Gravitational Time Dilation Calculator

What Is Gravitational Time Dilation?

Gravitational time dilation is a prediction of Einstein's general relativity: clocks run slower in stronger gravitational fields. The effect is tiny in everyday life but measurable with atomic clocks on Earth and essential for GPS satellites, which orbit farther from Earth's mass and tick slightly faster than clocks on the ground.

The Schwarzschild Time Dilation Formula

For a stationary clock at distance \(r\) from the center of a spherical mass \(M\):

Here \(\Delta t\) is proper time far from gravity, \(\Delta t'\) is elapsed time near the mass, \(G\) is the gravitational constant, and \(c\) is the speed of light. The ratio approaches zero as \(r\) approaches the Schwarzschild radius \(2GM/c^2\).

Altitude Comparison on Earth

Comparing sea level to an altitude \(h\) uses Earth's mass and radius \(R\). Evaluate the square-root factor at \(r = R\) and at \(r = R + h\). GPS orbits at roughly 400 km and gain about 45 microseconds per day relative to sea level before velocity corrections are applied.

Related tools: Earth Orbit Calculator and Black Hole Calculator.