Gravitational Time Dilation Calculator

From a body mass and distance from its center, compute the gravitational time dilation ratio t0/t using the Schwarzschild metric. See the Schwarzschild radius, yearly time loss, and the idea behind GPS clock corrections.

Input

Enter a body mass and the distance from its center to compute the gravitational time dilation ratio from the Schwarzschild metric.

Body preset

Selecting one fills in the mass and distance. Editing a value switches to custom.

Body mass M

Mass of the body that is the source of gravity.

Distance from center r

Distance from the body center to the point where time is measured.

Result

Time dilation ratio t0 / t

0.999999999

Slowdown fraction 1 − t0/t = 6.9608e-10

Schwarzschild radius r_s

0.008869407 m

Time loss per year

0.021966505 s

r / r_s

718,311,821.631035

Formula t0/t = sqrt(1 − 2GM/(rc²)) = sqrt(1 − r_s/r), using G = 6.674e-11 and c = 299792458 m/s.

GPS satellites orbit where gravity is weaker and their clocks run faster, so this gravitational dilation must be corrected for positioning.

How it works

The time dilation ratio is t0/t = sqrt(1 − 2GM/(rc²)) = sqrt(1 − r_s/r). Here t0 is the proper time at distance r and t is the coordinate time far from the gravitational field.
r_s = 2GM/c² is the Schwarzschild radius. As r approaches r_s the slowdown grows, and at r = r_s time appears frozen to a distant observer. When r is at or below r_s the formula breaks down and an error is shown.
The constants used are the gravitational constant G = 6.674e-11 m³ kg⁻¹ s⁻² and the speed of light c = 299792458 m/s. Enter mass in kilograms and distance in meters.
The yearly time loss is an approximation: the slowdown fraction 1 − t0/t multiplied by the number of seconds in a year. In the weak field at a planet surface the value is extremely small.
GPS satellites orbit higher up where gravity is weaker, so their clocks run faster than clocks on the ground. Without correcting this gravitational dilation and the special relativity effect, positioning would drift every day.

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