Gravitational Time Dilation Calculator

A clock higher up runs faster. This page works the Schwarzschild factor for two clocks at two radii and gives you the microseconds per day between them — then adds the speed correction for an orbiting clock, which is how the famous 38 microseconds a day of the GPS correction is actually built.

Last updated 11 Nov 2026

Two radii around a body → how fast the high clock gains

√(1 − r_s/r), and the GPS correction built from it

Mass of the bodyWith “Earth masses” selected, leave this at 1. Jupiter is 317.8, the Sun 333 000 Earth masses — or switch the unit and type 1 solar mass.

That mass is inThe first two options use the IAU 2015 nominal mass parameters, (GM)_E = 3.986 004×10¹⁴ and (GM)_☉ = 1.327 124 4×10²⁰ m³s⁻², which are known to ten figures. The kilograms option multiplies by G = 6.674 30(15)×10⁻¹¹, which is known to five — so if you can express the body in Earth or solar masses, do, and your answer is better by four orders of magnitude.

Radius of the LOWER clock, from the centreIn kilometres, measured from the CENTRE of the body and not from its surface. The default is 6378.1 km, the IAU nominal equatorial radius of the Earth. This is the mistake to watch for: an altitude typed here instead of a radius makes the clock sit nearly at the centre and the answer is nonsense.

Height of the UPPER clock above the lower oneIn kilometres, measured upwards from the lower clock — so this one IS an altitude. The default 20 184 km puts the upper clock in the GPS orbit, at a radius of 26 562 km. Try 0.3 km for the twenty-two-floor office block whose top and bottom have been shown to disagree with an optical clock.

Two clocks on one vertical line above a curved surface. The upper clock really does move as you change the height — it climbs with h/r₁ and stops at the top of the frame. The red arrow on the right is how fast it gains on the ground clock, and that arrow is drawn on a LOGARITHMIC scale, because a real rate difference of five parts in ten billion cannot be drawn to scale next to anything. The number beside it is the honest one.

45.6524µs/dayExample

A GPS satellite 20 184 km above the Earth’s surface

The exact factor, the school shortcut, and why they agree for the Earth

dτ/dt = √(1 − r_s/r) · r_s = 2GM/c² · rate ratio = √(1 − r_s/r₂) / √(1 − r_s/r₁) · weak field: Δf/f ≈ GM(1/r₁ − 1/r₂)/c² ≈ gh/c²

r_s = 2GM/c²: the Schwarzschild radius. For the Earth it is 8.87 mm and for the Sun 2.95 km, which is why the factor is so close to 1 everywhere in the solar system
r: the radial coordinate FROM THE CENTRE of the body, not the height above its surface. Confusing the two is the error this page’s input hint warns about, and it changes the answer by orders of magnitude
dτ/dt: the rate at which a clock at r ticks, compared with a clock infinitely far away. Always less than 1 — every clock in a gravitational well runs slow compared with a clock outside it
gh/c²: the school version, good when the height is small compared with the radius and the field is uniform. g = 9.80665 m/s² exactly by definition, h in metres; it gives 1.09×10⁻¹⁶ per metre of height, or 9.4 ps/day per metre

Worked example

A GPS satellite 20 184 km above the Earth's surface
SCHWARZSCHILD RADIUS OF THE EARTH. r_s = 2GM/c² = 2 × 3.986004×10¹⁴ / 8.98755×10¹⁶ = 8.870 mm. That is the whole Earth's worth of gravity expressed as a length, and it is the reason every factor on this page is 1 to nine decimal places.
THE TWO RADII. The lower clock is on the surface at r₁ = 6378.1 km. The upper is 20 184 km higher, so r₂ = 26 562.1 km. Notice that the altitude had to be added to the Earth's radius; using 20 184 km as the radius would put the satellite inside the planet.
THE TWO FACTORS. r_s/r₁ = 1.3907×10⁻⁹ and r_s/r₂ = 3.3399×10⁻¹⁰. Both are far too small for a calculator to square-root honestly, so use √(1−x) ≈ 1 − x/2: the factors are 1 − 6.9534×10⁻¹⁰ and 1 − 1.6700×10⁻¹⁰.
THE RATIO. Dividing and keeping first order, the upper clock runs fast by (1.3907 − 0.3340)×10⁻⁹/2 = 5.2835×10⁻¹⁰ of its rate.
PER DAY. 5.2835×10⁻¹⁰ × 86 400 s = 4.5652×10⁻⁵ s = 45.6524 µs/day, which is the published 45 µs/day figure.
NOW SUBTRACT THE SPEED TERM. The orbital speed at that radius is √(GM/r₂) = 3873.8 m/s, and special relativity takes v²/2c² = 8.3484×10⁻¹¹ off the satellite's rate, which is −7.2130 µs/day. The NET is 45.6524 − 7.2130 = +38.44 µs/day — the number everybody quotes as 38.

How big the effect is, from a staircase to a neutron star

The two orbit rows differ in sign of the NET effect and that is the thing worth taking away. A low orbit is only 400 km up but is moving at 7.7 km/s, so the speed term wins and an ISS clock loses about 25 µs per day. GPS is thirty times higher and only half as fast, so the height term wins and its clock gains. Somewhere around 3200 km of altitude the two cancel exactly, and a clock there would keep the same time as one on the ground.
Clocks at	Height difference	High clock gains	Comment
Floor and ceiling of a room	3 m	19 ps/day	optical lattice clocks resolve shifts over well under a metre of height
Ground and the top of a tower	300 m	2.8 ns/day	a little over 1 µs a year
Sea level and Mount Everest	8.85 km	83 ns/day	about 30 µs per year — a person living at the summit ages measurably faster
Ground and a low-Earth orbit	400 km	3.5 µs/day	but the orbital term is −28.3 µs/day here, so the ISS clock runs NET SLOW by 24.7
Ground and GPS	20 184 km	45.7 µs/day	net +38 µs/day after the speed term; 38 µs is 11 km of position error
Ground and geostationary	35 786 km	51.0 µs/day	higher than GPS, so a larger gain
Sun’s surface and far away	—	0.183 s/day	a fractional rate of 2.12×10⁻⁶ — the solar gravitational redshift
Neutron star surface and far away	—	rate about 0.81	1.4 solar masses in 12 km gives rₛ/r ≈ 0.34; no weak-field shortcut survives here

Higher clocks run faster, and the GPS has to pay for it every day

Gravity slows clocks, so being further out speeds them up. The Schwarzschild factor √(1 − r_s/r) is the rate of a clock sitting still at radius r, compared with one far away. It is less than 1 everywhere, and it gets closer to 1 as you go out — so the higher of two clocks ticks faster. For the Earth the whole effect lives in the ninth decimal place, because the Earth’s Schwarzschild radius is 8.87 millimetres.

Why GPS is the example that makes it real. A satellite clock is affected by two things pulling in opposite directions. Being 20 000 km up makes it gain about 45.7 µs a day. Moving at 3.87 km/s makes it lose about 7.2 µs a day, by ordinary special-relativistic time dilation. The net is about +38 µs a day. Light travels 300 m in a microsecond, so 38 µs of unaccounted drift is 11 km of position error — per day, accumulating. The satellites’ oscillators are therefore set slightly low before launch so that they run at the right rate once in orbit, which is as literal a piece of applied relativity as exists.

The school shortcut and when it dies. For small heights in a uniform field the factor reduces to Δf/f ≈ gh/c², which is 1.09×10⁻¹⁶ per metre of height near the Earth’s surface. That is the form used for tower experiments, and it is accurate as long as h is small compared with the Earth’s radius. For GPS it is not — you need the GM(1/r₁ − 1/r₂)/c² form, which is in the rows above and agrees with the exact answer to nine figures here. In a strong field neither approximation survives and only √(1 − r_s/r) is meaningful.

Where this goes wrong. Typing an altitude where the page asks for a radius from the centre: that is the single most destructive input error here, and it makes the lower clock sit near the centre of the planet. Forgetting that the speed effect exists at all, and quoting 45 µs/day as the GPS correction when the applied figure is the net 38. Getting the sign backwards — remember that the HIGH clock gains, so a satellite clock runs fast while a clock down a mine runs slow. And using gh/c² for a satellite, where it is wrong by about a factor of six because g is nowhere near 9.81 at 20 000 km.

Frequently asked questions

Which clock runs faster, the high one or the low one?

The high one, always. Deeper in a gravitational well means a smaller √(1 − r_s/r) and a slower clock. A clock at the bottom of a mine runs slow compared with one at the surface; a clock on a satellite runs fast. The memory hook that survives an exam is that gravity drags time, so the more gravity you are sitting in, the slower you go.

Is the GPS correction 45 or 38 microseconds a day?

Both, for different quantities, and the question is designed to catch you. The GRAVITATIONAL part is about +45.7 µs/day. The SPECIAL-RELATIVISTIC part, from the satellite’s orbital speed, is about −7.2 µs/day. The NET, which is what the system actually corrects for, is about +38 µs/day. This page gives all three.

Why does the page use GM rather than G times the mass?

Because GM is measured far more precisely than either factor. Spacecraft tracking gives the Earth’s GM to about ten significant figures, while G itself is known to five — it is the worst-determined of the fundamental constants, at 6.674 30(15)×10⁻¹¹. So the Earth-masses and solar-masses options use the IAU 2015 nominal mass parameters directly, and only the kilograms option goes through G.

Has this actually been measured on Earth?

Repeatedly, and at increasingly absurd precision. The Pound–Rebka experiment at Harvard in 1959 measured the shift over 22.5 m of tower using the Mössbauer effect. Hafele and Keating flew caesium clocks round the world in 1972. Optical lattice clocks have since resolved the shift over height differences of a few centimetres. None of those primary papers was read for this page and no number here is taken from them; the experiments are mentioned to say the effect is observed, not to quote a result.

What happens as the lower clock approaches the Schwarzschild radius?

√(1 − r_s/r) goes to zero, so the clock’s rate as seen from far away goes to zero and the ratio between it and any clock above it goes to infinity. At or inside r_s there is no static clock at all — nothing can hover there — so the page stops returning a number and says so instead of printing a huge one. Black hole radii themselves are worked out on a separate page in the astrophysics category.

Does this have anything to do with the twin paradox?

It is the other half of it. A twin who travels fast ages less by special-relativistic time dilation; a twin who sits lower in a gravitational field ages less by this effect. For a real journey both contribute, and for a satellite they partly cancel. The constant-velocity version of the travelling twin is worked out on the space travel page.

Related calculators

References

CODATA Internationally Recommended 2022 Values of the Fundamental Physical Constants, NIST Standard Reference Database 121. Read on 11 November 2026 from the machine-readable table at physics.nist.gov/cuu/Constants/Table/allascii.txt, which prints “speed of light in vacuum 299 792 458 (exact)” — exact because the 1983 definition of the metre fixes it, so c² = 8.9875517873681764×10¹⁶ m²/s² carries no uncertainty either. The same table gives G = 6.674 30(15)×10⁻¹¹ m³kg⁻¹s⁻², which is the least precisely known of the constants on this page by a factor of ten thousand, standard acceleration of gravity 9.806 65 (exact), and the rest energies used here: electron 0.510 998 950 69(16) MeV, proton 938.272 089 43(29) MeV, muon 105.658 3755(23) MeV.
International Astronomical Union, Resolution B3 on recommended nominal conversion constants for selected solar and planetary properties, XXIX General Assembly, 2015. Read on 11 November 2026 from the IAU’s own PDF. The values this page uses are the nominal terrestrial mass parameter (GM)^N_E = 3.986 004×10¹⁴ m³s⁻², the nominal solar mass parameter (GM)^N_☉ = 1.327 124 4×10²⁰ m³s⁻² and the nominal terrestrial equatorial radius 6.3781×10⁶ m. Using GM rather than G×M matters here: GM is known to ten significant figures from spacecraft tracking while G itself is known to five, so a Schwarzschild factor built from GM is four orders of magnitude better determined than one built from a mass times G.
The Ohio State University, Department of Astronomy, course note GPS and Relativity (astronomy.ohio-state.edu/thompson.1847/1144/gps.pdf), read 11 November 2026: GPS satellites at about 20 000 km altitude and 14 000 km/h, with the gravitational effect running their clocks 45 µs/day fast, the speed effect 7 µs/day slow, and a net 38 µs/day. Those are the round numbers everybody quotes and this page reproduces them from the orbit. N. Ashby’s Relativity in the Global Positioning System, Living Reviews in Relativity 6 (2003) 1, is the standard detailed treatment; it could not be fetched for this build (the publisher returned a client error), so nothing here is taken from it.
J. C. Hafele and R. E. Keating, “Around-the-World Atomic Clocks: Predicted Relativistic Time Gains”, Science 177 (1972) 166–168, and “…Observed Relativistic Time Gains”, ibid. 168–170. Neither paper could be read for this build — both are behind a paywall and the US Naval Observatory anniversary PDF that summarises them returned a client error. The figures quoted on this page (eastward −40 ± 23 ns predicted against −59 ± 10 ns observed; westward +275 ± 21 ns predicted against +273 ± 7 ns observed) come from a secondary summary and have not been checked against the papers themselves. Treat the sign and the order of magnitude as solid and the last digit as unverified.
AQA, AS and A-level Physics (7407, 7408) specification, section 3.12.3 Special relativity, inside the option 3.12 Turning points in physics (A-level only). Read from aqa.org.uk on 11 November 2026. It is worth knowing exactly what a board asks for: 3.12.3.3 requires proper time and time dilation in the form t = t₀ / √(1 − v²/c²), with evidence from muon decay; 3.12.3.4 requires length contraction as l = l₀√(1 − v²/c²); and 3.12.3.5 requires mass-energy equivalence, the form E = m₀c²/√(1 − v²/c²), graphs of mass and kinetic energy against speed, and Bertozzi’s experiment as direct evidence. Note what that implies: AQA still teaches relativistic mass. This page cites the specification; it does not claim the board endorses it.