Formula and Method for Gravitational Time Dilation
General relativity predicts that clocks run at different rates depending on how deep they sit in a gravitational field. For a non-rotating, spherically symmetric mass M, the exact relationship — the time component of the Schwarzschild metric — is Δt' = Δt√(1 − 2GM/(rc²)), where Δt is the time elapsed on a clock infinitely far from the mass, Δt' is the proper time elapsed on a clock held fixed at radial distance r from the mass's center, G is the gravitational constant, and c is the speed of light. Because Δt' is always smaller than Δt, a clock closer to a massive body ticks slower than one farther away.
Understanding the formula
The dimensionless term 2GM/(rc²) grows as you get closer to the mass (smaller r) or as the mass M increases. It stays below 1 as long as r is larger than the Schwarzschild radius, rs = 2GM/c² — the radius at which escape velocity would equal the speed of light. At r = rs the formula predicts Δt' = 0 (an infinitely dilated clock, i.e. an event horizon); for r < rs the square root becomes imaginary and the formula no longer describes a physical static clock. For ordinary planets and stars, 2GM/(rc²) is minuscule — about 1.4 × 10⁻⁹ at Earth's surface — so the effect shows up as a tiny fractional slowing rather than a dramatic factor. This calculator reports that slowing directly as "time lost" in whatever unit fits best (nanoseconds up to days).
Common mistakes
- Confusing gravitational and velocity time dilation: a moving clock also runs slow by the special-relativistic factor √(1 − v²/c²); GPS satellites experience both effects at once, and they act in opposite directions (gravity speeds the satellite clock up relative to the ground, motion slows it down).
- Using altitude instead of distance from the center: r in the formula is measured from the center of mass, so an object at altitude h above a planet of radius R uses r = R + h, not h alone.
- Ignoring the Schwarzschild-radius limit: plugging in a distance at or inside rs (as can happen with a compact object like a neutron star or black hole) makes the formula invalid; this calculator checks for that and reports an error instead of a nonsensical result.
Real-world applications
- GPS and other GNSS satellites apply a relativistic correction combining this gravitational effect (about +45 microseconds/day at orbital altitude) with the velocity-based effect (about −7 microseconds/day) — without it, position errors would accumulate by several kilometers per day.
- The 1959 Pound–Rebka experiment measured gravitational time dilation (via gravitational redshift) over a 22.5 m tower at Harvard, an early direct test of general relativity in a lab.
- Optical atomic clocks are now precise enough to detect gravitational time dilation from height differences of about one meter, letting physicists effectively "weigh" changes in elevation with a clock.
- Near a black hole's Schwarzschild radius the same formula predicts extreme time dilation — the physical basis for the near-horizon time-slowing effects sometimes shown in science fiction.