Time Dilation and Length Contraction

Einstein's two postulates lead to consequences that clash with common sense, but which have been verified countless times in the laboratory and in nature. The first is time dilation: a clock moving relative to an observer ticks more slowly than if it were at rest relative to them. This is neither an optical illusion nor a defect of the clock: time itself flows differently depending on the chosen frame of reference.

Take a concrete, verifiable example: muons, unstable particles created by cosmic rays in the upper atmosphere, have a proper lifetime of about 2.2 microseconds. Without relativistic effects, they should decay long before reaching the ground. Yet they are detected in abundance at sea level, because their speed close to c, about 0.995 c, dilates their apparent lifetime, measured from Earth, to about 22 microseconds.

The second consequence is length contraction: a moving object, measured along its direction of motion, appears shorter to an observer at rest than its proper length, the one measured in its own frame. A 100-meter ruler moving at 0.9 c would measure only about 43.6 meters for a stationary observer. Conversely, a passenger on the ship notices no shortening of their own ruler: it is the outside space that would seem contracted to them.

The following diagram takes up the Minkowski diagram again, adding two worldlines: that of a stationary observer, vertical, and that of a moving observer, inclined.

        ct
         ^        worldline of (B, moving)
        4|- - - - - - - -*  <- tick 4 (proper) of B
         |             /
        3|- - - - - -*     <- tick 3 (proper) of B
         |         /
        2|- - - -*         <- tick 2 (proper) of B
         |     /
        1|- -*             <- tick 1 (proper) of B
         | /
        0*_____________________> x
         O     worldline of (A, at rest, x=0)

(B's ticks are separated by equal proper time along its worldline)
(projected onto A's ct axis, these same ticks appear more widely spaced)
(this is the geometric expression of time dilation)

On the inclined worldline, the regularly spaced ticks mark equal proper durations for B. Projected onto the ct axis of the stationary observer A, these same ticks appear more widely spaced: this is exactly time dilation. Watch out for this classic pitfall: each observer sees the other's clock running slow compared with their own, without any contradiction, because the very notion of simultaneity depends on the frame of reference.