The Setup

Bob and Alice are eager to elope. Bob is responsible for the ladder, while Alice will provide the second-story window. A challenge arises with the barn situated between the road and Alice's window, as the ladder is significantly longer than the barn itself. The barn features both a front and a back door, but only one of these can be open at any given time.

How Can Bob Maneuver the Ladder?

Alice proposes an ingenious solution. She realizes that, according to the principles of special relativity, if Bob runs fast enough, the ladder will appear to contract in length. The ladder has 12 rungs, while the barn measures approximately 9 rungs long. Alice quickly calculates the necessary speed for Bob to achieve this contraction.

Calculating Velocity

To determine Bob's required speed, Alice uses the following equations:

1. The barn's length is ¾ the length of the ladder when stationary.
2. To fit the ladder into the barn, Bob must contract it to ¾ of its length.

A bit of algebra reveals that Bob needs to run at just under ⅔ the speed of light to make this work. "You better start far back to gain enough speed," Alice suggests.

From Alice's Perspective

As Bob accelerates to the necessary velocity, Alice observes that her calculations are accurate; the ladder indeed fits inside the barn. Bob approaches the barn, and from his viewpoint, the situation seems dire. At ⅔ the speed of light, the barn itself appears to have contracted to ¾ its original length, while the ladder remains unchanged.

Will the ladder fit through the barn, or will he collide with the back door?

Description: This video explains Einstein's Ladder Paradox, clarifying the concepts of length contraction and time dilation.

The Key to the Paradox

The crux of the paradox lies in the statement: "Only one door can be open at a time." The concept of simultaneity is not absolute. It's crucial to differentiate between Bob's frame of reference and Alice's. Events that occur simultaneously in Alice's frame may not be so in Bob's if they are spatially separated.

Examining Alice's Frame

Let’s consider the barn's length from Alice’s perspective. If the barn is 60 light-seconds long, the ladder measures 80 light-seconds long when at rest.

The approximate time for the front of the ladder to traverse the barn is calculated as follows:

Synchronized Clocks

Before Bob begins, he and Alice synchronize four clocks: two at the barn's front and back, and two at the front and back of the ladder. They ensure that all but the rear ladder clock read 0 when the front of the ladder arrives at the barn.

The rear clock is set ahead by a factor determined by the formula involving their respective velocities.

Bob's Perspective

In Bob's frame, the barn approaches at about ⅔ light speed. Just like in Alice's frame, when the front of the ladder meets the barn's front, the clocks in those positions read 0. However, the rear clock on the barn is ahead.

Description: This video challenges viewers to solve the Ladder Paradox, enhancing understanding of relativistic effects.

Conclusion: A Harmonious Resolution

Both Bob and Alice experience the effects of special relativity, albeit differently. Alice witnesses the ladder's entry into the barn due to length contraction, while Bob experiences time dilation and the loss of simultaneity that keeps the doors open long enough for him to pass through.

Ultimately, the paradox resolves in a manner that allows Bob and Alice to achieve their goal, demonstrating that neither space nor time is disrupted by their actions.

For now, the mysteries of relativity remain intact.