RELATIVISTIC TIME DILATION

INTRODUCTION: This week's "experiment" is a simulation of a famous experiment which conclusively disproves the classical theory of relativity by showing that moving muons -- subatomic particles with a mean lifetime of 2.2 microseconds -- have significantly longer lifetimes when they move at speeds close to the speed of light. BEFORE LAB: Skim David Frisch and James Smith, Am. J. Phys., 31, pp. 342-355 (1963).

THE MOVIE: This week you will be watching a film of the experiment (Frisch and Smith, above) designed to measure time dilation using muon decay. After the film, please answer questions 1-4 below in your lab notebooks, which you should turn in at the end of lab. After these preliminaries, we will start our experiment, which is described below.

PRELIMINARY QUESTIONS:
  1. DESCRIBE THE EXPERIMENT: Briefly describe the procedure used in the experiment. Your description should answer all of the following:
    • What do the two counts (blips) recorded on the oscilloscope represent?
    • How do the researchers know that they are only counting muons?
    • What is the purpose of the iron, and why is less used at sea level?
  2. TIME DILATION FACTOR: The article mentioned above states (p. 351) that the muons measured in their experiment had velocities of v =0.9950c to 0.9954c for the measurements on top of Mt. Washington (6300ft), but 0.9881c to 0.9897c in Cambridge, MA. So, overall, we can say that v = 0.992c 0.003c.
    • Calculate the time-dilation factor, γ (gamma), including uncertainty, for v = 0.992c 0.003c. (Make use of your lab text --- D. C. Baird's Experimentation --- if you don't already know how to calculate uncertainty.)
  3. LAB FRAME: (Answer the following for an observer on the ground.)
    • Convert Mt. Washington's height to meters (1m = 3.28 ft)
    • How long does it take the muons to travel this distance?
    • If the muons' mean lifetime is 2.20 μs in their own frame, what is it in the lab frame? (Include uncertainty.)
    • How many lifetimes does it take the muons to reach sea level? (Include uncertainty)
  4. MUONS' FRAME: (Answer the following from the muons' perspective.)
    • To the muons, how high is Mt. Washington? (Include uncertainty.) How long does it take to travel this distance?
    • What is the muons' mean lifetime in their own frame?
    • How many lifetimes does it take the muons to reach sea level? (Include uncertainty)
THE EXPERIMENT: You can now use the computer simulation Muon (Copy Muon to your P: drive) to calculate the length contraction (or time dilation) factor by counting incident muons at two or more altitudes. From your results you should calculate a value of γ (gamma) and its associated uncertainty,δγ. (Notice that the simulation may describe an experiment in which the iron shielding is a different thickness, so that gamma is different than what you calculated above.)

To make this experiment more interesting, I will ask you to help some colleagues of mine in Costa Rica who might want to recreate this experiment, taking measurements at their home laboratory (1000m elevation), at the volcano Irazú (3432m), and at sea level.

You may take 150 total "hours" of data, and you may only take data at these three (virtual) locations. [While you might be tempted to run more than one copy at a time on the same computer, experience shows that this gives bogus data (nasty little program bug). Beware.] You may take your data in whatever 'packets' -- 1 hour runs, 2 hour runs, 50 hour runs, etc. -- but you must have at least four separate data points to be able to calculate an uncertainty in gamma. Analyze your data by plotting count rate vs altitude and fitting the data to an exponential function. You can use Excel, Kaleidagraph, or the computer program SLUDGE to do this.

ANALYSIS:
If No is the decay rate at some arbitrary altitude, the decay rate, N, at the bottom of the mountain will be

N = Noe-t/<τ>,

where t is the time for the muon to travel the distance and <τ> (tau) is the muon's mean lifetime. Now, if we choose our arbitrary altitude to be h=0, and if both t and <τ> are measured in the muon's rest frame, then this time will be given by


where h is the altitude as measured in the lab frame, γ is the time dilation factor, and v is the speed of the muons, which to first order equals the speed of light, c. Use the preceding analysis, and your exponential fit to your own data, calculate γ, and, from it a revised, second order value for the muon's speed, v. Calculate uncertainties!

FYI: The mean lifetime, <τ>, and the 'half-life' , τ1/2, represent two different ways of quantifying the statistics of the range of lifetimes that a radioactive particle like a muon exhibits. They are related by τ1/2 = 0.693 <τ>.