## Problems with Length Contraction
## The speedy rocket shipLC is normally not described using single particles but with larger
objects such as a high-speed rocket. The claim is that when the rocket is still it has a
certain length, and when the rocket moves at 0.87 Given that a rocket is not a single particle but made of many particles
(atoms) the above claim appears to raise an anomaly. What it is saying is that not only do
the atoms shrink but the space between atoms also shrinks, resulting in a shorter rocket.
Somehow the LT has been applied to the space between the atoms. But how can empty space
shrink, especially when space is not a physical object and therefore cannot be considered
to be moving?
## ProblemsLC seems easy to picture when a single particle is involved. However a problem emerges when several moving particles are considered. Let’s say we have two electrons a fixed distance apart and initially at rest. They might look like this: Now the electrons begin to accelerate at an identical rate until they
reach a speed of 0.87 The correct answer must be B. If the distance from the front to the back
of each electron decreases by half, then the distance between the two electrons must also
shrink by half. Both electrons are at rest relative to one another and, like the rocket
above, they form a moving ‘super particle’ which also needs to reduce its length
by half. A small tube sits within the accelerator – represented by the red section at the base. The tube has a length that covers the spacing of two electrons such that as a new electron enters, an old one exits, and a third electron sits at the mid point. When the electrons are at rest the tube situation looks like this (for convenience, the tube is shown flattened out): Now let’s power up the accelerator; applying equal acceleration to
all the electrons until they reach a speed of 0.87 (A) shows the electrons shrinking but their spacing staying the same.
## Which direction?A second question regards the direction of shrinkage. LC tells us that shrinkage should happen along the direction of travel. Consider the following diagram: Three balls A, B and C are arranged on the points of an equal-sided triangle and move at equal speeds away from its centre. Suppose we wish to know the contraction of ball C. First let’s consider A and C in isolation. Without having B available as a reference, A and C are moving directly away from each other. So from the point of view of A, ball C should look like this: Now consider B and C in isolation. Without having A available as a reference, B and C are moving directly away from each other. So from the point of view of B, ball C should look like this: This is obviously a problem, since ball C cannot be simultaneously shaped like both of the above diagrams.
## By how much?A third question relates to the amount of shrinkage relative to an observer.
According to SR all velocities are relative and therefore the amount of shrinkage can be
different for different observers. According to the rules of LC, the first observer will see this: While the second observer sees this: Again we have a problem because the rod can’t have both these lengths.
## Length Contraction and RepositioningWe’ll now get to the nuts and bolts of what is said to be causing LC.
Under SR a moving particle will appear to reposition itself according to the formula: To see how this leads to LC, consider a rod of length Initially the rod is not moving, so SR doesn’t apply and everything
appears normal. We now set the rod in motion at 0.87 As can be seen, both the distance to the rod and the rod itself has
contracted by a factor of 2. The same applies to all the particles (atoms) within the rod:
they reposition themselves and give the impression that the rod overall has shrunk. For
this reason ‘relativistic length contraction’ should perhaps more properly be
referred to as ‘relativistic repositioning’. Note that the contraction and repositioning occurs along the direction of travel rather than in the direction of the observer.
## Is the Rocket Near or Far?An interesting story can be told about repositioning.
## Experimental evidenceNow let’s consider what experiments have to say about LC. Measuring
LC is no easy task because SR tells us that a moving object’s length immediately
returns to normal once it stops. And observing a particle in motion is also difficult as
that would normally require bouncing a light beam off the object; and the beam would have
trouble reaching a particle moving at near light speeds.
## ConclusionThe phenomenon of length contraction is said to be responsible for a number
of effects; some of which might be possible from the viewpoint of a single observer, but
would be contradictory from the viewpoint of multiple observers.
[1] Actually the equation is normally expressed in the
form vt)/γ where x
is the initial position, _{0}v is velocity and t is the time taken to get to the
current position.[2] Wood, A.B., Tomlinson, G.A., and Essen, L. 1937. "The Effect of the Fitzgerald-Lorentz Contraction on the Frequency of Longitudinal Vibration of a Rod," Proc. Royal Soc., 158, 606-633. [3] Renshaw, C. 1999. "Space Interferometry Mission as a Test of Lorentz Length Contraction," Proc. IEEE Aerospace Conf., 4, 15-24. |

Copyright © 2011 Bernard Burchell, all rights reserved.