Length contraction, also known as Lorentz contraction
, is a physical consequence of Einstein's special theory of relativity
. The basic interpretation is that moving objects are observed to be contracted
. In order to understand why this is the case, we'll need to understand a few premises:
1) The speed of light is the same in all reference frames.
2) The progression of time is relative from one reference frame to another, but we can calculate the exact ratio by which time is dilated (I recommend reading about time dilation for details).
Given these two premises, we can devise a system for measuring length which transforms properly from one reference frame to another. This method is similar to the light-clock from the time dilation node, but essentially works "backwards". By measuring the travel time for a beam of light, we should be able to accurately calculate the distance it travels. Since we know the speed of light is the same in different reference frames, and we know exactly how time transforms from one reference frame to another, we should be able to easily calculate how length transforms from one reference frame to another:
This is a crude model for our "ruler", which is really nothing more than two mirrors
. The length between the two mirrors is determined by measuring the amount of time a beam of light takes to make the round trip
2d0 = cΔt0
Or, Δt0 = 2d0/c
Now that we have a system for measuring distance, let's see what happens when we look at it from a different reference frame. If we put our "ruler" on a train moving along the ground (to the right), we notice from the ground that because the train is moving, the far side of the car (right side) will have changed its position by the time the light reaches it, lengthening the travel time. Likewise, on the return trip, the near side of the car (left side) will move, shortening the travel time.
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What we need now is to mathematically relate our previous d0
with our new measured distance, d, using what we know about the changes in travel time.
The new travel times Δt1 and Δt2 can be calculated by simply adding the motion of the train to the distance in the equation above:
Δt1 = (d + vΔt1)/c
Δt2 = (d - vΔt2)/c
Δt1 = d/(c-v)
Δt2 = d/(c+v)
The total time for the round trip, Δt:
Δt = Δt1 + Δt2
= d(1/(c-v) + 1/(c+v))
= d(2c/(c2 - v2))
Δt = (d/c)(2/(1-v2/c2))
Now, we need a formula which relates Δt0 to Δt. This we can determine from time dilation, the calculation of which has been done in the time dilation node. The net result of this was:
Δt = Δt0/√(1-v2/c2)
Using this equation, we can see that
Δt0 = (2d/c)(1/√(1-v2/c2))
Finally, we put it all together with our first equation for Δt0, and find that
d0 = d/√(1-v2/c2).
Distances in moving reference frames are contracted along the direction of motion. Thus, a meterstick on the train parallel to the track would be shorter than a meterstick on the ground, from the perspective of an observer on the ground.
Now we see a paradox analogous to the twin paradox of time dilation. If all observers are on equal standing, we could do this same experiment, but reversing the roles of the observers, and the observer on the train should observe that the ruler on the ground is shorter than his own. Each observer measures that the other's meterstick is shorter. This seems to be contradictory information. How is this resolved?
As in the case with time dilation, the paradox is resolved with the relativity of simultaneity. Let's imagine we have a regular meterstick and want to measure the length of another object with it. How exactly is this measurement carried out? Presumably, one reads off the values on the meterstick at both ends of the object, and subtracts to find the difference. If the object you're measuring is moving, however, one must be sure to measure both ends of the object simultaneously, or else the object will have changed its position on the meterstick between measurements. Here is where the ambiguity comes in. Since simultaneity is relative, the observer on the ground watching the observer on the train carry out his measurement will see him take his measurements at different times, invalidating his measurement of length, from the ground observer's perspective. It's no wonder to the grounded observer that the train-observer's measurement says his meterstick is shorter, since he measured improperly, from his reference frame. Likewise, the observer on the train sees a similar story while watching the ground observer carry out his measurement. Both observers properly calculate that the other's ruler is shorter, and furthermore, they observe that the other carries out his measurement improperly. The moral of the story: all of these observations are relative. For a very interesting and more thorough discussion, read about The Barn And The Pole: A Relativity Paradox .
One final interesting point regarding length contraction revolves around the difference between what one sees and what one observes. In a moving reference frame, an observer with all the laws of physics at his disposal can calculate properly that the length of an object becomes contracted. However, this is only after this observer takes into account the physics of how light rays can get from the object to his field of vision. Such a calculation with all of these concepts taken into account is what we can call an observation. What the observer sees is an entirely different picture. Specifically, instead of seeing an object contracted, the observer sees an object rotated. Here's why:
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moves through space at a velocity
close to the speed of light. Imagine there is an observer directly beneath the cube, with respect to this picture. The cube's length is contracted:
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Now, this cube is moving so fast that light from the top right corner of the cube will not be able to get past the bottom corner unless it is moving at an angle
with respect to the right face of the cube. The exact angle would be such that tan
(θ) = v/c. From the perspective of an observer at this angle, the face appears to be pointed along this trajectory
. For the right edge, the opposite happens. Light rays from the top left corner can be pointed into
the cube, at this same angle with respect to the face, and it can still be seen by an observer. Thus an observer at right angles with the cube's velocity can easily see all of the left face, but the right face remains hidden. The picture now looks something like this:
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Now, if we assume that the observer does not have an acute sense of depth perception
(we could assume that the cube was too far away to tell the difference between distances on the cube), we can distort
the distances so that our picture now looks like:
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After drudging through all of this ASCII art
, I hope that I have shown that it is at least plausible
that what one sees is in fact a rotation, even though the actual physical effect is a contraction. The difference is that the rotation is illusory
, while the contraction is physical
In principle, when one takes into account time dilation, length contraction, and the relativity of simultaneity, one unfolds the entirety of Einstein's Special Theory of Relativity.