Question

A rod of proper length \(10 \mathrm{~cm}\) moves longitudinally along the \(x\)-axis of S at speed \(\frac{1}{2} c\). How long (in S) does it take a particle, moving oppositely at the same speed, to pass the rod?

Short answer

Answer: No, it is not possible to calculate the time required for the particle to pass the rod in this scenario using special relativity principles as the relative velocity between the rod and the particle equals the speed of light (c), and the Lorentz transformation cannot be applied at this speed.

Step-by-step solution

Calculate the contracted length of the rod

According to the length contraction formula in special relativity, the length of an object moving at a relativistic speed will appear to be shorter in the observer's frame. The contracted length (L') of the rod can be calculated by the following formula: \(L' = L\sqrt{1 - \frac{v^2}{c^2}}\) Where: - L is the proper length of the rod, which is 10 cm - v is the speed of the rod, which is 0.5c - c is the speed of light Plugging in the values and calculating L': \(L' = 10\sqrt{1 - \frac{(0.5c)^2}{c^2}} = 10\sqrt{1 - 0.25} = 10\sqrt{0.75} \approx 8.66 \mathrm{~cm}\)

Apply the Lorentz transformation for space coordinates

As both the rod and the particle are moving along the x-axis, we will apply the Lorentz transformation for the x-component of position: \(x = \gamma(x' + vt')\) Where: - x is the position in the observer's frame (S) - x' is the position in the moving frame (S') in which the rod is at rest - v is the relative velocity between the frames (S and S') - t' is the time in the moving frame (S') - γ (gamma) is the Lorentz factor, given by the relation: ℝ^{1/2} Since the particle is moving oppositely at the same speed and approaching the rod, the relative velocity between the particle and the rod is (0.5c - (-0.5c)) = c. Now, let's find γ at v=c: \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{1}{\sqrt{1 - 1}}\) Unfortunately, calculating the Lorentz factor for v=c results in division by zero, which is not allowed. Therefore, the Lorentz transformation cannot be applied in this scenario.

Interpret the result

Since the Lorentz transformation cannot be applied in this case, it is an indication that such a scenario is not physically possible in the context of special relativity. Objects with mass (rod and particle in this case) cannot reach the speed of light, as it would require infinite energy. So, it is not possible to calculate the time required for the particle to pass the rod. However, for objects moving at speeds very close to the speed of light (but not exactly c), the Lorentz transformations can still be applied, and length contraction and time dilation will occur accordingly.

Key concepts

Length Contraction

In the realm of special relativity, length contraction is a fascinating phenomenon. This concept proposes that objects moving at a significant fraction of the speed of light appear shorter from the perspective of a stationary observer.
The proper length of an object, which is its length in its rest frame, is always longer compared to what a moving observer would measure.
This is computed using the formula:

• The contracted length, denoted as \(L'\), is determined by \(L' = L\sqrt{1 - \frac{v^2}{c^2}}\).
• Here, \(L\) represents the proper length, \(v\) is the velocity of the object, and \(c\) stands for the speed of light.

When analyzing the given problem, the rod's proper length stands at 10 cm.
With a velocity of \(0.5c\), the contracted length calculates to approximately 8.66 cm.
So, as the rod moves rapidly along the x-axis, it will appear shorter, providing an intriguing visual effect in the frame of the observer (S).

Proper Length

The proper length is a straightforward yet important concept in relativity. It refers to the length of an object measured in the frame where the object is at rest.
In this frame, no motion affects the measurement, ensuring the object's length is at its maximum possible value.
To clarify:

• Proper length is denoted by \(L\) and serves as a baseline measurement.
• For our rod moving along the x-axis, the proper length is 10 cm.

Understanding the proper length is essential before considering how objects transform under relative motion.
As motion begins to play a role, observers in different frames measure varying lengths.
The concept of proper length anchors us, giving a clear picture of what the unaltered length is in the object's rest frame.

Lorentz Transformation

The Lorentz Transformation is a pivotal tool in special relativity that allows us to relate the coordinates of events as seen from two different frames of reference.
Especially when these frames are in relative motion close to the speed of light.
Here's its mathematical essence:

• The transformation utilizes the Lorentz factor \(\gamma\), defined by \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\).
• Coordinates in different frames transform as \(x = \gamma(x' + vt')\).
• This formula links spatial and temporal coordinates in moving frames.

In your exercise, however, there's a twist. An attempt to apply the transformation for relative velocity equal to the speed of light \(v = c\) results in an undefined scenario since the Lorentz factor becomes infinite.
This physically impossibility highlights a fundamental aspect: massive objects cannot reach the speed of light.
Thus, setting a limit on applying the transformation for such velocities.

Relative Velocity

Relative velocity is a crucial concept in relativity and helps us understand how objects move in relation to one another.
It describes the velocity of one object in the rest frame of another.

• In special relativity, when dealing with velocities nearing the speed of light, interactions become non-intuitive.
• In the exercise, the rod moves at \(0.5c\) and the particle at \(-0.5c\), leading to a relative velocity of \(c\).

This interaction underscores another key principle in relativity: as the relative velocity approaches the speed of light, the dynamics between time and space become convoluted.
The formulas involving square roots emphasize the limitation placed by the speed of light.
Although theoretical in some cases, practical examples show that objects with mass experience realistic constraints and infinite energy demands when attempting to bridge this speed.