Gravitational Time Dilation Calculator

Calculator Inputs

Select body preset

Preset fills realistic mass and base radius.

Mass

For presets, mass is locked for accuracy.

Duration to compare

This is the shared coordinate interval being compared.

Use altitude above the selected body's surface (recommended for planets)

If off, you can enter radii directly from the center.

Altitude 1 (lower clock)

Typically 0 for surface.

Altitude 2 (higher clock)

Try 1000 m or a satellite height.

Altitude unit

Altitudes are added to the preset radius.

Example Data Table

Sample comparisons for Earth using altitude mode and a one-day duration.

Body	Altitude 1	Altitude 2	Duration	Expected sign	What it means
Earth	0 m	1,000 m	1 day	Positive	Higher clock runs slightly faster.
Earth	0 m	20,200 km	1 day	Positive	Satellite clock runs faster due to weaker gravity.
Sun	1.0 R☉	2.0 R☉	1 day	Positive	Time dilation decreases rapidly with distance.

Formula Used

For a non-rotating, spherically symmetric mass, the Schwarzschild metric gives the gravitational time dilation factor:

dτ = dt · √(1 − 2GM / (r c²)) = dt · √(1 − r_s/r)

dτ is proper time on a clock at radius r.
dt is coordinate time (ideal clock far away).
r_s = 2GM/c² is the Schwarzschild radius.
To compare two clocks at r1 and r2 for the same dt: τ1 = dt·f(r1), τ2 = dt·f(r2), and the relative rate is τ1/τ2.

How to Use This Calculator

Select a preset body for realistic parameters, or choose Custom.
Pick altitude mode to enter heights above the surface, or enter radii from the center.
Enter a duration to compare (day, year, or any interval).
Press Submit to view the factors, proper times, and differences.
Use Download CSV or Download PDF to save your results.

FAQs

1) What does a time dilation factor represent?

It is the fraction of far-away coordinate time experienced locally. A factor of 0.999 means your clock ticks 0.1% slower than a clock at infinity.

2) Why must radius be larger than the Schwarzschild radius?

The Schwarzschild formula applies outside the event horizon. At or inside the Schwarzschild radius, the square-root term becomes zero or negative, so this simple static calculation no longer applies.

3) What is the difference between radius and altitude?

Radius is measured from the mass center. Altitude is measured above the surface of a chosen body. Altitude mode adds your height to the body’s reference radius automatically.

4) Does this include special relativity from motion?

No. This calculator focuses on gravitational time dilation only. If your clocks are moving relative to each other, you would also add velocity-based (special relativistic) time dilation.

5) Why do GPS satellites need time corrections?

GPS clocks are higher in a weaker gravitational field, so they tick faster gravitationally. They also move quickly, which slows them slightly by special relativity. The combined effect must be corrected for accurate positioning.

6) Can I use this for black holes?

You can explore behavior outside the Schwarzschild radius, but near-horizon physics requires care. For rotating or charged black holes, different metrics apply, so treat results as a simple approximation.

7) What units should I use for best stability?

Use meters for radii and kilograms for mass when possible. Very large or tiny inputs can make differences extremely small, so scientific notation in the results helps interpretation.

8) Why is “coordinate time” used in the calculations?

Schwarzschild time is defined so that far away from the mass, it matches a stationary observer’s clock. Using the same coordinate interval lets you compare two different radii consistently.