Special-relativity postulates

A frame is not an observer's eye; it is a coordinate system, a lattice of rulers filling space with a clock at every node, all of them reading the same time. The frame is inertial if Newton's first law holds in it: a body with no force on it moves in a straight line at constant speed. That is a testable condition, and plenty of laboratories fail it. A rotating turntable fails, because a free puck curves across it. A braking car fails, because loose objects slide forward. The Earth's surface fails only mildly: at the equator the centripetal acceleration is ω²R = (7.29 × 10⁻⁵ s⁻¹)² × 6.38 × 10⁶ m = 0.034 m s⁻², about 0.35% of g, so for a bench experiment lasting a second the laboratory counts as inertial. Notice what that definition has already smuggled in. 'All of them reading the same time' is a claim about clocks in different places, and nothing so far says how they were set. That gap is where relativity walks in.

Galileo's principle says no experiment inside a smoothly moving ship reveals its motion, and mechanics obeys it: under x' = x − vt with t' = t, velocities shift by v but accelerations do not, so F = ma reads identically in both frames. Einstein's first postulate upgrades 'no mechanical experiment' to 'no experiment at all'. Electromagnetism immediately objects. Maxwell's equations predict waves at 1/√(ε₀μ₀) = 1/√(8.854 × 10⁻¹² × 1.257 × 10⁻⁶) = 3.00 × 10⁸ m s⁻¹, a number assembled from two bench constants with no frame attached to it. Nineteenth-century physics answered by giving light a medium, the aether, and reading Maxwell's speed as the speed in the aether's rest frame. That is a testable claim, because the Earth's 30 km s⁻¹ orbit should sweep the laboratory through the aether at β = 1.0 × 10⁻⁴, reversing every six months.

Michelson and Morley ran that test in 1887. Split a beam, send the halves along perpendicular 11 m arms, recombine them and rotate the table: the predicted fringe shift 2Lv²/(λc²), with λ = 590 nm and v = 30 km s⁻¹, is about 0.4 of a fringe, and the apparatus could resolve a fortieth of that. They found nothing, and every repetition since has found nothing. The other loophole, that light might be carried along by its source the way a ball is by a moving truck, was shut at CERN in 1964: neutral pions flying at 0.99975c decayed into photons, and those photons were clocked at 2.998 × 10⁸ m s⁻¹, not the 5.995 × 10⁸ m s⁻¹ that adding the source's speed predicts. So the second postulate is an experimental result promoted to an axiom. Read it precisely: one particular speed has the same value in every inertial frame. The ban on exceeding that speed is a consequence drawn later, not part of the postulate.

Put a headlight on a train moving at 0.60c. Galileo says the ground frame measures the beam at c + 0.60c = 1.60c; the light postulate says c. They cannot both be right, and the experiments side against Galileo. The two frames still disagree about how far the beam travelled, so if they are to agree about its speed they must also disagree about how long it took: the assumption t' = t has to go. What replaces the addition rule is u = (u' + v)/(1 + u'v/c²). Try it on a probe launched forward at 0.80c from a ship that itself passes a station at 0.60c: the station measures (0.80 + 0.60)/(1 + 0.48) = 0.946c = 2.84 × 10⁸ m s⁻¹, not the 1.40c Galileo predicts, and still under c. Try it on light, u' = c: (c + 0.60c)/(1 + 0.60) = c exactly, for any v at all. The rule is built so that c is the one speed nobody can argue about.

Before you can say that two distant events happened at the same time, you must have set the clocks. Einstein's recipe leans on the second postulate: send a pulse from A at t₁, bounce it off B, receive it back at t₂, and set B's clock to t₁ + (t₂ − t₁)/2, which is legitimate only because the pulse takes the same time each way in this frame. Now run the flash experiment. A rocket 240 m long in the station frame passes at 0.40c, and a flash at its midpoint sends pulses toward the nose and the tail. In the station frame each pulse moves at c while the tail advances into one and the nose retreats from the other, so the tail is hit at 120 m ÷ 1.40c = 286 ns and the nose at 120 m ÷ 0.60c = 667 ns, a gap of 381 ns. In the rocket's frame the ends are equidistant and the arrivals coincide. The same breakdown shows up in clocks rather than in pulses: two clocks synchronised a distance L₀ apart in their own rest frame are read, from a frame moving at v along that line, as offset by vL₀/c², with the trailing clock ahead.

An event is a happening with one place and one instant: a spark, a detector click, a pulse leaving a lamp. Frames give it different coordinates (t, x), and as we have just seen they can even disagree about the order of two events. What every inertial frame does agree on is the interval, (Δs)² = (cΔt)² − (Δx)². Take two laboratory clicks 200 m and 1.00 μs apart: cΔt = 300 m, so (Δs)² = 300² − 200² = +5.00 × 10⁴ m², positive. A positive interval is called timelike, and it means a signal at 200 m ÷ 1.00 μs = 2.00 × 10⁸ m s⁻¹ = 0.667c, comfortably under light speed, can run from the first click to the second. Every frame then agrees which came first, so cause can safely precede effect. The frame moving at that same 0.667c finds both clicks at one place, and there cΔt = √(5.00 × 10⁴ m²) = 224 m. Let (Δx)² be the larger term instead and (Δs)² comes out negative; the pair is then spacelike, no signal at or below c joins it, some frame calls the two events simultaneous and faster frames reverse their order, with no causal link at risk because none could exist. When (Δs)² = 0 a light pulse joins them exactly, which is the light postulate written as geometry.