How four clocks turn time into a place

A blue dot settles onto a street in Delhi. It feels as though something in orbit has found you and pointed down. That is almost the reverse of what happened.

The satellites did not ask who you were. They did not receive your position. Each repeatedly broadcast a message containing where it was and when the message left. Your phone listened. From several travel times, it calculated where the listener must be.

There is a catch. Distance equals travel time multiplied by the speed of light, so an error of one microsecond, a millionth of a second, corresponds to nearly 300 metres of light travel. The satellites carry atomic clocks; a phone does not. GPS works by treating the receiver's bad clock as another unknown to solve.

The receiver therefore solves four things together: its east–west position, north–south position, height and clock error. Three numbers describe a place, and a fourth describes how wrong “now” is.

The satellites are lighthouses, not observers

Every GPS signal identifies its transmission time and supplies orbital information. A receiver compares the transmitted time with its reception time and multiplies the difference by the speed of light. The result is called a pseudorange, an apparent distance. It includes the actual satellite-to-receiver distance plus errors from the clocks and other effects.

One distance places the receiver on a sphere around a satellite. Further distances narrow the intersection. With a clock perfectly matched to GPS time, three suitable measurements could in principle solve the three position coordinates.

A phone’s clock is not that precise. Its inexpensive oscillator supplies regular timing ticks, but its error can exceed the signal’s travel time. A fourth satellite supplies another measurement. Together, four suitable measurements let the receiver solve longitude, latitude, altitude and clock bias, how far its clock is ahead or behind. Neil Ashby's technical review explains why four measurements determine three position components plus the local clock error. In practice, receivers usually use more signals and a richer error model.

The phone’s clock error affects all its measurements. Treating that shared error as an unknown lets the receiver calculate it along with position.

An atom makes time into a ruler

The measurement still needs much better clocks in space. Atomic clocks do not count tiny mechanical ticks. Atoms can change between particular energy states. Electronics use the very stable frequency associated with one of those transitions as a timing reference. GPS satellites carry multiple atomic clocks, while ground control compares and corrects them against reference clocks. NIST describes the broadcast of satellite time and position and the continuing role of ground-based clock corrections.

The satellites therefore distribute more than coordinates. They distribute a shared time scale. GPS timing helps synchronize telecommunications, electrical networks and financial systems; GPS.gov calls time the system's fourth dimension.

But putting a precise clock in orbit creates a deeper problem. A clock is not outside physics. Its motion and the strength of gravity affect how quickly it ticks compared with a clock on the ground.

Two effects pull the clock in opposite directions

Special relativity says a moving clock accumulates less time than a clock at rest in the chosen reference frame. A GPS satellite travels at roughly 3.9 kilometres per second, so this motion effect makes its clock run slower than a comparable ground clock.

General relativity supplies the opposite sign. Gravity is weaker at the satellite's altitude than near Earth's surface, and a clock higher in the gravitational field accumulates more time. For a GPS-like circular orbit, this gravity effect is larger than the motion effect.

In the simplified comparison, gravity makes the satellite clock gain about +45.7 microseconds per day relative to the ground clock. Motion contributes about −7.2 microseconds per day, slowing it in the opposite direction. The net satellite clock gain is about +38.5 microseconds per day. Light travels roughly 11.5 kilometres in that interval. That indicates the scale of distance error. It does not predict a phone drifting exactly that far along a straight line. ESA notes that relativistic corrections can reach about 12 kilometres after a day.

A satellite was allowed to disagree with Einstein

The correction was not installed merely because a famous theory demanded respect. Engineers created an experiment inside an early navigation satellite.

NTS-2, launched on 23 June 1977, carried the first caesium atomic clock placed in orbit. The system included a frequency synthesizer, electronics that could adjust the clock’s output to apply the predicted correction later. For about 20 days, the clock ran without it while engineers measured the rate. Ashby's review reports a measured difference of +442.5 parts in a trillion against a relativistic prediction of +446.5 parts in a trillion. The measured and predicted rates agreed at about the one-percent level. The synthesizer was then turned on.

This episode matters because it turns the slogan “GPS proves relativity” into something specific that engineers tested. The clock was allowed to be wrong in a controlled way. Its disagreement with ground clocks had a predicted sign and size. The result determined how the system should be operated.

The correction is a ledger, not one magic number

The +38.5-microsecond figure is the memorable entry, not the whole account. Real orbits are slightly eccentric, stretched rather than perfectly circular. Satellite speed and distance from Earth therefore change around the orbit, changing both relativistic effects. Broadcast navigation messages carry clock parameters, and receivers apply additional relativistic corrections. Ground stations track satellite orbits and clocks. Earth rotates while a signal travels. The ionosphere, an upper region containing charged particles, and the troposphere, the lower atmosphere where most weather occurs, delay radio waves. Buildings can reflect them. Satellite geometry changes how timing errors map into position.

That is why GPS accuracy is not a single property of “the satellite.” It is the result of a ledger in which orbit, clock, atmosphere, receiver and map each contribute corrections and uncertainty. ESA's navigation reference describes broadcast clock offsets and relativistic terms. Galileo similarly uses multiple space-qualified clock technologies and regularly synchronizes them with ground references; ESA explains how its hydrogen-maser and rubidium clocks work.

The blue dot is added only after the receiver has solved its position and a mapping service has placed those coordinates on a road. NIST separates the satellite calculation from the map shown on screen. A misplaced dot can therefore come from signal blockage, reflections, atmosphere, poor geometry, receiver design or the map itself. It is not simply a satellite “seeing” the wrong street.

Your phone does not need to know the correct time beforehand. Four or more satellite signals let it calculate both its position and its local clock error. The blue dot begins with carefully corrected time.