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

CERN insists that you maintain your geometry in your own reference frame no matter how fast you go or what you’re made of. Otherwise it would be a dead give-away for detecting your own motion at cruising speed with the curtains closed. But the worldline x=L+vt transforms to x’=gamma*L so material properties have to intervene in an unspecified way to ensure that x’=L at whatever cruising speed you achieve. Conventional wisdom says Mother Nature is just opposed to unilateral acceleration. She insists that you accelerate different parts of yourself at different rates for maximum comfort. We have no idea if that’s true though because no one has ever put it to the test. We usually just grin and bear during take-off and assume it will be business as usual at cruising speed. It seems unlikely though because you get spagehettified when you’re falling in a gravitational field unless you acquire some tension to keep yourself together. Why doesn’t she maximize comfort in that case?

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, just as a real gravitational field manifests as a temporal gradient: https://youtu.be/F5PfjsPdBzg.

A Relativistic Speedometer

If 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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