☰ Contents · Physics

Foundations of special relativity

Lessons 32 · 1 lessons · N. Sh. Turdiyev, K. A. Tursunmetov, A. G. Ganiyev, K. T. Suyarov, J. E. Usarov, A. K. Avliyoqulov. Physics Grade 11, 1st edition. Niso Poligraf Publishing House, Tashkent, 2018
32

Foundations of special relativity. The relativistic law of velocity addition

Textbook: pp. 124–127
GoalKnow Einstein's postulates and apply their consequences (time dilation, length contraction, the relativistic law of velocity addition) in simple problems.
New words
inertial frame of reference · inersial sanoq sistemasipostulate · postulatproper time · xususiy vaqtLorentz contraction · Lorens qisqarishi
Explanation

In classical mechanics velocities simply add: a passenger walking in a train moves relative to the ground at u₁ + u. But Maxwell's equations imply that the speed of light c is the same for every observer, and experiments (Michelson–Morley) showed that it does not depend on the Earth's motion. In 1905 Einstein built special relativity on two postulates: (1) all laws of physics are the same in all inertial frames; (2) the speed of light in vacuum is the same in all inertial frames and does not depend on the motion of the source or the observer. It follows that a moving clock runs slower, τ = τ₀/√(1 − υ²/c²), where τ₀ is the proper time measured by an observer moving with the clock; length along the direction of motion contracts, l = l₀·√(1 − υ²/c²); and velocities add as u = (u₁ + υ)/(1 + u₁υ/c²). When all speeds are much smaller than c these formulas reduce to the classical ones, so relativity does not cancel Newtonian mechanics; it marks its limit of validity.

Worked examples
A clock on a rocket measured 12 minutes (proper time) while the rocket moved at 0.8 c relative to the Earth. Then √(1 − 0.64) = 0.6, so an Earth observer measures τ = 12/0.6 = 20 minutes.
Two ships go in the same direction: the first moves at 0.6 c relative to a station and a second ship is launched ahead of it at 0.6 c relative to the first. The classical law would give 1.2 c (impossible). Correct: u = (0.6 + 0.6)/(1 + 0.36) = 1.2/1.36 ≈ 0.88 c < c.
Class activity

In your notebook make a table of 1/√(1 − β²) for speeds 0.1 c, 0.6 c, 0.8 c, 0.9 c and 0.99 c, and decide from which speed the effect becomes noticeable. Use a calculator; no experiment is needed.

Practice
1
State Einstein's two postulates.
2
A rod of proper length 50 m flies at 0.6 c. What length in m does an observer measure?
3
A particle is shot from a rocket moving at 0.4 c, in the direction of motion, at 0.5 c relative to the rocket. What is its speed relative to the Earth?
4
Why is simple addition of velocities nearly correct for trains but wrong for light?