Time Dilation
The measurement of time involves the concept of clocks attached at each position co-ordinate of a Lorentz frame. As a consequence of Lorentz transformations, it is said (though incorrectly) that a moving clock runs slower than a stationary one. Let us discuss this phenomenon, which is known as time dilation.
Consider two events occurring at same position x', at different times t_{1}' and t_{2}' as observed in a clock attached at x' in frame S'. When observed from stationary frame S, the two events are observed to occur at different space-time points (x_{1},t_{1}) and (x_{2}, t_{2}). The time co-ordinates of the two events, according to clocks fixed in S at points x_{1} and x_{2}, are related to S'. Hence, we gett_{2} - t_{1} = (t_{2}' - t_{1}') , (x_{2}' = x_{1}' = x')or, Δ t' = 1/ Δ t = Δ t (1 - v^{2}/c^{2})^{1/2}That is, the time interval observed by the moving clock is less than the interval observed by stationary clocks. The time scale is apparently stretched out or dilated as observed by the moving clock. Hence the phrase, "the moving clock runs slower."However, all clocks run at the same rate in all Lorentz frames.For an infinitesimal time interval, we havedt' = 1/ dtWhile dt' is the time interval according to (same) clock associated with moving position co-ordinate (or the moving object to which this clock is attached), dt is the time interval as observed by two neighbouring clocks associated with the stationary frame. Remember that t_{1} and t_{2} are the readings by clocks at x_{1} and x_{2} in frame S.The time interval measured by the clock attached to a moving object is called proper or intrinsic time interval of the object. It is denoted by d τ; hence,d τ = dt' = 1/ dtwhere dt is the corresponding interval recorded by a stationary observer.Now, the space-time interval ds' for the object at rest in S' is,ds'^{2} = c^{2} dt'^{2} - dl'^{2}= c^{2} dt'^{2} = c^{2} d τ^{2}Therefore, d τ =ds'/cis an invariant of Lorentz transformations. That is, in a sense, is trivial because d τ is always the time interval measured in the frame attached to the object.