How To Calculate Time Dilation

Time dilation says that the interval between two events depends on who is measuring. A clock carried by a moving observer ticks more slowly than the same clock left behind, and both observers are correct in their own frames.

The effect is described by the Lorentz factor, which is one over the square root of one minus the velocity squared over the speed of light squared. At rest the factor is exactly one and there is no dilation at all.

The growth is not linear. Half the speed of light gives a factor of 1.154701, but ninety percent gives 2.294157, and ninety nine percent gives 7.0888. The curve bends sharply upward as the velocity approaches light speed.

Nothing with mass can reach the speed of light, because the factor grows without bound and the energy needed grows with it. The calculator clamps the velocity just below light speed to keep the result finite.

Proper time is the time measured by a clock that is present at both events. A passenger on a fast ship measures proper time; the observer left at the origin measures the longer dilated time.

Muons created in the upper atmosphere are the classic laboratory evidence. Their proper lifetime is about 2.2 microseconds, yet they reach the ground because their internal clocks run slow from our point of view.

The same factor governs length contraction. A fast ship is shortened along its direction of travel by exactly the same ratio, which keeps the speed of light constant for every observer.

Satellite navigation needs this correction. A GPS satellite orbits at about 1.2922 percent of light speed, which stretches each day by roughly 7.1712 microseconds before the general relativistic correction is added.

The effect is symmetric. Each observer sees the other's clock running slow, and the apparent paradox is resolved only when the two clocks are brought back together and one of them has accelerated.

Time dilation is measured directly and routinely. Atomic clocks flown on aircraft agree with the prediction to a fraction of a percent, and particle accelerators see unstable particles live far longer than their rest-frame lifetime.

General relativity adds a second effect. Deeper in a gravitational field a clock runs slower, so a clock on the ground and one in orbit disagree for two reasons that partly cancel in the case of the GPS satellites.

The muon evidence is worth spelling out. Created at about fifteen kilometres altitude with a lifetime of 2.2 microseconds, a muon at light speed covers only 660 metres before decaying. Detectors at sea level count far more of them than that non-relativistic fraction would allow.

The twin paradox is not a paradox. The travelling twin has to accelerate to leave, to turn around and to stop, so the two situations are not symmetric, and only the traveller comes back younger.