# Length Contraction

By

Dr Lim Zhi Han

,

Koh Zhi Yong

,

Gerald Kang Joon Kiat

●

Published 2022-09-25

The postulates of relativity have led us to conclude that time is relative - the time between two events is different when measured in different inertia frames. This, in turn, implies that the length of an object is likewise dependent on the frame in which it is measured.

To see this, imagine now you (still on the train) set up the "light-clock" such that the photon source is at one end of the train carriage while the mirror is at the other end, such that a photon can be sent down and back. (Figure 1 a)

Question: How long does the photon takes to complete one round trip? To you, the answer is

\begin{align}
\Delta t'=\frac{2\Delta L'}{c},
\end{align}

where \Delta L' is the length of the train as measured by you in S'. However, to your friend, the situation is a little more complicated since the train is moving (Figure 1 b). Let \Delta t_{1} be the time taken for the photon to reach the front end while \Delta t_{2} be the return time. Then geometrically, from Figure 1b, we have

\begin{align}
\Delta t_{1}=\frac{\Delta L + v\Delta t_{1}}{c} \nonumber \\
\Delta t_{2}=\frac{\Delta L - v\Delta t_{2}}{c},
\end{align}

where \Delta L is the length of the train as measured by your friend in S. Note that once again, we have used the second postulate. Solving for \Delta t_{1} and \Delta t_{2}, we have

\begin{align}
\Delta t_{1}=\frac{\Delta L}{c - v} \nonumber \\
\Delta t_{2}=\frac{\Delta L}{c + v}.
\end{align}

So, your friend will measure the round-trip time as

\begin{align}
\Delta t = \Delta t_{1} + \Delta t_{2} = 2\frac{\Delta L}{c}\frac{1}{1-v^{2}/c^{2}}.
\end{align}

Finally, recoginising that \Delta t and \Delta t' are related through time dilation TD (3) and using Equation LC (1), we arrive at

\begin{align}
\Delta L' = \frac{1}{\sqrt{1-v^{2}/c^{2}}}\Delta L = \gamma \Delta L. 
\end{align}

For v<c, \gamma>1 and \Delta L < \Delta L'. In other words, your friend measures a shorter train¹