Gravitational Time Dilation Calculator Formula
Understand the math behind the gravitational time dilation calculator. Each variable explained with a worked example.
Formulas Used
Time at Infinity
dilated_time = proper_time / sqrt(1 - 2 * 6.674e-11 * mass / (radius * pow(2.998e8, 2)))Extra Time per Second (ns)
time_difference_ns = (1 / sqrt(1 - 2 * 6.674e-11 * mass / (radius * pow(2.998e8, 2))) - 1) * 1e9Variables
| Variable | Description | Default |
|---|---|---|
mass | Central Mass(kg) | 5.972e+24 |
radius | Distance from Centre(m) | 6371000 |
proper_time | Proper Time (at surface)(s) | 1 |
How It Works
Gravitational Time Dilation
General relativity predicts that clocks run slower in stronger gravitational fields.
Formula
t_far = t_near / sqrt(1 - 2GM / (rc^2))
where the Schwarzschild factor 2GM/(rc^2) is the ratio of the Schwarzschild radius to r.
On Earth's surface, clocks run about 0.7 nanoseconds per second slower than clocks far from any gravity.
Worked Example
On Earth's surface (M = 5.972e24 kg, r = 6.371e6 m).
- 012GM/(rc^2) = 2 * 6.674e-11 * 5.972e24 / (6.371e6 * 8.988e16)
- 02= 7.972e14 / 5.726e23 = 1.392e-9
- 03sqrt(1 - 1.392e-9) = 1 - 6.96e-10
- 04t_far = 1 / (1 - 6.96e-10) = 1 + 6.96e-10 s
- 05Difference: 0.696 ns per second
Frequently Asked Questions
How does this affect GPS?
GPS satellite clocks are in weaker gravity and run about 45 microseconds per day faster. Without correction, GPS positions would drift by about 10 km per day.
What happens at a black hole?
As r approaches the Schwarzschild radius (2GM/c^2), the time dilation becomes infinite. Time appears to stop at the event horizon as seen from outside.
Has gravitational time dilation been measured?
Yes. The Pound-Rebka experiment (1959) measured the gravitational redshift over just 22.5 metres. Modern atomic clocks can detect differences over a height of 1 metre.
Ready to run the numbers?
Open Gravitational Time Dilation Calculator