Time Dilation Calculator

Proper time (symbol: tau or t) is the time measured by a clock that travels with the moving object, sometimes called the 'wristwatch time'. It is the shortest possible time interval between two events that occur at the same location in some reference frame. Proper time is invariant: every observer calculates the same proper time for a given spacetime path, even though they may disagree on the coordinate time between those events.

The twin paradox imagines one twin leaving Earth at high speed, travelling to a distant star, and returning. On return, the travelling twin is younger than the Earth-bound twin. It appears paradoxical because special relativity seems to say each twin should see the other's clock running slow. The resolution is that the situation is not symmetric: the travelling twin must accelerate and decelerate (turn around), which involves changing inertial frames. This asymmetry means the travelling twin genuinely accumulates less proper time and is measurably younger on reunion.

GPS satellites orbit at approximately 3.87 km/s (about 0.0000129c). Special relativistic time dilation causes satellite clocks to run about 7.2 microseconds slower per day relative to ground clocks. However, the satellites are also at higher altitude and in a weaker gravitational field, so general relativistic gravitational time dilation makes their clocks run faster by about 45.9 microseconds per day. The net effect is that satellite clocks run fast by roughly 38.4 microseconds per day, which GPS receivers compensate for. Without these corrections, GPS positions would drift by about 10 kilometres per day.

Time dilation is a real, measured physical effect, not science fiction. It means clocks in relative motion genuinely tick at different rates. In a practical sense, a high-speed traveller 'time travels' into the future: they age less and arrive in a future moment where more time has passed on Earth. However, this is strictly one-way: you cannot use special relativistic time dilation to travel backward in time. Each clock accumulates its own proper time monotonically forward.

The muon experiment is among the clearest demonstrations. Muons created by cosmic rays in the upper atmosphere (at roughly 10 km altitude) have a rest-frame half-life of about 2.2 microseconds. At rest they should decay before reaching Earth. But muons travel at about 0.998c, giving gamma around 15. Their proper lifetime is still 2.2 microseconds, but in the Earth's frame they live about 33 microseconds, long enough to reach the ground in measurable numbers. This was confirmed experimentally by Rossi and Hall in 1941 and has been reproduced many times since. Precision atomic clock experiments on aircraft (Hafele and Keating, 1971) also directly measured the predicted time dilation.