Last updated: 2026-06-16

Gravitational Time Dilation

Gravitational time dilation is a consequence of Einstein's general theory of relativity: clocks tick more slowly in stronger gravitational fields. A clock deep in a gravity well runs behind an identical clock far away. The effect is minute on Earth but becomes dramatic near very dense objects such as neutron stars and black holes.

This calculator takes the mass $M$ of a body, a distance $r$ from its centre, and a proper time interval $t_0$ measured at that radius, and returns the dilation factor and the time $t$ that a distant observer records while $t_0$ elapses close to the mass.

How gravitational time dilation works

In general relativity, mass curves spacetime, and that curvature slows the passage of time. The closer a clock sits to a massive body, the more its rate falls behind clocks far from any gravity. Because the distant observer's clock runs comparatively fast, they record a longer interval than the deep-gravity clock — the clock near the mass appears to run slow.

The effect is governed by the ratio of the Schwarzschild radius $2GM/c^2$ to the actual radius $r$. When $r$ is far larger than the Schwarzschild radius the correction is negligible; as $r$ shrinks toward it the dilation diverges.

Formula

The factor is always greater than one. As $r \to \infty$ it approaches one and the dilation vanishes; as $r$ approaches the Schwarzschild radius $2GM/c^2$ it grows without bound.

Worked example

Consider a clock at the surface of the Sun: $M = 1.989 \times 10^{30}\ \text{kg}$ and $r = 6.9634 \times 10^8\ \text{m}$.

Step 1 — Schwarzschild ratio:

Step 2 — Dilation factor:

A clock on the Sun's surface runs slow by about two parts in a million compared with a distant clock. Over one year, the distant observer records roughly 67 extra seconds. Enter these values in the calculator to reproduce the result.

Dilation at different field strengths

Real-world relevance

Gravitational time dilation is not just theoretical. GPS satellites, orbiting where gravity is weaker, gain about 45 microseconds per day relative to ground clocks; this must be corrected or positions would drift by kilometres each day. The Pound–Rebka experiment measured the shift over a 22-metre tower, and atomic clocks today detect the difference across height changes of a fraction of a metre.

Limitation: static, non-rotating mass

This calculator uses the Schwarzschild metric, which describes a spherical, non-rotating, uncharged mass. It requires the radius to lie outside the Schwarzschild radius $2GM/c^2$. Rotation (the Kerr metric), charge, and the velocity-based time dilation of special relativity are separate effects not included here.