Time Dilation Calculator

Time Dilation

Special relativity says a moving clock runs slow from the point of view of someone standing still. You write it as Δt = γ Δt₀. Here Δt₀ is the proper time, meaning the time the moving clock reads for itself, and γ = 1/√(1 - v²/c²) is the Lorentz factor. When v is far below c, γ barely differs from 1 and you can ignore the whole thing. As v climbs toward c, though, γ shoots up without limit.

In 1971 the Hafele-Keating experiment put this to the test. Four caesium atomic clocks rode commercial flights around the world, going both eastward and westward, while a fifth clock stayed put at the U.S. Naval Observatory. When the travelling clocks came back, the gaps between them and the reference clock came out to a few hundred nanoseconds, close enough to the special and general relativity predictions to land inside a few percent.

Applications

Take GPS. Those satellites circle the Earth at roughly 14,000 km/h and sit in a weaker gravitational potential than a receiver down on the ground. Put the two relativistic effects together and the onboard clocks need a correction of about 38 μs per day (-7 μs from special relativity, +45 μs from general relativity). Skip it and your reported position would wander off by something like 10 km every day. The twin paradox is just this same idea stretched out over an imaginary interstellar trip.

FAQ

Which twin actually ages less? The traveller, because that's the one who accelerates. The acceleration breaks the symmetry between the two frames, which is what clears up the apparent paradox.

Is the dilation real or just an illusion? It's real, and you can measure it. Atomic clocks on planes, GPS satellites, and cosmic-ray muons have all confirmed it in the lab and in the field.

Why include Δt₀ (proper time)? It's the time read in the clock's own rest frame, the one quantity that stays the same no matter which observer you ask.