The Brass Tacks of the Ladder Paradox

Lorentz Contraction and Perceptions of Simultaneity

Back Story

In the Ladder Paradox (aka. Pole-Barn Paradox), we are given a ladder (or pole), which is too long to fit inside our barn when both are at rest. A smart aleck physicist, who knows a thing or two about Lorentz contraction, claims he can make it fit (if only for an instant) by throwing it through the barn at a sufficiently high speed. We presume the ladder is perfectly rigid so it won't be contorted by the force of acceleration. That would be cheating. In fact, let's put rockets on both ends just to be sure. Here we go:
Hmmm. What went wrong? We followed the recipe, but the ladder didn't shrink. In fact, just the opposite. In its moving reference frame, the ladder actually thinks it has gotten longer (by a factor of γ.) Our smart aleck friend is undaunted though because, he says, the rest length of the ladder must be measured from the moving frame of reference. Its prelaunch length is therefore shorter so it will fit inside the barn. OK, so here we go again:

But wait a tick! That's shenanigans. We could have pushed that shorter ladder into the barn at a more mundane pace and saved ourselves a lot of bother and expense with the rockets. Even so, smart aleck has a point. What happens to the ladder while it's getting underway? Does it actually stretch out or does the rest of the universe shrink around it? To find out, we have to do a Doppler analysis.

The Reality of Perception

You may think that the ladder should perceive itself in its natural colours at all times because all of its various parts are stationary before the launch and co-moving afterwards. But light speed is finite so the front and the back perceive one another to be red-shifted after the local launch event until light from the the other end arrives. That's the shockwaves (black dashed lines) in the illustrations.

That red-shift makes a clock at the other end appear to run slow until the light catches up. It's an illusion of sorts, but the lost time persists for as long as the ladder maintains cruising speed. The effect is tempered somewhat by local time dilation, but there's still a net loss and that's the ladder's new reality as the rest of the universe whizzes by:

For all intents and purposes, the ladder is actually longer so the atomic and subatomic processes that keep it together must compensate accordingly. Everything else in the universe closes in due to Lorentz contraction. On the other hand, the bystander (i.e. barn) sees all points on the ladder launch at the same time so there's no shrinkage from that perspective. You'd be surprised how many people get that part wrong.

Resources

Check out these videos on this paradox here:










Comments

Post a Comment

Popular posts from this blog

The Brass Tacks of the Ehrenfest Paradox

Image
Coordinate Transforms for Rotating Reference Frames   Back Story Wikipedia explains the conundrum in the Ehrenfest paradox thusly. Imagine a disk of radius  R  rotating with constant angular velocity  . The reference frame is fixed to the stationary centre of the disk. Then the magnitude of the relative velocity of any point in the circumference of the disk is  . So the circumference will undergo  Lorentz contraction  by a factor of  . However, since the radius is perpendicular to the direction of motion, it will not undergo any contraction. So {\displaystyle {\frac {\mathrm {circumference} }{\mathrm {diameter} }}={\frac {2\pi R{\sqrt {1-(\omega R)^{2}/c^{2}}}}{2R}}=\pi {\sqrt {1-(\omega R)^{2}/c^{2}}}.} This is paradoxical, since in accordance with  Euclidean geometry , it should be exactly equal to  π . (Image credit: By Geek3 - Own work, CC BY 3.0,  https://commons.wikimedia.org/w/index.php?curid=24038062 ) Conventional Wisdom I won't bore you with the details, but Wikipedia goe
Read more

The Brass Tacks of the Twin Paradox

Image
The Brass Tacks of the Twin Paradox Back Story Remember Einstein’s Twin Paradox? Someone goes walkabout at near light speed and winds up younger than their twin who stayed at home. It seems like a conundrum wrapped in an enigma because the the experts insist that moving  clocks tick more slowly than one another. It's patent nonsense of course because you can't be both older and younger than your twin when you're standing face-to-face at the family reunion. So what's up with that? Are they pulling our proverbial legs? Here's the geometry in question: Conventional Wisdom You can find several explanations in the literature and on YouTube (see links below for example.) The general consensus is that  the symmetry is broken by the traveller's antics at the waypoint. The about face at that location is supposed to be the give-away. That's how we know whose clock is running slow. But naysayers with an ounce of common sense are quick to point out that we're being
Read more

The Brass Tacks of the Andromeda Paradox

Image
The Subjective "Now" Interpretation of Special Relativity Back Story In the Andromeda paradox (aka. Rietdijk–Putnam argument), we are given to understand that there's a certain faction of hostile actors in the Andromeda galaxy who want to invade the Earth. They have presented their case to their local council and the matter has been put to a vote. An astronomer on Earth observes the verdict through a telescope and, after correcting for the propagation delay, works out the time at which our fate was sealed. A passerby sees the same image at the same time, but having a skewed perception of simultaneity, arrives at a completely different conclusion. So when exactly did the verdict make the transition from the uncertain future to the certain past? i.e. When is "now" over there? Conventional Wisdom Conventional wisdom says the question is invalid. What you see is what you get. All perceptions of "now" over there are real because, unlike regular spacetime ev
Read more