Hong Life CalculatorsUnderstanding the gravitational time dilation calculator — concept and principle
This calculator finds how much slower a clock runs close to a massive body than one far away, according to general relativity. It gives the rate, the time lost per day and per year, the Schwarzschild radius, and how much faster a clock at a second, higher distance runs. It is used for physics courses and to understand why GPS satellites need relativistic corrections.
How it is calculated
dτ ÷ dt = √(1 − 2GM ÷ (r c²)) = √(1 − r_s ÷ r)
Schwarzschild radius r_s = 2GM ÷ c²
- r is the distance from the centre in km; the mass can be in Earth masses, solar masses or kg.
- GM_Earth = 3.986004418 × 10¹⁴ m³/s², GM_Sun = 1.32712440018 × 10²⁰ m³/s², c = 299,792,458 m/s.
- Time lost per day = (1 − dτ/dt) × 86,400 s; the gain at a second distance is the difference of the two slowdowns.
The principle
In general relativity, gravity is the curvature of spacetime, and clocks deeper in a gravitational well tick more slowly as seen from far away. For a non-rotating spherical mass the Schwarzschild solution gives the factor √(1 − r_s/r). Near Earth the effect is tiny, about 7 parts in 10¹⁰, but atomic clocks measure it easily. At the Schwarzschild radius the factor reaches zero, which is the event horizon of a black hole.
Worked example
Default inputs: 1 Earth mass, clock at 6,371 km, second clock at 26,560 km (GPS orbit radius).
- r_s = 2 × 3.986 × 10¹⁴ ÷ c² = 0.00887 m.
- Slowdown at the surface ≈ GM ÷ (r c²) = 0.6961 parts per billion; rate = 0.9999999993.
- Time lost per day = 6.961 × 10⁻¹⁰ × 86,400 s = 60.15 µs; per year 21.97 ms.
- At 26,560 km the slowdown is 1.670 × 10⁻¹⁰, so that clock gains 45.72 µs per day relative to the surface.
How to read the result
Positive gain at the second distance means the higher clock runs faster. The GPS figure of about 45.7 µs per day from gravity alone is partly offset by the satellites' speed, which slows their clocks by roughly 7 µs per day; the net correction is built into the system. Without it, position errors would grow by kilometres per day.
Common mistakes
- Entering altitude above the surface instead of distance from the centre.
- Forgetting that the speed effect of special relativity is not included.
- Expecting noticeable effects on Earth in daily life.
Frequently asked questions
Do clocks run slower in the basement?
Yes, very slightly; atomic clocks have measured the difference over height changes of less than a metre.
Why do GPS clocks run fast overall?
The gravitational speed-up of about 45.7 µs per day outweighs the slowing from orbital speed of about 7 µs per day.
What happens at the Schwarzschild radius?
The factor reaches zero; for a black hole this is the event horizon.
Sources: K. Griest, Physics 161: Black Holes, Lecture 5 (UC San Diego, 2011)