Gravitational Time Dilation Calculator
Calculates how gravity slows time near massive objects using Einstein's general relativity. Use it when comparing elapsed time at different distances from a planet, star, or black hole.
About this calculator
According to Einstein's general theory of relativity, a clock ticks more slowly when it is deeper in a gravitational well. The dilated time t experienced by a distant observer relates to the proper time τ measured locally by: t = τ × √(1 − 2GM / rc²), where G = 6.674×10⁻¹¹ m³ kg⁻¹ s⁻² is the gravitational constant, M is the mass of the massive object in kg, r is the distance from its center in meters, and c = 299,792,458 m/s is the speed of light. When r approaches the Schwarzschild radius (2GM/c²), the factor under the square root approaches zero, meaning time effectively stops for a distant observer. This effect is not merely theoretical — GPS satellites must correct for gravitational time dilation to maintain positional accuracy.
How to use
Suppose you want to find how much time passes for an observer 10,000 km from Earth's center (mass M = 5.972×10²⁴ kg), given a proper time of 3,600 seconds (1 hour). Step 1: Compute 2GM/rc² = 2 × 6.674×10⁻¹¹ × 5.972×10²⁴ / (10,000,000 × (299,792,458)²) ≈ 8.87×10⁻¹⁰. Step 2: Dilated time = 3,600 × √(1 − 8.87×10⁻¹⁰) ≈ 3,599.9999984 s. The difference is tiny near Earth but becomes enormous near neutron stars or black holes.
Frequently asked questions
How does gravitational time dilation differ from velocity-based time dilation?
Gravitational time dilation arises from differences in gravitational potential, as described by general relativity, while velocity-based (kinematic) time dilation arises from relative motion, as described by special relativity. Both slow clocks, but through different mechanisms. Near a black hole, gravitational time dilation dominates and can become arbitrarily large. In everyday life, both effects are combined — for example, GPS satellites experience both because they move fast and orbit at altitude.
Why does time pass slower closer to a massive object?
Mass curves spacetime, and clocks deeper in that curvature tick at a slower rate relative to clocks further away. This is a fundamental prediction of Einstein's general theory of relativity, confirmed by precision atomic-clock experiments. The greater the gravitational potential difference between two locations, the greater the time discrepancy. At the event horizon of a black hole, a distant observer sees a local clock freeze entirely.
How significant is gravitational time dilation for GPS satellites?
GPS satellites orbit at about 20,200 km altitude where Earth's gravity is weaker, causing their clocks to tick roughly 45 microseconds per day faster than clocks on the surface. A velocity correction subtracts about 7 microseconds per day, yielding a net gain of approximately 38 microseconds daily. Without correcting for this, GPS position errors would accumulate at roughly 10 km per day. Engineers pre-program satellite clocks to run slightly slow before launch to compensate.