Question

Like time dilation, length contraction cannot be seen directly by a single observer. To explain this claim, imagine a rod of proper length \(l_{\mathrm{o}}\) moving along the \(x\) axis of frame \(\mathcal{S}\) and an observer standing away from the \(x\) axis and to the right of the whole rod. Careful measurements of the rod's length at any one instant in frame \(\mathcal{S}\) would, of course, give the result \(l=l_{\mathrm{o}} / \gamma\). (a) Explain clearly why the light which reaches the observer's eye at any one time must have left the two ends \(A\) and \(B\) of the rod at different times. (b) Show that the observer would see (and a camera would record) a length more than \(l\). [It helps to imagine that the \(x\) axis is marked with a graduated scale.] ( \(\mathbf{c}\) ) Show that if the observer is standing close beside the track, he will see a length that is actually more than \(l_{\mathrm{o}}\); that is, the length contraction is distorted into an expansion.

Short answer

Light emitted at different times causes perception of expanded length, which is longer than both contracted and proper lengths due to relativity and viewing geometry.

Step-by-step solution

Understanding the Problem

We need to explain why the light from the ends of the moving rod reaches an observer at different times, show the apparent length as more than the contracted length, and prove that the rod appears longer than its proper length due to viewing dynamics.

Light Emission and Reception

The rod moves along the x-axis at speed v. If light waves from both ends reach the observer at the same time, they must have left the ends of the rod at different times. Light from point A (front of the rod) covers a shorter distance while the rod moves than light from point B (back of the rod), since the rod is moving towards the observer.

Apparent Length Calculation

Consider the observed reception times at the observer. If they record the position of both ends simultaneously, due to the different emission times, the length observed (calculated by considering these positions at the instant as seen) will appear longer than the contracted length.

View from Close Proximity

When the observer is close, the time difference between the light emissions making contact with their eyes diminishes the effect of contraction. Further, angles can cause the observer to perceive an expanded image due to speed and proximity, thus observing the rod as longer than the proper length.

Using Lorentz Contraction

The Lorentz factor, \ \( \gamma = \frac{1}{\sqrt{1-v^2/c^2}} \ \), shows the perceived contraction. However, due to relativistic Doppler effects and observer geometry, this contraction can be perceived as an expansion when viewing conditions enable the entire rod to be perceived as longer.

Final Calculation

By considering the relativistic effects and geometry (angles between viewer's line of sight and rod), calculate the length as perceived under various conditions. Here, apparent length is not \( l_o/\gamma \) but longer, due to relative motion and perspective effects.

Key concepts

Time Dilation

In the fascinating world of relativity, time dilation is a phenomenon where time appears to pass at different rates for observers in different frames of reference. Imagine you're watching a train go by while standing on a platform. For someone on the train, time ticks at a normal pace. However, for you, observing from the outside, time on the train seems to slow down.
This effect becomes significant at speeds close to the speed of light. It's described by the equation:
\[\Delta t' = \frac{\Delta t}{\sqrt{1 - \frac{v^2}{c^2}}}\]

• \(\Delta t'\) is the time interval measured by an observer in motion.
• \(\Delta t\) is the time interval measured by an observer at rest.
• \(v\) is the velocity of the moving frame.
• \(c\) is the speed of light.

As speeds increase, the factor \(\sqrt{1 - \frac{v^2}{c^2}}\) makes time appear to "dilate" or stretch. This instance is why astronauts on fast-moving spacecraft age slightly slower compared to people on Earth.

Lorentz Contraction

A key concept in Einstein's theory of relativity is Lorentz contraction, also known as length contraction. It explains how objects moving at a significant fraction of the speed of light change in length from the perspective of a stationary observer.
When viewing a fast-moving rod from a distance, its length along the direction of motion appears shorter. This contraction is only apparent to observers in a different frame than that of the rod. The formula for this is:
\[l = l_0 \sqrt{1 - \frac{v^2}{c^2}}\]

• \(l\) is the contracted length.
• \(l_0\) is the proper length (length of the rod in its rest frame).
• \(v\) is the speed of the object.
• \(c\) is the speed of light.

From an observer's viewpoint, if the rod is moving towards them, different light travel times from each end lead to a perceived length that might look longer than expected. This is because light from the far end has to travel further than from the near end, causing the complexities faced in observing true contraction.

Relativistic Doppler Effect

The relativistic Doppler effect describes how the frequency of light or other waves changes for observers moving relative to the source of the waves. This is different from the classical Doppler effect because it also accounts for time dilation effects.
Imagine a star moving away from Earth; its light shifts towards the red part of the spectrum. When it moves towards us, its light shifts blue. This shift can be calculated using:
\[u' = u \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}\]

• \(u'\) is the observed frequency.
• \(u\) is the emitted frequency.
• \(v\) is the relative velocity.
• \(c\) is the speed of light.

This effect is especially noticeable in astronomical observations and explains how light from stars can give scientists clues about their velocities and directions in space. It highlights the intricate interplay between motion, light, and the observer's perspective.