Proper time and coordinate time

Proper time Δτ is what one clock records between two events on its worldline. For a constant-speed segment relative to another inertial frame, that frame assigns Δt=γΔτ, with γ=1/√(1−v²/c²). Einstein’s 1905 paper in English translation develops clock synchronization by light signals and Lorentz transformations. The Max Planck Institute’s light-clock explanation derives why an inertial observer judges a uniformly moving clock to run slower.

Worked example at 0.6c

Suppose a clock records 4 years of proper time between two events, while moving uniformly at 0.6c relative to a laboratory frame. Then γ=1/√(1−0.6²)=1.25, so the lab assigns Δt=1.25×4=5 years to those same events. This compares one uniform-motion segment; it is not a complete age prediction after a round trip.

For a calculation, first identify the two events and the clock that records both; its reading is Δτ. Then name the inertial frame and relative speed used for γ. This prevents mixing a clock’s proper time with another frame’s coordinate interval.

A Lorentz transformation changes space and time coordinates together. For motion along one axis, x′=γ(x−vt) and t′=γ(t−vx/c²). Events at different positions that are simultaneous in one frame need not be simultaneous in another. The 0.6c calculation compares two events on one clock; it does not make distant-event simultaneity universal.

Reciprocity and limits

Two inertial observers can each calculate that the other’s moving clock runs slow because their frames define distant-clock synchronization and simultaneity differently. A departure-and-reunion comparison includes turning or acceleration, so one constant-velocity frame cannot describe the whole journey. This example assumes flat spacetime and uniform relative speed. Gravitational time dilation belongs to general relativity, and reunion comparisons require the full worldline.

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