Measuring Mesons and Length Contraction

Experiments on the variation of cosmic-ray intensity with altitude provided one of the first confirmations of time dilation [1,2].  Rossi and Hall [1] found an absorption anomaly of “mesotrons” due to spontaneous decay and used relativity theory to show that the effect must be more pronounced for particles of relatively low energy because of their shorter lifetimes.  They also made a connection with relativistic length variations.  Specifically, they pointed out that the “average range before decay” L, i.e. the average distance traveled by the particles before disintegrating, must be proportional to the observed lifetime τ and their speed β = v/c: L = β τ.  This formula shows that the distance traveled by the particles has a different value for two observers in relative motion.  Specifically, it shows that the observer with the slower clock (and therefore the shorter corresponding measured lifetime τ) must find a smaller value for the distance L traveled by the particles before decay.  This conclusion follows from the fact that the speed β is the same for all observers, as expected from application of Einstein’s velocity transformation (VT [3]).
  
The above experience has been discussed in various textbooks in connection with length contraction.  Weidner and Sells [4] considered a case in which the decaying particles move with speed β=0.98 at a height of L=2260 m toward the earth’s surface.  The authors point out that the measured lifetime of the particles is therefore 5.0 τ₀ in this rest frame (S) which is long enough so that exactly one-half of them reach the earth.  An observer traveling with the particles (in S’) must also find the same fraction, even though the lifetime from his vantage point is much shorter (τ₀).  The explanation is that the corresponding distance traveled is also smaller by the same factor of 5.0, in agreement with the above range formula [1].  The authors [4] conclude: “because of the space-contraction phenomenon, the Earth’s distance from him is contracted.”  Another version of the same argument is given in an introductory textbook [5].  In this case, the example of a rocket ship passing between two fixed points in space is used.  Consistent with the VT, it is assumed that two observers agree on the velocity (speed and direction) of the rocket.  It is concluded that the observer in S’ with the slower clock measures the smaller value for this distance.  The following equations in standard notation for the two rest frames S and S’ summarize these results [γ= (1-v²/c²)^-0.5 ]: t’= t/γ, x’=x/γ and y’= y/γ.  Note that the direction of travel is immaterial in computing the distance. 

In his 1905 paper [3], Einstein derived the length contraction effect and time dilation from the Lorentz transformation (LT).  The following equations summarize these results: t’= t/γ, x’= γx and y’= y.  Although both textbooks [4,5] conclude that their example serves as a verification of the phenomenon of relativistic length contraction, comparison of the above two sets of formulas shows that the opposite is the case.  In Einstein’s result lengths measured by the observer in S in the x direction are contracted, whereas in the textbook examples, lengths measured by the observer in S’ are contracted in all directions.  It is clearly necessary to resolve this discrepancy.

1)            B. Rossi and D. B. Hall, Phys. Rev. 59, 223 (1941).

2)            B. Rossi, K. Greisen, J. C. Stearns and D. K. Froman, Phys. Rev. 61, 675 (1942).

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

4)            R. T. Weidner and R. L. Sells, Elementary Modern Physics (Allyn and   Bacon, Boston, 1962), p. 410.

5)             R. A. Serway and R. J. Beichner, Physics for Scientists and Engineers,  5^th Edition (Harcourt, Orlando, 1999), p.1262.