Time Dilation
1.3 Time Dilation
Learning Objectives
- Explain how time intervals can be measured differently in different reference frames.
- Describe how to distinguish a proper time interval from a dilated time interval.
- Describe the significance of the muon experiment.
- Explain why the twin paradox is not a contradiction.
- Calculate time dilation given the speed of an object in a given frame.
The analysis of simultaneity shows that Einstein’s postulates imply an important effect:
Time intervals have different values when measured in different inertial frames.
Time Dilation
Time dilation is the lengthening of the time interval between two events for an observer in an inertial frame that is moving with respect to the rest frame of the events (in which the events occur at the same location).
Practice!
| Practice 1.3.1 |
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| Consider two identical light clocks. Bob has one and Alice takes the other on her spaceship and flies by Bob at speed V. Bob observes Alice’s clock as Alice flies by. Which of the following statements is true? |
| A. Bob observes Alice’s clock to tick at the same rate as his clock |
| B. Bob observes Alice’s clock to tick faster than his clock |
| C. Bob observes Alice’s clock to tick slower than his clock |
| Practice 1.3.2 |
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| Consider two identical light clocks. Bob has one and Alice takes the other on her spaceship and flies by Bob at speed V. Alice observes Bob’s clock as she flies past him. Which of the following statements is true? |
| A. Alice observes Bob’s clock to tick at the same rate as her clock |
| B. Alice observes Bob’s clock to tick slower than her clock |
| C. Alice observes Bob’s clock to tick faster than her clock |
| Practice 1.3.3 |
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 Two identical clocks are set to the same time as one passes the other at high velocity. Which of the figures represents a possible observation of the clocks at some later time in the frame of the fixed clock? |
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| Practice 1.3.4 |
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| True or False: The Lorentz factor is always greater than or equal to 1. |
| A. True |
| B. False |
Discuss!
The Lorentz factor is given as:
where v is the relative speed between observers’ reference frames and c is the speed of light.
Calculate the Lorentz factor for different speeds, v. Start with a speed of about 1 km/s and increase from there. Create of plot of the Lorentz factor versus v. For what value of v will the Lorentz factor be significant? In other words, what speed v should be considered “relativistic”?
The proper time interval Δ𝜏 between two events is the time interval measured by an observer for whom both events occur at the same location.
Practice!
| Practice 1.3.5 |
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| Given that muons travel at 0.998 times the speed of light when they are created in the upper atmosphere by cosmic rays, why is it surprising that they reach the surface of Earth? |
| A. Because they’re traveling so fast that they go right through the Earth |
| B. Because muons are repelled by Earth’s magnetic field |
| C. Because their mean lifetime would seem to be too short, i.e., even at that high speed, they would not make it to the surface of the Earth before decaying into other particles |
| Practice 1.3.6 |
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| From the perspective of a person at the surface of Earth, how does the special theory of relativity explain how the muons make it down to the surface? |
| A. The person observes the muon’s velocity to speed up by the Lorentz factor, so that it gets down to the surface of the Earth more quickly than it otherwise would |
| B. The person observes the distance from the upper atmosphere to the surface of the Earth to be shortened, due to length contraction, so the muon doesn’t have to travel as far as it otherwise would |
| C. The person observes the muon’s internal “clock” running more slowly than when the muon is at rest, due to time dilation, so the moving muon stays intact longer than it otherwise would |
| Practice 1.3.7 |
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| From the perspective of a muon, how does the special theory of relativity explain how it makes it down to the surface? |
| A. In the muon’s frame of reference, its internal “clock” runs more slowly than when the muon is at rest, due to time dilation, so the moving muon stays intact longer than it otherwise would |
| B. In the muon’s frame of reference, the distance from the upper atmosphere to the surface of the Earth is shortened, due to length contraction, so the muon doesn’t have to travel as far as it otherwise would |
| C. In the muon’s frame of reference, the muon’s velocity increases by the Lorentz factor, so that it gets down to the surface of the Earth more quickly than it otherwise would |
| Practice 1.3.8 |
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| An unstable particle (i.e. a radioactive nucleus) has a half life of 10-8 s when it is at rest. If we produce a beam of these particles traveling at 4/5 times the speed of light, how far on average will the particles travel before decaying? |
| A. About 1.4 m |
| B. About 2.4 m |
| C. About 4 m |
| D. They won’t decay once they are moving |
Discuss!
It should be noted that the same equations apply to your everyday life. The reason that you don’t notice them is that objects in your everyday life move much slower than the speed of light. Now let’s look at the differences in measurements between two frames moving relative to one another at a speed of 30 m/s (about 67 mph). Your calculator may not be able to store enough digits to work these problems accurately, so you may need to use the approximations from the binomial expansion.
What would be the difference between the time t measured by an observer moving at 30 m/s and a proper time t0 for a proper time interval of 1 hour (3600 s)? The answer is small but non-zero. You will need to find an expression for the time difference using the approximation from the binomial expansion before you substitute in the numbers; otherwise your calculator will just give zero.
Check your answer: 1.8 x 10-11 s
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