The Relativistic Hookean Spring: A Lorentz-Covariant Force

[atyy]

What is the simplest Lorentz covariant force - the relativistic version of an ideal Hookean spring in Newtonian mechanics?

"Simplest" as an easy to do an ideal calculation with.

 

[bcrowell]

I don't think there's anything simple about relativistic elasticity: http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/SimpleElasticity.html

 

[atyy]

I don't think there's anything simple about relativistic elasticity: http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/SimpleElasticity.html

Yes. I meant: what is a force that is as simple as Hooke's law?

Wow, thanks for the link! I didn't know Greg Egan wrote science! (No I haven't read him but my brother is a fan).

What? This is the same Greg Egan: http://arxiv.org/abs/gr-qc/0208010? I'm a dummy!

 

[pervect]

I would say that a Lagrangian density of -rho is the simplest relativistic equivalent to a spring - but I also suspect that's not quite what you're after, it's not formulated as a particle-particle force, and to turn it into one, you'd have to define the motion of the spring which starts to get non-simple.

http:http://www.gregegan.net/SCIENCE/Rindler/SimpleElasticity.html talks about it a bit, though he doesn't use Lagrangians.

I think https://www.physicsforums.com/showpost.php?p=1365762&postcount=157 was what I finally came up with for the Lagrangian density of a hoop, not that you're doing hoops. It should also work for a wire. You'll see some added terms there, related to the terms due to the motion of the wire, and the energy involved in stretching it.

Getting the volume element right was a painful process. And the stretch-factor computation isn't trivial, either - as you'll see if you have the patience to read the very very long thread. And the last caveat is that the speed of sound in the wire goes up when you stretch it, and when that speed gets to be faster than 'c' bad things happen, bad things including singular Lagrangians.

On the other hand,I suspect it's not quite what your'e after, so you might not want to bother.

 

[atyy]

@pervect, thanks too!

I guess what I was after is trying to think about this thread https://www.physicsforums.com/showthread.php?t=508190&page=2 .

In the cleanest version of the story, we set up the balls to have equal and opposite velocities in the train frame, and then just use velocity addition. In a sense, the balls then have always been moving that way, and the asymmetry judged from the platform has always been present.

In hprog's set-up, he says the man on the train pushes with "equal force". My intuition is that the final velocity of the balls is determined by force times time. The balls will leave the force field at the same distance (in both frames) from the center, but at the same time in the train frame, and not in the platform frame. So I wanted to know a "simple force" to enter into the calculation.