Gravitational Time Dilation Calculator
Time flows slower closer to a massive object. This calculator uses the exact Schwarzschild formula from general relativity to show how much time slows down at a given distance from any mass, compared to an observer far from any gravitational field. Enter the mass and radius of the object, a proper time interval, and optionally a second reference location to find the elapsed time at each point and the time difference between them. Presets for common bodies, show-your-work steps, and a reference table of dilation factors for famous objects are included.
What is gravitational time dilation?
Gravitational time dilation is a consequence of Einstein's general theory of relativity: a clock in a stronger gravitational field runs slower than one in a weaker field, as measured by a distant observer. The effect arises because massive objects curve spacetime, and a photon or clock traveling through curved spacetime takes a different path than it would in flat space. The phenomenon has been confirmed by atomic clocks at different altitudes, by the Pound-Rebka experiment using gamma rays, and by the precise timing corrections required for GPS satellites.
The Schwarzschild formula
For a static, non-rotating, spherically symmetric mass, the exact formula is: t_0 = t_f * sqrt(1 - 2GM / rc^2), where t_0 is the proper time measured by an observer at distance r from the center of a mass M, t_f is the coordinate time measured by an observer infinitely far away, G is the gravitational constant (6.674e-11 m^3 kg^-1 s^-2), and c is the speed of light. The term 2GM/c^2 is the Schwarzschild radius r_s. When r equals r_s, the dilation factor reaches zero and time stops, which is the definition of a black hole event horizon. As r approaches infinity, the factor approaches 1 and clocks run at the same rate as deep space.
GPS satellites and everyday applications
GPS satellites orbit at about 20 200 km altitude, where Earth's gravitational field is weaker than at the surface. Each satellite clock therefore runs about 45 microseconds fast per day due to gravitational time dilation. Partially offsetting this is a 7-microsecond slowdown per day from their orbital velocity (special relativistic time dilation). The net effect is that without relativistic corrections, GPS position errors would accumulate at about 10 kilometres per day. The corrections are baked into the satellite firmware, and every navigation fix you obtain silently relies on Einstein's equations.
Comparing locations: Earth vs. Sun vs. neutron stars
At Earth's surface the dilation factor is about 0.999 999 302 888, meaning a ground-level clock loses roughly 22 seconds per year compared to one in deep space. At the Sun's surface the factor drops to about 0.999 997 88, giving a loss of about 66 seconds per year. A typical neutron star with 1.4 solar masses crammed into a 10-kilometre radius has a dilation factor of only about 0.77, so a clock on its surface loses more than 7 million seconds per year versus deep space. Black holes represent the extreme limit: at exactly the event horizon the dilation factor is zero and time effectively stops. This creates the famous "frozen star" appearance, where in-falling matter appears to slow to a halt at the horizon as seen from far away.
Gravitational time dilation for known bodies
| Object | Surface / orbit | Dilation factor | Seconds lost per year |
|---|---|---|---|
| Deep space | reference | 1.000 000 000 000 | 0 |
| GPS orbit (Earth) | 26 571 km | 0.999 999 999 674 | 0.010 |
| Earth surface | 6 371 km | 0.999 999 302 888 | 22.0 |
| Sun surface | 695 700 km | 0.999 997 880 000 | 66.4 |
| White dwarf | ~7 000 km (0.6 M_sun) | 0.999 913 000 000 | 2 700 |
| Neutron star | ~10 km (1.4 M_sun) | ~0.769 000 000 000 | ~7.2 million |
| Black hole event horizon | r = r_s | 0 (time stops) | infinity |
Dilation factor at the surface (or characteristic radius) of various objects. A factor of 0.9999997 means a clock there runs at 99.99997% of the deep-space rate.
Frequently asked questions
What does a dilation factor of 0.9999993 mean?
It means a clock at that location ticks at 99.99993% of the rate of a clock in deep space, or equivalently loses about 22 seconds per year compared to a reference clock infinitely far from any mass. The further the factor is from 1.0, the stronger the gravitational field.
What is the Schwarzschild radius?
The Schwarzschild radius (r_s = 2GM/c^2) is the radius at which the escape velocity equals the speed of light. If a mass is compressed inside this radius, it becomes a black hole. For Earth, r_s is about 8.87 mm; for the Sun it is about 2.95 km. Any object at exactly r_s experiences infinite time dilation, meaning time stops relative to a distant observer.
Does this calculator include special relativistic (velocity) time dilation?
No. This calculator computes only gravitational time dilation from general relativity. If you also need velocity-based time dilation (from special relativity), you would multiply the two dilation factors together. For GPS satellites, gravitational dilation adds 45 microseconds per day and velocity dilation subtracts 7 microseconds per day, giving a net gain of 38 microseconds per day.
Why must the radial distance be greater than the Schwarzschild radius?
Inside the event horizon of a black hole, the formula produces a value under the square root that is negative, which has no real solution. Physically, no observer can remain at rest inside an event horizon, so the Schwarzschild metric does not apply in the same way. The calculator returns no result when r is at or below r_s.
Can I use this for the time dilation in a black hole movie like Interstellar?
Yes, approximately. In the film, Miller's planet orbits very close to the supermassive black hole Gargantua. At a distance just above the innermost stable circular orbit (3 r_s), the dilation factor is roughly 0.943, which corresponds to about 1 hour on the planet per 1.06 hours for a distant observer. The extreme ratio seen in the film (1 hour = 7 years) requires the black hole to be spinning extremely fast (a Kerr metric), which this calculator does not model. For a Schwarzschild (non-rotating) black hole, even at 1.5 r_s the factor is about 0.816.
Has gravitational time dilation been experimentally confirmed?
Yes, many times. The Pound-Rebka experiment at Harvard in 1959 measured the gravitational redshift of gamma rays over a 22-metre height difference and agreed with the prediction to within 10%. Later experiments using hydrogen masers on aircraft and atomic clocks on the Gravity Probe A rocket reached 0.01% agreement. GPS satellites provide continuous operational confirmation every day.