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Logical Basis of Fitzgerald-Lorentz Length Contraction (Robert J. Buenker, April 24, 2011)

Fitzgerald-Lorentz length contraction (FLC) is derived from the Lorentz transformation (LT) of the special theory of relativity (SR¹). The following discussion analyzes the logical basis for the FLC prediction.

Einstein used two postulates in his derivation of the LT: 1) the relativity principle (RP) and 2) the constancy of the speed of light in free space (c m/s) independent of the state of motion of the observer and the light source. The resulting equations lead to the conclusion that time dilation (the slowing down of clocks) in a moving rest frame is accompanied by the FLC, i.e., contraction of the lengths of objects located there. The amount of the contraction varies with orientation of the object (stationary in rest frame S’) to the observer (stationary in rest frame S): it is maximal along a line parallel to the relative velocity of S and S’, while no change is observed in a transverse direction.

The relationship between length variations and time dilation can be obtained directly from the above two postulates, however, without making use of the LT. To demonstrate this, consider the following example in which the length of a metal bar is determined under two different circumstances by measuring the elapsed time required for a light pulse to traverse it. Initially, the metal bar and two identical (proper) clocks are stationary in rest frame S. The elapsed time for light to traverse the metal bar is found to be ΔT=L/c s, showing that its length is L m at this stage in the experiment.

The bar and one of the clocks are then accelerated until they attain constant velocity relative to S and are thereafter stationary in S’. At this point in the argument, the RP is invoked. Accordingly, no change in either the length of the metal bar (L m) or the elapsed time for its light traversal (L/c s) is found by the observer in S’. Next we assume that time dilation has occurred, causing the clock in S’ to run Q>1 times slower than its counterpart left behind in S. The question is thus what this tells us about any possible change, if any, in the length of the metal bar that accompanies the time dilation in S’. Clearly, the corresponding elapsed time in S must have increased to QL/c s because of the aforementioned difference in rates of the two clocks. According to the second postulate this means that the observer in S now finds that the length of the metal bar has also increased by the same factor. It has changed from its initial value of L m to its current value while stationary in S’ to QL m. Moreover, the increase in length is the same in all directions because the local time measurement in S’ is completely independent of the orientation of the metal bar to the observer in S.

The above example indicates that isotropic length expansion accompanies time dilation, not the anisotropic length contraction of the FLC predicted by Einstein on the basis of the LT¹. The deduction of length expansion is based solely on the two relativistic postulates (RP and the constancy of the speed of light). Since the latter have received extensive experimental verification, there is no reason to doubt the correctness of this conclusion.

What it shows is that the theory of special relativity is not internally consistent. If one uses the above postulates directly to predict changes in length upon acceleration, the answer is opposite to what is deduced on the basis of the Lorentz transformation (LT). There is a simple explanation for this discrepancy. Einstein made an additional assumption in deriving the LT which was not declared as such. He claimed (see the four equations at the bottom of p. 900 in his original paper¹ that a function φ defined there only depends on v, the relative speed of S and S’. One obtains a qualitatively different result if one chooses φ instead so as to satisfy the basic assumption of time dilation, namely that the rates of clocks in relative motion are strictly proportional to one another, i.e. t=Qt’ in the notation used in the above example. The resulting alternative LT still satisfies Einstein’s two postulates, but predicts that the lengths of objects expand isotropically rather than contract when clock rates slow as a result of acceleration. More details may be found elsewhere.²

¹A. Einstein, Ann. Physik 17, 891 (1905).

²R. J. Buenker, Apeiron 15, 382 (2008).

--Rjbuenker (talk) 22:22, 3 May 2011 (UTC)

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