Imagine carrying a clock from one meeting to another. The elapsed time shown by that clock is the proper time along its path between the two meetings. The definition does not require the clock to be on Earth, inside a spacecraft, or at rest in some preferred frame.
In special relativity, an inertial frame assigns coordinates to events using a system of synchronised clocks at rest in that frame. The coordinate time between two events can therefore involve readings from different clocks. Proper time is the elapsed time recorded along a particular clock's own path.
Connecting a moving clock to a frame
Suppose a clock moves at constant speed v in an inertial frame. If the frame assigns a coordinate time interval Δt to two events on the clock's path, the clock records a proper time interval Δτ given by
The speed in this expression is measured in that chosen inertial frame. The formula compares the moving clock with the frame's coordinated timing system.
At v = 0.8c, the square-root factor is 0.6. If the frame assigns ten years to the journey, the travelling clock records six years. Neither clock is faulty. The intervals answer different operational questions.
Why the route matters
Two clocks can leave the same event and meet again later. Each records a proper time, but the two values need not agree because the clocks can follow different paths through spacetime.
For motion with a changing speed in an inertial frame, divide the trip into short intervals and add the proper time accumulated in each interval. In the continuous limit, this becomes an integral.
An outward-and-return journey cannot be described as the traveller staying at rest in one inertial frame throughout. The traveller changes inertial frames during the turnaround. That change removes the apparent symmetry in the usual twin scenario.
Proper time belongs to a path
It is misleading to ask which person has the proper time as if only one clock could measure it. Both people measure proper time along their own worldlines. If they separate and reunite, the readings can be compared directly at the reunion.
The reliable question is: Which two events are we comparing, and what path does each clock take between them? Once those are identified, the equations become a way of calculating a clearly defined measurement.
