Gravitational Time Dilation Calculator

Calculate how strongly gravity slows a local clock relative to a distant observer using the Schwarzschild solution.

Gravity and elapsed time
Enter a spherical body's mass, the clock's distance from its center, and the distant observer's elapsed time.

About gravitational time dilation

Gravitational time dilation is the difference in elapsed time measured by clocks at different gravitational potentials. In general relativity, mass curves spacetime, and a clock deeper in a gravitational field runs more slowly when compared with a clock very far from the mass. This calculator uses the Schwarzschild solution, which describes spacetime outside a spherical, nonrotating, uncharged body. It is a useful approximation for planets, ordinary stars, and idealized compact objects when rotation can be ignored. The calculation begins with the Schwarzschild radius, equal to two times the gravitational constant times mass divided by the speed of light squared. The local time-rate factor is the square root of one minus the Schwarzschild radius divided by the radial distance. Multiplying a distant observer's coordinate time by this factor gives the proper time recorded by a stationary local clock. A factor near one means that the difference is tiny; a smaller factor means stronger gravitational slowing. For Earth, the Schwarzschild radius is only about nine millimeters while the surface lies more than six million meters from the center. The resulting factor is extremely close to one, yet precision clocks can measure the accumulated difference. Satellite navigation systems must account for both gravitational and velocity-related relativistic effects. Near a neutron star or an idealized black hole, the ratio between Schwarzschild radius and distance is much larger, so the difference becomes dramatic. The radial distance must be measured from the center of mass, not from the object's surface. The formula also requires the clock to remain outside the Schwarzschild radius. At that radius the static-coordinate factor reaches zero, and below it a stationary observer cannot exist in this model. Real astronomical bodies may rotate, depart from spherical symmetry, or have surrounding matter, so advanced work can require a different spacetime metric. Use kilograms, meters, and seconds together for direct SI results. Scientific notation is convenient for astronomical masses, such as 5.972e24 kilograms for Earth. The displayed Schwarzschild radius provides a useful scale check, while the dimensionless time-rate factor makes comparisons straightforward. This tool supports education and estimation; it does not model acceleration, orbital motion, special-relativistic velocity effects, frame dragging, or the internal gravitational field of an extended body.

Gravitational time dilation examples

These examples compare weak planetary gravity with increasingly compact objects.

InputsResultInterpretation
Earth, surface, 86,400 sAbout 86,399.99994 sA surface clock loses only a tiny fraction of a second per distant day.
Mass 1e30 kg, radius 2,969.34 m, 100 sAbout 70.71 sAt twice the Schwarzschild radius, the time-rate factor is about 0.7071.
Sun, surface, 3,600 sAbout 3,599.992 sSolar surface gravity produces a measurable but still small difference.

How to calculate gravitational time dilation

  1. Enter the central body's mass in kilograms.
  2. Enter the clock's radial distance from the body's center in meters.
  3. Enter the elapsed time measured by a distant observer in seconds.
  4. Select Calculate time dilation and compare the local time, factor, and Schwarzschild radius.

Gravitational time dilation FAQ

Why does gravity slow time?

General relativity describes gravity as curved spacetime rather than an ordinary force. Clocks deeper in that curvature accumulate less proper time than clocks farther away.

What is the Schwarzschild radius?

It is the radius at which the escape speed in the corresponding idealized geometry reaches light speed. For a nonrotating black hole, it identifies the event horizon.

Should I enter altitude or distance from the center?

Enter distance from the center of mass. If you know altitude above a planet, add the planet's mean radius first.

Does this include motion-based time dilation?

No, this calculator isolates the static gravitational effect. Moving clocks can also experience special-relativistic time dilation that must be evaluated separately.

Can I calculate a value inside a black hole?

No, the stationary-clock interpretation used here applies only outside the Schwarzschild radius. Interior calculations require different coordinates and a more complete physical model.