Gravitational Time Dilation Calculator

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, so clocks run slow by about 60 μs/day. Over a year that is 22 milliseconds
• At GPS (Global Positioning System) satellite altitude (r ≈ 26,559 km): gravitational dilation is +45.7 μs/day (clocks run faster, being further out of the well)
• GPS also carries velocity-based dilation of −7.2 μs/day, for a net +38.5 μs/day. Left uncorrected that is a positioning error of about 10 km per day, so the correction is built into the satellites
• At the ISS (International Space Station) (400 km altitude): the gravitational effect is only +3.6 μs/day, because 400 km is barely out of the well, while the orbital speed of 7.66 km/s costs −28.2 μs/day. Net −24.6 μs/day, so ISS clocks lose time. That works out to roughly 9 milliseconds a year of aging saved, which is where the “astronauts return very slightly younger” line comes from
• 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.