The twin paradox is a famous thought experiment in special relativity where one twin stays on Earth while the other travels in a fast rocket and returns younger. It matters because it shows that time is not universal for all observers. Each twin measures their own proper time along a path through spacetime.
The surprising result is that the path involving high speed travel and turnaround contains less elapsed proper time.
Key Facts
- Time dilation for constant speed is Δt = γΔτ, where γ = 1/sqrt(1 - v^2/c^2).
- Proper time Δτ is the time measured by a clock moving along with an observer.
- For an inertial segment, the traveling twin ages by Δτ = Δt sqrt(1 - v^2/c^2).
- The Earth twin and traveling twin do not follow the same spacetime path between departure and reunion.
- The traveling twin changes inertial frames during the turnaround, creating the key asymmetry.
- Acceleration is not the direct cause of less aging, but it marks the frame switch needed for the traveler to return.
Vocabulary
- Proper time
- Proper time is the time interval measured by a clock that is present at both events being compared.
- Time dilation
- Time dilation is the effect where a moving clock is measured to tick more slowly than a clock at rest in an observer's frame.
- Lorentz factor
- The Lorentz factor γ is the multiplier 1/sqrt(1 - v^2/c^2) that describes how strongly relativistic effects appear.
- Worldline
- A worldline is the path an object traces through spacetime as its position changes over time.
- Inertial frame
- An inertial frame is a non-accelerating reference frame in which objects with no net force move at constant velocity.
Common Mistakes to Avoid
- Saying both twins must age less than each other is wrong because they are not in symmetric situations. The traveling twin changes inertial frames while the Earth twin remains approximately in one inertial frame.
- Treating acceleration as the whole cause is wrong because most of the age difference can occur during long constant-speed coasting segments. Acceleration is important because it makes the trip path asymmetric and allows the traveler to return.
- Using Δt = Δτ/γ for the wrong clock is wrong because Δτ must be the proper time of the clock present at both events. Always identify which clock is moving with the events being timed.
- Assuming the paradox violates relativity is wrong because relativity compares spacetime paths, not just relative speeds. When both twins reunite, the elapsed proper times along their different worldlines can be directly compared.
Practice Questions
- 1 A rocket travels away from Earth at 0.80c for 5.0 years as measured in Earth's frame, then instantly turns around and returns at 0.80c for 5.0 more Earth years. How much does the traveling twin age during the whole trip?
- 2 A spaceship moves at 0.60c relative to Earth. If the Earth twin measures the trip duration as 20 years, how much proper time passes for the traveling twin during the constant-speed parts?
- 3 Explain why the two twins cannot both use the same simple time dilation argument to claim the other twin is younger when they reunite.