Time is what clocks measure. Measured differently.
Two clocks: one stays home, one rides a ship. Move the velocity slider and watch the ship's clock fall behind. Switch on the twin paradox and a worldline draws itself below — the geometry of spacetime, doing the explaining for you.
Velocity of Ship B
Spacetime diagram
Home frame · ct vs xHow to read it
Time runs up. Space runs sideways. A clock at rest traces a straight vertical line — same place, all times. A ship moving at velocity v tilts: forward in space, forward in time. The steeper the tilt toward the light cone, the more your worldline stretches in this diagram while your own clock ticks off less proper time.
In the twin paradox, Ship B accelerates away, coasts to a star, turns around, and comes home. Its worldline is two slanted legs meeting at a peak. The home twin's worldline is the straight vertical between the same two events. The vertical line is the longest one in proper time, not the shortest. That inversion — straight lines maximize, not minimize, the duration — is special relativity in one picture.
§ Appendix · the equations behind the picture
Lorentz factor
Time on the moving clock advances by Δτ = Δt / γ. As v → c, γ → ∞ and the moving clock approaches a standstill in the home frame.
Lorentz transformation
Different observers slice spacetime into different "now"s. The dashed line on the diagram is Ship B's plane of simultaneity — what B calls "the present" tilts up with v.