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Section 52.5 Einstein's Clock

Einstein's clock consists of an evacuated chamber in which light reflects back and forth between two mirrors M\(_1\) and M\(_2\) placed at a fixed distance \(L\) from the source as shown in Figure 52.5.1.

In each trip, light travels from M\(_1\) back to M\(_1\text{,}\) which is a distance of \(2L\) ,at constant speed \(c\text{.}\) You might say one tick of the clock will occur every \(2L/c\) unit of time.

\begin{equation*} \text{One tick } = \frac{2L}{c}. \end{equation*}

Thus, if \(L=1.5\text{ m}\) and \(c=3\times 10^{8}\text{ m/s}\text{,}\) then each tick will take \(10^{-8}\text{ s}\) or \(0.1\text{ ns}\text{.}\) By reducing \(L\) arbitrarily, we can obtain clocks that tick at arbitrarily small intervals. Net counts of these ticks gives interval of time.

Figure 52.5.1.

In this clock time is obtained by analyzing three events. E1 : Light leaving M\(_1\text{,}\) E2: Light reflecting at M\(_2\text{,}\) and E3: Light arriving at M\(_1\text{.}\) By analyzing these three events in two frames, we can find time duration between events E3 and E1 in any frame.

Subsection 52.5.1 Synchronizing Clocks in One Frame

In a frame we would like time to be well-defined and same at all points of space at same instance. How can we make sure of this? Einstein suggested that to synchronize time at two points A and B in space we place two identical clocks at the two points and send a light signal from one to another, say from A to B as shown in Figure 52.5.2.

Suppose signal from A was sent when \(t = 0\) at A. Then, when light arrives at B, we reset the clock at B to a time adjusted for the travel time of light.

\begin{equation*} t_B = t_A + \frac{d_{AB}}{c}. \end{equation*}

Figure 52.5.2.

The procedure outlined here makes sure that at the instant light was sent from A the time at B must also have been \(t_A\text{,}\) i.e., whatever was the reading at A, even though we didn't know it at the time. synchronizing clocks is an important way to get local times which are guaranteed to be universal time in the frame we are working.