t are the space-time coordinates of the stationary frame S and x',y',z',t' the ones for the moving frame S'. It is assumed that at t=0 the origins coincide and that S' moves in the x direction.
$\gamma$ is an important factor that depends on the speed:
$$\gamma=\frac{1}{\sqrt{1-v^2/c^2}}$$

So for example, consider a stick in S' with the length L in the x direction. What's its length as measured from S? Well, take the coordinate of the right side and subtract the on on the left side at the same instant of time to find the length. You'll find it's independent of time and equal to $L/\gamma$.

 

[yogi]

Galileo - length contraction is not considered real in SR by most authorities

Here is the epitome of quotes by Eddington re contraction: "Contraction is true, but not really true"

 

[JesseM]

Galileo - length contraction is not considered real in SR by most authorities

I think that's misleading--"real" doesn't have any technical meaning, it's really just a question of how you choose to talk about it, and it's certainly not true that "most authorities" would say flat out that it's "not real". On this sci.physics thread I found a sampling of quotes by different physicists on the "interpretation" of length contraction:

> Kenneth Krane, in his textbook "Modern Physics" states that the
>length contraction, like time dilation, is a real effect. Does anybody
>in this newsgroup take issue with this statement? What does
>Krane mean by his statement?


Go to my home page
http://members.aol.com/mluttgens/index.htm
and click on "On the nature of relativistic effects":
<< On the nature of relativistic effects

The reciprocal effect of length contraction and time
dilation, which appears by logical necessity to emerge from
the kinematic part of the special theory of relativity, has
been variously explained as

1. true but not really true (guess who)
2. real
3. not real
4. apparent
5. the result of the relativity of simultaneity
6. determined by measurement
7. a perspective effect
8. mathematical.

Here is a small selection from the literature; references
are found at the end of Part 2. Unless placed in quotation
marks, authors' assessments are summarized.

1. Effects are true but not really true:

Pride of place goes to Eddington [1928, 33-34]:

"The shortening of the moving rod is true , but it is not
really true."
(Thanks to Prof. I. McCausland, Toronto, for contributing
this gem.)

2. Effects are real:

Arzelies [1966, 120-121]:

The Lorentz Contraction is a Real Phenomenon. ...
Several authors have stated that the Lorentz contraction
only seems to occur, and is not real. This idea is false.
So far as relativistic theory is concerned, this
contraction is just as real as any other phenomenon.
Admittedly ... it is not absolute, but depends upon the
system employed for the measurement; it seems that we might
call it an apparent contraction which varies with the
system. This is merely playing with the words, however. We
must not confuse the reality of a phenomenon with the
independence of this phenomenon of a change of system. ...
The difficulty arises because we have become accustomed to
the geometrical concept of a rigid body with a definite
shape, whatever the measuring system. This idea must be
abandoned. ... We must use the term "real" for every
phenomenon which can be measured ... The Lorentz
Contraction is an Objective Phenomenon. ...
We often encounter the following remark: The length of a
ruler depends upon its motion with respect to the observer.
... From this, it is concluded once again that the
contraction is only apparent, a subjective phenomenon. ...
such remarks ought to be forbidden.

Krane [1983, 23-25]:

It must be pointed out that time dilation is a real effect
that applies not only to clocks based on light beams but to
time itself. All clocks will run more slowly as observed
from the moving frame of reference. ...
The length measured by the moving observer is shorter. It
must be emphasized that this is a real effect.

Matveyev [1966, 305]:

The dimensions of bodies suffer contraction in the
direction of motion ... A body is, therefore, "flattened"
in the direction of motion. This effect is a real effect
...

Møller [1972, 44]:

Contraction is a real effect observable in principle by
experiment. It expresses, however, not so much a quality of
the moving stick itself as rather a reciprocal relation
between measuring-sticks in motion relative to each other.
... According to relativistic conception, the notion of the
length of a stick has an unambiguous meaning only in
relation to a given inertial frame. ... This means that the
concept of length has lost its absolute meaning.

Pauli [1981, 12-13]:

We have seen that this contraction is connected with the
relativity of simultaneity, and for this reason the
argument has been put forward that it is only an "apparent"
contraction, in other words, that it is only simulated by
our space-time measurements. If a state is called real only
if it can be determined in the same way in all Galilean
reference systems, then the Lorentz contraction is indeed
only apparent, since an observer at rest in K' will see the
rod without contraction. But we do not consider such a
point of view as appropriate, and in any case the Lorentz
contraction is in principle observable. ... It therefore
follows that the Lorentz contraction is not a property of a
single rod taken by itself, but a reciprocal relation
between two such rods moving relatively to each other, and
this relation is in principle observable.

Schwinger [1986, 52]:

Each will observe the other clock to be running more
slowly. This is an objective fact. It is not a property of
clocks but of time itself.

Tolman [1987, 23-24]:

Entirely real but symmetrical.

3. Relativistic effects are not physically real:

Taylor & Wheeler [1992, 76]:

Does something about a clock really change when it moves,
resulting in the observed change in the tick rate?
Absolutely not! Here is why: Whether a clock is at rest or
in motion ... is controlled by the observer. You want the
clock to be at rest? Move along with it. ... How can your
change of motion affect the inner mechanism of a distant
clock? It cannot and it does not.

4. Relativistic effects are apparent:

Aharoni [1985, 21]:

The moving rod appears shorter. The moving clock appears to
go slow.

Cullwick [1959, 65, 68]:

[A] rod which is at rest in S' ... appears to the observer
O to be contracted ... Similarly, a rod at rest in S will
appear in S' to be contracted...

Jackson [1975, 520]:

The time as seen in the rest system is dilated.

Joos [1958, 243-244]:

The interval appears to the moving observer to be
lengthened. A body which appears to be spherical to an
observer at rest will appear to a moving observer to be an
oblate spheroid.

McCrea [1954, 15-16]:

The apparent length is reduced. Time intervals appear to be
lengthened; clocks appear to go slow.

Nunn [1923, 43-44]:

A moving rod would appear to be shortened. An interval is
always less than measured by the other observer.

Whitrow [1980, 255]:

Instead of assuming that there are real, i.e. structural,
changes in length and duration owing to motion, Einstein's
theory involves only apparent changes, and these are
independent of the microscopic constitution and hidden
mechanisms controlling the structure of matter. [Unlike]...
real changes, these apparent phenomena are reciprocal.

5. Relativistic effects are the result of the
relativity of simultaneity:

Bohm [1965, 59]:

When measuring lengths and intervals, observers are not
referring to the same events.

French [1968, 97],
Rosser [1967, 37],
Stephenson & Kilmister [1987, 38-39]:

Measurements of lengths involve simultaneity and yield
different numerical values.

6. Relativistic effects are determined by
measurements:

Schwartz [1972, 113]:

Each observer determines distances to be foreshortened.

7. Relativistic effects are comparable to perspective
effects: Rindler [1991, 25-29]:

Moving lengths are reduced, a kind of perspective effect.
But of course nothing has happened to the rod itself.
Nevertheless, contraction is no illusion, it is real.
Moving clocks go slow, a 'velocity-perspective' effect.
Nothing at all happens to the clock itself. Like
contraction, this effect is real.

8. Relativistic effects are mathematical:

Eddington [1924, 16-18]:

The connection between lengths and intervals are problems
of pure mathematics. A traveling clock gives a low
reading.

Minkowski [1908, 81]:

[The] contraction is not to be looked upon as a consequence
of resistances in the ether, or anything of that kind, but
simply as a gift from above, - as an accompanying
circumstance of the circumstance of motion.

Rogers [1960, 496]:

Thus we have devised a new geometry, with our clocks and
scales conspiring, by their changes, to present us with a
universally constant speed of light.

I would say that the statement "moving objects contract in length" is analogous to the statement "the speed of light is the same for all observers"--both statements depend on using the set of inertial coordinate systems provided by the Lorentz transformation and could be false if you used some other set of coordinate systems for inertial observers (if you used a different clock synchronization convention than Einstein's, for example), but since the fundamental laws of physics are all Lorentz-invariant the coordinate systems provided by the Lorentz transformation are the most "natural" ones to use since they're the only ones that will insure the laws of physics obey the same equations in each coordinate system, even though you are free to use others. But it should be pointed out that all measurements of length are dependent on your choice of coordinate systems in this way--even the physicists who say that Lorentz contraction is not fully "real" are not claiming that the length of an object "really" doesn't change, I think they're making a more general point about the "unreality"/coordinate-dependence of any statements about length whatsoever.

 

[Galileo]

Look, when I say contraction is real. I mean it's not an optical effect or something of that sort. If you do, then you must also think time dilation is an apparent effect, since you can't have one without the other.

If you think time dilation is not 'real', then how how can my clock run behind an identical one if I go to the sun and back at the sped of light? Or, to take something observed, how can the lifetime of unstable particles be longer when traveling at high speed?

 

[Anders Lundberg]

... For example, light takes a while to reach your eye and such so you probably see a fast moving object somewhere else than it really is. A measurement takes that in account. All the contraction and dilation in relativity deal with what IS, not with what SEEMS.

When it comes to sound, I agree that eg. the sound I hear from an aeroplane high above is not at all located in the same direction as the aeroplane. But in that case I can measure the angle to the sound and compare it to the measured angle of the aeroplane.
But when it comes to light and astronomical distances and speeds this is not the case. There is no comparing measurements you can do that will tell "where Jupiter really is". If I see Jupiter in the evening sky why should I not suppose that this is the true position of Jupiter? Eg. if I could measure the gravitational pull from Jupiter, and the direction of that pull, I would see that Jupiter gravity pulls towards that position (for the same reason that I see Jupiter in that position). In this aspect, I would say, Jupiter really IS where I see it.
But then, here is another case: Let's say that X is a guy located on Mars. His spaceship has the remarkable ability to reach lightspeed in an instant. Now, Y is a guy located on Earth. His telecope has the remarkable ability to enlarge the picture to the extent that Y clearly can see X when he enters his spaceship and takes away towards Earth. Now, since X is traveling at the speed of light, he will reach Earth in the same instant as the light showing his takeoff. This means that Y will see both the arrival and the takeoff in the same instant. From Y's point of view, X has made an instantaneous jump from Mars to Earth!
Now, both X and Y will agree that his journey was instantaneous. X saw the distance "shrink" so that he never exceeded the speed of light. Y, though, saw him travel much faster than light. Not that I see any paradox in this relationship, but if X means that "what he sees really is real" (like in the Jupiter-case above) then, in this aspect, Y really has traveled faster than light.
So, when should we use the IS and when should we use SEEMS?

 

[JesseM]

When it comes to sound, I agree that eg. the sound I hear from an aeroplane high above is not at all located in the same direction as the aeroplane. But in that case I can measure the angle to the sound and compare it to the measured angle of the aeroplane.

But when it comes to light and astronomical distances and speeds this is not the case. There is no comparing measurements you can do that will tell "where Jupiter really is".

Why not? You can just see how far away Jupiter appears at a given moment, then subtract the time light would take to travel that distance moving at c, then you can conclude that you are seeing the position that Jupiter was at X minutes ago in your frame.

If I see Jupiter in the evening sky why should I not suppose that this is the true position of Jupiter?

Because that would contradict the assumption that light waves travel at c.

But then, here is another case: Let's say that X is a guy located on Mars. His spaceship has the remarkable ability to reach lightspeed in an instant.

It's impossible for a massive object to reach lightspeed, according to relativity. However, you can imagine looking at light from a distant flashlight, for example.

Now, Y is a guy located on Earth. His telecope has the remarkable ability to enlarge the picture to the extent that Y clearly can see X when he enters his spaceship and takes away towards Earth. Now, since X is traveling at the speed of light, he will reach Earth in the same instant as the light showing his takeoff. This means that Y will see both the arrival and the takeoff in the same instant. From Y's point of view, X has made an instantaneous jump from Mars to Earth!

Instead, let's say that Y is a guy on Earth who can see the moment X turns on his flashlight, and of course he will see the light from the flashlight arrive at the same moment he sees the flashlight turned on. So Y sees the departure of the light beam and the arrival on Earth at the same instant--but it would be absurd to interpret this to mean the light beam itself was traveling faster than light! Of course he realizes that the moment he sees the flashlight being turned on is a delayed image, and that if Mars was located at a distance of 20 light-minutes away from Earth at the moment the flashlight was turned on, then the event of the flashlight being turned on really happened 20 minutes ago.

In the way Einstein originally formulated relativity, you don't even need to worry about subtracting light signal delays. Instead, he suggested that each observer have a network of rulers filling space and at rest in his own frame, and that a clock should be attached to each marker on the ruler, with all the clocks synchronized with each other in that observer's frame. That way, if you want to mark down the position and time of a distant event, you can rely purely on local measurements by a nearby clock and ruler-marking--if I see a distant explosion in my telescope, I can look at the marking on the ruler right next to that explosion, and look at the reading on the clock at that marker at the moment the explosion was happening, and thus figure out the actual position and time of that explosion in my reference frame. So length contraction depends on comparing local measurements of the position of the front and back of an object which were made at the same time according to local clocks--for example, if the back of a rod passes by the 1 meter mark when the clock at that mark reads 3:00, and the front of the rod passes by the 4 meter mark when the clock at that mark also reads 3:00, then we say the clock is 3 meters long in the reference frame defined by this set of rulers and clocks.

The tricky part here is what it means for different clocks to be "synchronized"--Einstein proposed a synchronization method based on the assumption that light travels at the same speed in all directions in any given frame, which means if you set off a light flash at the midpoint of two clocks, they are defined as "synchronized" if they both read the same time at the moment the light from the flash reaches them. But this synchronization method means that clocks which are synchronized in one frame will appear out-of-sync in another. For example, say I'm in a rocket traveling forward at great speed in your frame, and I set off a flash at the midpoint of the rocket, and set clocks at the front and back of the rocket to read 12:00 at the moment the light from the flash reaches them. From your point of view, the front of the rocket is traveling away from the point where the flash was set off, while the back of the rocket is traveling towards that point, so if you assume light travels at the same speed in all directions in your frame, you will conclude that the light hits the clock at the back before it hits the clock at the front, and thus the two clocks will be out-of-sync in your frame. This is known as "the relativity of simultaneity", the idea that events which happen at the "same time" in one frame happen at different times in another, and it's a consequence of the assumption that each observer should define the speed of light to be constant in their own rest frame.

 

[yogi]

Hey Jesse - I like your post 9 - some months ago I endeavored to post the same material and Janus terminated the thread - guess when it comes from you its ok - but not from a skeptic.

My interpretation of real is based upon what Lorentz would call real - an actual physical foreshortening in the frame of measurement - when i said relativists don't buy that - that is what I meant.

 

[russ_watters]

But time dilation doesn't happen in your local frame either, yogi - does that make it not real?

 

[yogi]

Russ - perhaps we need a different lexocography - or better spelling. Resnic uses this idea - time dilation and length contraction as measured relative to a moving frame - are real in the sense that the measurements are real.

 

[russ_watters]

In your previous post, you said "in your frame of measurement" - in this post, you said "measured relative to a moving frame". So which is it?

Time dilation is best measured cumulatively, which makes it easier than length contraction (which can still be measured cumulatively with an odometer-type device, but people don't think of it that way), but it is still not measured in your frame, it is what you measure (rather, observe to be happening) in someone else's frame.

(incidentally, your "as measured relative to a moving frame" is what appears to be the prevailing view, while your "in your frame of measurement" is not.)

 

[Karthikeyan]

Look, when I say contraction is real. I mean it's not an optical effect or something of that sort. If you do, then you must also think time dilation is an apparent effect, since you can't have one without the other.
If you think time dilation is not 'real', then how how can my clock run behind an identical one if I go to the sun and back at the sped of light? Or, to take something observed, how can the lifetime of unstable particles be longer when traveling at high speed?

Can u please explain more on the lifetime of unstable particles?

 

[Galileo]

Taken from this site:

Many kinds of elementary particles are unstable. After some time has passed, they decay into certain other elementary particles. The average time until decay is called the particle's life-time. When particles of the same kind that are either at rest or speeding around in a particle accelerator are compared, it turns out that the life-time of unstable particles at rest is significantly shorter. Experiments of this kind are in excellent agreement with the predictions of special relativity - from the point of view of an external observer, the "inner clock" of such particles slows down when such particles are accelerated to high speeds. Correspondingly, on average, they can take much longer to decay than particles at rest.

Do a search about the lifetime of muons entering our atmosphere. There's tons of pages about it and it's one of the experimental verifications of time dilation.

 

[Karthikeyan]

Taken from this site:
Do a search about the lifetime of muons entering our atmosphere. There's tons of pages about it and it's one of the experimental verifications of time dilation.

Do u mean to say that particle A , B having same mean life time with respect their own reference frame, but moving with different velocities will disintegrate at different times with respect to some other stationary frame of reference ??

 

[Galileo]

If I understand your question correctly, then yes. The mean lifetime of a particles (say muons) moving wrt your frame of reference will have a longer lifetime then stationay muons.

 

[jtbell]

When I was in graduate school, one of my friends worked on an experiment that used beams of hyperons (sigmas and xis, I think). They have such a short lifetime when at rest that if it had not been for time dilation, even at the velocity the particles were traveling the beams would have decayed after such a short distance as to be useless. But time dilation extended their lifetimes and made the beams long enough (several meters) that the experimenters could build detectors around them and study the particles' properties.

 

[Karthikeyan]

Hi all,
When we r speaking about the unstable particles, I believe we r speaking with our frame of reference. What do we mean when we say that "moving clock runs slower" ? Say two persons ve their clock perfectly synchronised. One of them travels in a spaceship (with his clock) with a speed that is appreciable with respect to the speed of light. He returns to Earth after some time 'X' hours (measured with the clock he is carrying). Now, if we compare this time 'X' with the time 'Y' (measured with the clock possesed by the man who is stationary). Will 'X' will be less than 'Y' ?

 

[Karthikeyan]

Also, when we speak about the length contraction, i believe it is the length of something which is moving measured from a stationary reference. Say, I place a 1 m rod ( measured with stationary reference frame) in a spaceship and the spaceship is set off. Now, if i measure the length of the rod from the stationary frame of referece, the length will be less. ( Note: rod parallel to the direction of motion). But when measured from the spaceship it will be 1 m. That is nothing happens physically. Its the frame of reference that makes the contraction. CORRECT ME IF I AM WRONG. If this is the case, then how come the person returning after the space journey looks younger than his twin in the earth?

 

[Galileo]

When we speak of length and time(duration) of somehting you must always consider with respect to which frame of reference this is referred since these quantities are relative. Just like left and right depends on which direction you are facing. In relativity, unless mentioned otherwise, we are always referring to the frame which is stationary with repect to 'you'.

Moving clocks run slower means that if some time, like a day, has passed for you, less time will have passed on a clock that's moving relative to you. It will appear to 'run slow'. As seen from the moving clock the situation is reversed (it is symmetrical) and your clock, which is now moving will run slow.
In the unstable particle's scenario, the particle's clock goes slower when moving fast, so the lifetime is increased.
In the twin paradox scenario, the person going in the spaceship will be younger than his brother when he gets back to earth. Not the other way around, since the traveling brother had to accelerate (break and return) to go back to Earth and thus changed his frame of reference.
For the brother on earth, the 'clock' on board his twin brother's ship runs slow. So less time passes for his brother who is therefore younger upon return.
From the spaceguy's point of view his clock runs normal, but the distance he travels will be different. The distance from the point of return to Earth will get shorter because of the length contraction. (It's not only objects that contract, it's space itself).

When you say nothing is happening physically it's a little unclear what you mean. The stick certainly doesn't notice being shorter, since he's at rest wrt himself (as always).