Page
51531
|
8/21/04 |
|
|
Time Dilation |
Consider two arbitrary events located in space and time. Let Dx and Dt be the distance and time between the two events as measured in the rest frame and Dx’ and Dt’ the distance and time between the same two events as measured in the moving frame. Then the last inverse Lorentz transformation equations give the relationship
Dt = g (Dt’
+ vDx’/c2) .
If the two events occur at the same place but at different times in the moving frame, then
Substitution immediately yields
|
The Time Dilation Equation |
t = g to |
Time dilation is not just simply an illusion. It is a real phenomenon, verified quantitatively through numerous experiments performed during the last century.
See Serway pp.14-17 and Tipler pp.33-36 for more information on time dilation. Also look at the problems at the end of Chapter 1 in both books for additional problems on this topic.
Q1. Which of the
following verify experimentally the phenomenon of time dilation? (A) Muon decay
in the earth’s atmosphere. (B) Neutron decay in a nuclear reactor. (C)
Biological clocks in airplanes traveling east and west on the earth’s
equator. (D) Two of these. (E) Three of these.
P1. Muon Decay
Muons are unstable elementary particles that decay with a half-life of approximately 1.5 ms in their own proper frame. When cosmic rays bombard the earth, muons are created in the upper atmosphere and travel downward toward the surface of the earth at speeds approaching that of light. (a) If time dilation did not occur and they traveled at the speed of light, how far would they travel into the atmosphere before half of them are gone? (b) Suppose 108 muons are detected at an altitude of 9 km and no more are created on the way down, how many would be detected at sea level? (c) Now assume the muons are traveling at a speed of 0.995 c with relativistic time dilation in effect. What would be the half-life of muons in the earth frame of reference? (d) How far would they travel before half were gone? (e) How many would remain to be detected at sea level? (f) Is the difference between answers (b) and (d) large enough to be noticed? Note: Experiments of this type have been performed and confirm the reality of the time dilation phenomenon. (See Serway pp.16-17 and Tipler pp. 40-41.)
P2. Atomic Clocks
Today’s atomic clocks have extreme accuracy and can be used to test the time dilation phenomenon on a macroscopic scale. (a) Suppose a standard atomic clock is located on the earth’s equator whose radius is 6.37x106 m. Relative to a clock at rest with respect to the center of the earth, how many nanoseconds would the standard clock lose during a period of 24 hours? (b) Suppose an atomic clock is located in a jet aircraft traveling eastward along the equator at the speed of sound (345 m/s). How many nanoseconds would this eastward moving clock lose in 24 hours? (c) Suppose the airplane reversed its path and traveled westward. How many nanoseconds would its atomic clock lose in 24 hours? (d) Compare these three answers to determine how much time the eastbound clock would lose relative to the reference clock on the earth’s equator and how much the westbound clock would gain relative to the reference clock. Note: This type of experiment has also been performed with the result confirming the time dilation phenomenon. (See Serway pp. 17.)
|
|