Overview
This lecture covers the concept of time dilation in special relativity, including its experimental verification, mathematical formulation, and the twin paradox.
Time Dilation Concept
- Time dilation means that moving clocks are observed to run slower compared to stationary clocks.
- The effect has been confirmed experimentally.
- The faster an object moves relative to an observer, the slower its time appears to pass.
Light Clock Thought Experiment
- A light clock measures time by bouncing light between two mirrors.
- For a stationary observer, round-trip time is ΔTs = 2D / c (D = mirror distance, c = speed of light).
- For a moving observer, the light travels a longer diagonal path due to the horizontal motion of the clock.
- As a result, the moving observer sees the clock ticking more slowly.
Time Dilation Formula
- Time measured in the moving frame: ΔTm = ΔTs / sqrt(1 - v²/c²).
- ΔTs is the proper time (measured by an observer at rest relative to the clock).
- ΔTm ≥ ΔTs; time in the moving frame is always longer.
- As velocity v approaches c, time dilation becomes more significant.
Worked Examples
- Example: Astronaut's pulse of 75 beats/min at v = 0.10c has a period of 1.33 × 10⁻² min per beat, relativistic period ≈ 1.34 × 10⁻² min, pulse rate ≈ 74.6 beats/min.
- At v = 0.90c, relativistic period ≈ 3.05 × 10⁻² min per beat, pulse rate ≈ 33 beats/min.
Practical Evidence
- Time dilation is observed in particle decay rates (halflife increases at high speed).
- Experiments with atomic clocks on fast airplanes confirm time dilation.
Twin Paradox
- Twin travels away from Earth at high speed and returns to find the stationary twin has aged more.
- From each perspective, the other's time should slow, apparently paradoxical.
- Solution 1: Shortest spatial path equals longest time path; twin on Earth experiences more proper time.
- Solution 2: The traveling twin changes inertial frames when turning around, causing the difference in elapsed time.
- Only the twin who accelerates (changes direction/frames) ages less.
Key Terms & Definitions
- Time Dilation — Phenomenon where time runs slower for moving objects relative to a stationary observer.
- Proper Time (ΔTs) — Time interval measured by an observer at rest relative to the event/clock.
- Inertial Frame — Frame of reference moving at a constant velocity (no acceleration).
- Twin Paradox — Scenario showing asymmetrical aging due to time dilation when one twin travels at relativistic speeds.
Action Items / Next Steps
- Try assigned homework problems on time dilation.
- Review textbook derivation of the time dilation formula.
- Prepare for a deeper discussion of relativity in the next lesson.