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  4. Relativity
  5. Time dilation

5.3 Time dilation  (Page 6/8)

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In 1971, American physicists Joseph Hafele and Richard Keating verified time dilation at low relative velocities by flying extremely accurate atomic clocks around the world on commercial aircraft. They measured elapsed time to an accuracy of a few nanoseconds and compared it with the time measured by clocks left behind. Hafele and Keating’s results were within experimental uncertainties of the predictions of relativity. Both special and general relativity had to be taken into account, because gravity and accelerations were involved as well as relative motion.

Check Your Understanding a. A particle travels at 1.90 × 10 8 m/s and lives 2.10 × 10 −8 s when at rest relative to an observer. How long does the particle live as viewed in the laboratory?

a. Δ t = Δ τ 1 − v 2 c 2 = 2.10 × 10 −8 s 1 − ( 1.90 × 10 8 m/s) 2 ( 3.00 × 10 8 m/s) 2 = 2.71 × 10 −8 s .

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b. Spacecraft A and B pass in opposite directions at a relative speed of 4.00 × 10 7 m/s . An internal clock in spacecraft A causes it to emit a radio signal for 1.00 s. The computer in spacecraft B corrects for the beginning and end of the signal having traveled different distances, to calculate the time interval during which ship A was emitting the signal. What is the time interval that the computer in spacecraft B calculates?

b. Only the relative speed of the two spacecraft matters because there is no absolute motion through space. The signal is emitted from a fixed location in the frame of reference of A , so the proper time interval of its emission is τ = 1.00 s . The duration of the signal measured from frame of reference B is then
Δ t = Δ τ 1 − v 2 c 2 = 1.00 s 1 − ( 4.00 × 10 7 m/s) 2 ( 3.00 × 10 8 m/s) 2 = 1.01 s.

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Summary

  • Two events are defined to be simultaneous if an observer measures them as occurring at the same time. They are not necessarily simultaneous to all observers—simultaneity is not absolute.
  • Time dilation is the lengthening of the time interval between two events when seen in a moving inertial frame rather than the rest frame of the events (in which the events occur at the same location).
  • Observers moving at a relative velocity v do not measure the same elapsed time between two events. Proper time Δ τ is the time measured in the reference frame where the start and end of the time interval occur at the same location. The time interval Δ t measured by an observer who sees the frame of events moving at speed v is related to the proper time interval Δ τ of the events by the equation:
    Δ t = Δ τ 1 − v 2 c 2 = γ Δ τ ,

    where
    γ = 1 1 − v 2 c 2 .
  • The premise of the twin paradox is faulty because the traveling twin is accelerating. The journey is not symmetrical for the two twins.
  • Time dilation is usually negligible at low relative velocities, but it does occur, and it has been verified by experiment.
  • The proper time is the shortest measure of any time interval. Any observer who is moving relative to the system being observed measures a time interval longer than the proper time.

Conceptual questions

(a) Does motion affect the rate of a clock as measured by an observer moving with it? (b) Does motion affect how an observer moving relative to a clock measures its rate?

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To whom does the elapsed time for a process seem to be longer, an observer moving relative to the process or an observer moving with the process? Which observer measures the interval of proper time?

The observer moving with the process sees its interval of proper time, which is the shortest seen by any observer.

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(a) How could you travel far into the future of Earth without aging significantly? (b) Could this method also allow you to travel into the past?

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Problems

(a) What is γ if v = 0.250 c ? (b) If v = 0.500 c ?

a. 1.0328; b. 1.15

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(a) What is γ if v = 0.100 c ? (b) If v = 0.900 c ?

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Particles called π -mesons are produced by accelerator beams. If these particles travel at 2.70 × 10 8 m/s and live 2.60 × 10 −8 s when at rest relative to an observer, how long do they live as viewed in the laboratory?

5.96 × 10 −8 s

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Suppose a particle called a kaon is created by cosmic radiation striking the atmosphere. It moves by you at 0.980 c , and it lives 1.24 × 10 −8 s when at rest relative to an observer. How long does it live as you observe it?

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A neutral π -meson is a particle that can be created by accelerator beams. If one such particle lives 1.40 × 10 −16 s as measured in the laboratory, and 0.840 × 10 −16 s when at rest relative to an observer, what is its velocity relative to the laboratory?

0.800 c

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A neutron lives 900 s when at rest relative to an observer. How fast is the neutron moving relative to an observer who measures its life span to be 2065 s?

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If relativistic effects are to be less than 1%, then γ must be less than 1.01. At what relative velocity is γ = 1.01 ?

0.140 c

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If relativistic effects are to be less than 3%, then γ must be less than 1.03. At what relative velocity is γ = 1.03 ?

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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