Gravitational Time Dilation Calculator
Einstein's general theory of relativity predicts that clocks in stronger gravitational fields tick more slowly. This effect, called gravitational time dilation, has been experimentally confirmed by comparing atomic clocks at different altitudes and is essential for the operation of GPS satellites. The Schwarzschild metric formula for a non-rotating mass gives t0 = tf times the square root of (1 - 2GM/rc squared), where t0 is proper time at radius r from the center of mass M and tf is coordinate time for a distant observer. Enter the mass and radius of interest to compute the time dilation factor and how much time passes locally compared to a distant clock.
Gravitational time dilation formula
t0 = tf * sqrt(1 - 2GM / (r * c^2))
rs = 2GM / c^2 (Schwarzschild radius)
Constants: G = 6.67430 x 10 to the -11th m cubed kg to the -1 s to the -2; c = 299,792,458 m/s (NIST). The dilation factor ranges from 1 (at infinite distance, no dilation) down to 0 at the event horizon (r = rs). Time dilations greater than 1 second per day occur within a few Schwarzschild radii of a stellar-mass black hole.
Time dilation by location
Earth's surface: factor = 0.9999999993. GPS satellite altitude (20,200 km): factor = 0.9999999997. The Sun's surface: factor = 0.9999979. A neutron star surface (r = 10 km, M = 1.4 solar masses): factor = about 0.77. At r = 1.5 Schwarzschild radii: factor = about 0.58. The effect grows dramatically as you approach the event horizon.
Gravitational time dilation: frequently asked questions
What is gravitational time dilation?
According to Einstein's general theory of relativity, time passes more slowly in stronger gravitational fields. A clock near a massive body (like a black hole) ticks more slowly than a clock far from the body. This is gravitational time dilation.
What is the formula for gravitational time dilation?
t0 = tf * sqrt(1 - 2GM/rc squared), where t0 is the proper time measured by a clock at radius r from the center of mass M, tf is the coordinate time measured by a distant observer, G is Newton's gravitational constant, and c is the speed of light.
How significant is gravitational time dilation on Earth?
At Earth's surface, the factor is sqrt(1 - 2GM/(R_earth * c squared)) = approximately 1 - 6.96 x 10 to the -10th. This means clocks on Earth's surface tick about 0.7 microseconds per day slower than clocks in deep space. GPS satellites must account for this effect to maintain accuracy.
What happens at the Schwarzschild radius?
At r = 2GM/c squared (the Schwarzschild radius), the time dilation factor goes to zero, meaning t0 = 0. No time passes for an object at the event horizon of a black hole as seen by a distant observer. This is the mathematical singularity of the Schwarzschild metric.
Are GPS satellites affected by both gravitational and velocity time dilation?
Yes. Gravitational time dilation makes GPS satellite clocks tick faster (because they are farther from Earth's center). Special relativistic time dilation from their orbital speed makes them tick slower. The net effect is that GPS clocks gain about 38.4 microseconds per day relative to clocks on Earth.
Official sources
- NIST CODATA 2018 physical constants: physics.nist.gov/cuu/Constants/.
- NASA JPL relativistic effects in GPS: gps.gov.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.