Simultaneity

Section 52.9 Simultaneity

Simultaneous events refer to two or more events happening at the same time, say two lightning occurring at the same time or two trains arriving at the station at the same time, etc.

With the implicit assumption of a “universal time” in the Newtonian thinking, two events that were simultaneous in one frame would be automatically simultaneous in all frames. That is, simultaneity was an absolute concept.

In special relativity, there is no notion of a “universal time”, which makes it possible that two events that are simultaneous in one frame will not be simultaneous in another frame. Our analysis will show that simultaneity is not absolute but a relative concept.

Consider two events \(E_1\) and \(E_2\text{,}\) whose coordinates and times in the the coordinate system for the S' frame will be denoted by \((t_1^{\prime}, x_1^{\prime}, y_1^{\prime}, z_1^{\prime})\) and \((t_2^{\prime}, x_2^{\prime}, y_2^{\prime}, z_2^{\prime})\) respectively. The corresponding quanties in S frame have unprimed symbols.

As a concrete example, let us consider the following simultaneous events in S'.

\begin{equation*} t'_1 = 0,\ \ t'_2=0,\ \ x'_1 = 0,\ \ x'_2 = a. \end{equation*}

We ignore \(y\) and \(z\) coordinates. Using Lorentz transformation, we get times and positions of these events in frame S.

\begin{align*} \amp E_1:\ \ t_1 = 0,\ x_1 = 0.\\ \amp E_2: \ \ t_2 = \gamma \dfrac{V}{c^2}\: a,\ x_2 = \gamma\: a. \end{align*}

Since times \(t_1 \ne t_2\text{,}\) the two events will not be simultaneous in S frame even though they are simultaneous in the \(S6{\:\prime}\) frame! As a matter of fact, for \(a>0\text{,}\) the second event will occur later than the first event, and for \(a\lt 0\text{,}\) i.e. an event occurring on the negative \(x'\)-axis, the second event will occur earlier than the first event.

Note: some authors state this result as “the second event will appear to be earlier (or later) than the first event”, which is somewhat misleading since it will indeed be earlier than the first event, not just appear to be so. Special theory of relativity embodied in the Lorentz transformations clearly leads to an unmistakable conclusion that simultaneity is not an absolute concept. Simultaneous events in one frame will not be simultaneous in another frame.

Subsection 52.9.1 Order of Events

If simultaneity is a relative concept, then how about the order of events? If one event occurs later than the other in one frame, could the time order of events be opposite in some other frame? Let us see what the Lorentz transformations tell us for two events that are not simultaneous in the S' frame.

1. Events at the origin of one system

For simplicity let us consider two events that occur at the origin of S' frame at times \(t' = 0\) and \(t' = \tau\text{.}\)

\begin{align*} \amp E_1:\ \ t_1^{\prime} = 0,\ x_1^{\prime} = 0.\\ \amp E_2: \ \ t_2^{\prime} = \tau,\ x_2^{\prime} = 0. \end{align*}

According to the Lorentz transformations, the coordinates and times for these events in the frame S would be

\begin{align*} \amp E_1:\ \ t_1 = 0,\ x_1 = 0.\\ \amp E_2: \ \ E_2: \ \ t_2 = \gamma\: \tau,\ x_2 = \gamma\: V \tau. \end{align*}

Thus, if \(\tau>0\text{,}\) then both \(t_2^{\prime}>0\) and \(t_2>0\text{,}\) and if \(\tau\lt 0\text{,}\) then both \(t_2^{\prime}\lt 0\) and \(t_2\lt 0\text{.}\) The order of events is same in the two frames. You might say, this is particular to the events at the origin.

2. Events at the same location of one system

Let us check out two events at a place that is not at the origin of the coordinate system, but let them again occur at the same space point, say \(x^{\prime} = a\text{.}\)

\begin{align*} \amp E_3:\ \ t_3^{\prime} = 0,\ x_3^{\prime} = a,\\ \amp E_4: \ \ t_4^{\prime} = \tau,\ x_4^{\prime} = a. \end{align*}

Now the Lorentz transformations give

\begin{align*} \amp E_3:\ \ t_3 = \gamma\:\dfrac{V}{c^2}\: a,\ x_3 = \gamma a,\\ \amp E_4: \ \ t_4 = \gamma \left( \tau + \dfrac{V}{c^2}\: a\right),\ x_4 = \gamma\left( a + V \tau\right). \end{align*}

Again, we find that if \(t_4^{\prime}-t_3^{\prime}=\tau>0\) then \(t_4 - t_3 = \gamma \tau >0\) and if \(t_4^{\prime}-t_3^{\prime}=\tau\lt 0\) then \(t_4 - t_3 = \gamma \tau \lt 0\text{.}\) The order of events in not changed. You might now say that, this and the example at the origin are particular to the events at one location.

3. Events at the two locations of one system

Let us check out two events at two places, say one event at \(t' = 0\) at \(x' = 0\) and the later event at \(t' = \tau\) at \(x' = a\text{.}\)

\begin{align*} \amp E_5:\ \ t_5^{\prime} = 0,\ x_5^{\prime} = 0,\\ \amp E_6: \ \ t_6^{\prime} = \tau,\ x_6^{\prime} = a. \end{align*}

Now the Lorentz transformations give

\begin{align*} \amp E_5:\ \ t_5 = 0,\ x_5 = 0,\\ \amp E_6: \ \ t_6 = \gamma \left( \tau + \dfrac{V}{c^2}\: a\right),\ x_6 = \gamma\left( a + V \tau\right). \end{align*}

This will give

\begin{equation} t_6 - t_5 = \gamma \left( \tau + \dfrac{V}{c^2}\: a\right).\tag{52.9.1} \end{equation}

We see that for \(t_6^{\prime} - t_5^{\prime} = \tau > 0\text{,}\) the order of events will depend on the relative velocity \(V\) of the frames.

\begin{align*} \amp \textrm{If}\ \ V \gt - \dfrac{c^2\tau}{a}\: a, \ \ \textrm{then}\ \ t_6 - t_5 \gt 0,\ \ \textrm{Order of events same}\\ \amp \textrm{If}\ \ V \lt - \dfrac{c^2\tau}{a}\: a, \ \ \textrm{then}\ \ t_6 - t_5 \lt 0,\ \ \textrm{Order of events opposite}\\ \amp \textrm{If}\ \ V = - \dfrac{c^2\tau}{a}\: a, \ \ \textrm{then}\ \ t_6 - t_5 = 0,\ \ \textrm{Events simultaneous in S}. \end{align*}

Three examples in this subsection illustrate that the order of events in two frames may or may not be same which depends on the particulars of the two events in question.