Alice's rocket moves past Bob at 0.6c (γ = 1.25). Bob observes Alice's clock running slow, ticking at 80% of his own clock's rate. What does Alice observe about Bob's clock?
AAlice sees Bob's clock running fast, at 125% of her own clock's rate — because she is 'actually' the one in motion
BAlice sees Bob's clock running slow, at 80% of her own clock's rate — because in her frame, Bob is moving at 0.6c
CAlice sees Bob's clock running at the same rate as her own, because both clocks keep absolute time
DAlice cannot determine Bob's clock rate without knowing which frame is 'really' stationary
In special relativity there is no preferred frame — 'in motion' is a frame-dependent description. From Alice's perspective, she is at rest and Bob is moving at 0.6c in the opposite direction. By the same time dilation formula, she observes Bob's clock running at 80% of her own rate. Both Bob's and Alice's observations are correct within their respective frames. This is not a contradiction; it reflects the frame-dependence of simultaneity. Each is comparing clock readings at different pairs of events, and those pairs are not the same.
Question 2 Multiple Choice
A muon created in the upper atmosphere has a half-life of 2.2 μs in its rest frame, but reaches Earth's surface (roughly 15 km away) in large numbers. Which calculation correctly uses the Lorentz factor?
AThe muon travels 15 km in about 50 μs of Earth time; with γ ≈ 22, only 50/22 ≈ 2.3 μs of proper time elapses on the muon — within its half-life
BThe muon's half-life increases in the Earth frame by factor γ; with γ ≈ 22, the effective half-life is 2.2 × 22 ≈ 48 μs — long enough to reach the surface
CBoth A and B describe the same physical fact: the muon ages more slowly, consistent with γ ≈ 22, whether calculated as less proper time or longer lab-frame half-life
DTime dilation only applies to artificial clocks, not to decay rates of elementary particles
Both option A and B are correct descriptions of the same physical reality from different frames — which is why option C is the best answer. In the Earth frame, time dilation extends the muon's lab-frame half-life by γ ≈ 22, giving ~48 μs to traverse the ~15 km. In the muon's rest frame, length contraction reduces the 15 km to ~680 m, which takes only ~2.3 μs of proper time — within the half-life. Both calculations give the same answer (the muon survives) because proper time is invariant. This is the canonical experimental confirmation of relativistic time dilation.
Question 3 True / False
If Alice observes Bob's clock running slow, and Bob simultaneously observes Alice's clock running slow, then one of them should be making an error.
TTrue
FFalse
Answer: False
False. This symmetry is real and contains no contradiction. The apparent paradox arises from assuming there is a single objective 'present moment' shared by both observers — that there exists a unique instant where both clocks can be directly compared. Special relativity denies this: simultaneity is frame-dependent. When Bob says 'at this moment, Alice's clock reads T while mine reads T₀,' he is referring to events that are not simultaneous in Alice's frame. Both observers are correct about what they each observe, but they are observing different pairs of events.
Question 4 True / False
Proper time — the time measured by a clock traveling with an object — is invariant: every inertial observer calculates the same proper time accumulated between two events on the object's worldline.
TTrue
FFalse
Answer: True
True. Proper time τ = t/γ (where t is coordinate time in the 'stationary' frame) is a Lorentz-invariant quantity — it is the same number regardless of which frame you use to calculate it. This is the deep resolution of the symmetry puzzle: while coordinate time measurements differ between frames, proper time is the objective, frame-independent measure of 'how much a clock aged.' The twin paradox is resolved because the traveling twin's worldline has less proper time than the stay-at-home twin's, and every frame agrees on both proper time values.
Question 5 Short Answer
Why doesn't the symmetry of time dilation lead to a logical contradiction — if Alice sees Bob's clock running slow, and Bob sees Alice's clock running slow, how can both be correct?
Think about your answer, then reveal below.
Model answer: There is no contradiction because the two observers are comparing different pairs of events. 'Bob's clock reads 8 while Alice's clock reads 10' is a statement about two events — one on Bob's worldline and one on Alice's — that are simultaneous in one frame but not in the other. Simultaneity is frame-dependent in special relativity, so each observer picks a different pair of events to compare, and there is no single absolute comparison. Proper time resolves any apparent paradox: if both observers actually meet at two specific events, every frame agrees on how much proper time each accumulated between those events.
The symmetry becomes paradoxical only if you smuggle in an assumption of absolute simultaneity — that there is one objective 'now' at which both clocks can be compared. Once you accept that simultaneity is frame-dependent (a direct consequence of the two postulates of special relativity), the symmetry is not paradoxical but expected. Each observer's statement about the other's clock refers to a relativized comparison, not an absolute one. The twin paradox appears because one twin accelerates, breaking the symmetry: the non-inertial path accumulates less proper time, and all frames agree on the total difference when they reunite.