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Length Contraction |
Several years before Einstein published his special theory of relativity, Hendrik Lorentz and George FitzGerald independently explained the null result of the Michelson-Morley experiment by assuming that the length of the interferometer in the direction of motion contracted by a factor of gamma. For this reason, the relativistic phenomenon of length contraction is often called the Lorentz-FitzGerald contraction or simply the Lorentz contraction.
One way to derive the length contraction equation is to apply the Lorentz transformation equations to two arbitrary events. Let Dx and Dx' be the distance between the two events as measured in the rest frame and the moving frame, respectively, and Dt and Dt' be the corresponding time intervals. Then the first and last Lorentz transformation equations become
Dx' = g (Dx
– vDt)
Dt' = g (Dt
– vDx/c2)
If we now let the two events under consideration occur at the two ends of the rod at the same instant of time as measured in the rest frame, then we obtain:
Dx = L = the length of the
rod as measured in the rest frame
Dt = 0 = the time difference in
the rest frame between the two end measurements
Dx' = Lo = the length
of the rod measured in the moving frame = proper length of the rod
Dt' = the unknown time
difference in the moving frame between the two end measurements
Substitution and a little algebra render
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The Length Contraction Equation: |
L = Lo/g |
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The Moving Simultaneity Equation |
Dt' = -vLo/c2 |
According to the moving observer, the two end measurements were not made at the same time. This is fine from the moving observer’s viewpoint. Because the rod is not moving in his frame of reference, he will always obtain the same proper length measurement regardless of when the two end measurements are made.
But the rod is moving relative to the rest observer, so the timing of the two end measurements is critically important. Therefore, in the rest frame, the two measurements must be made simultaneously. But according to the moving observer, the two end measurements were not made simultaneously. Therefore, he claims that the rest observer’s measurement is distorted. Conversely, the rest observer says the moving observer’s measurement is distorted because his measuring rod is distorted.
Again we see that each observer asserts that the other observer’s world-view is distorted and that the other observer’s measurements are distorted. But they give different explanations for the distortions. The rest observer says the distortion is a consequence of the relativity of space while the moving observer says it is the result of the relativity of simultaneity.
The poll and barn paradox illustrate how easily the length contraction phenomenon can be misunderstood. The resolution of this paradox demonstrates again how length contraction and simultaneity work together to keep the theory of relativity self-consistent.
Q1. Length Contraction Paradox
Consider the case of two identical spaceships in relative motion. According to the
theory of relativity, the observers in each spaceship say the other spaceship
is shorter. But if spaceship A is shorter than B and B is shorter than A, then
spaceship A must be shorter than itself. This conclusion is logically
unacceptable. Therefore, there must be something wrong with the previous
reasoning. What is it? (A) Since the spaceships are
identical, they are really the same length. (B)
Spaceship A is really shorter than spaceship B. (C)
Spaceship B is really shorter than spaceship A. (D) The
moving spaceship is really shorter than the one at rest. (E)
The paradox is the result of trying to view reality from two incompatible
viewpoints.
Q2. Why does length contraction
occur – i.e. why is the moving rod measured by the rest observer to be
shorter than normal? (A) The rest observer says the result is a consequence of
the relativity of space. (B) The moving observer says the result is a
consequence of the relativity of time (simultaneity). (C) It doesn’t
occur – it is just an illusion – the moving object only looks
shorter to the rest observer because light from both ends of the object do not
reach the observer at the same time. (D) Two of these. (E) Three of these.
P1. Derivation of Length Contraction
Equation
Make the substitutions and do the algebra to derive the length contraction
equation L = Lo/g
and the time interval equation Dt' =
-vLo/c2
discussed above.
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