Chapter 4: Classic Paradoxes
The counterintuitive predictions of special relativity lead to apparent paradoxes. Resolving them deepens our understanding and shows how relativity is internally consistent. These aren't flaws in the theory—they're features that reveal our Galilean intuitions are wrong.
The Twin Paradox
The Puzzle
Twin A stays on Earth. Twin B travels to a star at 0.9c and returns. By time dilation, A sees B's clocks running slow, so B ages less. But B sees A moving, so shouldn't A age less?
The Resolution
The situation is NOT symmetric! Twin B must accelerate to leave, turn around, and return. B's worldline involves non-inertial motion. On a spacetime diagram:
- • A's worldline is straight (inertial)
- • B's worldline has a "kink" at turnaround
- • The twin with the straight worldline ages more (maximizes proper time)
Numerical Example
Star is 4 light-years away. At v = 0.8c: γ = 5/3. Earth time for round trip: 10 years. Traveler's proper time: 10/γ = 6 years. The traveling twin returns 4 years younger!
The Barn-Pole (Ladder) Paradox
The Puzzle
A 10m pole runs through an 8m barn at high speed (γ = 2). Barn frame: pole contracts to 5m, fits easily. Pole frame: barn contracts to 4m, pole can't fit. Can both doors be closed simultaneously?
The Resolution
"Simultaneously" means different things in different frames!
- • Barn frame: Both doors close at the same time; pole fits
- • Pole frame: Front door closes first, then back. The pole is never fully inside, but also never crashes through a closed door
The relativity of simultaneity resolves the apparent contradiction. There's no physical paradox—just our failure to account for simultaneity being frame-dependent.
The Train-Tunnel Paradox
The Puzzle
A train (proper length L) passes through a tunnel (proper length L). In the tunnel frame, the train is contracted and fits inside. In the train frame, the tunnel is contracted and the train doesn't fit. Can a bomb at each end of the tunnel explode simultaneously when triggered "while the train is fully inside"?
The Resolution
Again, simultaneity! In the tunnel frame, the bombs explode simultaneously while the train is inside. In the train frame, one bomb explodes before the front enters (or after the back exits). The physical events (explosions) happen—but their timing relative to the train's position differs by frame.