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Elapsed time between two events as measured by a clock that passes through both events
In relativity, proper time (from Latin, meaning own time) along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time, which is independent of coordinates, and is a Lorentz scalar. The interval is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line. The proper time interval between two events depends not only on the events, but also the world line connecting them, and hence on the motion of the clock between the events. It is expressed as an integral over the world line (analogous to arc length in Euclidean space). An accelerated clock will measure a smaller elapsed time between two events than that measured by a non accelerated (inertial) clock between the same two events. The twin paradox is an example of this effect. By convention, proper time is usually represented by the Greek letter τ (tau) to distinguish it from coordinate time represented by t. Coordinate time is the time between two events as measured by an observer using that observer's own method of assigning a time to an event. In the special case of an inertial observer in special relativity, the time is measured using the observer's clock and the observer's definition of simultaneity. The concept of proper time was introduced by Hermann Minkowski in 1908, and is an important feature of Minkowski diagrams.

The formal definition of proper time involves describing the path through spacetime that represents a clock, observer, or test particle, and the metric structure of that spacetime. Proper time is the pseudo Riemannian arc length of world lines in four dimensional spacetime. From the mathematical point of view, coordinate time is assumed to be predefined and an expression for proper time as a function of coordinate time is required. On the other hand, proper time is measured experimentally and coordinate time is calculated from the proper time of inertial clocks. Proper time can only be defined for timelike paths through spacetime which allow for the construction of an accompanying set of physical rulers and clocks. The same formalism for spacelike paths leads to a measurement of proper distance rather than proper time. For lightlike paths, there exists no concept of proper time and it is undefined as the spacetime interval is zero. Instead, an arbitrary and physically irrelevant affine parameter unrelated to time must be introduced.

For a twin paradox scenario, let there be an observer A who moves between the A coordinates (0,0,0,0) and (10 years, 0, 0, 0) inertially. This means that A stays at for 10 years of A coordinate time. The proper time interval for A between the two events is then

So being "at rest" in a special relativity coordinate system means that proper time and coordinate time are the same. Let there now be another observer B who travels in the x direction from (0,0,0,0) for 5 years of A coordinate time at 0.866c to (5 years, 4.33 light years, 0, 0). Once there, B accelerates, and travels in the other spatial direction for another 5 years of A coordinate time to (10 years, 0, 0, 0). For each leg of the trip, the proper time interval can be calculated using A coordinates, and is given by

So the total proper time for observer B to go from (0,0,0,0) to (5 years, 4.33 light years, 0, 0) and then to (10 years, 0, 0, 0) is

Thus it is shown that the proper time equation incorporates the time dilation effect. In fact, for an object in a SR (special relativity) spacetime traveling with velocity for a time , the proper time interval experienced is

which is the SR time dilation formula.

The difference between SR and general relativity (GR) is that in GR one can use any metric which is a solution of the Einstein field equations, not just the Minkowski metric. Because inertial motion in curved spacetimes lacks the simple expression it has in SR, the line integral form of the proper time equation must always be used.