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.
Time Dilation
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.