State the time dilation formula and explain the meaning of 'proper time.'

Δt₀ = γΔt — proper time is longer than the dilated time Δt = γΔt₀ — proper time (Δt₀) is the time measured by a clock that is present at both events; any other observer measures a longer dilated time Δt = γΔt₀ Δt = Δt₀/γ — moving clocks run faster by the Lorentz factor Time dilation only applies to electromagnetic phenomena — mechanical clocks are unaffected

A muon travels at v = 0.99c. Its proper lifetime is 2.2 μs. What lifetime does an Earth observer measure?

2.2 μs — the muon's lifetime is fixed and doesn't change About 15.6 μs — Δt = γΔt₀, γ = 1/√(1−0.99²) ≈ 7.09, so Δt = 7.09 × 2.2 ≈ 15.6 μs About 0.31 μs — the muon's clock runs faster at high speed About 220 μs — speed squared gives a factor of 100 × 2.2 μs

What is length contraction, and in which direction does it occur?

Moving objects expand in all directions — length contraction is a misnomer Moving objects are contracted along the direction of motion by factor 1/γ: L = L₀/γ. Dimensions perpendicular to motion are unchanged. Objects contract in all three spatial dimensions equally when moving Length contraction affects only the transverse dimensions, not the direction of motion

A spaceship is 100 m long in its rest frame and travels at 0.8c relative to Earth. How long does an Earth observer measure it to be?

100 m — length contraction only applies to the time dimension, not physical size 125 m — the ship appears stretched when moving fast 60 m — L = L₀/γ = 100/1.667 = 60 m (γ at 0.8c = 1/√(1−0.64) = 1/√0.36 = 1/0.6 ≈ 1.667) 80 m — the contraction equals the fraction of c (80% of 100 m)

The twin paradox: one twin (Alice) travels at high speed to a distant star and returns. Is Alice genuinely younger than her twin (Bob) who stayed on Earth?

No — special relativity says the situation is symmetric; both twins age equally Yes — Alice is genuinely younger. The 'paradox' is resolved because Alice (not Bob) experienced acceleration (turning around), breaking the symmetry between the two frames Bob is younger — the stay-at-home twin ages less because he didn't use energy The paradox is unsolvable — special relativity gives contradictory predictions for the twin scenario

What is spacetime, and why is it more fundamental than space and time separately?

Spacetime is just a way of drawing space and time on the same graph — they remain fundamentally separate Spacetime is the 4D union of space (3D) and time — Minkowski showed that the spacetime interval ds² = c²dt² − dx² is invariant for all observers, while space and time separately are observer-dependent Spacetime was Einstein's name for the ether — the medium through which light travels Spacetime is only relevant in general relativity (gravity) — not in special relativity

At the Large Hadron Collider (LHC), protons are accelerated to v ≈ 0.999999991c with γ ≈ 7461. What does this mean for the proton's energy?

The proton's mass increases to 7461 proton masses, but its energy stays at 938 MeV The proton's total energy is E = γmc² ≈ 7461 × 938 MeV ≈ 6.99 TeV — 7461 times its rest mass energy At such high speed, the proton has zero energy — it moved all its mass-energy to kinetic form The proton moves at exactly c inside the LHC — its energy is infinite

Is time dilation only a prediction of special relativity, or has it been directly measured?

Time dilation is still theoretical — no experiment has ever measured it directly Directly measured — the Hafele-Keating experiment (1971) flew atomic clocks on commercial aircraft; the recorded time differences agreed with SR and GR predictions to within experimental error Time dilation has only been measured indirectly — no direct clock comparison has been done Time dilation is measured only at speeds greater than 0.99c — at everyday speeds, it cannot be detected even in principle

From the muon's frame, why does it manage to travel 15 km to Earth's surface in only 2.2 μs proper time?

In the muon's frame, 2.2 μs is sufficient because time runs faster for the muon Length contraction: from the muon's frame, the atmosphere is Lorentz-contracted by factor γ ≈ 7 to ~2 km — which the muon crosses easily in 2.2 μs In the muon's frame, the distance to Earth is 15 km but it travels faster than c to cross it in 2.2 μs The muon's frame is identical to Earth's frame — both agree on a 15 km distance and 15 μs travel time

What does the invariance of the spacetime interval mean physically?

All observers agree on the spatial distance between events but not on the time separation All inertial observers agree on ds² = c²dt² − dx² − dy² − dz² for any pair of events — even though they disagree on dt and dx separately; spacetime geometry is objective even when space and time are not The spacetime interval is a theoretical construct — it has no physical meaning The spacetime interval is only invariant in general relativity, not special relativity