State Einstein's two postulates of special relativity.

1) Mass and energy are equivalent. 2) The speed of light depends on the observer's speed 1) The laws of physics are the same in all inertial frames. 2) The speed of light in vacuum is the same (c ≈ 3 × 10⁸ m/s) for all observers, regardless of relative motion 1) Time passes equally for all observers. 2) The speed of light increases with temperature 1) All motion is relative. 2) Space is curved by massive objects

What was the Michelson-Morley experiment, and what did its null result imply?

An experiment that measured the speed of light for the first time — the result was 3 × 10⁸ m/s An 1887 experiment searching for the ether (medium for light propagation) — its failure to detect any ether drift showed light's speed does not depend on Earth's motion through space A test of general relativity — it detected the curvature of space near the Sun An experiment that showed the speed of light is slightly slower in glass than in vacuum

What is an inertial reference frame?

A frame attached to an accelerating rocket — special relativity applies here A frame of reference moving at constant velocity (not accelerating) — in such a frame, Newton's first law holds: objects at rest stay at rest unless acted on by a force The frame of absolute space — the unique reference frame in which light always travels at c A frame tied to Earth's surface — since Earth is approximately non-accelerating for everyday purposes

A spaceship travels at 0.9c and fires a photon forward. What speed does the photon travel at, measured by a stationary observer?

1.9c — the spaceship's speed adds to the photon's speed 0.1c — the spaceship is 'using up' most of the available speed, leaving little for the photon c — the speed of light is c in all inertial frames; the photon's speed does not add to the spaceship's speed c + 0.9c = 1.9c for the photon, but only c relative to the ship — the observer sees 1.9c

What is the Lorentz factor γ (gamma), and when is it important?

γ = v/c — the ratio of the object's speed to the speed of light γ = 1/√(1−v²/c²) — at low speeds (v << c), γ ≈ 1 and classical physics is recovered; near c, γ → ∞, making relativistic effects dramatic γ = mc² — the relativistic energy of the particle γ = √(1−v²/c²) — it decreases from 1 at rest to 0 at the speed of light

What does it mean that simultaneity is relative in special relativity?

All observers agree on which events are simultaneous — relativity only affects time duration Two events that are simultaneous in one inertial frame are NOT simultaneous in another frame moving relative to the first — there is no universal 'now' for all observers Simultaneity is defined only relative to the speed of light — events are simultaneous if light from both reaches an observer at the same time Relativity says nothing about simultaneity — only about time dilation and length contraction

Why can no material object be accelerated to the speed of light?

Air resistance becomes infinite near the speed of light — this is what prevents objects from reaching c As v → c, relativistic momentum p = γmv → ∞ — an infinite force for infinite time would be needed to reach c; it is kinematically impossible for massive objects Objects can reach the speed of light — it just requires a nuclear reactor to provide enough power Objects cannot exceed c because of air resistance and gravity slowing them down

How does GPS rely on special (and general) relativity to provide accurate positioning?

GPS doesn't need relativistic corrections — the effects are too small to matter in practice GPS satellites move fast (special relativity: clocks run slow by ~7 μs/day) and are high up (general relativity: clocks run fast by ~45 μs/day) — without corrections totalling ~38 μs/day, GPS would accumulate ~10 km of position error per day GPS corrects only for special relativity — general relativity has no practical effect GPS requires Newtonian mechanics — relativity would reduce its accuracy

Cosmic ray muons produced at 15 km altitude reach Earth's surface despite having a proper half-life of ~1.5 μs. How?

The muons have more energy than expected, slowing their decay — this is purely energy conservation, not relativity Muons travel at v ≈ 0.998c; their γ ≈ 15 — time dilation extends their lifetime by 15× from Earth's frame, allowing them to travel 15 km before most decay The atmosphere is thinner at high altitudes — muons don't decay until they hit dense air near the surface Muons travel at the speed of light — at c, time stops, so they never decay

Two spaceships approach each other, each travelling at 0.7c relative to Earth. What is their speed relative to each other?

1.4c — in their own reference frames, they see each other approaching faster than light Exactly c — any two velocities below c combine to exactly c About 0.94c — using relativistic velocity addition: u = (0.7c + 0.7c)/(1 + 0.7×0.7) = 1.4c/1.49 ≈ 0.94c 0.7c — the ships cancel each other's velocity due to their equal and opposite motions