Time Dilation

Special relativistic time dilation

For an observer moving at speed v relative to a clock:

Δt_observed = γ · Δt_proper

where Δt_proper is the time interval measured by the clock at rest in its own frame, and γ = 1/√(1-v²/c²).

v/c	γ	Time dilation
0.001 (300 km/s)	1.0000005	0.5 ppb (negligible)
0.01 (3 × 10⁶ m/s)	1.00005	50 ppm
0.1	1.005	0.5%
0.5	1.155	15.5%
0.866	2	2× slower
0.99	7.09	7× slower
0.9999	70.7	71× slower

Gravitational time dilation

Clocks run slower in stronger gravitational fields. For weak fields:

Δt' / Δt ≈ 1 - GM/(r·c²)

So a clock at radius r runs slower than a clock at infinity by a factor (1 - GM/(rc²)).

Examples:

Location	Time difference vs infinity
Earth's surface	~7 × 10⁻¹⁰ slower than infinity
Sun's surface	~2 × 10⁻⁶ slower
Just outside black hole event horizon	Approaches zero (infinite slowdown)
Earth's center vs surface	Earth's center clocks run ~~3 × 10⁻¹⁰ slower per s

Confirmations

Experiment	Year	Result
Hafele-Keating (clocks on jets)	1971	SR + GR effects measured
Pound-Rebka (gravity redshift)	1959	22.5 m vertical Δf measured
Muon time dilation (cosmic rays)	Continuous	Confirms γ for muons
GPS satellites	1980s+	Need both SR and GR corrections
Atomic clock altitude (Pound-Rebka 2.0)	2010s	10⁻¹⁸ precision tests
Astronaut twins (Scott Kelly)	2015-2016	5 ms time difference detected

JavaScript — time dilation

const c = 299792458;
const G = 6.674e-11;

// Special relativistic
function specialTimeDilation(time_proper, v) {
  const gamma = 1 / Math.sqrt(1 - (v/c)**2);
  return gamma * time_proper;
}

// 1 year at 0.95c
console.log(`1 yr at 0.95c: ${specialTimeDilation(1, 0.95*c).toFixed(2)} yr in lab frame`);

// Muon flight (rest lifetime 2.2 µs at 0.999c)
const muon_life_rest = 2.2e-6;
const muon_v = 0.999 * c;
const muon_life_lab = specialTimeDilation(muon_life_rest, muon_v);
const muon_distance = muon_v * muon_life_lab;
console.log(`Muon at 0.999c: lab lifetime ${(muon_life_lab * 1e6).toFixed(1)} µs`);
console.log(`Muon travel distance: ${muon_distance.toFixed(0)} m = ${(muon_distance/1000).toFixed(2)} km`);
// ~14.7 km — way more than 660 m without dilation; reaches Earth surface

// Gravitational time dilation (weak field)
function gravTimeDilation(M, r) {
  return Math.sqrt(1 - 2 * G * M / (r * c * c));
}

const M_earth = 5.972e24;
const R_earth = 6.371e6;
console.log(`Earth surface vs infinity: ${gravTimeDilation(M_earth, R_earth).toFixed(12)}`);

// GPS satellite at 20,200 km altitude
const r_GPS = R_earth + 20.2e6;
const v_GPS = 3.87e3;  // m/s
const tau_GPS_SR = 1 - 1/(2 * (c*c)) * v_GPS * v_GPS;
const tau_ground_GR = gravTimeDilation(M_earth, R_earth);
const tau_GPS_GR = gravTimeDilation(M_earth, r_GPS);

const SR_per_day = (1 - tau_GPS_SR) * 86400 * 1e6;
const GR_per_day = (tau_GPS_GR/tau_ground_GR - 1) * 86400 * 1e6;
console.log(`GPS SR slowdown: -${SR_per_day.toFixed(1)} µs/day`);
console.log(`GPS GR speedup: +${GR_per_day.toFixed(1)} µs/day`);
console.log(`Net: +${(GR_per_day - SR_per_day).toFixed(1)} µs/day`);
// ~+38 µs/day matches GPS engineering requirement

// Twin paradox: 1 year on Earth = ? on traveler
function twinParadox(earth_years, v_fraction_c) {
  const gamma = 1 / Math.sqrt(1 - v_fraction_c * v_fraction_c);
  return earth_years / gamma;  // traveler experiences less time
}

console.log(`Earth 100 yr at 0.99c traveler: ${twinParadox(100, 0.99).toFixed(2)} yr`);
console.log(`Earth 100 yr at 0.999c: ${twinParadox(100, 0.999).toFixed(2)} yr`);

// Black hole: time dilation factor at distance r outside event horizon
function bhTimeDilation(M, r) {
  const r_s = 2 * G * M / (c * c);
  if (r < r_s) return 0;
  return Math.sqrt(1 - r_s/r);
}

// Just outside Sgr A* (4.3M solar mass)
const M_sgr = 4.3e6 * 1.989e30;
const r_s_sgr = 2 * G * M_sgr / (c * c);
console.log(`Sgr A* event horizon: ${(r_s_sgr / 1e9).toFixed(2)} Gm`);
console.log(`Time at 1.01 × r_s: ${bhTimeDilation(M_sgr, r_s_sgr * 1.01).toFixed(3)}× normal`);

Where time dilation matters

• GPS. Daily corrections for both SR and GR effects.
• Particle physics. Particle lifetimes, decay chains, accelerator beam dynamics.
• Astrophysics. Pulsars, neutron star precision, gravitational lensing.
• Cosmology. Cosmological redshift partly time dilation in expanding spacetime.
• Atomic clocks. 10⁻¹⁸ precision clocks measure altitude differences via gravity.
• Spaceflight. Astronauts age slightly differently from people on Earth.
• Education. Most accessible relativistic effect; demonstrates "spacetime."

Common mistakes

• Thinking only "moving" clocks dilate. Both SR (motion) AND GR (gravity) cause dilation. They can add or partially cancel (GPS).
• Confusing observer effects with mechanics of clock. Clock isn't physically slowed; spacetime structure makes time intervals frame-dependent.
• Forgetting reciprocity in SR. If A sees B's clock slow, B sees A's clock slow (mutual). Asymmetry comes from acceleration (twin paradox).
• Ignoring direction. Eastward jet at Earth's surface has Earth-spin velocity ADDED → more SR slowdown. Westward subtracts → less.
• Treating it as small effect always. At v near c or near black hole, dilation is enormous. At everyday conditions, microseconds at most.
• Confusing apparent and proper time. Each observer's wristwatch is correct in their own frame ("proper time"). Different observers measure different durations between events.