Twin Paradox

The setup

Two identical twins, Alice and Bob. Bob stays on Earth; Alice climbs aboard a rocket, accelerates briskly to cruising speed v, coasts to a distant star, decelerates and re-accelerates to head home, coasts back, then docks at Earth. The question physicists were asked from 1911 onwards: when she steps off the ship, are they the same age?

Special relativity's answer is unambiguous. Alice's clock has run more slowly than Bob's; she is biologically younger than him. By how much depends on v.

The numbers

Cruise speed v	Lorentz γ	10 Earth years aboard	20 Earth years aboard
0.1c	1.005	9.95 yr	19.9 yr
0.5c	1.155	8.66 yr	17.3 yr
0.8c	1.667	6.00 yr	12.0 yr
0.9c	2.294	4.36 yr	8.72 yr
0.95c	3.203	3.12 yr	6.24 yr
0.99c	7.089	1.41 yr	2.82 yr
0.999c	22.37	0.45 yr	0.89 yr
0.9999c	70.71	0.14 yr	0.28 yr

At 0.99c, ten years aboard ship is only one year and five months — and the destination 4.95 light-years away (in Earth frame) becomes a contracted 0.70 light-years in the ship frame, which the ship covers at 0.99c in about 8.5 months ship-time each way.

Why it isn't really a paradox

The apparent contradiction is: "If motion is relative, why not flip it? From Alice's view, Bob receded at v and came back; shouldn't he be the younger one?"

The flaw is the assumption of symmetry. Bob sits in a single inertial frame the whole time. Alice does not. To leave, turn around, and come back she must accelerate — and acceleration is absolute, not relative (you can detect it with an accelerometer that doesn't need to know about any external frame). Bob's world line in a spacetime diagram is straight; Alice's is bent at the turnaround.

In Minkowski geometry, straight world lines between two given events have maximum proper time — opposite the familiar Euclidean rule. Bend the path and you lose proper time. So Alice, with the bent path, is younger.

Proper time along a world line

τ = ∫ √(1 − v(t)²/c²) dt

The integral runs along the world line of an observer; v(t) is their speed (in some inertial frame) at each moment; τ is the time their own wristwatch records. For an inertial observer at rest in that frame, v(t) = 0 and τ = t — clock-time equals coordinate-time. For an observer moving at constant v, τ = t/γ — moving clocks tick slowly. For an observer with a complicated trajectory, you integrate piece by piece.

Worked example — a 10-year round trip at 0.99c

Alice accelerates quickly to v = 0.99c, coasts to a star 4.95 light-years away (in Earth frame), turns around, coasts back. We'll idealize the turnaround as instantaneous to keep the arithmetic clean.

1. Earth-frame outbound time: 4.95 ly / 0.99c = 5.00 years.
2. Earth-frame inbound time: another 5.00 years. Total Earth time = 10.00 years.
3. Lorentz factor at 0.99c: γ = 1/√(1 − 0.9801) = 1/√0.0199 = 7.089.
4. Alice's proper time = 10.00 yr / γ = 10/7.089 = 1.41 years.
5. When she steps off the ship, Bob has aged 10 years; she has aged 1.41 years. They differ by 8.59 years.

From Alice's frame the calculation looks different but lands at the same answer. The destination is length-contracted to 4.95/γ = 0.698 ly. She crosses 0.698 ly at 0.99c in 0.706 years. Round trip = 1.41 ly-worth at 0.99c = 1.41 years. The two frames agree on the only invariant question: how much did each wristwatch tick between the meet-up events?

JavaScript — proper time calculator

const c = 299792458;

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

// Idealized round trip with instantaneous turnaround
function twinRoundTrip(distance_ly, v_fraction_c) {
  const g = gamma(v_fraction_c * c);
  const t_earth = 2 * distance_ly / v_fraction_c;  // years
  const t_traveller = t_earth / g;
  return {
    distance_ly,
    v_fraction_c,
    gamma: g,
    Earth_years: t_earth,
    Traveller_years: t_traveller,
    age_gap_years: t_earth - t_traveller,
  };
}

console.log(twinRoundTrip(4.95, 0.99));
// { gamma: 7.089, Earth: 10.00 yr, Traveller: 1.41 yr, gap: 8.59 yr }

console.log(twinRoundTrip(4.37, 0.95));  // Alpha Centauri
// { gamma: 3.203, Earth: 9.20 yr, Traveller: 2.87 yr, gap: 6.33 yr }

// Numerical proper-time integral for an arbitrary v(t)
function properTimeIntegral(v_of_t, t_start, t_end, steps = 10000) {
  const dt = (t_end - t_start) / steps;
  let tau = 0;
  for (let i = 0; i < steps; i++) {
    const t = t_start + (i + 0.5) * dt;
    const v = v_of_t(t);
    tau += Math.sqrt(1 - (v * v) / (c * c)) * dt;
  }
  return tau;
}

// Smooth acceleration profile — accelerate first half, decelerate second
function smoothProfile(t, T_half, v_peak) {
  if (t < T_half) return (v_peak * t) / T_half;
  if (t < 2 * T_half) return v_peak * (2 - t / T_half);  // outbound deceleration
  if (t < 3 * T_half) return -v_peak * (t / T_half - 2); // inbound acceleration
  return -v_peak * (4 - t / T_half);                     // inbound deceleration
}

// Hafele–Keating-style: 41-hour eastward jet flight at 250 m/s
const HK_seconds = 41 * 3600;
const HK_v = 250;
const HK_dilation_ns = HK_seconds * (1 - 1/gamma(HK_v)) * 1e9;
console.log(`H-K eastward SR offset: ${HK_dilation_ns.toFixed(0)} ns`);
// ~50 ns (combined with GR effects, observed −59 ns)

History — from Langevin to Hafele–Keating

Paul Langevin posed the scenario in 1911 with rocket-borne travellers; it became a textbook puzzle through the 1910s and 20s. Einstein revisited it several times to insist there was no contradiction. Herbert Dingle published a famous mistaken objection in 1956–1972, generating decades of pedagogical rebuttals. The empirical settlement came in 1971: Joseph Hafele and Richard Keating flew four caesium-beam atomic clocks on commercial jets eastward and westward around the world and compared them to identical clocks at the US Naval Observatory. The observed offsets matched the relativistic prediction (combined SR + GR) within the experimental error of about 10%, removing all reasonable doubt that moving clocks really tick more slowly.

The cleanest modern demonstration is GPS. Each satellite's clock runs ~7 µs/day slower than ground (SR — they're moving at ~14,000 km/h) and ~45 µs/day faster (GR — they're 20,200 km higher in Earth's gravity well). Net: +38 µs/day. Without that correction, GPS would drift by ~10 km/day. The correction is wired into the satellites' rest-frequency tuning.

Common misconceptions

• "Acceleration causes the time dilation." No — acceleration only breaks the symmetry. The age gap can be reproduced without any acceleration at all by using a three-clock relay, where a returning clock takes over from an outbound clock at the turnaround event. The proper-time integral over the bent path still gives less.
• "From the traveller's frame, Earth's clock runs slow — so Bob should be younger." True during each coast leg in isolation, but the turnaround sweeps Alice's plane of simultaneity forward through huge amounts of Earth-time. A consistent calculation in Alice's frame returns the same answer as in Bob's frame.
• "Special relativity can't handle acceleration." It can. SR handles any trajectory through flat spacetime, including curved world lines from accelerating rockets. General relativity is required only for gravitation (curved spacetime), not for acceleration.
• "The twin paradox needs general relativity to resolve." It doesn't. Pure SR gives the right answer. GR is consistent with SR's answer but adds no new content to the basic version of the paradox.
• "At everyday speeds the effect doesn't exist." It exists — atomic clocks have measured it on airliners and high-speed trains. It is just numerically tiny (nanoseconds over days) until v gets close to c.
• "The traveller experiences time passing slowly." No — Alice's biology and her wristwatch run normally to her. She just spends less elapsed time between the two meet-up events than Bob does.

Twin paradox vs related effects

Effect	Source	Symmetric?	Confirmed by	Order of magnitude
Twin paradox (kinematic)	Bent world line in SR	No (only one bends)	Hafele–Keating, GPS, ISS	γ−1 per trip
Coast-leg time dilation	Inertial v ≠ 0	Yes (mutual)	Muon flight, accelerator decays	γ factor
Gravitational time dilation	Spacetime curvature (GR)	No (deeper well = slower)	Pound–Rebka, GPS	GM/(rc²)
Doppler shift (longitudinal)	Relative motion of source	Reciprocal pair	Stellar spectroscopy, redshifts	√((1±β)/(1∓β))
Length contraction	Lorentz boost	Yes (mutual)	Muon range, accelerator focusing	1/γ
Relativity of simultaneity	Lorentz boost	Reciprocal	Standard derivations; implicit in tests	γvΔx/c²

Where the twin paradox shows up

• GPS. Permanent +38 µs/day twin-style correction baked into satellite oscillators; without it, position would drift ~10 km/day.
• Atomic timekeeping. Caesium and optical-lattice clocks transported between labs accumulate measurable time differences; international timescales (TAI) include relativistic corrections.
• Muon detection. Cosmic-ray muons created ~15 km up reach the ground only because their proper lifetime is dilated — same physics as the twin's slowed clock.
• Particle accelerators. Storage-ring muons live γ times longer in lab frame; CERN's muon g−2 ring exploited this directly.
• Astronaut biology. NASA's Twins Study (Scott and Mark Kelly) measured a ~5 ms age difference after one year on ISS — within special-relativity prediction.
• Hypothetical interstellar travel. Sets the trade-off: getting to Alpha Centauri in a human lifetime requires v close enough to c that γ is large.
• Pedagogy. The cleanest single demonstration that simultaneity and proper time are frame-dependent, while invariant quantities (like the age gap at reunion) are absolute.