Illustrated Physics
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Series

Time dilation — the series

11 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A light clock at β = 0.6. The same clock at rest and moving. Light covers the hypotenuse rather than the height, and since its speed is the same for both observers, the moving clock must take longer to tick.

    The clock that has to slow, and why no clock can refuse

    One constant speed and one right-angled triangle force a moving clock to tick slower. The argument is Pythagoras, which is what makes it inescapable rather than merely surprising.

    part 1 · relativity
  2. A spacetime diagram at β = 0.6. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.6 of the speed of light.

    The twin who comes back younger

    If motion slows a clock, and motion is relative, each twin should find the other younger — and yet when they meet, one of them has aged less. The asymmetry is not in the speed and not in the acceleration; it is in which worldline is straight.

    part 2 · relativity
  3. Two corrections, opposite in sign and different in size. How fast a clock in a circular orbit runs compared with one on the ground, in microseconds a day, against the height of the orbit — with the two effects drawn apart rather than added. Being high speeds a clock up, by an amount that saturates: the potential term is bounded because there is only so much potential to climb out of. Moving slows it down, and a higher orbit is a slower one, so that term shrinks toward zero. They cancel at 3186 km — a radius of exactly 1.5 Earth radii, which follows from setting the sum to zero and contains neither G, nor the Earth's mass, nor the speed of light. At 20200 km the gravitational term is 45.7 µs a day and the speed term −7.2, leaving 38.5. Left uncorrected, that is 11.5 km of position error a day, growing without limit, from a clock that is working perfectly.

    The clock that is wrong in two directions

    A satellite clock loses 7.2 microseconds a day to its speed and gains 45.9 to its height. The two effects have opposite signs, different sizes and different dependence on the orbit, so there is exactly one altitude where they cancel — and 38.6 microseconds a day, left alone, is eleven and a half kilometres of position error.

    part 3 · relativity
  4. The clock that gains going one way and loses going the other. The rate at which a flown clock gains on a clock left at 30° latitude, in nanoseconds per hour, against the aeroplane's ground speed, with east taken as positive. Two terms are drawn and then their sum. Height alone gives 3.5 nanoseconds an hour at 9 km and does not care which way the aircraft is pointed. Motion costs time, and because the ground is already moving eastward at 402 metres a second, flying east adds to that speed and flying west subtracts from it — so the kinematic term is much larger going east and can change sign going west. The sum crosses zero at 180 metres a second eastward, which is the ground speed at which an aeroplane's clock keeps the time of the airfield it left. Over the two flights Hafele and Keating actually made, this simple model gives -61 nanoseconds eastward and +304 westward, against their own predictions of -40 and +275 and their measurements of -59 and +273. The model here uses one average altitude, one average speed and one latitude, where the real prediction integrated the flight logs; getting the signs and the rough sizes out of three lines of arithmetic is the point, and the last twenty per cent is what the logs are for. What no amount of arithmetic supplies is the thing the experiment settled: that the effect is real, that it acts on a caesium clock in a passenger seat, and that a difference of a few hundred nanoseconds after two days is measurable.

    The two clocks that flew in opposite directions

    Two caesium clocks were flown round the world in 1971, one each way, and came back disagreeing with the clock left behind — one having lost 59 nanoseconds and the other gained 273. Height alone would have made both gain. The sign flip comes from the ground already moving eastward at 400 metres a second before the aircraft took off.

    part 4 · relativity
  5. Every detour costs time. 4 routes between the same two events, 10 seconds apart in the frame drawn, each swinging out and back 1 time on the way. The proper time each carries is the integral of the square root of one minus the speed squared, computed by Simpson's rule along each curve: the straight route, 10.0000 s; wandering 1 light-seconds, 9.7485 s; wandering 2 light-seconds, 8.9245 s; wandering 3 light-seconds, 7.0935 s. The straight one carries the most, and every other one carries less — checked, on each drawn route. That is the opposite of what a length behaves like on paper, where the straight line is the shortest, and the whole difference is the minus sign in front of the space term.

    The longest way round is the shortest clock

    Of all the routes between two events, the one with no acceleration in it carries the most time on its own clock. That is the opposite of the Euclidean statement about straight lines, it comes entirely from one minus sign, and in a gravitational field it is why a thrown ball follows the path it does.

    part 5 · relativity
  6. The paths in space: orbits of one period, and a throw straight up. The same free falls drawn in space around the Earth, which is the filled disc. All start at the marked point 2 Earth radii from the centre. The circle is the circular orbit. The ellipses, of eccentricity 0.2 and 0.4, have the same period, so they come back to the start at the same moment. The straight line is the thrown clock's path: straight up to 4.46 Earth radii and back down the same line, arriving as the orbits complete one revolution. The Earth's rotation is ignored and it is treated as a point mass for the paths that pass close to it.

    The orbit that ages less than a throw

    A clock in orbit and a clock thrown straight up leave the same point at the same moment and meet there again one period later. Both fall freely the whole way, so both follow paths of stationary proper time — and the thrown clock comes back 4.1 microseconds older. Even a clock held still by a rocket, which is not falling at all, beats the orbit. Free fall picks out a path that is stationary, not one that is longest.

    part 6 · relativity
  7. How much slower a warm atom ticks. The fraction by which a free atom's clock runs slow on average because of its thermal motion, (3/2)kT/mc², against temperature on logarithmic axes, for five atoms used in clocks or tests of relativity: hydrogen 4.1 × 10⁻¹¹ at 300 K, aluminium ion 1.5 × 10⁻¹² at 300 K, strontium 4.8 × 10⁻¹³ at 300 K, caesium 3.1 × 10⁻¹³ at 300 K, mercury ion 2.1 × 10⁻¹³ at 300 K. Every line falls in proportion to the temperature and to one over the mass. A clock aiming at one part in 10¹⁸ must hold strontium below 0.63 mK and hydrogen below 7.3 µK — or know their temperature well enough to correct for it. Pound and Rebka's gravitational shift across 22.5 m, 2.46 × 10⁻¹⁵, is marked for scale.

    The clock that runs slow because it is warm

    Every atom in a gas is moving, and every moving clock runs slow. The first-order Doppler shifts of a hot gas cancel on average; the time dilation does not, because it has only one sign. A hydrogen maser at 40 °C runs slow by four parts in a hundred billion from this alone, a nucleus in a crystal runs slow even at absolute zero, and the first test of gravitational time dilation nearly drowned in a one-kelvin temperature difference.

    part 7 · relativity
  8. Two straight worldlines that meet twice. A spacetime diagram of a universe whose space is a circle 10 light-years round, drawn unrolled: the shaded strip is the whole of space, and the strips beside it are the same space again, so the dashed lines are the same stay-at-home twin A seen in the copies. Twin B leaves A at 0.6c and never changes speed or direction. After 16.67 years by A's clock B has gone once round and meets A again — the line reaches A's copy at 10 light-years. Neither twin has accelerated at any moment; both worldlines are straight. A has aged 16.67 years and B 13.33, less by the factor 1/γ = 0.800. The usual resolution of the twin paradox, that the travelling twin is the one who turned round, is not available: here nobody turned.

    The twin who never turns round

    The twin paradox is usually settled by pointing at the turn: the travelling twin is the one who fired the engines and came back, and so the one whose worldline is not straight. Put the twins in a universe whose space closes on itself, like the surface of a cylinder, and the traveller can come home by going straight on. Neither twin ever accelerates, both worldlines are straight, and the traveller still comes back younger. What decides it is not any motion but the shape of space, which picks out one state of motion as special — and hands every observer a way to measure it.

    part 8 · relativity
  9. Two clocks that slow together. For a muon circling a 1.45 T magnetic field, its mean life as the laboratory measures it and the time it takes to go round once, against its Lorentz factor γ, on logarithmic scales. Time dilation stretches the life to γ × 2.197 μs: 6.6 μs at γ = 3, 64.4 μs at γ = 29.3, the muon g − 2 experiments' value. The orbit grows with the momentum, so the revolution takes γ times longer too: 15.3 ns at γ = 3, 149 ns at γ = 29.3. The two lines are parallel. Their ratio, the number of turns in one mean life, is 431 at every energy — set by the field and the muon's own properties, eBτ/2πm, with γ cancelled out.

    The turns a muon makes at any energy

    A muon lives two microseconds, and in a storage ring it lives longer the faster it goes, because its clock runs slow. Its orbit runs slow too: a faster muon carries more momentum, bends less in the same field and takes longer to go round. The two slowings are the same factor of gamma, so they cancel. A muon in a 1.45 tesla ring makes 431 turns in a mean life whether it is moving at nine-tenths of the speed of light or at nine nines, and the only way to more turns is a stronger magnet. The same cancellation is why the first cyclotrons stopped working near twenty million electronvolts.

    part 9 · relativity
  10. The tick an elliptical orbit adds to a clock. The periodic part of the time a satellite clock gains or loses against one on the ground, after its constant rate offset is removed, against hours from closest approach, for the two Galileo satellites stranded in 2014 on orbits of eccentricity 0.156 and period 12.9 hours, a GPS satellite at 0.01 and a Galileo satellite on its intended near-circular orbit. The stranded satellites' clocks run ahead and behind by ±367 ns every orbit: slow near closest approach, where they are deepest and fastest, fast near the far point. A GPS clock wobbles by ±23 ns, and a satellite on a circular orbit by almost nothing. The wobble is −2√(GMa) e sin E/c², fixed by the orbit alone, which is what makes it a test.

    The satellites launched into the wrong orbit

    In August 2014 two European navigation satellites were left on the wrong orbit by a frozen fuel line in their rocket's upper stage: an ellipse instead of a circle. For navigation they were nearly useless. For relativity they were a gift. A clock on an elliptical orbit dips deeper into the Earth's gravity and speeds up as it falls, then climbs and slows, and both effects make it lose time near the bottom of the ellipse and gain it back near the top — a wobble of 367 nanoseconds every 12.9 hours, set by the orbit alone. Watching that wobble for three years checked the gravitational slowing of clocks five times more closely than anything had in the forty years before.

    part 10 · relativity
  11. Two effects that have to add up to nothing. The rate of a clock at sea level, against latitude, relative to a clock at the pole. Being carried at 465 m/s at the equator slows a clock by 1.20 parts in 10¹². The equator's greater height in the well, on the ellipsoid and with the Earth's flattened gravity, makes it run fast by 1.20. Their sum, on this two-term model of the Earth's field, is flat to 9 × 10⁻¹⁵, which is about the size of the terms left out. The international value for that shared rate, computed here for checking, is 6.969290 × 10⁻¹⁰.

    The clocks the sea keeps together

    A clock at the equator is moving at 465 m/s and a clock at the pole is not. Einstein said in 1905 that the moving one must run slow. It does not, at sea level, by any amount anyone has detected. The planet has bulged by exactly what it takes to make the difference vanish, and the agreement is not a coincidence. A sea has run downhill until the speed term and the height term add up to the same thing everywhere.

    part 11 · relativity

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