GPS clocks gain 38 microseconds a day: time does not pass the same for everyone

Four atomic clocks in the cabin, and a time that does not add up

In 1971 four atomic clocks boarded ordinary scheduled flights and circled the globe twice: once eastward, once westward. On their return they were compared with the clocks that had stayed on the ground. Westward the difference was 273 nanoseconds, against the 275 the calculation predicted ¹¹. A nanosecond is a billionth of a second. Set beside the duration of an intercontinental flight, that number is a crumb: none of the passengers seated next to the instruments came back younger in any sense that could be recounted over dinner. And yet it is all the time travel humanity knows how to do today, and it was clocked.

The question that opens this issue is not whether the phenomenon exists — we will come back to that with the error bars in hand — but where it comes from. Why should time depend on how fast one runs? The answer begins long before the airplanes, in a Cleveland basement, in front of a measurement that refused to give the expected result. At the end of the nineteenth century physics took for granted that light, like sound, needed a medium to travel through: they called it the ether. If the ether existed, the Earth had to move through it as it circles the Sun, at nearly 30 kilometres per second ²¹.

From that motion followed a precise, testable prediction. Light sent along the direction of the Earth's course should have taken a different time from light sent crosswise, and the difference should have changed with the seasons, because in six months the Earth reverses the direction of its course around the Sun ¹. The measurement was made. It found nothing: the speed of light turned out to be the same in every direction, and the same at every time of year ². If the speed never changes — not even when the one measuring is in motion — then something else has to give. This issue tells what gives, by how much, and how precisely we know it.

The constancy of light is not a postulate: it is a measurement, repeated for a century

The apparatus that in 1887 was supposed to reveal the ether was an interferometer. If the Earth was running through the ether, those fringes had to shift when the instrument was rotated. The shift observed was about one fortieth of the one expected, that is, within the instrument's margins of error, consistent with zero ¹. It is a formulation worth unpacking: it does not mean the result was zero, it means the apparatus was not sensitive enough to tell it apart from zero.

The null result was not accepted at once, and that was the best thing that could have happened. Between 1902 and 1904 Morley and Miller repeated the campaign, expressly looking for a daily or annual pattern in the data: those replications too gave negative results within the margins of error ¹. The method has never been shelved since; it has only been refined. Modern versions replace the optical arms with cryogenic optical resonators.

With those instruments, anisotropy is today ruled out at the level of 10⁻¹⁷ ³. Put less drily: if light travelled even one hundred-billionth of a billionth faster in one direction than in another, we would notice today. Between the sensitivity of the 1887 apparatus and that of today's resonators lie some fifteen orders of magnitude. This is the point popular accounts tend to skip: the constancy of the speed of light is not an elegant assumption the theory starts from, it is one of the best-verified quantities in experimental physics. And so the books have to be balanced somewhere else. If two observers in relative motion measure the same identical value for the same beam of light, and speed is distance divided by time, one of the two terms of the fraction must behave differently from what intuition suggests.

A compact formula has been circulating in popular accounts for years to explain all this: every object would move through spacetime at a fixed total speed, equal to that of light, divided between motion through space and motion through time; the more one spends in one direction, the less is left for the other, and time slows down. It is an effective image, and the section that follows puts it to the test in its exact form. It should be said at once that none of the scientific sources consulted for this issue measures a quantity called “total speed through spacetime”: it is a geometric restatement adopted for teaching, not a laboratory value.

Opinion reported as such, not verified by the editorial team.

The books balance, but in quadrature

The formula in circulation has a solid basis, provided it is written out in full. The exact relation between the share of motion spent in space and the share left to time is a quadraturea sum that closes on the squares of the two shares, not on the shares themselves: as in the Pythagorean theorem, where the legs add up squared and not in a straight line: (v/c)² + (1/γ)² = 1. The letter γ, gamma, is the Lorentz factor. Its definition is γ = 1/√(1−v²/c²) ⁶.

The term 1/γ is the rate of time on board: how much the traveller's clock counts while one full unit goes by on the ground. The term v/c is the fraction of the speed of light at which one is travelling. The identity says that their squares, added together, make exactly one: the books balance, always, with nothing left over. The difference between adding the shares and adding their squares is not an exam-hall detail. It changes the shape of the phenomenon completely: at half the speed of light the share spent in space is already 25%, but the rate of time on board is still 87% of the rate back home. The slowdown is not proportional to speed.

The effect stays hidden until one comes close to the limit. At 0.99c the share of motion spent in space reaches 98%, and the rate of time on board drops to 14.1% — one day on board while seven pass on the ground. At the speed of an airliner that share is of the order of 10⁻¹¹ per cent: hence the nanoseconds, instead of years. It is worth pausing on this asymmetry, because it is the thing the popular image fails to convey. Put in words, “more speed through space, less through time” suggests a linear trade, a slider that moves. The real split is almost entirely crammed into the last sliver of the scale, and the figure below shows it number by number.

The knee of the curve: where the phenomenon stops hiding

If γ is plotted against speed, the curve tells a story in two acts. For most of the run almost nothing happens: up to half the speed of light the dilation factor is a mere 1.155, a 15% stretch that no human experience could notice without instruments. Then the curve bends. At 0.9c the factor is already 2.294; at 0.99c it reaches 7.089 ⁶; at 0.999999c it touches 707.107 ⁸. Between the last two of these values the speed has grown by barely a hundredth, while the factor has multiplied a hundredfold. All the work is done by the final decimals.

The curve has a twin that falls where the other rises. It is length contraction: for the traveller, the distance to be covered shortens by exactly the same factor, L = L₀/γ ¹⁰. These are not two separate phenomena to be learned by heart, they are the same relation seen from either side. This symmetry is what keeps the account coherent. A crew that covers ten light years in little more than a year has not exceeded the speed of light: in its own frame the distance is no longer ten light years, it has shortened along with time. Those who stay on the ground see the journey last more than ten years; those who leave see a shorter route.

The factor that appears in the two formulas is literally the same number ^{9 10}. That is why a single quantity, γ, is enough to describe both the clocks and the rulers, and why every experimental test of one of the two effects automatically holds for the other as well. What remains is the reason none of this belongs to ordinary experience. For γ to depart perceptibly from 1, speeds are needed that no object built by human hands, and no body with mass that we have ever accelerated apart from particles, has ever reached. The next section lines up those speeds.

The real speeds, set side by side

An airliner at cruise does about 900 kilometres an hour, that is, a quarter of a kilometre a second. It seems fast because it is judged from the window; on the scale that matters here it is a fraction of the speed of light with eleven zeros after the decimal point. The reader, however, is already in motion even while sitting still, and at a far more serious clip. The Earth runs around the Sun at a mean orbital speed of 29.78 kilometres per second ²¹, a value confirmed by the NASA Goddard fact sheet ²². That is a hundred and twenty times the airplane, and it still is not enough.

The Sun does not stand still either. Its speed around the centre of the Milky Way is of the order of 230 kilometres per second, a value that sits within the range of accepted estimates: the historical standard of the International Astronomical Union gives 220 km/s ²³, the measurements from the Gaia and RAVE surveys give 240 km/s ²⁴, the solar parameters sheet reports 251 km/s ²⁵. The only truly relativistic objects ever clocked carry no crew. In CERN's storage ring, muons were made to circulate at a γ factor of 29.33 ¹⁴. That is the regime in which the effect stops being a correction and becomes the main phenomenon.

The starships of the standard examples all lie beyond that. At 0.99c a ten-light-year journey lasts 10.10 years for those who stay home and about 1.43 years for those who leave; at 0.999999c the Earth sees 10.00 years go by and the crew lives through 0.0142, that is, a little more than five days. These two on-board numbers are the real surprise of the scale. The two terrestrial columns are practically identical — ten years and ten years — while the on-board ones differ by two orders of magnitude. The last 0.999% of speed does all the work, and it should be said plainly that no public source publishes those two journeys already worked out: they are direct derivations from the formula, made here.

Predicted against measured: the agreement is judged on the error bars

Let us go back to the airplanes of 1971, because that is where the comparison between theory and stopwatch truly exists. The experiment produced four numbers, not one: a prediction and a measurement for the eastward trip, a prediction and a measurement for the westward trip. Eastward the prediction was −40 nanoseconds, with an uncertainty of 23; the measurement gave −59 nanoseconds, with an uncertainty of 10 ¹¹. The two central values sit further apart than one would like at first sight, but their respective error bands overlap widely: the agreement is there, and it is good without being spectacular.

Westward the picture is of another quality. The prediction was +275 nanoseconds with an uncertainty of 21, the measurement +273 with an uncertainty of 7 ¹². Two nanoseconds of discrepancy out of two hundred and seventy-five: here the overlap is almost total. The opposite sign between the two trips is not a whim of the data, it is the signature of the experiment. Flying eastward the airplane adds its own speed to that of the Earth's rotation, flying westward it subtracts it, and the two clocks end up on opposite sides of the reference left on the ground. There is a second test bench, a harsher one, where the speeds are not those of a jet. In CERN's muon storage ring the mean lifetimes of the two species of particle were measured: 64.419 microseconds for positive muons and 64.368 for negative ones, at a γ factor of 29.33 ¹⁴.

The agreement with the relativistic prediction came out within a fractional error of 2×10⁻³ ¹⁴, and CERN's final report later put it at (0.8 ± 0.7)×10⁻³ ¹⁵. Translated: the formula that stretches time by a factor of twenty-nine works to two parts in a thousand. The same apparatus also verified the asymmetry that makes the twin paradox possible ¹⁶. The contrast between the two experiments is the instructive part. On the airplanes the effect is so small that an atomic clock is needed to see it; in the muon ring it is so large that, without accounting for it, the particles would not even reach the end of the loop.

The number that is no longer measured, and the 38 microseconds that settle the bill every day

There is a subtlety almost no one recounts, and it changes the meaning of the most quoted figure of them all. The speed of light in vacuum is exactly 299,792,458 metres per second — but since 1983 that value is no longer the result of a measurement: it is the definition ⁴. The BIPM defines the metre by fixing the numerical value of c. Before that date the metre was defined in its own right and the speed of light was measured; the number chosen in 1983 was the one that best agreed with the measurements available until then ⁵.

The reversal is sharp and worth stating in full: today a physicist who “measures the speed of light” is in fact measuring the length of their own metre. That is why c no longer carries an error bar beside it: it does not have one, by construction. From theory to daily life the step is shorter than it looks, and it is called satellite navigation. The atomic clocks aboard GPS satellites do not tick at the rate of those on the ground, and the difference has to be corrected every day. The bill has two entries of opposite sign. Orbital speed makes the clocks run late by about 7 microseconds a day, through the same effect measured on the airplanes of 1971; the weaker gravity at that altitude makes them run ahead by about 45 microseconds a day ¹⁸. The net balance is a gain of 38 microseconds a day ¹⁷.

Without the correction the system would not degrade: it would stop working. Position errors would pile up at a rate of about 10 kilometres a day, and a computed position would already be false after two minutes ¹⁷. The systematic treatment of these effects is the subject of a reference work by Neil Ashby ²⁰. An honest caveat about this example: the larger part of the correction, those 45 microseconds, comes not from speed but from gravity, and is therefore general relativity ¹⁹. GPS demonstrates the whole of relativity, not just the piece this issue is about — but within that balance the 7 microseconds of speed are there, and they have to be paid.

Curiosities from the notebook

The chronology, lined up, reverses the order everyone imagines. In the popular telling Einstein thinks, predicts, and then someone goes off to check. In fact the measurement that did not add up arrives in 1887 and the explanation eighteen years later, in 1905: the theory did not anticipate the anomaly, it tidied it up. The direct confirmations of time dilation came later still, when the person who had predicted them had been dead for sixteen years. Then there is the title of the paper that changes everything and promises nothing. On 30 June 1905, in the Annalen der Physik, “Zur Elektrodynamik bewegter Körper” appears, that is, “On the Electrodynamics of Moving Bodies” ²⁸. Neither the word time nor the word relativity shows up in the headline: it looks like a technical note on moving conductors, and it is the text that introduces special relativity.

The place it came out of carries a precise job title worth reporting. Einstein had joined the Swiss Patent Office in Bern on 23 June 1902 as a “technical expert, third class”, and in the year of the four papers he went on working there full time ²⁷. 1905 was not a sabbatical: it was a year of double shifts. In the material behind this fact-check there is also a gap that deserves to be declared. Hafele and Keating's original paper came out in Science in 1972 ¹³ and the muon one in Nature in 1977 ¹⁴, but neither would open in full: one answers with an access error, the other with a subscription. The numbers in this piece come from secondary sources that agree with one another, and it is only right that the reader should know.

A similar case concerns the Sun's speed within the galaxy. There is no single official value: it ranges from the 220 km/s of the IAU standard ²³ to the 240 of the Gaia measurements ²⁴, up to the 251 of other data sheets ²⁵. It depends on how the local standard of rest is fixed and on how far away the galactic centre is placed. The 230 km/s that circulate everywhere are a reasonable choice among discordant numbers, not a constant. The most elusive detail, though, is the smallest. At an airliner's cruising speed, time dilation is commonly quantified as a few nanoseconds per hour: it is a plausible order of magnitude, consistent with the tens and hundreds of nanoseconds accumulated in the 1971 experiment over an entire trip around the world ²⁶, but none of the sources gathered publishes that hourly figure. That a number so often quoted has no public reference is, in its own way, a piece of information.

One drawing remains that we would have liked to make and did not. The GPS satellite that carries the clocks of this story deserved a cross-section to scale, with the dimensions of the body and the span of the panels: in the verified material of this issue its measurements are absent, and a stock drawing would depict an object that is not the one in the account. Also missing are the Minkowski diagram, the only honest geometric way to show the vector rotating between space and time, and the light clock from which the formula for γ is derived in three lines: neither exists in our library, and we tell them in words instead of faking them. One last number, the one that closes the circle opened in the cabin. Every GPS satellite passing overhead as you read this is correcting its own clocks, because up there time does not run as it does here. The only time machine ever built is in orbit, it works every day, and its job is to tell us where we are.

Scientific American summed the matter up in a formula worth reporting in full: in Einstein's universe, airplanes and staircases are time machines ²⁶. It is a rhetorical choice, not a laboratory result — but it points well to the real scale of the phenomenon, which on Earth is measured in tens or hundreds of nanoseconds accumulated over an entire trip around the world.