Special Relativity

The two postulates

1. Principle of Relativity: The laws of physics are the same in all inertial reference frames.
2. Constancy of Light Speed: The speed of light in vacuum c is the same for all inertial observers, regardless of the motion of the source or observer.

From these (with consistency required), all of special relativity follows.

The Lorentz factor

γ = 1 / √(1 - v²/c²)

v/c	γ
0	1
0.1	1.005
0.5	1.155
0.866	2
0.99	7.09
0.999	22.4
0.9999	70.7
1	∞

Relativistic effects

Effect	Formula	Description
Time dilation	Δt' = γ·Δt	Moving clock runs slow
Length contraction	L' = L/γ	Moving objects shorter (along motion)
Relativistic mass (old form)	m_rel = γm₀	Mass effectively increases (modern physics avoids this terminology)
Energy-momentum	E² = (pc)² + (mc²)²	Including rest energy
Relativistic momentum	p = γmv	Replaces Newtonian mv
Velocity addition	u' = (u-v)/(1-uv/c²)	Doesn't exceed c
Doppler effect (longitudinal)	f' = f·√((1+β)/(1-β))	Stronger than classical

JavaScript — relativistic calculations

const c = 299792458;

function lorentzFactor(v) {
  const beta = v / c;
  if (beta >= 1) return Infinity;
  return 1 / Math.sqrt(1 - beta * beta);
}

console.log(`γ at 0.5c: ${lorentzFactor(0.5*c).toFixed(4)}`);  // 1.155
console.log(`γ at 0.99c: ${lorentzFactor(0.99*c).toFixed(2)}`); // 7.09
console.log(`γ at 0.999c: ${lorentzFactor(0.999*c).toFixed(1)}`); // 22.4

// Time dilation
function timeDilation(t_at_rest, v) {
  return t_at_rest * lorentzFactor(v);
}

// 1 second on Earth → time on a craft at 0.99c
console.log(`1 sec at 0.99c: ${timeDilation(1, 0.99*c).toFixed(2)} sec on Earth`);

// Twin paradox: traveler ages less
function travelerAging(distance_ly, v_fraction_c) {
  // From Earth: time = distance/v
  const t_earth = distance_ly / v_fraction_c;
  // From traveler: distance is contracted; time = (distance/γ)/v
  const gamma = lorentzFactor(v_fraction_c * c);
  const t_traveler = t_earth / gamma;
  return { Earth_time: t_earth, Traveler_time: t_traveler, age_diff: t_earth - t_traveler };
}

// 4.2 light-years (Alpha Centauri) at 0.95c
console.log(travelerAging(4.2, 0.95));

// Length contraction
function lengthContraction(L_at_rest, v) {
  return L_at_rest / lorentzFactor(v);
}

console.log(`1m ruler at 0.99c: ${lengthContraction(1, 0.99*c).toFixed(2)} m`);

// Velocity addition (relativistic)
function relAdd(u, v) {
  return (u + v) / (1 + (u * v) / (c * c));
}

console.log(`0.5c + 0.5c = ${(relAdd(0.5*c, 0.5*c) / c).toFixed(2)}c`); // 0.8c
console.log(`0.9c + 0.9c = ${(relAdd(0.9*c, 0.9*c) / c).toFixed(3)}c`); // 0.994c
console.log(`0.99c + 0.99c = ${(relAdd(0.99*c, 0.99*c) / c).toFixed(4)}c`); // 0.9999c — never exceeds c

// Energy of moving particle
function relEnergy(rest_mass_kg, v) {
  return lorentzFactor(v) * rest_mass_kg * c * c;
}

console.log(`Electron rest E: ${(relEnergy(9.11e-31, 0) / 1.602e-19 / 1e6).toFixed(3)} MeV`);
console.log(`Electron at 0.99c: ${(relEnergy(9.11e-31, 0.99*c) / 1.602e-19 / 1e6).toFixed(2)} MeV`);

// GPS clock correction
function gpsTimeDilation_special(orbital_v_kmh) {
  const v = orbital_v_kmh * 1000 / 3600;
  // Per second of "Earth time," moving clock loses time
  return (1 - 1/lorentzFactor(v)) * 1e6 * 86400;  // microseconds per day
}

console.log(`GPS SR slowdown: ${gpsTimeDilation_special(14000).toFixed(2)} µs/day`);  // ~7

Where special relativity matters

• GPS. Satellites run ~38 µs/day faster than Earth (combined SR + GR effects); without correction, errors accumulate to km/day.
• Particle accelerators. Particles at > 99% c need full relativistic mechanics.
• Cosmic rays. Muons created in atmosphere reach ground only because of time dilation (otherwise decay too quickly).
• Nuclear reactions. Mass-energy equivalence E = mc² explains binding energies.
• Astrophysics. Relativistic jets from black holes, quasars; cosmological redshifts.
• Engineering. Particle therapy (cancer), high-energy physics experiments.
• Foundations. Established space and time as components of unified spacetime.

Common mistakes

• Confusing reference frames. Each observer's measurements are correct in their own frame. They don't contradict; they describe different perspectives of the same physics.
• Thinking SR violates conservation laws. Energy and momentum still conserved (using relativistic forms). Total mass-energy conserved.
• Treating "twin paradox" as real paradox. Resolved by noting traveler accelerates (changes frames). Asymmetry between traveler and stay-at-home is real.
• Using Newtonian formulas at high v. p = mv, KE = ½mv² fail. Use p = γmv, KE = (γ-1)mc² instead.
• Believing nothing can move faster than c. True for objects with mass. Faster-than-light "things" (like shadows on a wall) can move arbitrarily fast — but they don't carry information or matter.
• Confusing time dilation with the universe slowing. Each observer's clock works fine in their frame. From OTHER frames, clocks appear to run slow (mutually).