How much slower does a clock run when it moves fast or sits in gravity?
Einstein’s relativity says time is not the same for everyone. A clock moving quickly runs slow compared with one at rest, and a clock deep in a gravitational field runs slow compared with one far away. This time dilation calculator handles speed (special relativity), gravity (near a planet, star, neutron star or black hole) and the twin paradox, showing each step so you can follow the working.
Both effects are real and measured: atomic clocks flown on aircraft, GPS satellite clocks and particle lifetimes all agree with the formulas below.
How to use the Time Dilation Calculator
- Choose Speed, Gravity or Twin paradox.
- For speed, enter the speed as a fraction of c or in km/s, m/s, km/h or mph, and the time interval.
- For gravity, pick a body or enter a mass and radius; add a second radius to compare two clocks.
- For the twin trip, enter the speed and the one-way distance in light years.
- Read the factor, the times on each clock and the working underneath.
Time Dilation Calculator features
Lorentz factor
γ with full precision even at everyday speeds, shown as 1 + a tiny number.
Gravity presets
Earth, GPS orbit, the Sun, a neutron star and a black hole, plus custom mass and radius.
Two-clock comparison
Reproduces the roughly 45.7 microseconds per day that GPS clocks gain from weaker gravity.
Twin paradox
Years on Earth and for the traveller on a round trip.
Worked steps
Each formula with the numbers substituted.
Formulas and worked examples
Speed: γ = 1 ÷ √(1 − v²/c²) and a moving clock records t′ = t ÷ γ. At 0.8 c, γ = 5/3, so one year for a stationary observer is 0.6 years (219 days) on the moving clock.
Gravity: t′ = t × √(1 − 2GM ÷ (r c²)) for a clock held at rest at radius r from mass M, compared with a clock far away. The term 2GM/c² is the Schwarzschild radius. On Earth’s surface the factor is 1 − 7×10⁻¹⁰, a loss of about 22 milliseconds a year; at twice a ten-solar-mass black hole’s Schwarzschild radius it is 0.707.
Twin paradox: a traveller flying 4.37 light years out and back at 0.8 c spends 2 × 4.37 ÷ 0.8 = 10.9 years by Earth’s clocks but 10.9 ÷ γ = 6.55 years by their own, so the twin on Earth is older.
Constants and sources
c = 299,792,458 m/s (exact). GM for the Sun, 1.3271244×10²⁰ m³/s², and for Earth, 3.986004×10¹⁴ m³/s², are the IAU 2015 Resolution B3 nominal values; the solar radius 6.957×10⁸ m and Earth’s equatorial radius 6.3781×10⁶ m are the same resolution’s nominal radii. Years are Julian years of 31,557,600 seconds, and a light year is the distance light travels in one.
Limits
- Gravity mode is for a clock at rest in the Schwarzschild solution: no spin, no orbital motion, and one mass only. The real GPS correction also subtracts about 7 µs per day for orbital speed.
- The twin trip uses instant turnarounds and constant speed; real acceleration phases change the numbers a little.
- See the light year converter for distances and travel times at ordinary speeds.
Time Dilation Calculator FAQ
What is the time dilation formula?
For speed, t′ = t ÷ γ where γ = 1 ÷ √(1 − v²/c²). For gravity, t′ = t × √(1 − 2GM ÷ (r c²)).
How much does time slow at 90% of the speed of light?
γ is about 2.29, so a moving clock runs at 44% of the normal rate: one year passes there for every 2.3 years outside.
Is time dilation real?
Yes. It has been measured with atomic clocks on aircraft and satellites, and in the lifetimes of fast muons. GPS must correct for it.
Which twin is older in the twin paradox?
The one who stayed on Earth, because only the traveller changes direction (accelerates) so the situation is not symmetric.
What happens to time at a black hole horizon?
The factor goes to zero at the Schwarzschild radius for a distant observer, but a clock cannot be held at rest there, so the calculator stops at the horizon.
Does gravity time dilation affect me on Earth?
Very slightly: a clock at sea level runs about 22 milliseconds a year slower than one far from Earth, and a clock a metre higher gains about a hundredth of a nanosecond a day.