Advanced Gravitational Time Dilation Calculation

Calculator inputs

Enter a spherical mass model. This calculator includes gravitational effects only and does not add special relativistic velocity corrections.

Central mass (kg)

Body radius (m)

Target altitude above surface (m)

Reference clock mode

Reference altitude above surface (m)

Elapsed time at reference clock

Elapsed time unit

Graph maximum altitude (m)

Graph points

Plotly graph

The graph shows gravitational clock drift per day across altitude, relative to your chosen reference clock.

Example data table

Scenario	Mass (kg)	Radius (m)	Target altitude (m)	Rate ratio	Drift per day (ns)	Difference per day (s)
Earth surface to ISS orbit	5.972000e+24	6.37e+6	408,000.00	1.000000000042	3,619.798861	0.000003619803
Earth surface to GPS orbit	5.972000e+24	6.37e+6	2.02e+7	1.000000000529	45,722.900666	0.000045722903
Jupiter cloud tops to 1,000 km	1.898000e+27	6.99e+7	1.00e+6	1.000000000284	24,564.913303	0.000024564913
Sun photosphere to 1,000 km	1.988470e+30	6.96e+8	1.00e+6	1.000000003041	262,744.136847	0.000262744143

Formula used

For a non-rotating spherical mass, the Schwarzschild time factor at radius r is:

dτ = dt × √(1 - 2GM / (rc²))

Here, dτ is proper time at the clock, dt is coordinate time far away, G is the gravitational constant, M is mass, r is distance from the center, and c is the speed of light.

To compare two clocks at radii r_target and r_reference:

τ_target / τ_reference = √[(1 - 2GM / (r_target c²)) / (1 - 2GM / (r_reference c²))]

If the reference clock measures τ_reference, then the target clock measures:

τ_target = τ_reference × (τ_target / τ_reference)

This model isolates gravitational time dilation only. It does not include velocity-based time dilation, frame dragging, atmospheric effects, or rotation of the central body.

How to use this calculator

Enter the central body mass in kilograms.
Enter the body radius from center to surface in meters.
Set the target clock altitude above the surface.
Choose whether the reference clock is at the surface, custom altitude, or infinity.
Enter the reference clock altitude if you selected custom mode.
Enter the elapsed time measured by the reference clock.
Select the time unit for that elapsed reference duration.
Set the graph altitude range and number of graph points.
Press calculate to show results above the form, beneath the header.
Use the download buttons to export the current result as CSV or PDF.

Frequently asked questions

1) What is gravitational time dilation?

It is the slowing of time deeper in a gravitational field. A clock closer to a massive object accumulates less proper time than a clock farther away.

2) What does this calculator compare?

It compares a target clock at one altitude with a reference clock at the surface, a custom altitude, or effectively at infinity.

3) Does this include orbital speed effects?

No. The calculation is gravitational only. Satellite applications usually need both gravitational and special relativistic velocity corrections for full accuracy.

4) Why must the radius stay above the Schwarzschild radius?

The Schwarzschild expression used here is real and meaningful only outside that boundary. At or below it, this simple calculator is not valid.

5) Why is the drift so small for planets?

Gravity-induced clock differences are usually tiny for planets across everyday distances. High precision navigation and timing systems still care about those small offsets.

6) Can I use this for stars or compact objects?

Yes, as long as the mass can be approximated as spherical and the chosen radii remain outside the Schwarzschild radius.

7) What does the graph show?

It plots altitude on the x-axis and clock drift per day on the y-axis, relative to your selected reference clock.

8) When should I choose infinity as reference?

Use infinity when you want the clock compared to an ideal distant observer, where the gravitational potential is taken as nearly zero.