Characteristics of Special Relativity due to Lorentz Transformation: Loss of Simultaneity
Characteristics of Lorentz Transformation
The transformation between two coordinate systems in special relativity is different from classical transformation due to the principle that “the speed of light is the same for all observers”. Derived with this condition in mind, we get the Lorentz transformation. The Lorentz transformation introduces three new phenomena that do not appear in classical physics.
- Loss of simultaneity
- Time dilation
- Length contraction
Loss of Simultaneity
Among the physical problems we commonly encounter from a young age, there’s this kind.
The most important concept in solving these problems is “simultaneity.” In Chinese characters, simultaneity is 同時, which means “the same time”. Then, readers would know what it means when we say “two events occur simultaneously.” However, when relativistic effects are considered, whether or not events are simultaneous can depend on the situation of the observer. An event that is simultaneous to $A$ is not simultaneous to $B$, and an event that is simultaneous to $B$ is not to $A$. Let’s say inertial coordinate system $A^{\prime}$ is moving at a speed of $v_{0}$ in the $x$ axis direction relative to inertial coordinate system $A$.
Suppose two events occur at time $t=0$ in the $A$ system.
At the origin $P =\begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}$, at point $L$ on the $x$ axis, $Q =\begin{pmatrix} 0 \\ L \\ 0 \\ 0 \end{pmatrix}$, let’s think about how these two events would appear from the $A^{\prime}$ system. Using the Lorentz transformation, we can calculate as follows.
$$ \begin{align*} P^{\prime} &= \begin{pmatrix} \gamma_{0} & -\gamma_{0}\beta_{0} & 0 & 0 \\ -\gamma_{0}\beta_{0} & \gamma_{0} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \\ Q^{\prime} &= \begin{pmatrix} \gamma_{0} & -\gamma_{0}\beta_{0} & 0 & 0 \\ -\gamma_{0}\beta_{0} & \gamma_{0} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0 \\ L \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} \color{red}{-\gamma_{0}\beta_{0}L } \\ \color{red}{\gamma_{0}L} \\ 0 \\ 0 \end{pmatrix} \end{align*} $$
Look at the area painted in red. Comparing events $Q$ and $Q^{\prime}$, it’s clear they are the same event but appear different. An event $Q$ observed in the $A$ system is simultaneous with event $P$, but it’s not in the $A^{\prime}$ system. It’s easier to understand with a diagram. The world lines of the two events from $A$ and $A^{\prime}$ systems are as follows.
It’s important to note that there’s no time difference in the direction perpendicular to the direction of motion of the coordinate system. A time difference occurs only in the direction of movement. For instance, in the above case, if event $Q$ in the $A$ system is at coordinate $y$ which is point $L$, then $P=\begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}$ and $Q=\begin{pmatrix} 0 \\ 0 \\ L \\ 0 \end{pmatrix}$, and observing the two events from the $A^{\prime}$ system yields the following.
$$ \begin{align*} P^{\prime} &= \begin{pmatrix} \gamma_{0} & -\gamma_{0}\beta_{0} & 0 & 0 \\ -\gamma_{0}\beta_{0} & \gamma_{0} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \\ Q^{\prime} &= \begin{pmatrix} \gamma_{0} & -\gamma_{0}\beta_{0} & 0 & 0 \\ -\gamma_{0}\beta_{0} & \gamma_{0} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0 \\ 0 \\ L \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ L \\ 0 \end{pmatrix} \end{align*} $$
In this case, the two events are simultaneous in both $A$ and $A^{\prime}$ systems.