Implications of Special Relativity due to Lorentz Transformation: Length Contraction
Characteristics of the Lorentz Transformation
In the theory of Special Relativity, the transformation between two coordinate systems is different from classical transformations. This is due to the fact that ’the speed of light is the same for all observers’. Considering this condition, the Lorentz transformation was derived. As a result of the Lorentz transformation, there are three new phenomena that do not appear in classical physics.
- Loss of simultaneity
- Time dilation
- Length contraction
Length Contraction
Length contraction is not actually separate from time dilation. It’s essentially the same phenomenon. It means that it’s impossible for time dilation to occur without length contraction. Let’s look at the picture below.
Assume there is a $A$frame and a $A$frame moving at a constant velocity of $v_{0}$ in the $x$direction. And let’s say there is a rod with length $L$ stationary in the $A$frame. This rod is seen from the $A$frame as in the picture below.
Then what would it look like from the $A^{\prime}$frame? If we find the coordinate of the left end of the rod, it would be as follows.
$$ \begin{pmatrix} ct^{\prime} \\ x^{\prime} \\ y^{\prime} \\ z^{\prime} \end{pmatrix} = \begin{pmatrix} \gamma_{0} & -\gamma_{0}\beta_{0} & 0 & 0 \\ -\gamma_{0}\beta_{0} & \gamma_{0} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} ct \\ 0 \\ 0 \\ 0 \end{pmatrix}= \begin{pmatrix} \gamma_{0}ct \\ -\gamma_{0}\beta_{0}ct \\ 0 \\ 0 \end{pmatrix} $$
Then, since $ct^{\prime}= \gamma_{0}ct$ and $x^{\prime}=-\gamma_{0}\beta_{0}ct$, by combining them we can get $x^{\prime}=-\beta_{0}ct^{\prime}$. Now, if we find the coordinate of the right end of the rod, it would be as follows.
$$ \begin{pmatrix} ct^{\prime} \\ x^{\prime} \\ y^{\prime} \\ z^{\prime} \end{pmatrix} =\begin{pmatrix} \displaystyle \gamma_{0} & -\gamma_{0}\beta_{0} & 0 & 0 \\ -\gamma_{0}\beta_{0} & \gamma_{0} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} ct \\ L \\ 0 \\ 0 \end{pmatrix}= \begin{pmatrix} \gamma_{0}ct-\gamma_{0}\beta_{0}L \\ -\gamma_{0}\beta_{0}ct+\gamma_{0}L \\ 0 \\ 0 \end{pmatrix} $$
Therefore,
$$ ct^{\prime}=\gamma_{0}ct-\gamma_{0}\beta_{0}L \implies \gamma_{0}ct=-ct^{\prime}-\gamma_{0}\beta_{0}L $$
And, since $x^{\prime}= -\gamma_{0}\beta_{0}ct+\gamma_{0}L$, by combining them we can get the following.
$$ x^{\prime}-\gamma_{0} L= \beta_{0} \left[ ct^{\prime} + \gamma_{0}\beta_{0} L \right] $$
It appears as a parallel translation by $\gamma_{0}L$ in the $x^{\prime}$ direction and by $-\gamma_{0}\beta_{0}L$ in the $ct^{\prime}$ direction. Looking at the picture, it would be as follows.
It would be easier to understand if we think that time dilation occurs due to the Lorentz transformation, and the observed length changes due to that time difference.
Lastly, let’s take a look at how the length changed by length contraction relates to the original length. Looking at the picture, it shows that when $x^{\prime}-\gamma_{0}L= -\beta_{0} \left[ ct^{\prime}-(-\gamma_{0}\beta_{0}L) \right]$, if $ct^{\prime}=0$ then $x^{\prime}=L^{\prime}$is true. If we substitute $ct^{\prime}=0$ directly and calculate, it would be as follows.
$$ x^{\prime}=\gamma_{0}L-\gamma_{0}{\beta_{0}}^2L=\gamma_{0}(1-{\beta_{0}}^2)L $$
At this time, the following equation holds.
$$ \gamma_{0}=\dfrac{1}{\sqrt{1-{\beta_{0}}^2}} \implies (1-{\beta_{0}}^2)=\dfrac{1}{{\gamma_{0}}^2} $$
Therefore, it follows that:
$$ x^{\prime}=\frac{1}{\gamma_{0}}L=L^{\prime} $$
However, since $L^{\prime} = \dfrac{L}{\gamma_{0}}$ and $\gamma_{0} \ge 1$, $L^{\prime} \le L$ is always true. Here lies the reason for length contraction. As the formula shows, it can never extend. Similar to time dilation, contraction does not occur in the direction that the $A^{\prime}$frame is not moving (in the perpendicular direction). If you’re curious, try calculating with different coordinates. It’s always in the direction parallel to the movement where simultaneity breaks, and time dilation and length contraction occur.