Paradox of Special Relativity
inconsistency in electric and magnetic phenomena

The Muon Experiment

In 1937, after nearly a decade of study of cosmic rays and H.~Yukawa's 1935 prediction of the existence of a mid-mass particle (meson) responsible as carrier of the nuclear force, S.H.~Neddermeyer and C.D.~Anderson found evidence in cosmic ray showers of “some particles less massive than protons but more penetrating than electrons.”~ By 1940 it had been determined that these mesotrons were not Yukawa's predicted particle, but rather an unstable particle that rapidly decays into electrons.~ In 1941 B.~Rossi and D.~Hall, attempting to measure the lifetime, measured the intensity of mesotrons from cosmic rays, shielded by iron from cosmic rays showers and soft components, both at the top of Mt. Washington, the highest peak in the northeastern United States, and at sea level.~ Rossi eventually determined that mesotrons (now called muons) have an at rest “mean lifetime of 2.3 ±~0.2 microseconds” (now accepted to be 2.2 x 10-6seconds). In 1963 David Frisch and James Smith repeated the Mt. Washington experiment as a demonstration of special relativity, recording the experiment and results in a movie.
The muons are short-lived particles which are created when higher energy cosmic rays strike the earth's atmosphere. Half of the muons decay every 1.56 x 10-6 seconds. (Note - this is the half-life which is related the the "mean lifetime", but not the same thing). While the muons are stopped or deflected only slightly by the earth's atmosphere, the muons spontaneously decay at this rate regardless of the physical conditions around them.  So their decay occurs as if governed by a very accurate clock that is oblivious to conditions such as temperature that affect many other types of clocks.  In this famous Mt. Washington experiment, the number of muon decays which they detected at an elevation of 1910 m, 568 per hour, was compared with the number they detected at an elevation of only a few meters, 412 per hour. The muons travel at nearly the speed of light but they still take a fraction of a second to travel the additional distance from the elevation of Mt. Washington to sea level. As a result some of the muons decay so fewer are counted at sea level.~ But since the muons are travelling nearly the speed of light, their internal clock is slowed by the amount accounted for by Einstein's special relativity so that more reach sea level than otherwise expected.

Pre-Lab

1.  Draw a sketch of the situation indicating each of the following:

                The distance traveled by the muons bewteen the top and the bottom of Mt. Washington

                The number of muons detected at the top, and the number of muons detected at the bottom

A good example of the experiment can be found at http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/muon.html#c1
(note that this does not have the actual data - see the summary above for that)

2.  Note the half-life (not mean life) of a muon.  How long is the half life of a muon?  What does this mean?  
        (reference:  http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)

3.  A sample of 1,000,000 muons is observed in a laboratory.   How many muons will remain after one half life?  How many after two half lives?

4.  How many muon half lives occur in 5 x 10 -6 seconds?

5.  Using the formula Nt = No (1/2)k, where k is the number of half lives,  and N0 is the original number of muons, find the remaining muons Nt, given an original sample of No = 1,000,000 muons and t = 5 x 10 -6 seconds. (perhaps helful hint: k = time elapsed/half life)

Calculations
1. Calculate the approximate time, t, needed for the muons to travel the vertical distance from the elevation of Mt. Washington to sea level in the Earth frame of reference assuming the travel at approximately the speed of light, 3 x 10 8 m/s
2. Using the formula Nt = No (1/2)k where the exponent, k, is the number of half lives, determine the number of muons after time t, Nt, that one might expect to count at sea level without the effects described by special relativity.  Use the decay rate measured at the top of the mountain as a measure of the initial number of mesons, No. Note the difference of Nt we have predicted compared to the actual sea level count.

Set up parts 3 and 4 as a computer spreadsheet in either Excel of Google.

3. Have four rows for the following speeds, v, that will be tested for the muons:
        a. 50% of the speed of light, c
        b. 90% of c
        c. 95% of c
        d. 99% of c
3. For each have four columns to calculate the following:
        a. the time, t, required to cover the surveyed distance in Earth frame of reference,
        b. the dilated time, t0 = t sqt( 1 - v2 / c2 ) measured in the muon frame of reference
c. and a revised number, Nt, that should be counted at sea level, now taking into account dilated elapsed time predicted by special relativity using the equation: Nt = No (1/2)k
4. Which speed gives the closest fit to the actual number of counts at sea level?
5. Determine the height of Mt. Washington, x, from the muon's perspective.
6.  Graph Nt vs. V in Excel - include a best fit line.
7.  Write a brief conclusion - start with stating the objective of this experiment then explain how the data supports (or not) time dilation as a real effect.

Record this in your journal or as a Google doc shared with mhirsh@g.needham.k12.ma.us



References
1. A. Summerfeld, The Principle of Relativity, a collection of original papers, Dover, 1923, including Einstein's “On the Electrodynamics of Moving Bodies,” translated from “Zur Elektrodynamik bewegter Körper” from Annalen der Physik, 1905._
2. L. Pearce Williams, The Origins of Field Theory, Random House, 1966._
3. Albert Einstein & Leopold Infeld, The Evolution of Physics, Simon & Schuster, 1938_
4. B. Rossi & D. Hall, (measurement of muon lifetime), Phys. Rev. 59 (1941) 223_
5.B. Rossi & N. Nereson, Experimental Determination of the Disintegration Curve of Mesotrons, Phys. Rev. 62 (1942) 417_
6. D. Frisch and J. Smith, Measurement of the Relativistic Time Dilation Using Mesons, Am. J. Phys. 31 (1963) 342.

created 24 April 2004
latest revision 3 May 2007
by D Trapp

Some references
Where I stole this from
http://web.me.com/dtrapp/ePhysics.f/labIV_13.html
Good background on the experiment and what you need to know about muons:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html



Lab Assignment:

Make a Google Doc and share it with my address: mikehirsh@gmail.com
State the members of the lab group - each member must be responsible for one section (say which)

1.  Introduction
– state a purpose for the lab (in this case testing the predictions of special relativity)
- provide the background necessary to understand the experiment, some possibilities:
        these are mandatory:
        - overview of the Mt. Washington experiment
        - predictions of special relativity and equations
        - frames of reference
        -which frame measures proper time and which frame measures proper length from top to bottom

        optional (but fun and extra credit):
        what is a muon
        where are they coming from in this experiment?

Cite your sources in parenthesis

2.  Procedure – briefly explain the experiment that was conducted (if you feel you already did this in the intro you can skip this)

3.  Data table and results –
        
        clearly label the content and units of each column – where an equation is used, include the equation at the head of the column or as a footnote to the table by column
        example:  col 2:  t = 1910m/v

4a. Example calculation – not all calculations must be shown but at least one sample of each.

4b.  Graph – a graph of N vs v in this lab would help.  It should include proper labels and units and a best fit line.

5.  Conclusion – restate the purpose and explain how this lab did or did not achieve its purpose.  Use specific data and results from the lab.  


6. Discuss major sources of error and how they influence the results and the ability to reach a conclusion.