A Length Contraction Paradox
How a relativistic length contraction may disagree with the limit of light speed
In this article we use Einstein’s method to measure the length contraction of a very long rod, and conclude that when shortened in a very small lapse of time, light speed is perceived to be surpassed.
Measuring the Length Contraction
In his publication Relativity: The Special and General Theory, Einstein explains the contraction of a rod when moving with constant linear speed v respect to a stationary frame of reference.
He measures the contraction, and deduces it is shortened by a factor of
respect than when at rest. As a result, the higher the speed v, the shorter the resulting rod, until it may become a 0-length rod when moving at light speed, c.
The Change in the Rod
At time t=0, let the remote end of the rod be placed at x= L (point B in the sketch below). As per the aforementioned effect, this end experiences a shift towards the end at the origin (point A in the sketch) when the speed of the rod increases.
The magnitude of the displacement being
Thus there are 2 variables affecting the displacement:
- the length of the rod when at rest, L,
- and its speed when in motion, v.
In both cases the higher their value, the longer the shift experienced.
The Question Arising
Now let’s assume the time elapsed between both states of the rod (from motionless to moving) is sufficiently small.
Of course Relativity detemines a speed limit at which the rod may be moving but, while the the rod shortens: what is the speed at which the end B is being displaced?
As prevoiusly said, Einstein used the Lorentz transformation to measure the length contraction. He found the position of both ends of the rod as perceived by the stationary reference frame at a given time, t=0 (the same would be achieved at any other time interval):
Arbitrarily High Length and Low Lapse
Now, for the stationary reference frame, let the static rod extend for a length L at time t=0, and be moving with shortened length at time t=1.
Then, since the shift perceived in point B is L-L·√(1-v²/c²), and the time interval is the unity, the speed perceived shall be
Thus, for an unitary time interval, this speed may range between 0 ㎧ when v=0, and L ㎧ when v=c. So when the length L is higher than the value c meters, the speed perceived surpasses the speed of light.
Therefore here is the paradox: nothing prevents neither the shortening to be arbitrarily high nor the time elapsed to be arbitrarily small, which means that while increasing the speed of an object always below light speed, another reference frame may observe a part of it surpassing this limit (and the space surrounding).
Condition Not to Exceed Light Speed
In order not to perceive the aforementioned effect, we need point B moving at a speed below c,
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