The Brass Tacks of the Spaceship Paradox

 

Does the Thread Break?

Back Story

Bell's Spaceship Paradox is a lesson in formation flying at near light speed. Here's the challenge:
  1. Point two spaceships in the same direction and line them up, one behind the other
  2. Tether them with a delicate thread and pull it taut
  3. Try to get the whole assembly up to speed without breaking the thread

On the one hand, the entire assembly is subject to Lorentz contraction so everything, including the thread, should survive. On the other hand, each component of the assembly is subject to Lorentz contraction so the thread should break as the spaceships shrink. It's a conundrum.

Here are the worldlines for the ends of the thread from the two perspectives of interest:


Rest Length

Note that Lorentz contraction is an unfortunate moniker because it goes both ways. Objects perceive themselves to grow longer as they pick up speed and shrink as they slow down. When physicists talk about "rest length", they are referring to perceived length at cruising speed, not the length that you measure on the launch pad when everyone is at rest.

Relativistic Stress

Everyone seems to agree that Lorentz contraction has real physical consequences. They called it "relativistic stress" back in the day. You'd be hard pressed to find a rigorous definition, but it is most certainly not the kind of stress you get from pushing or pulling. It has to do with perceptions of simultaneity. A more accurate term would be residual stress. It's called spaghettification when gravity is the cause, but it's essentially the same thing.

Conventional Wisdom

Getting back to the issue at hand, conventional wisdom says the thread is longer than you think. It's natural length is its "rest" length so it is already stressed on the launch pad in anticipation of the cruising speed it is going to achieve. I kid you not. The presumption seems to be that rest length is invariant. It's not.

You have to fork out $35 to see the actual AJP article (doi:10.1119/1.1996214 c. 1959), but the Wikipedia account is quite accurate. Invariant rest length was essentially shouted down at CERN in 1976 when they held an "informal and non-systematic survey of opinion" on the matter, but it seems to have gone mainstream anyway because the naysayers went silent. It's time to get down to the brass tacks.

Metal and Silk

First of all, SR doesn't distinguish between metal and silk so the spaceships are a distraction. We need to focus on the thread. In fact, let's replace it with a perfectly rigid rod so it can't go slack. The question is then, does the rod experience any ill effects from Lorentz contraction when it jumps frames? It most certainly does.

The Reality of Perception

Imagine yourself sitting on one end of the rod, watching a clock on the other end while the entire assembly is getting underway. Light speed is finite so you perceive a stationary, red-shifted clock until the image of its launch reaches your eyes. If the clock had eyes, it would perceive you in the same way.

The image of the clock eventually returns to its natural colours after the rod has achieved cruising speed and light from that event has had time to traverse its length. But a red-shifted clock appears to run slow. It's an illusion of sorts, but the lost time persists and becomes the rod's new reality as the rest of the universe whizzes by. Each end perceives the other to have slipped into the past. Nobody went backwards in time of course. It's just that the clock at the other end ran slow for a while.

The perceived time shift has consequences because light speed is not only finite. It is also invariant (i.e. the same for all inertial observers.) For all intents and purposes, the rod is actually longer from its own perspective so the electrostatic forces, which keep it together, have to compensate accordingly. A sufficiently delicate thread will therefore break, but it's not like it jumps frames and suddenly discovers that it has gotten longer. It breaks during the acceleration phase, as you would expect. The temporal gradient manifests as a fictitious gravitational field. I'll let the learned goofball at The Science Asylum explain: https://youtu.be/F5PfjsPdBzg.

A Relativistic Speedometer

Since your "rest" length is longer at cruising speed, you can use a laser range finder on your toes to measure the velocity of your head with respect to the launch pad. It smacks of absolute velocity, which is taboo, but it's really no different than reckoning speed by Doppler shift. In either case, you need an external reference point. It's a bit trickier for a co-moving reference point, but the principle (light speed invariance) is the same. I expect smart phones of the future will incorporate such a device to make intergalactic travel more convenient.

Digging Deeper

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Comments

  1. Chuck WApril 9, 2021 at 6:46 PM

    testing this link

    ReplyDelete
  2. UnknownMay 16, 2021 at 9:53 AM

    Great stuff! You might be interested in a crude but, I think, valid "solution" for the Bell Spaceship paradox that doesn't rely on "relativistic stress." I use a potential energy approach to point out that the acceleration at the tip of a long spaceship is not the same as at the tail. Just like with gravity, acceleration changes with altitude. If you use a 1g thrust drone at 10 foot altitude, so that a thing hovers, and then use the same 1g thrust at 10,000 feet, you discover the upper drone will pull away (because it weighs less then due to altitude). So it is a kind a nod to the equivalence principle, that if you don't want to stretch your spaceship or break your thread, you must have decreasing acceleration with height (or length, if you prefer). I also worked out the acceleration necessary to avoid breaking the thread and, lo and behold, it results in the exact same decreasing distance (seen by the still observer) as the lorentz contraction predicts! See https://arxiv.org/abs/2007.04186 where I wrote up the idea in detail. (I came to your site via your answer to a question at Quora. I am Ralph Berger, and I teach Nuclear Engineering at UC Berkeley.)

    ReplyDelete
    Replies
    1. Mike GaleMay 16, 2021 at 4:35 PM

      Thanks. I did read that one. Very good I think. I'm with you all the way.

      Delete

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