Time dilation
The relativistic effect that a clock measured by an inertial observer in motion ticks slower than an identical clock at rest in that observer's frame, by the Lorentz factor γ = 1/√(1 − β²). Symmetric between frames; not an illusion; verified by muon decay, atomic clocks, and GPS.
Definition
Time dilation is the relativistic kinematic effect that a clock measured by an observer in inertial motion ticks slower than an identical clock at rest in that observer's frame, by the Lorentz factor γ = 1/(1 − β²)^(1/2), where β = v/c. A proper time interval Δτ measured between two events on a single clock's worldline is related to the coordinate time interval Δt measured by an observer who sees that clock moving at velocity v by Δt = γ Δτ. The effect is symmetric: A's clock runs slow as measured from B's frame, and B's clock runs slow as measured from A's frame. There is no contradiction because the two measurements use different simultaneity slices to compare ticks — the relativity of simultaneity is what makes the symmetry consistent.
Time dilation is not an illusion or a measurement artefact. The 1941 Rossi-Hall cosmic-ray muon-decay experiment was the first direct verification: muons created at altitude by cosmic-ray showers and observed at sea level survive the trip in numbers far exceeding what their 2.2 μs proper lifetime would allow at non-relativistic speeds, because their lab-frame lifetime is dilated by γ ≈ 30 at typical cosmic-ray energies. Hafele-Keating (1971) flew caesium clocks around the world east and west on commercial airliners and measured the predicted nanosecond-level deviations from ground clocks. The Global Positioning System's atomic-clock corrections include a special-relativistic time-dilation contribution of roughly −7 μs/day (orbital motion, blue-shifting on-board time relative to the ground); without it, GPS positioning would drift by about 2 km in the first 24 hours of operation.