Data on pure time intervals obtained with respect to two relatively moving inertial frames of reference will differ and so will data on spatial distances. It is possible, however, to form from time intervals plus distances a single expression that will have the same value with respect to all inertial frames of reference. If the time interval between two distant events be denoted by T and their distance from each other by L, an expression involving a quantity symbolized by t can be derived in which t squared equals the square of the time interval minus the fraction of distance squared over speed of light squared: t² = T² - L²/c². This will have the same value as T² - L²/c², with T and L having been obtained in another inertial frame of reference. If t² is positive, then t is called the invariant (timelike) interval between the two events. If t² is negative, then the expression l, derived from the above as l² = L² - c²T², will be called the invariant (spacelike) interval.

The invariant interval between two instants in the history of one physical system equals the ordinary time lapse T measured by means of a clock at rest relative to that physical system, because, in such a comoving frame of reference, L vanishes. That is why such an invariant (timelike) interval is also referred to as the "proper time" elapsed between the two instants. Any clock will read its own proper time.

Given an inertial frame of reference and two similar material systems ("twins")--for instance, two atomic clocks of identical design--suppose that one of these clocks remains permanently at rest in the given frame, whereas the other clock is moved at a high speed first in one direction away from the first clock and subsequently in the opposite direction until the two clocks are again close to each other. According to the Lorentz transformation, the second clock has been slower than the first throughout its journey, and hence it shows a smaller lapse of time than the clock that has remained at rest. By reading the clocks, one can then tell which clock has remained at rest, which one has moved. This difference in behaviour of the two clocks has been called the clock paradox or the twin paradox.

The "paradox" supposedly consists of a violation of the principle of relativity, according to which no asymmetric distinctions exist between different inertial frames of reference. The fallacy of this argument lies in the fact that no inertial frame of reference is associated with the second clock, as it cannot have moved free of acceleration throughout its journey: at least once its velocity (i.e., the direction of its motion) must have been changed drastically, so as to enable it ever to return to its mate. Hence no violation of the principle of relativity; no paradox is involved. Various experiments on moving particles and atoms have indeed confirmed the predictions of the theory.