Gravitational Time Dilation - Outside A Non-rotating Sphere

Outside A Non-rotating Sphere

A common equation used to determine gravitational time dilation is derived from the Schwarzschild metric, which describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object. The equation is:

, where

• is the proper time between events A and B for a slow-ticking observer within the gravitational field,
• is the coordinate time between events A and B for a fast-ticking observer at an arbitrarily large distance from the massive object (this assumes the fast-ticking observer is using Schwarzschild coordinates, a coordinate system where a clock at infinite distance from the massive sphere would tick at one second per second of coordinate time, while closer clocks would tick at less than that rate),
• is the gravitational constant,
• is the mass of the object creating the gravitational field,
• is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object, but is actually a Schwarzschild coordinate),
• is the speed of light, and
• is the Schwarzschild radius of M.

To illustrate then, a clock on the surface of the Earth (assuming it does not rotate) will accumulate around 0.0219 seconds less than a distant observer over a period of one year. In comparison, a clock on the surface of the sun will accumulate around 66.4 seconds less in a year.