# Length contraction and special relativity

1. Jan 12, 2006

### 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 ??

Last edited: Jan 12, 2006
2. Jan 12, 2006

### 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.

3. Jan 12, 2006

### Karthikeyan

Thanks.Can u refer some materials for this???

4. Jan 12, 2006

### finchie_88

Just as a matter of interest, but if there are two people (A and B lets 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 lets 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?

5. Jan 12, 2006

### 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 lenght of the rod but a non-proper time interval. Observer B can measure alone the speed of rod obtaining a nonproper lenght but measuring a proper time interval. So we should have L(proper)/t(nonproper)=L(nonproper/t(proper)=v.

6. Jan 12, 2006

### 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.

7. Jan 12, 2006

### Galileo

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).

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')

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 $L/\gamma$.

Last edited: Jan 12, 2006
8. Jan 12, 2006

### 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"

9. Jan 12, 2006

### JesseM

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:
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.

Last edited by a moderator: May 2, 2017
10. Jan 13, 2006

### 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?

11. Jan 13, 2006

### Anders Lundberg

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: Lets 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 travelling 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 travelled faster than light.
So, when should we use the IS and when should we use SEEMS???

12. Jan 13, 2006

### JesseM

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.
Because that would contradict the assumption that light waves travel at c.
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.
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 travelling 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 travelling 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 travelling away from the point where the flash was set off, while the back of the rocket is travelling 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.

Last edited: Jan 13, 2006
13. Jan 13, 2006

### 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.

14. Jan 13, 2006

### Staff: Mentor

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

15. Jan 14, 2006

### 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.

16. Jan 14, 2006

### Staff: Mentor

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.)

Last edited: Jan 14, 2006
17. Jan 18, 2006

### Karthikeyan

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

18. Jan 18, 2006

### Galileo

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.

19. Jan 18, 2006

### Karthikeyan

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 ??

20. Jan 18, 2006

### 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.