Time Dilation
Proper time is measured by a clock that stays at one place in its own rest frame. Other inertial frames assign a larger coordinate-time interval between the same two ticks.
Why does a moving clock accumulate less proper time between the same events?
Use a light clock: in its own rest frame the light pulse moves straight between mirrors. A frame that sees the clock move sees a longer diagonal light path. Because both frames measure the same light speed, the moving-frame coordinate time must be larger.
is proper time: the interval read by one clock present at both events.
The same tick traces different spacetime geometry.
It is the time recorded by a clock carried along the path connecting the two events, not a universal master time.
At ordinary speeds γ is almost 1. As v approaches c, the difference between proper time and coordinate time grows rapidly.
Each inertial observer can describe the other's moving clock as accumulating less proper time between appropriately compared events. The asymmetry in the twin scenario comes from different worldlines, not a preferred inertial frame.
When v = 0, proper time and coordinate time agree.
Six seconds of proper time correspond to ten seconds in the frame where the clock moves at 0.8c.
A massive clock cannot reach c, but the dilation factor grows without bound as the speed approaches it.