Suppose two observers O and O’ in the inertial systems S and S‘ respectively are at rest with respect to each other. They synchronise their respective clocks and they agree that the time interval between any two events as measured by their own clocks is the same. Let the clock in the system S give signals at regular intervals and suppose the system S’ moves to the right
along the X-axis with a uniform velocity v with respect to S.
The observer O in the frame S keeps his clock at a fixed point x1 in his system and measures the time interval T0 that elapses between two events (y two signals) that occours at times t1 and t2 in this frame
$ \displaystyle \therefore {{T}_{0}}={{t}_{2}}-{{t}_{1}}$
Let the times registered by the observer O’ in the inertial system S’ therefore, according to Lorentz transformations
$ \displaystyle {{{{t}’}}_{1}}=\frac{{{{t}_{1}}-\frac{v}{{{{c}^{2}}}}{{x}_{1}}}}{{\sqrt{{1-\frac{{{{v}^{2}}}}{{{{c}^{2}}}}}}}}$
and
$ \displaystyle {{{{t}’}}_{2}}=\frac{{{{t}_{2}}-\frac{v}{{{{c}^{2}}}}{{x}_{2}}}}{{\sqrt{{1-\frac{{{{v}^{2}}}}{{{{c}^{2}}}}}}}}$
$ \displaystyle \therefore {{{{t}’}}_{2}}-{{{{t}’}}_{1}}=\frac{{{{t}_{2}}-{{t}_{1}}-\frac{v}{{{{c}^{2}}}}\left( {{{x}_{2}}-{{x}_{1}}} \right)}}{{\sqrt{{1-\frac{{{{v}^{2}}}}{{{{c}^{2}}}}}}}}$
The clock in the system S remains fixed at the point $ \displaystyle {{{x}_{1}}}$
$\displaystyle \therefore {{x}_{2}}={{x}_{1}}$
Hence,
$ \displaystyle {{{{t}’}}_{2}}-{{{{t}’}}_{1}}=\frac{{{{t}_{2}}-{{t}_{1}}}}{{\sqrt{{1-\frac{{{{v}^{2}}}}{{{{c}^{2}}}}}}}}$
Thus, the moving observer O’ in the inertial system S’ measures the time interval $ \displaystyle {{t}_{2}}-{{t}_{1}}=T$ between the same two events for which the observer O in the system S measures the time interval
$ \displaystyle {{t}_{2}}-{{t}_{1}}={{T}_{0}}$
$ \displaystyle \therefore T=\frac{{{{T}_{0}}}}{{\sqrt{{1-\frac{{{{v}^{2}}}}{{{{c}^{2}}}}}}}}$
As v is less than c, $\displaystyle T>{{T}_{0}}$. Hence the observer S’ measures a longer interval of time between the two events with his clock at the rest with respect to him. In other words, a clock in the frame S appears to go slow to an observer in the frame S’ who is in motion with respect to the frame S. This phenomenon is known as time dilation. If there are two observers in relative motion with respect to each other, each observer would find the other moving observer’s clock to run slow. This the consequence of time-dilation is reciprocal.
As an example of the phenomenon of time dilation, suppose that the velocity of the observer
$ \displaystyle v=0.98c,\text{ then}$
$\displaystyle \sqrt{{1-\frac{{{{v}^{2}}}}{{{{c}^{2}}}}}}=\sqrt{{1-\frac{{{{{0.98}}^{2}}{{c}^{2}}}}{{{{c}^{2}}}}}}=0.2$
$\displaystyle \therefore T=\frac{{{{T}_{0}}}}{{0.2}}=5{{T}_{0}}$
In other words , if the moving observer in the inertial frame S’ measures a time interval of 5 seconds between the two events by his own clock, the interval between the same two events, as measured by him on the clock of the stationary observer in the inertial frame S, will be 1 second and vice versa.
Proper and non-proper time.
We have seen that an observer O in the inertial frame S measures a time interval $ \displaystyle {{T}_{0}}={{t}_{2}}-{{t}_{1}}$ between two events by a clock fixed in hhis frame at a point, say $ \displaystyle {{x}_{1}}$, whereas the time interval between the same two events is measured as
$ \displaystyle {{T}_{0}}={{{{t}’}}_{2}}-{{{{t}’}}_{1}}=\frac{{{{t}_{2}}-{{t}_{1}}}}{{\sqrt{{1-\frac{{{{v}^{2}}}}{{{{c}^{2}}}}}}}}$
by the observer O’ in the inertial frame S’ moving with a velocity v in the +X direction with respect to the clock in the frame S’.
The time interval between two events which occur at the same position measured by a clock in the inertial frame in which the events occur is called proper time.
The time interval between the same two events measured by an observer from an inertial frame which is moving with respect to the clock is known as non-proper or relativistic time.
