Gravitational Time Dilation

Calculate how much slower time runs near a massive body, compared with a far-away observer, using the Schwarzschild solution of general relativity. From GPS satellites to neutron stars.

RelativityGravityBlack Holes

Inputs

Input examples

Click Solve next to the quantity you want to find: it moves into the result below. Then fill in the others. Example: solve r to find how close to a neutron star you lose 1 hour per day.

Mass M

Distance from centre r

Time interval far away Δt_∞

Time lost by the nearby clock Δt_∞ − Δt₀

What is Gravitational Time Dilation?

Einstein’s general relativity predicts that clocks tick slower deeper in a gravitational field. A clock on the ground loses time compared with one on a mountain — measurably so with modern atomic clocks, even over a height difference of a few centimetres.

The effect is set by the ratio r_s/r, where r_s = 2GM/c² is the Schwarzschild radius. For Earth r_s is only 9 mm, so the effect is tiny — yet GPS would drift by kilometres per day without correcting for it. At r = r_s, the event horizon of a black hole, time appears to stop.

r >> r_seffect is tiny

r → r_stime almost stops

r < r_sinside the event horizon

higher upclocks run faster

Live Graph

log–log

fraction of time lost: 1 − √(1 − r_s/r)

your value

reference bodies

Result

Enter the values above

How Slow do Clocks Run?

vs. far away

Location	r_s / r	Clock loses
GPS satellite orbit	3.3×10⁻¹⁰	14 µs / day
Earth’s surface	1.4×10⁻⁹	60 µs / day
Sun’s surface	4.2×10⁻⁶	0.18 s / day
White dwarf (Sirius B)	5.2×10⁻⁴	22 s / day
Neutron star surface	0.34	4.6 h / day
Black hole horizon	1	time stops

Key Ideas

Schwarzschild Radius

The radius to which a mass would have to be squeezed to become a black hole. It sets the natural scale for all relativistic effects around that mass.

Example:

Weak-Field Approximation

When r is much larger than r_s, the fractional slowdown equals the gravitational potential divided by c².

Example:

The GPS Correction

Satellite clocks run fast because they sit higher in Earth’s field, and slightly slow because they move at 3.9 km/s. Uncorrected, positions would drift by about 11 km per day.

Example:

Sources & Verification

References

J. B. Hartle, Gravity: An Introduction to Einstein’s General Relativity, Addison-Wesley, 2003 — Ch. 9 (The Geometry Outside a Spherical Star).
E. Tiesinga, P. J. Mohr, D. B. Newell & B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 2018”, Reviews of Modern Physics 93, 025010 (2021) — values of the constants used.