How this instrument works
The ratio dt' ⁄ dt compares two clocks: one held stationary at radial distance r from the center of a mass M, and one infinitely far away where spacetime is flat. It comes straight out of the Schwarzschild solution to Einstein's field equations, the exact description of spacetime around any non-rotating, spherically symmetric body. For a clock that is not moving through space, only through time, the metric collapses to exactly this square root — nothing is approximated to reach it.
The term inside the root, 2GM ⁄ rc², is the body's Schwarzschild radius r_s = 2GM ⁄ c² divided by r. Write it that way and the formula reads dt' ⁄ dt = √(1 − r_s ⁄ r): the closer r gets to r_s, the harder the ratio falls toward zero. For Earth, r_s works out to about 8.9 millimetres — a marble-sized sphere hiding under 6,371 kilometres of planet — so ordinary distances stay nowhere near it and the effect on a wristwatch is tiny. Push r below r_s, as at a black hole's event horizon, and the square root has no real value: no stationary clock can exist there at all.
The formula covers gravity alone. A clock in motion also runs slow for the separate, purely kinematic reason described by special relativity, and a real orbiting clock — a GPS satellite, say — experiences both effects at once, added rather than multiplied at these small magnitudes. Confusing the two is the most common misreading: a satellite clock is farther from Earth (gravitational effect: ticks faster) but also moving fast (kinematic effect: ticks slower), and the two corrections do not cancel.
- Enter the Central mass — the body whose gravity is slowing time nearby, in kilograms.
- Enter the Distance from center — the radial distance from that body's middle to the clock, in kilometres or metres, not height above the surface.
- Read the Time dilation factor (dt' ⁄ dt) — a number just under 1 for any real mass and distance.
- Subtract that reading from 1 to get the fractional slowdown, then multiply by any elapsed duration to see how far the local clock falls behind.
Worked example — a clock at Earth's surface versus one at infinity
Set the Central mass to Earth's value, 5.9722×10²⁴ kg, and the Distance from center to Earth's mean radius, 6,371,000 m. First 2GM: 2 × 6.6743×10⁻¹¹ × 5.9722×10²⁴ = 7.972×10¹⁴. Then rc²: 6,371,000 × (299,792,458)² = 5.726×10²³. Divide the two, 1.392×10⁻⁹, subtract from 1, and take the square root: dt' ⁄ dt = 0.999999999304 — precisely the golden figure this instrument is built to reproduce.
That ratio means a surface clock loses about 6.96×10⁻¹⁰ of every second compared to one sitting far from Earth's gravity — roughly 7 parts in 10 billion, which is a drift of about 22 milliseconds over a full year. It sounds negligible until you add the numbers up for a GPS satellite: sitting higher, in weaker gravity, its onboard clock gains around 45 microseconds a day from this same effect, partly offset by about 7 microseconds a day lost to its orbital speed, for a net gain near 38 microseconds a day that receivers must subtract or positions drift by kilometres.
Questions
Why is the result always less than 1?
Because 2GM ⁄ rc² is positive for any real mass and any positive distance, so 1 minus that term is always below 1, and its square root stays below 1 too. The ratio equals exactly 1 only in the limiting case of zero mass or infinite distance — the flat-spacetime baseline the formula measures every other clock against.
What does it mean if the calculator has no real answer?
It means the Distance from center you entered is smaller than the body's Schwarzschild radius, 2GM ⁄ c², so the term under the square root has gone negative. That boundary is the event horizon of a black hole; no clock can sit still at or inside it, which is exactly why the algebra refuses to produce a number.
Does this calculator include the speed-based time dilation satellites feel?
No — this formula isolates the gravitational effect only, for a clock at rest relative to the central mass. A moving clock also picks up special-relativistic time dilation from its velocity, which is a separate calculation; engineers add both effects together to get a satellite clock's true offset from the ground.
Why do GPS satellites care about a 7-parts-in-10-billion effect?
Because GPS timing has to be accurate to nanoseconds: light travels about 30 centimetres per nanosecond, so a clock error of microseconds becomes a position error of hundreds of metres. Left uncorrected, the combined relativistic drift would accumulate into kilometres of positioning error within a single day.
Has gravitational time dilation actually been measured on Earth?
Yes. Pound and Rebka confirmed it in 1959 by sending gamma rays up a 22.5-metre tower at Harvard and detecting the predicted frequency shift. Fifty years later, NIST researchers repeated the idea with optical atomic clocks and measured a rate difference between two clocks just 33 centimetres apart in height.
What counts as r in this formula?
The straight-line distance from the center of the mass to the clock, not altitude above a surface. For a clock on Earth's surface, r is Earth's radius, about 6,371 km; for a satellite, r is that radius plus the satellite's altitude, measured out to the satellite itself.