Question

Consider two events that occur simultaneously at \(t=0\) in frame \(\mathcal{S},\) both on the \(x\) axis at \(x=0\) and \(x=a\). (a) Find the times of the two events as measured in a frame \(\mathcal{S}^{\prime}\) traveling in the positive direction along the \(x\) axis with speed \(V\). (b) Do the same for a second frame \(\mathcal{S}^{\prime \prime}\) traveling at speed \(V\) but in the negative direction along the \(x\) axis. Comment on the time ordering of the two events as seen in the three different frames. This startling result is discussed further in Section 15.10 .

Short answer

Frame \( \mathcal{S}^{\prime} \): Event 2 occurs before Event 1. Frame \( \mathcal{S}^{\prime\prime} \): Event 2 occurs after Event 1.

Step-by-step solution

Understanding Simultaneity in Frame S

In the original frame \( \mathcal{S} \), two events occur simultaneously at \( t=0 \). Event 1 happens at \( x=0 \) and Event 2 at \( x=a \). Simultaneity means \( t_{\text{event 1}} = t_{\text{event 2}} = 0 \) in frame \( \mathcal{S} \).

Apply Lorentz Transformation for Frame S’

For a frame \( \mathcal{S}^{\prime} \) moving in the positive \( x \)-direction at speed \( V \), use the Lorentz transformations to find the times of the events:\[ t' = \gamma \left( t - \frac{Vx}{c^2} \right) \]where \( \gamma = \frac{1}{\sqrt{1-V^2/c^2}} \). Using \( t = 0 \) and \( x = 0 \) for Event 1, we get \( t_1' = 0 \). For Event 2 at \( x = a \), \( t_2' = -\gamma\frac{Va}{c^2} \).

Apply Lorentz Transformation for Frame S’’

For frame \( \mathcal{S}^{\prime\prime} \) moving in the negative \( x \)-direction at speed \( V \), again apply the Lorentz transformations:\[ t'' = \gamma \left( t + \frac{Vx}{c^2} \right) \]For Event 1 at \( x = 0 \), \( t_1'' = 0 \). For Event 2 at \( x = a \), \( t_2'' = \gamma\frac{Va}{c^2} \).

Analyze Time Order of Events in Different Frames

In frame \( \mathcal{S}^{\prime} \), Event 2 occurs before Event 1 because \( t_2' = -\gamma\frac{Va}{c^2} < 0 = t_1' \). In frame \( \mathcal{S}^{\prime\prime} \), Event 2 occurs after Event 1 since \( t_2'' = \gamma\frac{Va}{c^2} > 0 = t_1'' \). This demonstrates that the time order depends on the observer’s frame, highlighting the relativity of simultaneity.

Key concepts

Simultaneity

Simultaneity, at its core, means that two events occur at the exact same time as observed from a particular reference frame. In the original frame \( \mathcal{S} \), discussed in the exercise, both events occur simultaneously at \( t=0 \). So, if you're standing in frame \( \mathcal{S} \), events at different positions along the \( x \)-axis—specifically, \( x=0 \) and \( x=a \)—happen at the same moment. This is straightforward when the observer and events are in the same, stationary frame.

However, things become more interesting when you consider different frames of reference. Each frame may experience or "see" time differently due to motion. Thus, the idea of simultaneity can vary significantly between different observers. This sets the stage for understanding more complex concepts like the relativity of simultaneity.

Relativity of Simultaneity

The relativity of simultaneity is a crucial idea in Einstein’s theory of relativity. It states that whether two events are simultaneous is not absolute but depends on the observer's state of motion. So, in one frame, two events might occur simultaneously, but in another moving frame, they might happen at different times.

Using the exercise, in the prime frame \( \mathcal{S}^{\prime} \), moving at speed \( V \) in the positive \( x \)-direction, the times of the events as seen by an observer alter due to the application of the Lorentz transformation:

• Event 1 at \( x=0 \), remains at \( t_1' = 0 \).
• Event 2 at \( x=a \), now occurs at \( t_2' = -\gamma \frac{Va}{c^2} \). This shows Event 2 occurring before Event 1 in frame \( \mathcal{S}^{\prime} \).

In another frame \( \mathcal{S}^{\prime\prime} \), moving at the same speed \( V \), but in the negative \( x \)-direction, the opposite time order is observed. This variation underscores how the concept of simultaneity is relative to the observer's motion.

Reference Frames

Reference frames are an essential structure in understanding motion and observations. Think of a reference frame as an imaginary grid or coordinate system attached to an observer. The state of a reference frame—whether at rest or in motion—can profoundly affect observations, especially in the realm of relativity.

In the given exercise, there are three different frames:

• Frame \( \mathcal{S} \) is stationary and where the events are initially simultaneous.
• Frame \( \mathcal{S}^{\prime} \) moves at speed \( V \) in the positive \( x \)-direction.
• Frame \( \mathcal{S}^{\prime\prime} \) moves at the same speed but in the negative direction.

Each frame interprets the timing of events uniquely due to relative motion. By using Lorentz transformations, we see how the measurements of time and space are dynamic and interdependent across frames. This concept encapsulates that there isn't a single, privileged frame of reference, reinforcing the relativity principle that no observer's frame is "more correct" than another.