A spaceship travels past Earth at v = 0.866c (Lorentz factor γ = 2) for what Earth clocks measure as 10 seconds. How much time elapses on the spaceship's own clock?
A20 seconds — the moving clock ticks faster to compensate for the relative motion
B10 seconds — there is no time dilation at constant velocity
C5 seconds — the moving clock ticks slower, so less proper time elapses on the ship
DIt depends on which observer you ask, so no definite answer exists
The spaceship clock measures proper time Δt₀, since the clock is present at both events (ship at departure point and ship at arrival point) in its own rest frame. The relationship is Δt = γΔt₀, so Δt₀ = Δt/γ = 10/2 = 5 seconds. The moving clock runs slow — less proper time elapses along the ship's worldline than coordinate time elapses in the Earth frame. Option D is a common confusion: proper time is a Lorentz scalar, the same in all frames. Option A reverses the direction of time dilation.
Question 2 Multiple Choice
Muons created by cosmic rays at 15 km altitude travel at ~0.999c with a proper lifetime of 2.2 μs. Classical (non-relativistic) physics predicts they should decay after traveling only ~660 m. Yet they reach Earth's surface in large numbers. The special-relativistic explanation in the Earth frame is:
AThe muons' internal decay process is genuinely slowed by the energy of motion
BThe coordinate time in the Earth frame is stretched by γ ≈ 22, giving the muon an apparent lifetime of ~50 μs, long enough to cover ~15 km
CThe muons' mass increases at high speed, slowing their decay rate
DThe distance to the surface is length-contracted so the muon travels a shorter path
In the Earth frame, the muon's proper lifetime of 2.2 μs corresponds to a coordinate time of γ × 2.2 μs ≈ 22 × 2.2 μs ≈ 50 μs. At 0.999c, this is enough time to travel ~15 km. The muon does not actually live longer in any absolute sense — its proper lifetime (elapsed time in its own rest frame) is still 2.2 μs. But in the Earth frame, the time dilation factor γ ≈ 22 stretches the coordinate duration. Option D is the muon's-rest-frame explanation (length contraction), which is equally valid but describes the same physical outcome from a different frame. Option C describes relativistic mass, which is a deprecated concept — mass does not cause slower decay.
Question 3 True / False
Since each observer sees the other's clock running slow in special relativity, there is no frame-independent fact about how much time elapses along a worldline.
TTrue
FFalse
Answer: False
Proper time is a Lorentz scalar — it has the same value in every inertial frame. All observers agree on how much proper time elapses along a given worldline between two events, even though they may disagree on the coordinate time. The mutual time-dilation effect (each frame sees the other's clock slow) applies to coordinate time, not proper time. Proper time is precisely the frame-independent physical quantity: it is what a clock physically accumulates as it travels between two events. This is why proper time becomes the natural 'arc length' of a worldline in spacetime geometry.
Question 4 True / False
Time dilation is a real physical effect, confirmed by experiment, not merely a coordinate artifact of the reference frame chosen.
TTrue
FFalse
Answer: True
The muon decay experiment directly confirms this. Muons arrive at Earth's surface in quantities that are only consistent with time dilation — their apparent lifetime in the lab frame is ~22 times their proper lifetime. No coordinate choice or perceptual illusion can explain this; the muons physically survive a journey they should not survive based on their decay rate. GPS satellites also provide a real-world application: their atomic clocks accumulate proper time at a slightly different rate than Earth-surface clocks (both special and general relativistic effects), requiring daily corrections. Time dilation is as empirically real as any physical effect.
Question 5 Short Answer
What is proper time, why is it the minimum time elapsed between two events connected by a physical process, and what makes it more fundamental than coordinate time in special relativity?
Think about your answer, then reveal below.
Model answer: Proper time is the time measured by a clock that is physically present at both events — it travels with the process being timed and has no spatial displacement in its own rest frame. It is the minimum because any other clock, moving relative to the first, accumulates additional elapsed time due to its motion through space; the Lorentz factor γ ≥ 1 ensures that all other frames measure a longer interval Δt = γΔt₀. Proper time is more fundamental because it is a Lorentz scalar — frame-independent, the same for all observers. Coordinate time depends on the reference frame and is not invariant. In spacetime geometry, proper time is the natural arc length of a worldline, analogous to path length in ordinary space.
The deeper point is about what 'really happens' physically versus how we describe it in coordinates. Different observers use different coordinate systems, leading to different coordinate times — but they all agree on the proper time elapsed along any given worldline. This invariance makes proper time the physically meaningful quantity: it is what a biological process ages by, what a radioactive nucleus decays according to, what a clock physically displays. Coordinate time is merely a label. This perspective anticipates the spacetime geometry viewpoint, where proper time as arc length immediately explains the twin paradox: curved worldlines (accelerated motion) have shorter proper time than straight ones (inertial motion) between the same two events.