Question

A traveler in a rocket of proper length 2d sets up a coordinate system \(\mathcal{S}^{\prime}\) with its origin \(O^{\prime}\) anchored at the exact middle of the rocket and the \(x^{\prime}\) axis along the rocket's length. At \(t^{\prime}=0\) she ignites a flashbulb at \(O^{\prime} .\) (a) Write down the coordinates \(x_{\mathrm{F}}^{\prime}, t_{\mathrm{F}}^{\prime}\) and \(x_{\mathrm{B}}^{\prime}, t_{\mathrm{B}}^{\prime}\) for the arrival of the light at the front and back of the rocket. (b) Now consider the same experiment as observed from a frame \(\delta\) relative to which the rocket is traveling with speed \(V\) (with \(\delta\) and \(S\) ' in the standard configuration). Use the inverse Lorentz transformation to find the coordinates \(x_{\mathrm{F}}, t_{\mathrm{F}}\) and \(x_{\mathrm{B}}, t_{\mathrm{B}}\) for the arrival of the two signals. Explain clearly in words why the two arrivals are simultaneous in \(\mathcal{S}^{\prime}\) but not in \(\mathcal{S} .\) This phenomenon is called the relativity of simultaneity.

Short answer

In \( \mathcal{S}^{\prime} \), arrivals are simultaneous. In \( \mathcal{S} \), they are not due to the relativity of simultaneity.

Step-by-step solution

Identifying Events in Rocket's Frame

In the rocket's frame \( \mathcal{S}^{\prime} \), the traveler is exactly at the midpoint, so the distances to the front and back of the rocket are \( d \). Since light travels at speed \( c \), it will take time \( \frac{d}{c} \) to reach both the front and back, simultaneously. Thus, the coordinates for the front reaching event are \( x_{\mathrm{F}}^{\prime} = d \) and \( t_{\mathrm{F}}^{\prime} = \frac{d}{c} \). Similarly, for the back reaching event, \( x_{\mathrm{B}}^{\prime} = -d \) and \( t_{\mathrm{B}}^{\prime} = \frac{d}{c} \).

Inverse Lorentz Transformation

In a frame \( \delta \), the rocket is moving at velocity \( V \). The inverse Lorentz transformation for space is \( x = \gamma (x^{\prime} + Vt^{\prime}) \) and for time is \( t = \gamma (t^{\prime} + \frac{Vx^{\prime}}{c^2}) \), where \( \gamma = \frac{1}{\sqrt{1-V^2/c^2}} \).

Calculating Front Event in Stationary Frame

For the event at the front of the rocket in \( \delta \):\[ x_{\mathrm{F}} = \gamma (d + V\frac{d}{c}) \]\[ t_{\mathrm{F}} = \gamma \left( \frac{d}{c} + \frac{Vd}{c^2} \right) \]

Calculating Back Event in Stationary Frame

For the event at the back of the rocket in \( \delta \):\[ x_{\mathrm{B}} = \gamma (-d + V\frac{d}{c}) \]\[ t_{\mathrm{B}} = \gamma \left( \frac{d}{c} - \frac{Vd}{c^2} \right) \]

Explanation of Relativity of Simultaneity

In the frame \( \mathcal{S}^{\prime} \) (rocket's frame), the light reaches both ends simultaneously at \( t^{\prime} = \frac{d}{c} \). However, in frame \( \delta \), due to the inverse Lorentz transformation, the two events occur at different times \( t_{\mathrm{F}} eq t_{\mathrm{B}} \). This demonstrates that simultaneity is relative and depends on the observer's frame of reference.

Key concepts

Lorentz Transformation

The Lorentz transformation is a vital concept in the realm of special relativity. It explains how the measurements of time and space are related for observers moving relative to each other. This is especially important when discussing events happening in different frames of reference. The transformation equations show how coordinates in one frame adjust depending on the velocity concerning another frame. In essence, if an object moves at a significant fraction of the speed of light, the times and positions of events will differ for one observer compared to another.

In the context of the exercise, the formulae for Lorentz transformation are:

• Space: \( x = \gamma (x^{\prime} + Vt^{\prime}) \)
• Time: \( t = \gamma (t^{\prime} + \frac{Vx^{\prime}}{c^2}) \)

Where \( \gamma = \frac{1}{\sqrt{1-V^2/c^2}} \). These equations help in transforming the coordinates \((x^{\prime}, t^{\prime})\) from one frame into \((x, t)\) in another frame moving at velocity \( V \). This illustrates how events that are simultaneous in one frame may not be so in another, as seen in the rocket's scenario.

Speed of Light

Central to understanding relativity is the invariant nature of the speed of light. In any given frame of reference, light always travels at the speed \( c \), approximately 299,792,458 meters per second, regardless of the observer's velocity. This constancy leads to phenomena like time dilation and length contraction, pivotal in special relativity.

For the described experiment, the speed of light is crucial. In the traveler's rocket frame, light takes the same time to reach the front and back because the distances are equal, and light's speed is constant. This simultaneity arises directly from the consistent speed of light and how it operates in different frames. However, for an observer in a different inertial frame (e.g., watching the moving rocket), the difference arises in perceived timing, highlighting the relativity of simultaneity.

This unchanging speed of light implies that if two events seem simultaneous in one reference frame, they might not appear so in another. This surprises many but is a cornerstone of understanding how the universe operates.

Frames of Reference

Frames of reference are essential when discussing relativity. They are systems that specify the position and time of events. In relativity, moving frames compare observations from different perspectives. The concept allows us to understand how various observers perceive time and space differently, particularly at higher velocities.

In our exercise, two main frames of reference are used: the traveler's frame in the rocket \( \mathcal{S}^{\prime} \) and the stationary frame \( \delta \). In the rocket's frame, events at the front and back happen simultaneously because the observer is equidistant from both events at the midpoint. However, for observers in frame \( \delta \), the events appear at different times due to the relative motion of the rocket.

When switching frames of reference, especially at significant speeds close to light's speed, these differences in the perception of time and space lead to fascinating results, demonstrating the relativity of simultaneity. This concept underscores the importance of the observer's perspective in describing any physical phenomena accurately.