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Chapter 38: Problem 58

Review Problem 40 , in which we concluded that a limo of proper length \(30 \mathrm{~m}\) can fit into a garage of proper length \(6 \mathrm{~m}\) with room to spare. This result is possible because the speeding limo is observed by Garageman to be Lorentz -contracted. Carman protests that in the rest frame of the limo (in which the limo is its full proper length) it is the garage that is Lorentz-contracted. As a result, he claims, there is no possibility whatever that the limo can fit into the garage. What could be the possible basis for resolving this paradox? (Hint: Think about the space and time locations of two events: event A, front garage door closes and event \(\mathrm{B}\), rear garage door opens.)

Short Answer

Expert verified
The paradox is resolved by the relativity of simultaneity, with different timing of events in each frame.

Step by step solution

01

- Understand Lorentz Contraction

Lorentz contraction explains that an object moving at a significant fraction of the speed of light will appear contracted in the direction of its motion to an observer at rest. Garageman, observing the limo, sees it contracted to fit inside the garage.
02

- Identify the Two Reference Frames

Identify the two frames of reference: one is the Garageman's frame (where the garage is stationary and the limo is moving), and the other is Carman's frame (where the limo is stationary and the garage is moving).
03

- Event Analysis in Garageman's Frame

In Garageman's frame, the limo is contracted to less than 6 meters in length due to its high speed. Therefore, at some point, the entire limo fits inside the garage, and the front door can close while the rear door is still closed.
04

- Event Analysis in Carman's Frame

In Carman's frame, the garage appears Lorentz-contracted. However, the key to resolving the paradox is the timing of events. Due to the relativity of simultaneity, the events (A: front door closing and B: rear door opening) are not simultaneous in Carman's frame. The back door opens before the front door closes, allowing the limo to fit momentarily.
05

- Summary of Resolution

The paradox is resolved by recognizing the relativity of simultaneity - what is simultaneous in one frame is not necessarily simultaneous in another. In Garageman鈥檚 frame, both doors can be closed with the limo inside. In Carman鈥檚 frame, events are viewed at different times, ensuring the limo can pass through without contradiction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lorentz Contraction
Imagine you are watching a speeding limo from the sidewalk. The faster it moves, the shorter it appears to you. This is called Lorentz contraction. When objects travel close to the speed of light, they undergo this peculiar shortening in the direction they are moving. It鈥檚 a fundamental principle of Einstein鈥檚 theory of relativity.

Suppose a limo has a proper length of 30 meters when at rest. As it speeds to the garage, it seems to shrink. For someone standing next to the stationary garage (Garageman), this speeding limo appears shorter and can fit inside the 6-meter-long garage. Understanding this concept helps grasp why, to Garageman, the limo can fit inside the garage by appearing contracted.

However, if you think about it from the limo driver's perspective (Carman), the scenario flips. Everything else, including the garage, seems to be moving. Hence, to Carman, it鈥檚 the garage that鈥檚 Lorentz-contracted.
Relativity of Simultaneity
A crucial element in this arrangement is the relativity of simultaneity. This principle asserts that what appears to be simultaneous in one reference frame may not be so in another.

Garageman might observe two events: the front door of the garage closing, followed by the rear door opening. To Garageman, both actions occur almost at the same time. This makes sense in his frame because, to him, the limo is shortened and fits within the garage.

Carman, however, perceives these events differently. Due to the high speed, the rear door of the garage opens before the front door closes, ensuring that the limo (in its full proper length) fits momentarily in the garage. This apparent disagreement in the sequence of events is due to the relativity of simultaneity.

For Carman, the timing flows in such a way that the limo never seems cramped. This demonstrates that what seems to be a paradox in one frame can be resolved by recognizing the variance in perceived timing across different frames.
Reference Frames
Understanding the concept of reference frames is key to resolving the paradox in the original problem. A reference frame is essentially a point of view in which an observer stands still and looks at other objects.

In this problem, we have two main reference frames:
  • **Garageman's frame**: The garage is at rest, and the limo is moving.
  • **Carman's frame**: The limo is at rest, and the garage is moving.


Each frame observes events differently because of their unique velocities. In Einstein's theory, the laws of physics apply uniformly, but observations like lengths and times can differ depending on the perspective or the frame of reference.

By understanding how these frames interact, you can see that although Garageman and Carman perceive different realities, neither is wrong. Both perspectives are valid within their respective frames.

The concept of reference frames thus helps explain why objects might appear differently to different observers, as shown in the simplistic handling of moving objects like the limo and the garage in this context.

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Most popular questions from this chapter

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