Gravitational Time Dilation Calculator
Calculate how much slower time passes in a gravitational field.
From GPS satellites to black hole event horizons — general relativity in action.
Gravitational time dilation is a prediction of Einstein’s General Relativity: clocks run slower in stronger gravitational fields.
Formula:
t_local / t_∞ = √(1 - r_s/r) = √(1 - 2GM/(rc²))
Where:
- t_local = time elapsed near the mass
- t_∞ = time elapsed far from the mass (in flat spacetime)
- r_s = Schwarzschild radius = 2GM/c²
- r = distance from the center of the mass
Practical examples:
- At Earth’s surface (r = 6,371 km, M = 5.97×10²⁴ kg): ratio ≈ 0.9999999993 — clocks run slow by 69 μs/day
- At GPS satellite altitude (r ≈ 26,559 km): gravitational dilation is +45.9 μs/day (clocks run faster)
- (GPS also has velocity-based dilation of −7 μs/day, net +38.4 μs/day — this must be corrected!)
- At the ISS (400 km altitude): gravitational dilation of +45 μs/day minus velocity dilation −25 μs/day = net clock loses time
- Near a black hole at r = 1.1 × r_s: ratio ≈ 0.302 (clocks run at 30% the far-field rate)
- At r = r_s: ratio = 0 (time stops from the perspective of a distant observer)
Why GPS needs this correction: Without correcting for both gravitational and velocity time dilation, GPS position errors would accumulate at about 10 km per day. General relativity is not just an academic curiosity — it is built into every GPS calculation.
The Schwarzschild radius of Earth: only 8.87 mm. Earth is far from any relativistic effects.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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